Non-linear impulsive dynamical systems. Part I: Stability ... · In this paper we develop Lyapunov and invariant set stability theorems for non-linear impulsive dynamical systems.

Non-linear impulsive dynamical systems Part I Stability and dissipativity

WASSIM M HADDAD VIJAYSEKHAR CHELLABOINA and NATASIuml A A KABLAR

In this paper we develop Lyapunov and invariant set stability theorems for non-linear impulsive dynamical systemsFurthermore we generalize dissipativity theory to non-linear dynamical systems with impulsive e ects Speciregcally theclassical concepts of system storage functions and supply rates are extended to impulsive dynamical systems providing ageneralized hybrid system energy interpretation in terms of stored energy dissipated energy over the continuous-timesystem dynamics and dissipated energy over the resetting instants Furthermore extended KalmanplusmnYakubovichplusmnPopovconditions in terms of the impulsive system dynamics characterizing dissipativeness via system storage functions arederived Finally the framework is specialized to passive and non-expansive impulsive systems to provide a generalizationof the classical notions of passivity and non-expansivity for non-linear impulsive systems These results are used in thesecond part of this paper to develop extensions of the small gain and positivity theorems for feedback impulsive systemsas well as to develop optimal hybrid feedback controllers

1 Introduction

Modern complex engineering systems as well as bio-

logical and physiological systems typically possess amulti-echelon hierarchical hybrid architecture character-ized by continuous-time dynamics at the lower levels ofhierarchy and discrete-time dynamics at the higher levelsof the hierarchy Hence it is not surprising that hybridsystems have been the subject of intensive research overthe past recent years (see Branicky et al 1998 Ye et al1998 b Haddad and Chellaboina 2001 and referencestherein) Such systems include dynamical switchingsystems (Branicky 1998 Leonessa et al 2000) non-smooth impact and constrained mechanical systems(Back et al 1993 Brogliato 1996 Brogliato et al

1997) biological systems (Lakshmikantham et al1989) demographic systems (Liu 1994) sampled-datasystems (Hagiwara and Araki 1988) discrete-eventsystems (Passino et al 1994) intelligent vehiclehighwaysystems (Lygeros et al 1998) and macright control systems(Tomlin et al 1998) to cite but a few examples Themathematical descriptions of many of these systemscan be characterized by impulsive di erential equations(Simeonov and Bainov 1985 1987 Liu 1988Lakshmikantham et al 1989 1994 Bainov and

Simeonov 1989 1995 Kulev and Bainov 1989Lakshmikantham and Liu 1989 Hu et al 1989Samoilenko and Perestyuk 1995) Impulsive dynamicalsystems can be viewed as a subclass of hybrid systemsand consist of three elements namely a continuous-timedi erential equation which governs the motion of the

dynamical system between impulsive or resetting eventsa di erence equation which governs the way the systemstates are instantaneously changed when a resettingevent occurs and a criterion for determining when thestates of the system are to be reset As in classical dyna-mical systems theory it seems natural that dissipativitytheory should play a fundamental role in addressingrobustness disturbance rejection stability of feedbackinterconnections and optimality for hybrid dynamicalsystems

The key foundation in developing dissipativity

theory for general non-linear dynamical systems waspresented by Willems (1972 a b) in his seminal two-partpaper on dissipative dynamical systems In particular

Willems (1972 a) introduced a deregnition of dissipa-tivity for general dynamical systems in terms of aninequality involving a generalized system power inputor supply rate and a generalized energy function orstorage function Since Lyapunov functions can beviewed as generalizations of energy functions for non-linear dynamical systems the notion of dissipativitywith appropriate storage functions and supply ratescan be used to construct Lyapunov functions for non-linear feedback systems by appropriately combiningstorage functions for each subsystem Even though theoriginal work on dissipative dynamical systems was for-mulated in the state space setting describing the systemdynamics in terms of continuous macrows on appropriatemanifolds an inputplusmnoutput formulation for dissipativedynamical systems extending the notions of passivity

(Zames 1966) non-expansivity (Zames 1966) andconicity (Zames 1966 Safonov 1980) was presented inMoylan (1974) and Hill and Moylan 1976 1980 Morerecently the notion of dissipativity theory was general-ized in Chellaboina and Haddad (2000) to formalize theconcepts of the non-linear analogue of strict positiverealness and strict bounded realness In particular

International Journal of Control ISSN 0020plusmn7179 printISSN 1366plusmn5820 online 2001 Taylor amp Francis Ltdhttpwwwtandfcoukjournals

DOI 10108000207170110081705

INT J CONTROL 2001 VOL 74 NO 17 1631plusmn1658

Received 3 February 2000 Revised 20 June 2001 Author for correspondence e-mail wmhaddad

aerospacegatechedu School of Aerospace Engineering Georgia Institute of

Technology Atlanta GA 30332-0150 USA Mechanical and Aerospace Engineering University of

Missouri Columbia MO 65211 USA

using exponentially weighted system storage functionswith appropriate exponentially weighted supply ratesthe concept of exponential dissipativity was introducedin Chellaboina and Haddad (2000)

Dissipativity theory along with its connections toLyapunov stability theory has been extensively devel-oped for dynamical systems possessing continuousmacrows However in light of the increasingly complex nat-ure of the dynamical systems discussed above discontin-uous system macrows arise naturally Alternatively withinthe context of feedback control active energy macrow reset-ting control for interconnected subsystems also gives riseto discontinuous closed-loop system macrows Speciregcallyif a dissipative or lossless plant is at a high energy leveland a dissipative feedback controller at a low energylevel is attached to it then energy will generally tendto macrow from the plant into the controller decreasingthe plant energy and increasing the controller energy(Kishimoto et al 1995) Of course emulated energyand not physical energy is accumulated by the control-ler Conversely if the attached controller is at a highenergy level and a plant is at a low energy level thenenergy can macrow from the controller to the plant sincea controller can generate real physical energy to e ectthe required energy macrow Hence if and when the con-troller states coincide with a high emulated energy levelthen we can reset these states to remove the emulatedenergy so that the emulated energy is not returned to theplant In this case the overall closed-loop system con-sisting of the plant and the controller possesses discon-tinuous macrows characterized by impulsive di erentialequations (Lakshmikantham et al 1989) Within thecontext of vibration control using resetting virtualabsorbers these ideas were regrst presented in Bupp etal (2000)

Motivated by complex hybrid dynamical systemspossessing discontinuous macrows in this paper we developstability dissipativity and exponential dissipativityconcepts for non-linear impulsive dynamical systemsSpeciregcally we develop an invariance principle forimpulsive dynamical systems wherein system trajectoriesconverge to a largest invariant set contained in a hybridlevel surface composed of a union involving vanishingLyapunov derivatives and di erences of the continuous-time trajectories and resetting instants respectivelyFurthermore we extend the notions of classical dissipa-tivity theory using generalized storage functions andsupply rates for impulsive dynamical systems The over-all approach provides an interpretation of a generalizedhybrid energy balance for an impulsive dynamicalsystem in terms of the stored or accumulated general-ized energy dissipated energy over the continuous-timedynamics and dissipated energy at the resetting instantsFurthermore as in the case of dynamical systems pos-sessing continuous macrows (Willems 1972 a) we show that

the set of all possible storage functions of an impulsive

dynamical system forms a convex set and is bounded

from below by the systemrsquos available stored generalized

energy which can be recovered from the system and

bounded from above by the systemrsquos required general-ized energy supply needed to transfer the system from an

initial state of minimum generalized energy to a given

state In addition for two kinds of non-linear impulsive

dynamical systems namely time-dependent and state-dependent impulsive systems we develop extended

KalmanplusmnYakubovichplusmnPopov algebraic conditions in

terms of the system dynamics for characterizing dissipa-

tiveness via system storage functions for impulsive dyna-mical systems

Although the results of this paper are conregned to

analysis stability and optimality results of feedback

non-linear impulsive systems are discussed in the secondpart of this paper (Haddad et al 2001) The main con-

tribution of this two-part paper is to develop a unireged

framework for the analysis and control synthesis of

non-linear impulsive systems However since impulsive

dynamical systems involve a hybrid formulation of con-tinuous-time and discrete-time dynamics these papers

also provide a tutorial for stability dissipativity feed-

back interconnections and optimality of continuous-time and discrete-time dynamical systems which can be

viewed as a specialization of impulsive systems

The contents of the paper are as follows In 2 we

establish deregnitions notation and review some basic

results on impulsive dynamical systems In 3 we presentLyapunov asymptotic and exponential stability results

for impulsive dynamical systems Furthermore new

invariant set theorems are derived wherein system tra-

jectories converge to a largest invariant set contained ina hybrid Lyapunov level surface composed of a union

involving vanishing Lyapunov derivatives and di er-

ences of the hybrid system dynamics Then in 4 we

extend the notion of dissipative dynamical systems todevelop the concept of dissipativity for impulsive dyna-

mical systems In 5 we develop extended Kalmanplusmn

YakubovichplusmnPopov algebraic conditions in terms of

the hybrid system dynamics for characterizing dissipa-tiveness via system storage functions for impulsive

systems Furthermore a generalized hybrid energy bal-

ance interpretation involving the systemrsquos stored or

accumulated energy dissipated energy over the contin-

uous-time dynamics and dissipated energy at the reset-ting instants is given Specialization of these results to

passive and non-expansive impulsive systems is also pro-

vided In 6 we specialize the results of 5 to linearimpulsive systems to obtain extended hybrid Kalmanplusmn

YakubovichplusmnPopov equations for positive real and

bounded real impulsive systems Finally we draw con-

clusions in 7

1632 W M Haddad et al

2 Non-linear impulsive dynamical systems

In this section we establish deregnitions notation and

review some basic results on impulsive dynamical

systems (Simeonov and Bainov 1985 1987 Liu 1988Lakshmikanthan et al 1989 1994 Bainov and

Simeonov 1989 1995 Kulev and Bainov 1989

Lakshmikantham and Liu 1989 Hu et al 1989

Samoilenko and Perestyuk 1995) Let denote the set

of real numbers n denote the set of n 1 real column

vectors hellip daggerT denote transpose N denote the set of non-

negative integers n denote the set of n n symmetricmatrices n (resp n) denote the set of n n non-

negative (resp positive) deregnite matrices and let In or

I denote the n n identity matrix Furthermore let S

S8 and middotSS denote the boundary the interior and the clo-

sure of the subset S raquo n respectively We write k k for

the Euclidean vector norm Bhellipnotdagger not 2 n gt 0 for theopen ball centred at not with radius V 0hellipxdagger for the

FreAcirc chet derivative of V at x and M 0 (resp M gt 0)

to denote the fact that the Hermitian matrix M is non-

negative (resp positive) deregnite Finally let C0 denote

the set of continuous functions and Cr denote the set of

functions with r continuous derivatives

As discussed in the introduction an impulsive dyna-mical system consists of three elements

(1) a continuous-time dynamical equation which

governs the motion of the system between reset-

ting events

(2) a di erence equation which governs the way the

states are instantaneously changed when a reset-

ting event occurs and

(3) criterion for determining when the states of the

system are to be reset

For the characterization of an impulsive dynamical

system ~UU 7 ~UUc~UUd is an input space and consists of

bounded continuous U-valued functions on the semi-

inregnite interval permil0 1dagger The set U 7 Uc Ud where

Uc sup3 mc and Ud sup3 md contains the set of input

values that is for every u ˆ hellipuc uddagger 2 ~UU and

t 2 permil0 1dagger uhelliptdagger 2 U uchelliptdagger 2 Uc and udhelliptdagger 2 Ud

Furthermore ~YY 7 ~YYc~YYd is an output space and con-

sists of bounded continuous Y-valued functions on the

semi-inregnite interval permil0 1dagger The set Y 7 Yc Yd where

Yc sup3 lc and Yd sup3 ld contains the set of output values

that is for every y ˆ hellipyc yddagger 2 ~YY and t 2 permil0 1daggeryhelliptdagger 2 Y ychelliptdagger 2 Yc and ydhelliptdagger 2 Yd Thus an impulsive

dynamical system has the form

_xxhelliptdagger ˆ fchellipxhelliptdaggerdagger Dagger Gchellipxhelliptdaggerdaggeruchelliptdagger

xhellip0dagger ˆ x0 hellipt xhelliptdagger uchelliptdaggerdagger 62 S

9=

hellip1dagger

centxhelliptdagger ˆ fdhellipxhelliptdaggerdagger Dagger Gdhellipxhelliptdaggerdaggerudhelliptdagger hellipt xhelliptdagger uchelliptdaggerdagger 2 S

hellip2dagger

ychelliptdagger ˆ hchellipxhelliptdaggerdagger Dagger Jchellipxhelliptdaggerdaggeruchelliptdagger hellipt xhelliptdagger uchelliptdaggerdagger 62 S

hellip3dagger

ydhelliptdagger ˆ hdhellipxhelliptdaggerdagger Dagger Jdhellipxhelliptdaggerdaggerudhelliptdagger hellipt xhelliptdagger uchelliptdaggerdagger 2 S

hellip4dagger

where t 0 xhelliptdagger 2 D sup3 n D is an open set with 0 2 Dcentxhelliptdagger 7 xhelliptDaggerdagger iexcl xhelliptdagger uchelliptdagger 2 Uc sup3 mc udhelliptkdagger 2 Ud sup3

md tk denotes the kth instant of time at whichhellipt xhelliptdagger uchelliptdaggerdagger intersects S for a particular trajectoryxhelliptdagger and input uchelliptdagger ychelliptdagger 2 Yc sup3 lc ydhelliptkdagger 2 Yd sup3

ld fc D n is Lipschitz continuous and satisregesfchellip0dagger ˆ 0 Gc D n mc fd D n is continuousGd D n md hc D lc and satisreges hchellip0dagger ˆ 0Jc D lc mc hd D ld Jd D ld md and S raquopermil0 1dagger D Uc is the resetting set Here we assumethat uchellip dagger and udhellip dagger are restricted to the class of admis-sible inputs consisting of measurable functions such thathellipuchelliptdagger udhelliptkdaggerdagger 2 U c Ud for all t 0 and k 2 N permil0tdagger 7

fk 0 micro tk lt tg where the constraint set Uc Ud isgiven with hellip0 0dagger 2 Uc Ud We refer to the di erentialequation (1) as the continuous-time dynamics and werefer to the di erence equation (2) as the resetting law

For convenience we use the notation shellipt frac12 x0 udaggerto denote the solution xhelliptdagger of (1) (2) at time t gt frac12with initial condition xhellipfrac12dagger ˆ x0 where u ˆ hellipuc uddagger

T Uc Ud and T 7 ft1 t2 g Furthermorewe call the times tk the resetting times Thus the trajec-tory of the system (1) and (2) from the initial conditionxhellip0dagger ˆ x0 is given by Aacutehellipt 0 x0 udagger for 0 lt t micro t1 where

Aacutehellipt 0 x0 udagger denotes the solution to the continuous-timedynamics (1) If and when the trajectory reaches astate x1 7 xhellipt1dagger satisfying hellipt1 x1 u1dagger 2 S where u1 7

uchellipt1dagger then the state is instantaneously transferred toxDagger

1 7 x1 Dagger fdhellipx1dagger Dagger Gdhellipx1daggerud where ud 2 Ud is a giveninput according to the resetting law (2) The trajectoryxhelliptdagger t1 lt t micro t2 is then given by Aacutehellipt t1 xDagger

1 udagger and soon Note that the solution xhelliptdagger of (1) and (2) is left-continuous that is it is continuous everywhere exceptat the resetting times tk and

xk 7 xhelliptkdagger ˆ lim0Dagger

xhelliptk iexcl dagger hellip5dagger

xDaggerk 7 xhelliptkdagger Dagger fdhellipxhelliptkdaggerdagger Dagger Gdhellipxhelliptkdaggerdaggerudhelliptkdagger

ˆ lim0Dagger

xhelliptk Dagger dagger udhelliptkdagger 2 Ud hellip6dagger

for k ˆ 1 2 We make the following additional assumptions

A1 If hellipt xhelliptdagger uchelliptdaggerdagger 2 SnS then there exists gt 0such that for all 0 lt macr lt

shellipt Dagger macr t xhelliptdagger uchellipt Dagger macrdaggerdagger 62 S

Non-linear impulsive dynamical systems Part I 1633

A2 If helliptk xhelliptkdagger uchelliptkdaggerdagger 2 S S then there exists

gt 0 such that for all 0 micro macr lt andudhelliptkdagger 2 Ud

shelliptk Dagger macr tk xhelliptkdagger Dagger fdhellipxhelliptkdaggerdagger

DaggerGdhellipxhelliptkdaggerdaggerudhelliptkdagger uchelliptk Dagger macrdaggerdagger 62 S

Assumption A1 ensures that if a trajectory reachesthe closure of S at a point that does not belong to Sthen the trajectory must be directed away from S thatis a trajectory cannot enter S through a point thatbelongs to the closure of S but not to S FurthermoreA2 ensures that when a trajectory intersects the resettingset S it instantaneously exits S Finally we note thatif hellip0 x0 uc0dagger 2 S then the system initially resets toxDagger

0 ˆ x0 Dagger fdhellipx0dagger Dagger Gdhellipx0daggerudhellip0dagger which serves as theinitial condition for the continuous dynamics (1)

Remark 1 It follows from A2 that resetting removesthe pair helliptk xk uchelliptkdaggerdagger from the resetting set S Thusimmediately after resetting occurs the continuous-time

dynamics (1) and not the resetting law (2) becomesthe active element of the impulsive dynamical systemFurthermore it follows from A1 and A2 that no tra-

jectory can intersect the interior of S Speciregcally itfollows from A1 that a trajectory can only reach Sthrough a point belonging to both S and its boundary

And from A2 it follows that if a trajectory reaches apoint in S that is on the boundary of S then the tra-jectory is instantaneously removed from S Since a

continuous trajectory starting outside of S and inter-secting the interior of S must regrst intersect the bound-ary of S it follows that no trajectory can reach the

interior of S

To show that the resetting times tk are well deregnedand distinct assume that for a given input u 2 ~UU T ˆ infft Aacutehellipt 0 x0 udagger 2 Sg lt 1 Now ad absurdumsuppose t1 is not well deregned that is minft

Aacutehellipt 0 x0 udagger 2 Sg does not exist Since Aacutehellip 0 x0 udagger iscontinuous it follows that AacutehellipT 0 x0 udagger 2 S andsince by assumption minft Aacutehellipt 0 x0 udagger 2 Sg doesnot exist it follows that AacutehellipT 0 x0 udagger 2 SnS Note that

Aacutehellipt 0 x0 udagger ˆ shellipt 0 x0 udagger for every t such that

Aacutehellipfrac12 0 x udagger 62 S for all 0 micro frac12 micro t Now it follows fromA1 that there exists gt 0 such that shellipT Dagger macr 0 x0udagger ˆ AacutehellipT Dagger macr 0 x0 udagger macr 2 hellip0 dagger which implies thatinfft Aacutehellipt 0 x0 udagger 2 Sg gt T which is a contradictionHence AacutehellipT 0 x0 udagger 2 S S and infft Aacutehellipt 0 x0udagger 2 Sg ˆ minft Aacutehellipt 0 x0 udagger 2 Dg which implies thatthe regrst resetting time t1 is well deregned for all initialconditions x0 2 D Next it follows from A2 that t2 isalso well deregned and t2 6ˆ t1 Repeating the above argu-ments it follows that the resetting times tk are wellderegned and distinct

Since the resetting times are well deregned and distinctand since the solution to (1) exists and is unique itfollows that the solution of the impulsive dynamicalsystem (1) (2) also exists and is unique over a forwardtime interval However it is important to note that theanalysis of impulsive dynamical systems can be quiteinvolved In particular such systems can exhibitZenoness beating as well as conmacruence wherein sol-utions exhibit inregnitely many resettings in a regnite-time encounter the same resetting surface a regnite orinregnite number of times in zero time and coincideafter a given point in time In this paper we allow forthe possibility of conmacruence and Zeno solutionsHowever A2 precludes the possibility of beatingFurthermore since not every bounded solution of animpulsive dynamical system over a forward time intervalcan be extended to inregnity due to Zeno solutionswe assume that existence and uniqueness of solutionsare satisreged in forward time For details seeLakshmikantham et al (1989) and Bainov andSimeonov (1989 1995)

In Simeonov and Bainov (1985 1987) Liu (1988)Lakshmikantham et al (1989 1994) Bainov andSimeonov (1989) Kulev and Bainov (1989)Lakshmikantham and Liu (1989) and Hu et al (1989)the resetting set S is deregned in terms of a countablenumber of functions frac12k D hellip0 1dagger and is given by

S ˆ[

k

fhellipfrac12khellipxdagger x uchellipfrac12khellipxdaggerdaggerdagger x 2 Dg hellip7dagger

The analysis of impulsive dynamical systems with aresetting set of the form (7) can be quite involvedFurthermore since impulsive dynamical systems of theform (1)plusmn(4) involve impulses at variable times they aretime-varying systems Here we will consider impulsivedynamical systems involving two distinct forms of theresetting set S In the regrst case the resetting set isderegned by a prescribed sequence of times which areindependent of the state x These equations are thuscalled time-dependent impulsive dynamical systems Inthe second case the resetting set is deregned by a regionin the state space that is independent of time Theseequations are called state-dependent impulsive dynamicalsystems

21 Time-dependent impulsive dynamical systems

Time-dependent impulsive dynamical systems can bewritten as (1)plusmn(4) with S deregned as

S 7 T D Uc hellip8dagger

where

T 7 ft1 t2 g hellip9dagger


and 0 micro t1 lt t2 lt are prescribed resetting timesNow (1)plusmn(2) can be rewritten in the form of the time-dependent impulsive dynamical system

_xxhelliptdagger ˆ fchellipxhelliptdaggerdagger Dagger Gchellipxhelliptdaggerdaggeruchelliptdagger xhellip0dagger ˆ x0 t 6ˆ tk hellip10dagger

centxhelliptdagger ˆ fdhellipxhelliptdaggerdagger Dagger Gdhellipxhelliptdaggerdaggerudhelliptdagger t ˆ tk hellip11dagger

ychelliptdagger ˆ hchellipxhelliptdaggerdagger Dagger Jchellipxhelliptdaggerdaggeruchelliptdagger t 6ˆ tk hellip12dagger

ydhelliptdagger ˆ hdhellipxhelliptdaggerdagger Dagger Jdhellipxhelliptdaggerdaggerudhelliptdagger t ˆ tk hellip13dagger

Since 0 62 T and tk lt tkDagger1 it follows that the Assump-tions A1 and A2 are satisreged Since time-dependentimpulsive dynamical systems involve impulses at a regxedsequence of times they are time-varying systems

Remark 2 Standard continuous-time and discrete-time dynamical systems as well as sampled-datasystems can be treated as special cases of impulsivedynamical systems In particular setting fdhellipxdagger ˆ 0Gdhellipxdagger ˆ 0 hdhellipxdagger ˆ 0 and Jdhellipxdagger ˆ 0 it follows that(10)plusmn(13) has an identical state trajectory as the non-linear continuous-time system


xhellip0dagger ˆ x0 t 0 hellip14dagger

ychelliptdagger ˆ hchellipxhelliptdaggerdagger Dagger Jchellipxhelliptdaggerdaggeruchelliptdagger hellip15dagger

Alternatively setting fchellipxdagger ˆ 0 Gchellipxdagger ˆ 0 hchellipxdagger ˆ 0Jchellipxdagger ˆ 0 tk ˆ kT and T ˆ 1 and assuming fdhellip0dagger ˆ 0it follows that (10)plusmn(13) has an identical state trajectoryas the non-linear discrete-time system

xhellipk Dagger 1dagger ˆ fdhellipxhellipkdaggerdagger Dagger Gdhellipxhellipkdaggerdaggerudhellipkdagger

xhellip0dagger ˆ x0 k 2 N hellip16dagger

ydhellipkdagger ˆ hdhellipxhellipkdaggerdagger Dagger Jdhellipxhellipkdaggerdaggerudhellipkdagger hellip17dagger

Finally to show that (10)plusmn(13) can be used to representsampled-data systems consider the continuous-timenon-linear system (14) and (15) with piecewise constantinput uchelliptdagger ˆ udhelliptkdagger t 2 helliptk tkDagger1Š and sampled measure-ments ydhelliptkdagger ˆ hdhellipxhelliptkdaggerdagger Dagger Jdhellipxhelliptkdaggerdaggerudhelliptkdagger Deregning

xx ˆ permilxT uTc ŠT it follows that the sampled-data system

can be represented as

_xxxx ˆ ff hellipxxhelliptdaggerdagger t 6ˆ tk hellip18dagger

centxxhelliptdagger ˆ0 0

0 iexclI

xxhelliptdagger Dagger

0

I

udhelliptdagger t ˆ tk hellip19dagger

yhelliptdagger ˆ hhhellipxxhelliptdaggerdagger t 6ˆ tk hellip20dagger

ydhelliptdagger ˆ hhdhellipxxhelliptdaggerdagger Dagger JJdhellipxxhelliptdaggerdaggerudhelliptdagger t ˆ tk hellip21dagger

where

ff hellipxxdagger ˆfchellipxdagger Dagger Gchellipxdaggeruc

0

hhhellipxxdagger ˆ hchellipxdagger Dagger Jchellipxdaggeruc

hhdhellipxxdagger ˆ hdhellipxdagger JJdhellipxxdagger ˆ Jdhellipxdagger

and new input variable udhelliptkdagger

Remark 3 The time-dependent impulsive dynamicalsystem (10)plusmn(13) includes as a special case the impul-sive control problem addressed in Yang (1999) whereinat least one of the state variables of the continuous-time plant can be changed instantaneously to anyvalue given by an impulsive control at a set of controlinstants T

22 State-dependent impulsive dynamical systems

State-dependent impulsive dynamical systems can bewritten as (1)plusmn(4) with S deregned as

S 7 permil0 1dagger Z hellip22dagger

where Z 7 Zx Uc and Zx raquo D Therefore (1)plusmn(4) canbe rewritten in the form of the state-dependent impulsivedynamical system


xhellip0dagger ˆ x0 hellipxhelliptdagger uchelliptdaggerdagger 62 Z hellip23dagger

centxhelliptdagger ˆ fdhellipxhelliptdaggerdagger Dagger Gdhellipxhelliptdaggerdaggerudhelliptdagger

hellipxhelliptdagger uchelliptdaggerdagger 2 Z hellip24dagger

ychelliptdagger ˆ hchellipxhelliptdaggerdagger Dagger Jchellipxhelliptdaggerdaggeruchelliptdagger


ydhelliptdagger ˆ hdhellipxhelliptdaggerdagger Dagger Jdhellipxhelliptdaggerdaggerudhelliptdagger


We assume that if hellipx ucdagger 2 Z then hellipx Dagger fdhellipxdaggerDaggerGdhellipxdaggerud ucdagger 62 Z ud 2 Ud In addition we assume thatif at time t the trajectory hellipxhelliptdagger uchelliptdaggerdagger 2 ZnZ thenthere exists gt 0 such that for 0 lt macr lt hellipxhellipt Dagger macrdaggeruchellipt Dagger macrdaggerdagger 62 Z These assumptions represent the spec-ialization of A1 and A2 for the particular resetting set(22) It follows from these assumptions that for a par-ticular initial condition the resetting times frac12khellipx0 ucdaggerare distinct and well deregned Since the resetting set Zis a subset of the state space and is independent oftime state-dependent impulsive dynamical systems aretime-invariant systems Finally in the case whereS 7 permil0 1dagger D Zuc

where Zucraquo Uc we refer to

(23)plusmn(26) as an input-dependent impulsive dynamicalsystem while in the case where S 7 permil0 1dagger Zx Zuc

we refer to (23)plusmn(26) as an inputstate-dependent impul-sive dynamical system Both these cases represent a gen-


eralization to the impulsive control problem consideredin Yang (1999)

Remark 4 For the state-dependent impulsive dyna-mical system given by (23)plusmn(26) let x 2 n satisfyfdhellipx dagger ˆ 0 Then x 62 Zx To see this suppose x 2 ZxThen x Dagger fdhellipx dagger ˆ x 2 Zx which contradicts the as-sumption that if x 2 Zx then x Dagger fdhellipxdagger Dagger Gdhellipxdaggerud 62Zx ud 2 Ud since 0 2 Ud Speciregcally we note that0 62 Zx

3 Stability theory of impulsive dynamical systems

In this section we present Lyapunov asymptotic andexponential stability theorems for non-linear time-dependent and state-dependent impulsive dynamicalsystems Furthermore for state-dependent impulsivedynamical systems we present new invariant set stabilitytheorems that generalize the BarbashinplusmnKrasovskiiplusmnLaSalle invariance principle (Barbashin and Krasovskii1952 Krasovskii 1959 LaSalle 1960) to impulsivesystems Even though versions of the Lyapunov stabilityresults in this section have appeared in the literature(Bainov and Simeonov 1989 1995 Samoilenko andPerestyuk 1995) the invariant set stability theoremsare new to this paper Note that for addressing the stab-ility of the zero solution of an impulsive dynamicalsystem the usual stability deregnitions are valid

Theorem 1 Suppose there exists a continuously di er-entiable function V D permil0 1dagger satisfying Vhellip0dagger ˆ 0Vhellipxdagger gt 0 x 6ˆ 0 and

V 0hellipxdaggerfchellipxdagger micro 0 x 2 D hellip27dagger

Vhellipx Dagger fdhellipxdaggerdagger micro Vhellipxdagger x 2 D hellip28dagger

Then the zero solution xhelliptdagger sup2 0 of the undisturbedhellipuchelliptdagger udhelliptkdaggerdagger sup2 hellip0 0daggerhellip dagger system hellip10daggerhellip11dagger is Lyapunov

stable Furthermore if the inequality hellip27dagger is strict forall x 6ˆ 0 then the zero solution xhelliptdagger sup2 0 of the undis-turbed hellipuchelliptdagger udhelliptkdaggerdagger sup2 hellip0 0daggerhellip dagger system hellip10dagger hellip11dagger isasymptotically stable Alternatively if there exist scalarsnot shy gt 0 and p 1 such that

notkxkp micro Vhellipxdagger micro shy kxkp x 2 D hellip29dagger

V 0hellipxdaggerfchellipxdagger micro iexclVhellipxdagger x 2 D hellip30dagger

and hellip28dagger holds then the zero solution xhelliptdagger sup2 0 of theundisturbed (hellipuchelliptdagger udhelliptdaggerdagger sup2 hellip0 0dagger) system hellip10dagger hellip11dagger isexponentially stable Finally if D ˆ n and

Vhellipxdagger 1 as kxk 1 hellip31dagger

then the above results are global

Proof Prior to the regrst resetting time we can deter-mine the value of Vhellipxhelliptdaggerdagger as

Vhellipxhelliptdaggerdagger ˆ Vhellipxhellip0daggerdagger Daggerhellip t

0

V 0hellipxhellipfrac12daggerdaggerf hellipxhellipfrac12daggerdagger dfrac12

t 2 permil0 t1Š hellip32dagger

Between consecutive resetting times tk and tkDagger1 we candetermine the value of Vhellipxhelliptdaggerdagger as its initial value plus theintegral of its rate of change along the trajectory xhelliptdaggerthat is

Vhellipxhelliptdaggerdagger ˆ Vhellipxhelliptkdagger Dagger fdhellipxhelliptkdaggerdaggerdagger

Daggerhellipt

tk

V 0hellipxhellipfrac12daggerdaggerf hellipxhellipfrac12daggerdagger dfrac12 t 2 helliptk tkDagger1Š hellip33dagger

for k ˆ 1 2 Adding and subtracting Vhellipxhelliptkdaggerdagger toand from the right hand side of (33) yields

Vhellipxhelliptdaggerdagger ˆ Vhellipxhelliptkdaggerdagger Dagger permilVhellipxhelliptkdagger Dagger fdhellipxhelliptkdaggerdaggerdagger iexcl VhellipxhelliptkdaggerdaggerŠ

Daggerhellipt

tk


and in particular at time tkDagger1

VhellipxhelliptkDagger1daggerdagger ˆ Vhellipxhelliptkdaggerdagger Dagger permilVhellipxhelliptkdagger Dagger fdhellipxhelliptkdaggerdaggerdagger

iexcl VhellipxhelliptkdaggerdaggerŠ DaggerhelliptkDagger1

tk

V 0hellipxhellipfrac12daggerdaggerf hellipxhellipfrac12daggerdagger dfrac12 hellip35dagger

By recursively substituting (35) into (34) and ultimatelyinto (32) we obtain

Vhellipxhelliptdaggerdagger ˆ Vhellipxhellip0daggerdagger Daggerhellipt

0


DaggerXk

iˆ1

permilVhellipxhelliptidagger Dagger fdhellipxhelliptidaggerdaggerdagger iexcl VhellipxhelliptidaggerdaggerŠ

t 2 helliptk tkDagger1Š hellip36dagger

If we allow t0 7 0 andP0

iˆ1 7 0 then (36) is valid fork 2 N From (36) and (28) we obtain

Vhellipxhelliptdaggerdagger micro Vhellipxhellip0daggerdagger Daggerhellipt

0


t 0 hellip37dagger

Furthermore it follows from (27) that

Vhellipxhelliptdaggerdagger micro Vhellipxhellip0daggerdagger t 0 hellip38dagger

so that Lyapunov stability follows from standardarguments

Next it follows from (28) and (36) that

Vhellipxhelliptdaggerdagger iexcl Vhellipxhellipsdaggerdagger microhellipt

s

V 0hellipxhellipfrac12daggerdaggerf hellipxhellipfrac12daggerdagger dfrac12 t gt s hellip39dagger

and assuming strict inequality in (27) we obtain

Vhellipxhelliptdaggerdagger lt Vhellipxhellipsdaggerdagger t gt s hellip40dagger


provided xhellipsdagger 6ˆ 0 Asymptotic and exponential stabilityand with (31) global asymptotic and exponential stab-ility then follow from standard arguments amp

Remark 5 If in Theorem 1 the inequality (28) isstrict for all x 6ˆ 0 as opposed to the inequality (27)and an inregnite number of resetting times are used thatis the set T ˆ ft1 t2 g is inregnitely countable thenthe zero solution xhelliptdagger sup2 0 of the undisturbed system(10) (11) is also asymptotically stable A similar re-mark holds for Theorem 2 below

Remark 6 In the proof of Theorem 1 we note thatassuming strict inequality in (27) the inequality (40) isobtained provided xhellipsdagger 6ˆ 0 This proviso is necessarysince it may be possible to reset the states to theorigin in which case xhellipsdagger ˆ 0 for a regnite value of s Inthis case for t gt s we have Vhellipxhelliptdaggerdagger ˆ Vhellipxhellipsdaggerdagger ˆVhellip0dagger ˆ 0 This situation does not present a problemhowever since reaching the origin in regnite time is astronger condition than reaching the origin as t 1

Remark 7 Theorem 1 presents su cient conditions fortime-dependent impulsive dynamical systems in termsof Lyapunov functions that do not depend explicitlyon time Since time-dependent impulsive dynamicalsystems are time-varying Lyapunov functions that ex-plicitly depend on time can also be considered How-ever in this case the conditions on the Lyapunov func-tions required to guarantee stability are signiregcantlyharder to verify For further details see Bainov andSimeonov (1989) Samoilenko and Perestyuk (1995)and Ye et al (1998 a)

Next we state a stability theorem for non-linearstate-dependent impulsive dynamical systems


V 0hellipxdaggerfchellipxdagger micro 0 x 62 Zx hellip41dagger

Vhellipx Dagger fdhellipxdaggerdagger micro Vhellipxdagger x 2 Zx hellip42dagger

Then the zero solution xhelliptdagger sup2 0 of the undisturbedhellipuchelliptdagger udhelliptkdaggerdagger sup2 hellip0 0daggerhellip dagger system hellip23dagger hellip24dagger is Lyapunov

stable Furthermore if the inequality hellip41dagger is strict forall x 6ˆ 0 then the zero solution xhelliptdagger sup2 0 of the undis-turbed hellipuchelliptdagger udhelliptkdaggerdagger sup2 hellip0 0daggerhellip dagger system hellip23dagger hellip24dagger isasymptotically stable Alternatively if there exist scalars

not shy gt 0 and p 1 such that hellip29dagger holds

V 0hellipxdaggerfchellipxdagger micro iexclVhellipxdagger x 62 Zx hellip47dagger

and hellip42dagger holds then the zero solution xhelliptdagger sup2 0 of theundisturbed (hellipuchelliptdagger udhelliptdaggerdagger sup2 hellip0 0dagger) system hellip23dagger hellip23dagger isexponentially stable Finally if D ˆ n and hellip31dagger is satis-reged then the above results are global

Proof For S ˆ permil0 1dagger Zx it follows from Assump-tions A1 and A2 that the resetting times frac12khellipx0dagger arewell deregned and distinct for every trajectory of (23)(24) with hellipuchelliptdagger udhelliptkdaggerdagger sup2 hellip0 0dagger Now the proof fol-lows as in the proof of Theorem 1 with tk replaced byfrac12khellipx0dagger amp

Remark 8 To examine the stability of linear state-dependent impulsive systems set fchellipxdagger ˆ Acx andfdhellipxdagger ˆ hellipAd iexcl Indaggerx in Theorem 2 Considering thequadratic Lyapunov function candidate Vhellipxdagger ˆ xTPxwhere P gt 0 it follows from Theorem 2 that the con-ditions

xThellipATc P Dagger PAcdaggerx lt 0 x 62 Zx hellip44dagger

xThellipATd PAd iexcl Pdaggerx micro 0 x 2 Zx hellip48dagger

establish asymptotic stability for linear state-dependentimpulsive systems These conditions are implied byP gt 0 AT

c P Dagger PAc lt 0 and ATd PAd iexcl P micro 0 which can

be solved using a linear matrix inequality (LMI) feasi-bility problem (Boyd et al 1994)

Next we generalize the BarbashinplusmnKrasovskiiplusmnLaSalle invariance principle (Barbashin and Krasovskii1952 Krasovskii 1959 LaSalle 1960) to state-dependentimpulsive dynamical systems Recall that a state-dependent impulsive dynamical system is time-invariantand hence shellipt Dagger frac12 frac12 x0 0dagger ˆ shellipt 0 x0 0dagger for all x0 2 Dt frac12 2 permil0 1dagger For simplicity of exposition in the remain-der of this section we denote the trajectory shellipt 0 x0 0daggerby shellipt x0dagger and let the map st D D be deregned bysthellipxdagger 7 shellipt x0dagger x0 2 D for a given t 0 The followingderegnitions and key theorem are needed for this result

Deregnition 1 Consider the non-linear impulsivedynamical system G given by (23) (24) withhellipuchelliptdagger udhelliptkdaggerdagger sup2 hellip0 0dagger The trajectory xhelliptdagger 2 D sup3 nt 0 of G denotes the solution to (23) (24) corre-sponding to the initial condition xhellip0dagger ˆ x0 evaluatedat time t The trajectory xhelliptdagger t 0 of G is bounded ifthere exists reg gt 0 such that kxhelliptdaggerk lt reg t 0

Deregnition 2 Consider the non-linear impulsivedynamical system G given by (23) (24) withhellipuchelliptdagger udhelliptkdaggerdagger sup2 hellip0 0dagger A set M sup3 D is a positively in-variant set for the dynamical system G if sthellipMdagger sup3 Mfor all t 0 where sthellipMdagger 7 fsthellipxdagger x 2 Mg A setM sup3 D is an invariant set for the dynamical system Gif sthellipMdagger ˆ M for all t 0

Deregnition 3 p 2 middotDD raquo n is a positive limit point ofthe trajectory xhelliptdagger t 0 if there exists a monotonicsequence ftng1

nˆ0 of non-negative real numbers withtn 1 as n 1 such that xhelliptndagger p as n 1 Theset of all positive limit points of xhelliptdagger t 0 is the posi-tive limit set hellipx0dagger of xhelliptdagger t 0


The following key assumption is needed for thestatement of the next result

Assumption 1 Consider the impulsive dynamicalsystem G given by hellip23dagger hellip24dagger with hellipuchelliptdagger udhelliptkdaggerdagger sup2 hellip0 0daggerand let shellipt x0dagger t 0 denote the solution to hellip23dagger hellip24daggerwith initial condition x0 Then for every x0 2 D thereexists T x0

sup3 permil0 1dagger such that permil0 1daggernT x0is countable

and for every gt 0 and t 2 T x0 there exists

macrhellip x0 tdagger gt 0 such that if kx0 iexcl yk lt macrhellip x0 tdagger y 2 Dthen kshellipt x0dagger iexcl shellipt ydaggerk lt

Assumption 1 is a generalization of the standardcontinuous dependence property for dynamical systemswith continuous macrows to dynamical systems with dis-continuous macrows Speciregcally by letting T x0

ˆ T x0ˆ

permil0 1dagger where T x0denotes the closure of the set T x0

Assumption 1 specializes to the classical continuous de-pendence of solutions of a given dynamical system withrespect to the systemrsquos initial conditions x0 2 D(Vidyasagar 1993) If in addition x0 ˆ 0 shellipt 0dagger ˆ 0t 0 and macrhellip 0 tdagger can be chosen independent of tthen continuous dependence implies the classicalLyapunov stability of the zero trajectory shellipt 0dagger ˆ 0t 0 Hence Lyapunov stability of motion can be inter-preted as continuous dependence of solutions uniformlyin t for all t 0 Conversely continuous dependence ofsolutions can be interpreted as Lyapunov stability ofmotion for every regxed time t (Vidyasagar 1993)Analogously Lyapunov stability of impulsive dynami-cal systems as deregned in Lakshmikantham et al (1989)can be interpreted as quasi-continuous dependence of sol-utions (ie Assumption 1) uniformly in t for all t 2 T x0

For the next result note that p is a positive limit

point of the trajectory shellipt x0dagger t 0 if and only ifthere exists a monotonic sequence ftng1

nˆ0 raquo T x0 with

tn 1 as n 1 such that shelliptn x0dagger p as n 1 Tosee this let p 2 hellipx0dagger and let T x0

be a dense subset of thesemi-inregnite interval permil0 1dagger In this case it follows thatthere exists an unbounded sequence ftng1

nˆ0 such thatlimn1 shelliptn x0dagger ˆ p Hence for every gt 0 there existsn gt 0 such that kshelliptn x0dagger iexcl pk lt =2 Furthermoresince shellip x0dagger is left-continuous and T x0

is a dense subsetof permil0 1dagger there exists ttn 2 T x0

ttn micro tn such thatkshellipttn x0dagger iexcl shelliptn x0daggerk lt =2 and hence kshellipttn x0dagger iexcl pk microkshelliptn x0dagger iexcl pk Dagger kshellipttn x0dagger iexcl shelliptn x0daggerk lt Using thisprocedure with ˆ 1 1=2 1=3 we can constructan unbounded sequence fttkg1

kˆ1 raquo T x0 such that

limk1 shellipttk x0dagger ˆ p Hence p 2 hellipx0dagger if and only ifthere exists a monotonic sequence ftng1


tn 1 as n 1 such that shelliptn x0dagger p as n 1Next we state and prove a fundamental result on

positive limit sets for impulsive dynamical systemsThe result generalizes the classical results on positivelimit sets to systems with left-continuous macrows Forthe remainder of the paper the notation shellipt x0dagger

M sup3 D as t 1 denotes the fact that limt1 shellipt x0daggerevolves in M that is for each gt 0 there exists T gt 0such that disthellipshellipt x0dagger Mdagger lt for all t gt T wheredisthellipp Mdagger 7 infx2M kp iexcl xk

Theorem 3 Consider the impulsive dynamical system Ggiven by hellip23dagger hellip24dagger with hellipuchelliptdagger udhelliptkdaggerdagger sup2 hellip0 0dagger assumeAssumption 1 holds and suppose the trajectory xhelliptdagger of Gis bounded for all t 0 Then the positive limit set

hellipx0dagger of xhelliptdagger t 0 is a non-empty compact invariantset Furthermore xhelliptdagger hellipx0dagger as t 1

Proof Let shellipt x0dagger t 0 denote the solution to Gwith initial condition x0 2 D Since shellipt x0dagger is boundedfor all t 0 it follows from the BolzanoplusmnWeierstrasstheorem (Royden 1988) that every sequence in thepositive orbit regDaggerhellipx0dagger 7 fshellipt x0dagger t 2 permil0 1daggerg has atleast one accumulation point y 2 D as t 1 andhence hellipx0dagger is non-empty Furthermore since shellipt x0daggert 0 is bounded it follows that hellipx0dagger is bounded Toshow that hellipx0dagger is closed let fyig1

iˆ0 be a sequence con-tained in hellipx0dagger such that limi1 yi ˆ y Now sinceyi y as i 1 it follows that for every gt 0 thereexists i such that ky iexcl yik lt =2 Next since yi 2 hellipx0daggerit follows that for every T gt 0 there exists t T suchthat kshellipt x0dagger iexcl yik lt =2 Hence it follows that forevery gt 0 and T gt 0 there exists t T such thatkshellipt x0dagger iexcl yk micro kshellipt x0dagger iexcl yik Dagger ky iexcl yik lt which im-plies that y 2 hellipx0dagger and hence hellipx0dagger is closed Thussince hellipx0dagger is closed and bounded hellipx0dagger is compact

Next to show positive invariance of hellipx0dagger lety 2 hellipx0dagger so that there exists an increasing unboundedsequence ftng1

nˆ0 raquo T x0such that shelliptn x0dagger y as

n 1 Now it follows from Assumption 1 that forevery gt 0 and t 2 T y there exists macrhellip y tdagger gt 0 suchthat ky iexcl zk lt macrhellipy tdagger z 2 D implies kshellipt ydagger iexcl shellipt zdaggerk lt or equivalently for every sequence fyig

1iˆ1 converging

to y and t 2 T y limi1 shellipt yidagger ˆ shellipt ydagger Now since byassumption there exists a unique solution to G it followsthat the semi-group property shellipfrac12 shellipt x0daggerdagger ˆ shellipt Dagger frac12 x0daggerholds Furthermore since shelliptn x0dagger y as n 1 itfollows from the semi-group property that shellipt ydagger ˆshellipt limn1 shelliptn x0daggerdagger ˆ limn1 shellipt Dagger tn x0dagger 2 hellipx0dagger forall t 2 T y Hence shellipt ydagger 2 hellipx0dagger for all t 2 T y Nextlet t 2 permil0 1daggernT y and note that since T y is dense inpermil0 1dagger there exists a sequence ffrac12ng1

nˆ0 such that frac12n micro tfrac12n 2 T y and limn1 frac12n ˆ t Now since shellip ydagger is left-con-tinuous it follows that limn1 shellipfrac12n ydagger ˆ shellipt ydagger Finallysince hellipx0dagger is closed and shellipfrac12n ydagger 2 hellipx0dagger n ˆ 1 2 itfollows that shellipt ydagger ˆ limn1 shellipfrac12n ydagger 2 hellipx0dagger Hencesthelliphellipx0daggerdagger sup3 hellipx0dagger t 0 establishing positive invarianceof hellipx0dagger

Now to show invariance of hellipx0dagger let y 2 hellipx0dagger sothat there exists an increasing unbounded sequenceftng

1nˆ0 such that shelliptn x0dagger y as n 1 Next let

t 2 T x0and note that there exists N such that tn gt t


n N Hence it follows from the semi-group prop-erty that shellipt shelliptn iexcl t x0daggerdagger ˆ shelliptn x0dagger y as n 1Now it follows from the BolzanoplusmnWeierstass theorem(Royden 1988) that there exists a subsequence znk

of thesequence zn ˆ shelliptn iexcl t x0dagger n ˆ N N Dagger 1 suchthat znk

z 2 D and by deregnition z 2 hellipx0dagger Nextit follows from Assumption 1 that limk1 shellipt znk

dagger ˆshellipt limk1 znk

dagger and hence y ˆ shellipt zdagger which impliesthat hellipx0dagger sup3 sthelliphellipx0daggerdagger t 2 T x0

Next let t 2 permil0 1daggernT x0

let tt 2 T x0be such that tt gt t and consider y 2 hellipx0dagger

Now there exists zz 2 hellipx0dagger such that y ˆ shelliptt zzdagger and itfollows from the positive invariance of hellipx0dagger thatz ˆ shelliptt iexcl t zzdagger 2 hellipx0dagger Furthermore it follows fromthe semi-group property that shellipt zdagger ˆ shellipt shelliptt iexcl t zzdaggerdagger ˆshelliptt zzdagger ˆ y which implies that for all t 2 permil0 1daggernT x0

and for every y 2 hellipx0dagger there exists z 2 hellipx0dagger suchthat y ˆ shellipt zdagger Hence hellipx0dagger sup3 sthelliphellipx0daggerdagger t 0 Nowusing positive invariance of hellipx0dagger it follows thatsthelliphellipx0daggerdagger ˆ hellipx0dagger t 0 establishing invariance of thepositive limit set hellipx0dagger

Finally to show shellipt x0dagger hellipx0dagger as t 1 supposead absurdum shellipt x0dagger 6 hellipx0dagger as t 1 In this casethere exists an deg gt 0 and a sequence ftng1

nˆ0 withtn 1 as n 1 such that

infp2hellipx0dagger

kshelliptn x0dagger iexcl pk n 0

However since shellipt x0dagger t 0 is bounded the boundedsequence fshelliptn x0daggerg

1nˆ0 contains a convergent sub-

sequence fshelliptn x0daggerg1nˆ0 such that shelliptn x0dagger p 2 hellipx0dagger

as n 1 which contradicts the original suppositionHence shellipt x0dagger hellipx0dagger as t 1 amp

Remark 9 Note that the compactness of the positivelimit set hellipx0dagger depends only on the boundedness of thetrajectory shellipt x0dagger t 0 whereas the left-continuityand Assumption 1 are key in proving invariance of thepositive limit set hellipx0dagger In classical dynamical systemswhere the trajectory shellip dagger is assumed to be continuousin both its arguments both the left-continuity and As-sumption 1 are trivially satisreged Finally we note thatunlike dynamical systems with continuous macrows theomega limit set of an impulsive dynamical system maynot be connected

Henceforth we assume that fchellip dagger fdhellip dagger and Zx aresuch that Assumption 1 holds Su cient conditions thatguarantee that the non-linear impulsive dynamicalsystem G given by (23) (24) satisreges Assumption 1 aregiven in Chellaboina et al (2000) Next we present themain result of this section characterizing impulsivedynamical system limit sets in terms of C1 functionsFor this result deregne the notation Viexcl1hellipregdagger 7 fx 2 QVhellipxdagger ˆ regg where reg 2 Q sup3 D and V Q is a con-tinuously di erentiable function and let Mreg denote thelargest invariant set (with respect to G) contained inViexcl1hellipregdagger

Theorem 4 Consider the impulsive dynamical system Ggiven by hellip23dagger hellip24dagger with hellipuchelliptdagger udhelliptkdaggerdagger sup2 hellip0 0dagger assumeDc raquo D is a compact positively invariant set with respectto hellip23dagger hellip24dagger and assume that there exists a continuouslydi erentiable function V Dc such that

V 0hellipxdaggerfchellipxdagger micro 0 x 2 Dc x 62 Zx hellip46dagger

Vhellipx Dagger fdhellipxdaggerdagger micro Vhellipxdagger x 2 Dc x 2 Zx hellip47dagger

Let R 7 fx 2 Dc x 62 Zx V 0hellipxdaggerfchellipxdagger ˆ 0g [ fx 2 Dcx 2 Zx Vhellipx Dagger fdhellipxdaggerdagger ˆ Vhellipxdaggerg and let M denote thelargest invariant set contained in R If x0 2 Dc thenxhelliptdagger M as t 1

Proof Using identical arguments as in the proof ofTheorem 1 it follows that for all t 2 hellipfrac12khellipx0dagger frac12kDagger1hellipx0daggerŠ

Vhellipxhelliptdaggerdagger iexcl Vhellipxhellip0daggerdagger ˆhellipt

0

V 0hellipxhellipfrac12daggerdaggerfchellipxhellipfrac12daggerdagger dfrac12

DaggerXk

iˆ1

permilVhellipxhellipfrac12ihellipx0daggerdagger Dagger fdhellipxhellipfrac12ihellipx0daggerdaggerdaggerdagger

iexcl Vhellipxhellipfrac12ihellipx0daggerdaggerdaggerŠ

Hence it follows from (46) and (47) that Vhellipxhelliptdaggerdagger microVhellipxhellip0daggerdagger t 0 Using a similar argument it followsthat Vhellipxhelliptdaggerdagger micro Vhellipxhellipfrac12daggerdagger t frac12 which implies thatVhellipxhelliptdaggerdagger is a non-increasing function of time SinceVhellip dagger is continuous on a compact set Dc there existsshy 2 such that Vhellipxdagger shy x 2 Dc Furthermore sinceVhellipxhelliptdaggerdagger t 0 is non-increasing regx0

7 limt1 Vhellipxhelliptdaggerdaggerx0 2 Dc exists Now for all y 2 hellipx0dagger there exists anincreasing unbounded sequence ftng1

nˆ0 such thatxhelliptndagger y as n 1 and since Vhellip dagger is continuous itfollows that

Vhellipydagger ˆ V limn1

xhelliptndaggerplusmn sup2

ˆ limn1

Vhellipxhelliptndaggerdagger ˆ regx0

Hence y 2 Viexcl1hellipregx0dagger for all y 2 hellipx0dagger or equivalently

hellipx0dagger sup3 Viexcl1hellipregx0dagger Now since Dc is compact and posi-

tively invariant it follows that xhelliptdagger t 0 is boundedfor all x0 2 Dc and hence it follows from Theorem 3 that

hellipx0dagger is a non-empty compact invariant set Thus

hellipx0dagger is a subset of the largest invariant set containedin Viexcl1hellipregx0

dagger that is hellipx0dagger sup3 Mregx0 Hence for every

x0 2 Dc there exists regx02 such that hellipx0dagger sup3 Mregx0

where Mregx0

is the largest invariant set contained inViexcl1hellipregx0

dagger which implies that Vhellipxdagger ˆ regx0 x 2 hellipx0dagger

Now since Mregx0is an invariant set it follows that

for all xhellip0dagger 2 Mregx0 xhelliptdagger 2 Mregx0

t 0 and thus_VVhellipxhelliptdaggerdagger 7 dVhellipxhelliptdaggerdagger= dt ˆ V 0hellipxhelliptdaggerdaggerfchellipxhelliptdaggerdagger ˆ 0 for all

xhelliptdagger 62 Zx and Vhellipxhelliptdagger Dagger fdhellipxhelliptdaggerdaggerdagger ˆ Vhellipxhelliptdaggerdagger for allxhelliptdagger 2 Zx Thus Mregx0

is contained in M which is thelargest invariant set contained in R Hence xhelliptdagger Mas t 1 amp


Finally using Theorem 4 we provide a generalizationof Theorem 2 for local asymptotic stability of a non-linear state-dependent impulsive dynamical system

Corollary 1 Consider the impulsive dynamical systemG given by hellip23dagger hellip24dagger with hellipuchelliptdagger udhelliptkdaggerdagger sup2 hellip0 0dagger as-sume Dc raquo D is a compact positively invariant set withrespect to hellip23dagger hellip24dagger such that 0 2 8D and assume thatthere exists a continuously di erentiable functionV Dc permil0 1dagger such that Vhellip0dagger ˆ 0 Vhellipxdagger gt 0 x 6ˆ 0and hellip46dagger hellip47dagger are satisreged Furthermore assume thatthe set R 7 fx 2 Dc x 62 Zx V 0hellipxdaggerfchellipxdagger ˆ 0g [ fx 2 Dcx 2 Zx Vhellipx Dagger fdhellipxdaggerdagger ˆ Vhellipxdaggerg contains no invariant setother than the set f0g Then the zero solution xhelliptdagger sup2 0to hellip23dagger hellip24dagger is asymptotically stable and Dc is a subsetof the domain of attraction of hellip23dagger hellip24dagger

Proof Lyapunov stability of the zero solutionxhelliptdagger sup2 0 to (23) (24) follows from Theorem 2 Nextit follows from Theorem 4 that if x0 2 Dc thenhellipx0dagger sup3 M where M denotes the largest invariant setcontained in R which implies that M ˆ f0g Hencexhelliptdagger M ˆ f0g as t 1 establishing asymptoticstability of the zero solution xhelliptdagger sup2 0 to (23) (24) amp

Remark 10 Setting D ˆ n and requiring Vhellipxdagger 1as kxk 1 in Corollary 1 it follows that the zerosolution xhelliptdagger sup2 0 of the undisturbed system (23) (24)is globally asymptotically stable

4 Dissipative impulsive dynamical systems inputplusmnoutput and state properties

Many of the great landmarks of feedback controltheory are associated with the theory of absolute stab-ility and dissipativity The Aizerman conjecture and theLureAcirc problem as well as the circle and Popov criteriaare extensively developed in the classical monographs ofAizerman and Gantmacher (1964) Lefschetz (1965) andPopov (1973) Since absolute stability theory concernsthe stability of a dynamical system for classes of feed-back non-linearities which as noted in Zames (1966)Safonov (1980) and Haddad and Bernstein (1993) canreadily be interpreted as an uncertainty model it is notsurprising that dissipativity theory forms the basis ofmodern-day robust stability analysis and synthesis(Haddad and Bernstein 1993 1994 Hadded et al1994) Furthermore since Lyapunov functions can beviewed as generalizations of energy functions for generalnon-linear dynamical systems the notion of dissipativ-ity with appropriate storage functions and supply ratescan be used to construct Lyapunov functions for non-linear feedback systems by appropriately combiningstorage functions for each subsystem In this sectionwe extend the notion of dissipative dynamical systemsto develop the concept of dissipativity for impulsivedynamical systems

In this section we consider non-linear impulsivedynamical systems G of the form given by (1)plusmn(4) witht 2 hellipt xhelliptdagger uchelliptdaggerdagger 62 S and hellipt xhelliptdagger uchelliptdaggerdagger 2 S replacedby Xhellipt xhelliptdagger uchelliptdaggerdagger 6ˆ 0 and Xhellipt xhelliptdagger uchelliptdaggerdagger ˆ 0 respect-ively where X D Uc Note that settingXhellipt xhelliptdagger uchelliptdaggerdagger ˆ hellipt iexcl t1daggerhellipt iexcl t2dagger where tk 1 ask 1 (1)plusmn(4) reduce to (10)plusmn(13) while settingXhellipt xhelliptdagger uchelliptdaggerdagger ˆ Xhellipxhelliptdaggerdagger where X D D is a supportfunction characterizing the manifold Z (1)plusmn(4) reduce to(23)plusmn(26) Furthermore we assume that the system func-tions fchellip dagger fdhellip dagger Gchellip dagger Gdhellip dagger hchellip dagger hdhellip dagger Jchellip dagger and Jdhellip daggerare smooth (at least continuously di erentiable map-pings) In addition for the non-linear dynamical system(1) we assume that the required properties for the exist-ence and uniqueness of solutions are satisreged such that(1) has a unique solution for all t 2 (Lakshmikanthamet al 1989 Bainov and Simeonov 1995) For theimpulsive dynamical system G given by (1)plusmn(4) afunction helliprchellipuc ycdagger rdhellipud yddaggerdagger where rc Uc Yc and rd Ud Yd are such that rchellip0 0dagger ˆ 0 andrdhellip0 0dagger ˆ 0 is called a supply rate if rchellipuc ycdagger is locallyintegrable that is for all inputplusmnoutput pairs uchelliptdagger 2 Ucychelliptdagger 2 Yc rchellip dagger satisreges

bdquo tt

tjrchellipuchellipsdagger ychellipsdaggerdaggerj ds lt 1

t tt 0 Note that since all inputplusmnoutput pairsudhelliptkdagger 2 Ud ydhelliptkdagger 2 Yd are deregned for discrete instantsrdhellip dagger satisreges

Pk2N permiltttdagger

jrdhellipudhelliptkdagger ydhelliptkdaggerdaggerj lt 1 wherek 2 N permiltttdagger 7 fk t micro tk lt ttg

Deregnition 4 An impulsive dynamical system G of theform (1)plusmn(4) is dissipative with respect to the supply ratehelliprc rddagger if the dissipation inequality

0 microhellipT

t0

rchellipuchelliptdagger ychelliptdaggerdagger dt DaggerX

k2N permilt0 T dagger

rdhellipudhelliptkdagger ydhelliptkdaggerdagger

T t0 hellip48dagger

is satisreged for all T t0 with xhellipt0dagger ˆ 0 An impulsivedynamical system G of the form (1)plusmn(4) is exponentiallydissipative with respect to the supply rate helliprc rddagger ifthere exists a constant gt 0 such that the dissipationinequality (48) is satisreged with rchellipuchelliptdagger ychelliptdaggerdagger replacedby etrchellipuchelliptdagger ychelliptdaggerdagger and rdhellipudhelliptkdagger ydhelliptkdaggerdagger replaced byetk rdhellipudhelliptkdagger ydhelliptkdaggerdagger for all T t0 with xhellipt0dagger ˆ 0 Animpulsive dynamical system is lossless with respect tothe supply rate helliprc rddagger if the dissipation inequality (48)is satisreged as an equality for all T t0 withxhellipt0dagger ˆ xhellipTdagger ˆ 0

Next deregne the available storage Vahellipt0 x0dagger of theimpulsive dynamical system G by

Vahellipt0 x0dagger 7 iexcl infhellipuchellip daggerudhellip daggerdagger T t0

hellipT

t0

rchellipuchelliptdagger ychelliptdaggerdagger dt

DaggerX



hellip49dagger


where xhelliptdagger t t0 is the solution to (1)plusmn(4) with admis-sible inputs hellipuchellip dagger udhellip daggerdagger and xhellipt0dagger ˆ x0 Note thatVahellipt0 x0dagger 0 for all hellipt xdagger 2 D since Vahellipt0 x0dagger isthe supremum over a set of numbers containing thezero element hellipT ˆ t0dagger It follows from (49) that theavailable storage of a non-linear impulsive dynamicalsystem G is the maximum amount of generalized storedenergy which can be extracted from G at any time T Furthermore deregne the available exponential storage ofthe impulsive dynamical system G by


hellipT

t0

etrchellipuchelliptdagger ychelliptdaggerdagger dt

DaggerX

k2N permilt0 Tdagger

etk rdhellipudhelliptkdagger ydhelliptkdaggerdagger

hellip50dagger

where xhelliptdagger t t0 is the solution of (1)plusmn(4) with admis-sible inputs hellipuchellip dagger udhellip daggerdagger and xhellipt0dagger ˆ x0

Remark 11 Note that in the case of (time-invariant)state-dependent impulsive dynamical systems theavailable storage is time-invariant that is Vahellipt0 x0dagger ˆVahellipx0dagger Furthermore the available exponential storagesatisreges

Vahellipt0 x0dagger ˆ iexcl infhellipuchellip daggerudhellip daggerdagger T t0

hellipT

t0


DaggerX



ˆ iexclet0 infhellipuchellip daggerudhellip daggerdagger T 0

hellipT

0


DaggerX

k2N permil0T dagger


ˆ et0 VVahellipx0dagger hellip51dagger

where

VVahellipx0dagger 7 iexcl infhellipuchellip daggerudhellip daggerdagger T 0

hellipT

0


DaggerX

k2N permil0T dagger


hellip52dagger

Next we show that the available storage (respavailable exponential storage) is regnite if and only if Gis dissipative (resp exponentially dissipative) In orderto state this result we require two more deregnitions

Deregnition 5 Consider the impulsive dynamicalsystem G given by (1)plusmn(4) Assume G is dissipative with

respect to the supply rate helliprc rddagger A continuous non-negative-deregnite function Vs D satisfying

VshellipT xhellipTdaggerdagger micro Vshellipt0 xhellipt0daggerdagger DaggerhellipT

t0


DaggerX


rdhellipudhelliptkdagger ydhelliptkdaggerdagger hellip53dagger

where xhelliptdagger t t0 is a solution to (1)plusmn(4) withhellipuchelliptdagger udhelliptkdaggerdagger 2 U c Ud and xhellipt0dagger ˆ x0 is called a sto-rage function for G A continuous non-negative-deregnitefunction Vs D satisfying

eTVshellipT xhellipTdaggerdagger micro et0 Vshellipt0 xhellipt0daggerdagger

DaggerhellipT

t0


DaggerX


etk rdhellipudhelliptkdagger ydhelliptkdaggerdagger hellip54dagger

is called an exponential storage function for G

Note that Vshellipt xhelliptdaggerdagger is left-continuous on permilt0 1dagger andis continuous everywhere on permilt0 1dagger except on anunbounded closed discrete set T ˆ ft1 t2 g whereT is the set of times when the jumps occur for xhelliptdaggert 0

Deregnition 6 An impulsive dynamical system G givenby (1)plusmn(4) is zero-state observable if hellipuchelliptdagger udhelliptkdaggerdagger sup2hellip0 0dagger hellipychelliptdagger ydhelliptkdaggerdagger sup2 hellip0 0dagger implies xhelliptdagger sup2 0 An im-pulsive dynamical system G given by (1)plusmn(4) is stronglyzero-state observable if hellipuchelliptdagger udhelliptkdaggerdagger sup2 hellip0 0dagger ychelliptdagger sup2 0implies xhelliptdagger sup2 0 An impulsive dynamical system G iscompletely reachable if for all hellipt0 xidagger 2 D thereexist a regnite time ti micro t0 square integrable inputs uchelliptdaggerderegned on permilti t0Š and inputs udhelliptkdagger deregned onk 2 N permiltit0dagger such that the state xhelliptdagger t ti can be dri-ven from xhelliptidagger ˆ 0 to xhellipt0dagger ˆ x0 Finally an impulsivesystem G is minimal if it is zero-state observable andcompletely reachable

Remark 12 Note that strong zero-state observabilityis a stronger condition than zero-state observability Inparticular strong zero-state observability implies zero-state observability but the converse is not necessarilytrue

Theorem 5 Consider the impulsive dynamical system Ggiven by hellip1dagger-hellip4dagger and assume that G is completely reach-able Then G is dissipative (resp exponentially dissipa-tive) with respect to the supply rate helliprc rddagger if and only ifthe available system storage Vahellipt0 x0dagger given by hellip49dagger(resp the available exponential system storageVahellipt0 x0dagger given by hellip50dagger) is regnite for all t0 2 andx0 2 D Moreover if Vahellipt0 x0dagger is regnite for all t0 2and x0 2 D then Vahellipt xdagger hellipt xdagger 2 D is a storage


function (resp exponential storage function) for GFinally all storage functions (resp exponential storagefunctions) Vshellipt xdagger hellipt xdagger 2 D for G satisfy

0 micro Vahellipt xdagger micro Vshellipt xdagger hellipt xdagger 2 D hellip55dagger

Proof Suppose Vahellipt xdagger hellipt xdagger 2 D is regnite Nowit follows from (49) (with T ˆ t0) that Vahellipt xdagger 0hellipt xdagger 2 D Next let xhelliptdagger t t0 satisfy (1)plusmn(4)with admissible inputs hellipuchelliptdagger udhelliptkdaggerdagger t t0 k 2 N permilt0tdaggerand xhellipt0dagger ˆ x0 Since iexclVahellipt xdagger hellipt xdagger 2 Dis given by the inregmum over all admissible inputshellipuchellip dagger udhellip daggerdagger and T t0 in (49) it follows that forall admissible inputs hellipuchellip dagger udhellip daggerdagger and t 2 permilt0 T Š

iexclVahellipt0 x0dagger

microhellipT

t0




ˆhellipt

t0

rchellipuchellipsdagger ychellipsdaggerdagger ds DaggerX

k2N permilt0 tdagger


DaggerhellipT

t


k2N permiltT dagger


which implies

iexclVahellipt0 x0dagger iexclhellip t

t0


iexclX



microhellipT

t


k2N permiltT dagger


Hence

Vahellipt0 x0dagger Daggerhellipt

t0


DaggerX



iexcl infhellipuchellip daggerudhellip daggerdagger T t

hellipT

t

rchellipuchellipsdagger ychellipsdaggerdagger ds

DaggerX

k2N permiltT dagger


ˆ Vahellipt xhelliptdaggerdagger hellip56dagger

which shows that Vahellipt xdagger hellipt xdagger 2 D is a storagefunction for G

Conversely suppose G is dissipative with respect tothe supply rate helliprc rddagger and let t0 2 and x0 2 D Since Gis completely reachable it follows that there exists tt lt t0uchelliptdagger tt micro t lt t0 and udhelliptkdagger k 2 N permilttt0dagger such that xhellipttdagger ˆ 0and xhellipt0dagger ˆ x0 Hence since G is dissipative with respectto the supply rate helliprc rddagger it follows that for all T t0

0 microhellipT

tt


k2N permilttT dagger


ˆhellipt0

tt


k2N permilttt0dagger


DaggerhellipT

t0




and hence there exists W D such that

iexcl1 lt Whellipt0 x0dagger microhellipT

t0


DaggerX



Now it follows from (57) that for all hellipt xdagger 2 D

Vahellipt xdagger ˆ iexcl infhellipuchellip daggerudhellip daggerdagger T t0

hellipT

t0


DaggerX



micro iexclWhellipt xdagger hellip58dagger

and hence the available storage Vahellipt xdagger hellipt xdagger 2 Dis regnite

Next if Vshellipt xdagger hellipt xdagger 2 D is a storage functionthen it follows that for all T t0 and x0 2 D

Vshellipt0 x0dagger VshellipT xhellipTdaggerdagger iexclhellipT

t0


iexclX



iexcl hellipT

t0


DaggerX



which implies


Vshellipt0 x0dagger iexcl infhellipuchellip daggerudhellip daggerdagger T t0

hellipT

t0


DaggerX



ˆ Vahellipt0 x0dagger

Finally the proof for the exponentially dissipativecase follows a similar construction and hence isomitted amp

The following corollary is immediate from Theorem5

Corollary 2 Consider the impulsive dynamical systemG given by hellip1dagger-hellip4dagger and assume that G is completelyreachable Then G is dissipative (resp exponentiallydissipative) with respect to the supply rate helliprc rddagger if andonly if there exists a continuous storage function (respexponential storage function) Vshellipt xdagger hellipt xdagger 2 Dsatisfying hellip53dagger (resp hellip54dagger)

The next result gives necessary and su cient con-ditions for dissipativity exponential dissipativity andlosslessness over an interval t 2 helliptk tkDagger1Š involving theconsecutive resetting times tk and tkDagger1

Theorem 6 Assume G is completely reachable Then Gis dissipative with respect to the supply rate helliprc rddagger if andonly if there exists a continuous non-negative-deregnitefunction Vs D such that for all k 2 N

Vshelliptt xhellipttdaggerdagger iexcl Vshellipt xhelliptdaggerdagger microhellip tt

t


tk lt t micro tt micro tkDagger1 hellip59dagger

Vshelliptk xhelliptkdagger Dagger fdhellipxhelliptkdaggerdagger Dagger Gdhellipxhelliptkdaggerdaggerudhelliptkdaggerdagger

iexclVshelliptk xhelliptkdaggerdagger micro rdhellipudhelliptkdagger ydhelliptkdaggerdagger hellip60dagger

Furthermore G is exponentially dissipative with respect tothe supply rate helliprc rddagger if and only if there exists a con-tinuous non-negative-deregnite function Vs D such that

ettVshelliptt xhellipttdaggerdagger iexcl etVshellipt xhelliptdaggerdagger microhellip tt

t

esrchellipuchellipsdagger ychellipsdaggerdagger ds




Finally G is lossless with respect to the supply rate helliprc rddaggerif and only if there exists a continuous non-negative-deregnite function Vs D such that hellip59dagger and hellip60daggerare satisreged as equalities

Proof Let k 2 N and suppose G is dissipative withrespect to the supply rate helliprc rddagger Then there exists acontinuous non-negative-deregnite function Vs D such that (53) holds Now since for tk lt t micro tt micro tkDagger1N permiltttdagger ˆ 1 (59) is immediate Next note that

VshelliptDaggerk xhelliptDagger

k daggerdagger iexcl Vshelliptk xhelliptkdaggerdagger

microhelliptDagger

k

tk

rchellipuchellipsdagger ychellipsdaggerdagger ds Dagger rdhellipudhelliptkdagger ydhelliptkdaggerdagger hellip63dagger

which since N permiltk tDaggerk

dagger ˆ fkg implies (60)Conversely suppose (59) and (60) hold let tt t 0

and let N permiltttdagger ˆ fi i Dagger 1 jg (Note that if N permiltttdagger ˆ 1the converse is a direct consequence of (53)) In this caseit follows from (59) and (60) that

Vshelliptt xhellipttdaggerdagger iexcl Vshellipt xhelliptdaggerdagger

ˆ Vshelliptt xhellipttdaggerdagger iexcl VshelliptDaggerj xhelliptDagger

j daggerdagger Dagger VshelliptDaggerj xhelliptDaggerj daggerdagger

iexcl VshelliptDaggerjiexcl1 xhelliptDaggerjiexcl1daggerdagger Dagger VshelliptDagger

jiexcl1 xhelliptDaggerjiexcl1daggerdagger iexcl

iexcl VshelliptDaggeri xhelliptDaggeri daggerdagger Dagger VshelliptDagger

i xhelliptDaggeri daggerdagger iexcl Vshellipt xhelliptdaggerdagger


j daggerdagger Dagger Vshelliptj xhelliptjdagger Dagger fdhellipxhelliptjdaggerdagger

Dagger Gdhellipxhelliptjdaggerdaggerudhelliptjdaggerdagger iexcl Vshelliptj xhelliptjdaggerdagger Dagger Vshelliptj xhelliptjdaggerdagger

iexcl VshelliptDaggerjiexcl1 xhelliptDaggerjiexcl1daggerdagger Dagger Dagger Vshellipti xhelliptidagger Dagger fdhellipxhelliptidaggerdagger

Dagger Gdhellipxhelliptidaggerdaggerudhelliptidaggerdagger iexcl Vshellipti xhelliptidaggerdagger Dagger Vshellipti xhelliptidaggerdagger

iexcl Vshellipt xhelliptdaggerdagger

microhellip tt

tDaggerj

rchellipuchellipsdagger ychellipsdaggerdagger ds Dagger rdhellipudhelliptjdagger ydhelliptjdaggerdagger

Daggerhelliptj

tDaggerjiexcl1

rchellipuchellipsdagger ychellipsdaggerdagger ds Dagger Dagger rdhellipudhelliptidagger ydhelliptidaggerdagger

Daggerhellipti

t


ˆhellip tt

t


k2N permiltttdagger


which implies that G is dissipative with respect to thesupply rate helliprc rddagger

Finally similar constructions show that G is expo-nentially dissipative with respect to the supply ratehelliprc rddagger if and only if (61) and (62) are satisreged and Gis lossless with respect to the supply rate helliprc rddagger if andonly if (59) and (60) are satisreged as equalities amp

If in Theorem 6 Vshellip xhellip daggerdagger is continuously di erenti-able ae on permilt0 1dagger except on an unbounded closed dis-crete set T ˆ ft1 t2 g where T is the set of timeswhen jumps occur for xhelliptdagger then an equivalent statementfor dissipativeness of the impulsive dynamical system Gwith respect to the supply rate helliprc rddagger is


_VsVshellipt xhelliptdaggerdagger micro rchellipuchelliptdagger ychelliptdaggerdagger tk lt t micro tkDagger1 hellip64daggercentVshelliptk xhelliptkdaggerdagger micro rdhellipudhelliptkdagger ydhelliptkdaggerdagger k 2 N hellip65dagger

where _VVshellip dagger denotes the total derivative of Vshellipt xhelliptdaggerdaggeralong the state trajectories xhelliptdagger t 2 helliptk tkDagger1Š of theimpulsive dynamical system (1)plusmn(4) and

centVshelliptk xhelliptkdaggerdagger 7 VshelliptDaggerk xhelliptDagger

k daggerdagger iexcl Vshelliptk xhelliptkdaggerdaggerˆ Vshelliptk xhelliptkdagger Dagger fdhellipxhelliptkdaggerdagger

Dagger Gdhellipxhelliptkdaggerdaggerudhelliptkdaggerdagger iexcl Vshelliptk xhelliptkdaggerdaggerk 2 N

denotes the di erence of the storage function Vshellipt xdagger atthe resetting times tk k 2 N of the impulsive dynamicalsystem (1)plusmn(4) Furthermore an equivalent statementfor exponential dissipativeness of the impulsive dynami-cal system G with respect to the supply rate helliprc rddagger isgiven by

_VsVshellipt xhelliptdaggerdagger Dagger Vshellipt xhelliptdaggerdagger micro rchellipuchelliptdagger ychelliptdaggerdagger tk lt t micro tkDagger1

hellip66daggerand (65)

The following theorem provides su cient conditionsfor guaranteeing that all storage functions (resp expo-nential storage functions) of a given dissipative (respexponentially dissipative) impulsive dynamical systemare positive deregnite

Theorem 7 Consider the non-linear impulsive dynami-cal system G given by hellip1daggerplusmnhellip4dagger and assume that G iscompletely reachable and zero-state observable Further-more assume that G is dissipative (resp exponentiallydissipative) with respect to the supply rate helliprc rddagger andthere exist functions microc Yc Uc and microd Yd Ud suchthat microchellip0dagger ˆ 0 microdhellip0dagger ˆ 0 rchellipmicrochellipycdagger ycdagger lt 0 yc 6ˆ 0 andrdhellipmicrodhellipyddagger yddagger lt 0 yd 6ˆ 0 Then all the storage func-tions (resp exponential storage functions) Vshellipt xdaggerhellipt xdagger 2 D for G are positive deregnite that isVshellip 0dagger ˆ 0 and Vshellipt xdagger gt 0 hellipt xdagger 2 D x 6ˆ 0

Proof It follows from Theorem 5 that the availablestorage Vahellipt xdagger hellipt xdagger 2 D is a storage functionfor G Next suppose ad absurdum there exists hellipt xdagger 2

D such that Vahellipt xdagger ˆ 0 x 6ˆ 0 or equivalently

infhellipuchellip daggerudhellip daggerdagger T t0

hellipT

t0

rchellipuchelliptdagger ychelliptdaggerdagger dt Dagger

X



ˆ 0 hellip67dagger

Furthermore suppose there exists permilts tf dagger raquo suchthat ychelliptdagger 6ˆ 0 t 2 permilts tfdagger or ydhelliptkdagger 6ˆ 0 for somek 2 N Now since there exists microc Yc Uc and microdYd Ud such that rchellipmicrochellipycdagger ycdagger lt 0 yc 6ˆ 0 andrdhellipmicrodhellipyddagger yddagger lt 0 yd 6ˆ 0 the inregmum in (67) occurs ata negative value which as a contradiction Hence

ychelliptdagger ˆ 0 ae t 2 and ydhelliptkdagger ˆ 0 for all k 2 N Next since G is zero-state observable it follows thatx ˆ 0 and hence Vahellipt xdagger ˆ 0 if and only if x ˆ 0 Theresult now follows from (55) Finally the proof for theexponentially dissipative case is similar and hence isomitted amp

Next we introduce the concept of required supply ofa non-linear impulsive dynamical system given by (1)plusmn(4) Speciregcally deregne the required supply Vrhellipt0 x0dagger ofthe non-linear impulsive dynamical system G by

Vrhellipt0 x0dagger 7 infhellipuchellip daggerudhellip daggerdagger Tmicrot0

hellipt0

T


DaggerX

k2N permilT t0dagger


hellip68dagger

where xhelliptdagger t T is the solution of (1)plusmn(4) withxhellipTdagger ˆ 0 and xhellipt0dagger ˆ x0 It follows from (68) that therequired supply of a non-linear impulsive dynamicalsystem is the minimum amount of generalized energywhich can be delivered to the impulsive dynamicalsystem in order to transfer it from an initial statexhellipTdagger ˆ 0 to a given state xhellipt0dagger ˆ x0 Similarly deregnethe required exponential supply of the non-linear impul-sive dynamical system G by


hellipt0

T


DaggerX



hellip69dagger

where xhelliptdagger t T is the solution of (1)plusmn(4) withxhellipTdagger ˆ 0 and xhellipt0dagger ˆ x0

Next using the notion of required supply we showthat all storage functions are bounded from above bythe required supply and bounded from below by theavailable storage Hence as in the case of systems withcontinuous macrows (Willems 1972 a) a dissipative impul-sive dynamical system can only deliver to its surround-ings a fraction of its stored generalized energy and canonly store a fraction of the generalized work done to it

Theorem 8 Consider the non-linear impulsive dyna-mical system G given by hellip1daggerplusmnhellip4dagger and assume that G iscompletely reachable Then G is dissipative (resp expo-nentially dissipative) with respect to the supply ratehelliprc rddagger if and only if 0 micro Vrhellipt xdagger lt 1 t 2 x 2 DMoreover if Vrhellipt xdagger is regnite and non-negative for allhellipt0 x0dagger 2 D then Vrhellipt xdagger hellipt xdagger 2 D is astorage function (resp exponential storage function)for G Finally all storage functions (resp exponentialstorage functions) Vshellipt xdagger hellipt xdagger 2 D for G satisfy


0 micro Vahellipt xdagger micro Vshellipt xdagger micro Vrhellipt xdagger lt 1

hellipt xdagger 2 D hellip70dagger

Proof Suppose 0 micro Vrhellipt xdagger lt 1 hellipt xdagger 2 D Nextlet xhelliptdagger t 2 satisfy (1)plusmn(4) with admissible inputs hellipuchelliptdaggerudhelliptkdaggerdagger t 2 k 2 N permilt0tdagger and xhellipt0dagger ˆ x0 Since Vrhellipt xdaggerhellipt xdagger 2 D is given by the inregmum over all admissibleinputs hellipuchellip dagger udhellip daggerdagger and T micro t0 in (68) it follows that forall admissible inputs hellipuchellip dagger udhellip daggerdagger and T micro t micro t0

Vrhellipt0 x0dagger microhellipt0

T


DaggerX

k2N permilT t0 dagger


ˆhellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX

k2N permiltt0 dagger


and hence

Vrhellipt0 x0dagger micro infhellipuchellip daggerudhellip daggerdagger Tmicrot

hellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX



ˆ Vrhellipt xhelliptdaggerdagger Daggerhellipt0

t


DaggerX


rdhellipudhelliptkdagger ydhelliptkdagger hellip71dagger

which shows that Vrhellipt xdagger hellipt xdagger 2 D is a storagefunction for G and hence G is dissipative

Conversely suppose G is dissipative with respect tothe supply rate helliprc rddagger and let t0 2 and x0 2 D Since Gis completely reachable it follows that there existsT lt t0 uchelliptdagger T micro t lt t0 and udhelliptkdagger k 2 N permilT t0dagger suchthat xhellipTdagger ˆ 0 and xhellipt0dagger ˆ x0 Hence since G is dissi-pative with respect to the supply rate helliprc rddagger it followsthat for all T micro t0

0 microhellipt0

T




and hence

0 micro infhellipuchellip daggerudhellip daggerdagger Tmicrot0

hellipt0

T


DaggerX



hellip73dagger

which implies that

0 microVrhellipt0 x0dagger lt 1 hellipt0 x0dagger 2 D hellip74dagger

Next if Vshellip dagger is a storage function for G then itfollows from Theorem 5 that


Furthermore for all T 2 such that xhellipTdagger ˆ 0 it followsthat

Vshellipt0 x0dagger micro VshellipT 0dagger Daggerhellipt0

T


DaggerX



and hence

Vshellipt0 x0dagger micro infhellipuchellip daggerudhellip daggerdagger Tmicrot0

hellipt0

T


DaggerX



ˆ Vrhellipt0 x0dagger lt 1 hellip77dagger

which implies (70) Finally the proof for the exponen-tially dissipative case follows a similar construction andhence is omitted amp

Finally as a direct consequence of Theorems 5 and8 we show that the set of all possible storage func-tions of an impulsive dynamical system forms a convexset An identical result holds for exponential storagefunctions

Proposition 1 Consider the non-linear impulsive dy-namical system G given by hellip1daggerplusmnhellip4dagger with available storageVahellipt xdagger hellipt xdagger 2 D and required supply Vrhellipt xdaggerhellipt xdagger 2 D and assume that G is completely reach-able Then

Vshellipt xdagger 7 notVahellipt xdagger Dagger hellip1 iexcl notdaggerVrhellipt xdagger not 2 permil0 1Š hellip78dagger

is a storage function for G

Proof The result is a direct consequence of the dissi-pation inequality (53) by noting that if Vahellipt xdagger andVrhellipt xdagger satisfy (53) then Vshellipt xdagger satisreges (53) amp


5 Extended KalmanplusmnYakubovichplusmnPopov conditions forimpulsive dynamical systems

In this section we show that dissipativeness of animpulsive dynamical system can be characterized in

terms of the system functions fchellip dagger Gchellip dagger hchellip dagger Jchellip daggerfdhellip dagger Gdhellip dagger hdhellip dagger and Jdhellip dagger Here we concentrate on

the theory for dissipative time-dependent impulsive

dynamical systems Since in the case of dissipative

state-dependent impulsive dynamical systems it follows

from Assumptions A1 and A2 that for S ˆ permil0 1dagger Zthe resetting times are well deregned and distinct for every

trajectory of (23) (24) the theory of dissipative state-

dependent impulsive dynamical systems closely parallels

that of dissipative time-dependent impulsive dynamical

systems and hence many of the results are similar In the

case where the results for dissipative state-dependent

impulsive dynamical systems deviate markedly fromtheir time-dependent counterparts we present a thor-

ough treatment of these results For the results in this

section we consider the special case of dissipative im-

pulsive systems with quadratic supply rates and set

Uc ˆ mc and Ud ˆ md Speciregcally let Qc 2 lc

Sc 2 lc mc Rc 2 mc Qd 2 ld Sd 2 ld md and Rd 2 md

be given and assume rchellipuc ycdagger ˆ yTc Qcyc Dagger 2yT

c Scuc DaggeruT

c Rcuc and rdhellipud yddagger ˆ yTdQdyd Dagger 2yT

dSdud Dagger uTdRdud For

simplicity of exposition in the remainder of the paper

we assume that for time-dependent impulsive dynamical

systems the storage functions do not depend explicitly

on time This corresponds to the case in which G is time-

varying but the energy storage mechanism does not

remacrect this However this is not to say that systemenergy dissipation does not have a time-varying charac-

ter Furthermore we assume that there exist functions

microclc mc and microd ld md such that microchellip0dagger ˆ 0

microdhellip0dagger ˆ 0 rchellipmicrochellipycdagger ycdagger lt 0 yc 6ˆ 0 and rdhellipmicrodhellipyddagger yddagger lt 0

yd 6ˆ 0 so that the storage function Vshellipxdagger x 2 n is

positive deregnite and we assume that Vshellipxdagger x 2 n iscontinuously di erentiable

Theorem 9 Let Qc 2 lc Sc 2 lc mc Rc 2 mc

Qd 2 ld Sd 2 ld md and Rd 2 md If there exist

functions Vsn `c

n pc `d n pd Wcn pc mc Wd n pd md P1ud

n 1 md and

P2ud n md such that Vshellip dagger is continuously di eren-

tiable positive deregnite Vshellip0dagger ˆ 0

Vshellipx Dagger fdhellipxdagger Dagger Gdhellipxdaggeruddagger

ˆ Vshellipx Dagger fdhellipxdaggerdagger Dagger P1udhellipxdaggerud Dagger uT

d P2udhellipxdaggerud

x 2 n ud 2 md hellip79dagger

and for all x 2 n

0 ˆ V 0s hellipxdaggerfchellipxdagger iexcl hT

c hellipxdaggerQchchellipxdagger Dagger `Tc hellipxdagger`chellipxdagger hellip80dagger

0 ˆ 12V 0

s hellipxdaggerGchellipxdagger iexcl hTc hellipxdagger

hellipQcJchellipxdagger Dagger Scdagger Dagger `Tc hellipxdaggerWchellipxdagger hellip81dagger

0 ˆ Rc Dagger STc Jchellipxdagger Dagger JT

c hellipxdaggerSc

Dagger JTc hellipxdaggerQcJchellipxdagger iexcl WT

c hellipxdaggerW chellipxdagger hellip82dagger

0 ˆ Vshellipx Dagger fdhellipxdaggerdagger iexcl Vshellipxdagger

iexcl hTd hellipxdaggerQdhdhellipxdagger Dagger `T

d hellipxdagger`dhellipxdagger hellip83dagger

0 ˆ 12P1ud

hellipxdagger iexcl hTd hellipxdaggerhellipQdJdhellipxdagger Dagger Sddagger

Dagger `Td hellipxdaggerWdhellipxdagger hellip84dagger

0 ˆ Rd Dagger STd Jdhellipxdagger Dagger JT

d hellipxdaggerSd Dagger JTd hellipxdaggerQdJdhellipxdagger

iexcl P2udhellipxdagger iexcl WT

d hellipxdaggerWdhellipxdagger hellip85dagger

then the non-linear impulsive system G given by hellip10daggerplusmnhellip13daggeris dissipative with respect to the quadratic supply rate

helliprchellipuc ycdagger rdhellipud yddaggerdaggerˆ hellipyT

c Qcyc Dagger 2yTc Scuc Dagger uT

c Rcuc yTd Qdyd

Dagger2yTd Sdud Dagger uT

d Rduddagger

If alternatively

N chellipxdagger 7 Rc Dagger STc Jchellipxdagger Dagger JT

c hellipxdaggerSc Dagger JTc hellipxdaggerQcJchellipxdagger

gt 0 x 2 n hellip86dagger

and there exist a continuously di erentiable functionVs

n and matrix functions P1ud n 1 md and

P2ud n md such that Vshellip dagger is positive deregnite

Vshellip0dagger ˆ 0 hellip79dagger holds and for all x 2 n

N dhellipxdagger 7 Rd Dagger STd Jdhellipxdagger Dagger JT

d hellipxdaggerSd

Dagger JTd hellipxdaggerQdJdhellipxdagger iexcl P2ud

hellipxdagger gt 0 hellip87dagger

0 V 0s hellipxdaggerfchellipxdagger iexcl hT

c hellipxdaggerQchchellipxdagger

Dagger permil12V 0

s hellipxdaggerGchellipxdagger iexcl hTc hellipxdaggerhellipQcJchellipxdagger Dagger ScdaggerŠ

N iexcl1c hellipxdaggerpermil1

2V 0

s hellipxdaggerGchellipxdagger

iexcl hTc hellipxdaggerhellipQcJchellipxdagger Dagger ScdaggerŠT hellip88dagger

0 Vshellipx Dagger fdhellipxdaggerdagger iexcl Vshellipxdagger iexcl hTd hellipxdaggerQdhdhellipxdagger

Dagger permil12P1ud

hellipxdagger iexcl hTd hellipxdaggerhellipQdJdhellipxdagger Dagger SddaggerŠ

N iexcl1d hellipxdaggerpermil1

2P1ud

hellipxdagger iexcl hTd hellipxdaggerhellipQdJdhellipxdagger Dagger SddaggerŠT hellip89dagger

then G is dissipative with respect to the quadratic supplyrate


helliprchellipuc ycdagger rdhellipud yddaggerdagger ˆ hellipyTc Qcyc Dagger 2yT

c Scuc

Dagger uTc Rcuc yT

d Qdyd

Dagger 2yTd Sdud Dagger uT

d Rduddagger

Proof For any admissible input uchellip dagger t tt 2 tk ltt micro tt micro tkDagger1 and k 2 N it follows from (80)plusmn(82) that

Vshellipxhellipttdaggerdagger iexcl Vshellipxhelliptdaggerdagger ˆhellip tt

t

_VsVshellipxhellipsdaggerdagger ds

microhellip tt

t

_VsVshellipxhellipsdaggerdagger Dagger permil`chellipxhellipsdaggerdagger

Dagger WchellipxhellipsdaggerdaggeruchellipsdaggerŠTpermil`chellipxhellipsdaggerdagger

Dagger WchellipxhellipsdaggerdaggeruchellipsdaggerŠŠ ds

ˆhellip tt

t

permilV 0s hellipxhellipsdaggerdaggerhellipfchellipxhellipsdaggerdagger

Dagger Gchellipxhellipsdaggerdaggeruchellipsdaggerdagger Dagger `Tc hellipxhellipsdaggerdagger`chellipxhellipsdaggerdagger

Dagger 2`Tc hellipxhellipsdaggerdaggerWchellipxhellipsdaggerdaggeruchellipsdagger

Dagger uTc hellipsdaggerWT

c hellipxhellipsdaggerdaggerWchellipxhellipsdaggerdaggeruchellipsdaggerŠ ds

ˆhellip tt

t

permilhTc hellipxhellipsdaggerdaggerQchchellipxhellipsdaggerdagger

Dagger 2hTc hellipxhellipsdaggerdaggerhellipSc Dagger QcJchellipxhellipsdaggerdaggerdaggeruchellipsdagger

Dagger uTc hellipsdaggerhellipJT

c hellipxhellipsdaggerdaggerQcJchellipxhellipsdaggerdagger

Dagger STc Jchellipxhellipsdaggerdagger Dagger JT

c hellipxhellipsdaggerdaggerSc

Dagger RcdaggeruchellipsdaggerŠ ds

ˆhellip tt

t

permilyTc hellipsdaggerQcychellipsdagger Dagger 2yT

c hellipsdaggerScuchellipsdagger

Dagger uTc hellipsdaggerRcuchellipsdaggerŠ ds

ˆhellip tt

t

rchellipuchellipsdagger ychellipsdaggerdaggerds hellip90dagger

where xhelliptdagger t 2 helliptk tkDagger1Š satisreges (10) and _VsVshellip dagger denotesthe total derivative of the storage function along thetrajectories xhelliptdagger t 2 helliptk tkDagger1Š of (10) Next for anyadmissible input udhelliptkdagger tk 2 and k 2 N it followsthat

centVshellipxhelliptkdaggerdagger ˆ Vshellipxhelliptkdagger Dagger fdhellipxhelliptkdaggerdagger

Dagger Gdhellipxhelliptkdaggerdaggerudhelliptkdaggerdagger iexcl Vshellipxhelliptkdaggerdagger hellip91dagger

where centVshellip dagger denotes the di erence of the storage func-tion at the resetting times tk k 2 N of (11) Hence itfollows from (83)plusmn(85) the structural storage functionconstraint (79) and (91) that for all x 2 n andud 2 md

centVshellipxdagger ˆ Vshellipx Dagger fdhellipxdagger Dagger Gdhellipxdaggeruddagger iexcl Vshellipxdagger

ˆ Vshellipx Dagger fdhellipxdaggerdagger iexcl Vshellipxdagger Dagger P1udhellipxdaggerud

Dagger uTd P2ud

hellipxdaggerud

ˆ hTd hellipxdaggerQdhdhellipxdagger iexcl `T

d hellipxdagger`dhellipxdagger

Dagger 2permilhTd hellipxdaggerhellipQdJdhellipxdagger Dagger Sddagger iexcl `T

d hellipxdaggerWdhellipxdaggerŠud

Dagger uTd permilRd Dagger ST

d Jdhellipxdagger Dagger JTd hellipxdaggerSd

Dagger JTd hellipxdaggerQdJdhellipxdagger iexcl WT


ˆ rdhellipud yddagger iexcl permil`dhellipxdagger Dagger WdhellipxdaggerudŠT

permil`dhellipxdagger Dagger WdhellipxdaggerudŠ

micro rdhellipud yddagger hellip92dagger

Now using (90) and (92) the result is immediate fromTheorem 6

To show (88) (89) imply that G is dissipative withrespect to the quadratic supply rate helliprc rddagger note that(80)plusmn(85) can be equivalently written as

Achellipxdagger Bchellipxdagger

BTc hellipxdagger Cchellipxdagger

ˆ iexcl

`Tc hellipxdagger

WTc hellipxdagger

`chellipxdagger Wchellipxdaggerpermil Š

micro 0 x 2 n hellip93dagger

Adhellipxdagger Bdhellipxdagger

BTd hellipxdagger Cdhellipxdagger

ˆ iexcl

`Td hellipxdagger

WTd hellipxdagger

`dhellipxdagger Wdhellipxdaggerpermil Š


where

Achellipxdagger 7 V 0s hellipxdaggerfchellipxdagger iexcl hT


Bchellipxdagger 7 12V 0

s hellipxdaggerGchellipxdagger iexcl hTc hellipxdaggerhellipQcJchellipxdagger Dagger Scdagger

Cchellipxdagger 7 iexcl hellipRc Dagger STc Jchellipxdagger Dagger JT

c hellipxdaggerSc Dagger JTc hellipxdaggerQcJchellipxdaggerdagger

Adhellipxdagger 7 Vshellipx Dagger fdhellipxdaggerdagger iexcl Vshellipxdagger iexcl hTd hellipxdaggerQdhdhellipxdagger

Bdhellipxdagger 7 12P1ud


and

Cdhellipxdagger 7 iexcl hellipRd Dagger STd Jdhellipxdagger Dagger JT

d hellipxdaggerSd


hellipxdaggerdagger

Now for all invertible T c 2 hellipmcDagger1dagger hellipmcDagger1dagger andT d 2 hellipmdDagger1dagger hellipmdDagger1dagger (93) and (94) hold if and only ifT T

c hellip93dagger T c and T Td hellip94dagger T d hold Hence the equiva-

lence of (80)plusmn(85) to (88) and (89) in the case when(86) and (87) hold follows from the hellip1 1dagger block ofT T

c hellip93daggerT c where


T c 71 0

iexclCiexcl1c hellipxdaggerBT

c hellipxdagger Imc

and hellip1 1dagger block of T Td hellip94dagger T d where

T d 71 0

iexclCiexcl1d hellipxdaggerBT

d hellipxdagger Imd

amp

Remark 13 Note that Theorem 9 also holds for dissi-pative state-dependent impulsive dynamical systems In

this case however x 2 n is replaced with x 62 Zx for

(80)plusmn(82) and x 2 Zx for (79) and (83)plusmn(85) Similar re-

marks hold for the remainder of the theorems in this

section

Remark 14 The structural constraint (79) on the

system storage function is similar to the structural con-

straint invoked in standard discrete-time non-linear

passivity theory (Byrnes et al 1993 Byrnes and Lin1994 Lin and Byrnes 1994 1995 Chellaboina and

Haddad 1998) This of course is not surprising since

impulsive dynamical systems involve a hybrid formula-

tion of continuous-time and discrete-time dynamics In

the case where ud ˆ 0 or G is lossless with respect to a

quadratic supply rate or G is dissipative with respect

to a quadratic supply rate of the form helliprc 0dagger Con-dition (79) is necessary and su cient (see Theorems 10

and 11 below) and hence is automatically satisreged Si-

milarly in the case where G is linear and dissipative

with respect to a quadratic supply rate Condition (79)

is also necessary and su cient (see Theorem 14 below)

In general however it is extremely di cult if not im-

possible to obtain (algebraic) su cient conditions fordissipativity with respect to quadratic supply rates for

impulsive dynamical systems without the structural

constraint (79) Similar remarks hold for discrete-time

non-linear systems (see Byrnes et al 1993 Byrnes and

Lin 1994 Lin and Byrnes 1994 1995 Chellaboina and

Haddad 1998 for further details)

Remark 15 Note that it follows from (66) that if the

conditions in Theorem 9 are satisreged with (80) re-placed by

0 ˆ V 0s hellipxdaggerfchellipxdagger Dagger Vshellipxdagger iexcl hT


Dagger`Tc hellipxdagger`chellipxdagger x 2 n hellip95dagger

where gt 0 then the non-linear impulsive dynamical

system G is exponentially dissipative Similar remarks

hold for Corollaries 3 and 4 below

Using (80)plusmn(85) it follows that for tt t 0 andk 2 N permiltttdagger

hellip tt

t


k2N permiltttdagger


ˆ Vshellipxhellipttdaggerdagger iexcl Vshellipxhelliptdaggerdagger

Daggerhellip tt

t

permil`chellipxhellipsdaggerdagger Dagger W chellipxhellipsdaggerdaggeruchellipsdaggerŠT

permil`chellipxhellipsdaggerdagger Dagger WchellipxhellipsdaggerdaggeruchellipsdaggerŠ ds

DaggerX

k2N permiltttdagger

permil`dhellipxhelliptkdaggerdagger Dagger WdhellipxhelliptkdaggerdaggerudhelliptkdaggerŠT

permil`dhellipxhelliptkdaggerdagger Dagger WdhellipxhelliptkdaggerdaggerudhelliptkdaggerŠ hellip96dagger

which can be interpreted as a generalized energy balanceequation where Vshellipxhellipttdaggerdagger iexcl Vshellipxhelliptdaggerdagger is the stored or accu-mulated generalized energy of the impulsive dynamicalsystem the second path-dependent term on the rightcorresponds to the dissipated energy of the impulsivedynamical system over the continuous-time dynamicsand the third discrete term on the right correspondsto the dissipated energy at the resetting instantsEquivalently it follows from Theorem 6 that (96) canbe rewritten as

_VsVshellipxhelliptdaggerdagger ˆ rchellipuchelliptdagger ychelliptdaggerdagger

iexcl permil`chellipxhelliptdaggerdagger Dagger W chellipxhelliptdaggerdaggeruchelliptdaggerŠT

permil`chellipxhelliptdaggerdagger Dagger W chellipxhelliptdaggerdaggeruchelliptdaggerŠ

tk lt t micro tkDagger1 hellip97dagger

centVshellipxhelliptkdaggerdagger ˆ rdhellipudhelliptkdagger ydhelliptkdaggerdagger

iexcl permil`dhellipxhelliptkdaggerdagger Dagger WdhellipxhelliptkdaggerdaggerudhelliptkdaggerŠT

permil`dhellipxhelliptkdaggerdagger Dagger WdhellipxhelliptkdaggerdaggerudhelliptkdaggerŠ

k 2 N hellip98dagger

which yields a set of generalized energy conservationequations Speciregcally (97) shows that the rate ofchange in generalized energy or generalized powerover the time interval t 2 helliptk tkDagger1Š is equal to the gener-alized system power input minus the internal generalizedsystem power dissipated while (98) shows that thechange of energy at the resetting times tk k 2 N isequal to the external generalized system energy at theresetting times minus the generalized dissipated energyat the resetting times

Remark 16 Note that if G with hellipuchelliptdagger udhelliptkdaggerdagger sup2 hellip0 0daggerand a continuously di erentiable positive-deregniteradially unbounded storage function is dissipative withrespect to a quadratic supply rate where Qc micro 0Qd micro 0 it follows that _VVshellipxhelliptdaggerdagger micro yT

c helliptdaggerQcychelliptdagger micro 0t 0 and centVshellipxhelliptkdaggerdagger micro yT

d helliptkdaggerQdydhelliptkdagger micro 0 k 2 N Hence the undisturbed helliphellipuchelliptdagger udhelliptkdaggerdagger sup2 hellip0 0daggerdagger non-linear impulsive dynamical system (10)plusmn(13) is Lyapu-


nov stable Alternatively if G with hellipuchelliptdagger udhelliptkdaggerdagger sup2hellip0 0dagger and a continuously di erentiable positive-deregnite radially unbounded storage function is expo-nentially dissipative and Qc micro 0 Qd micro 0 it followsthat _VVshellipxhelliptdaggerdagger micro iexclVshellipxhelliptdaggerdagger Dagger yT

c helliptdaggerQcychelliptdagger micro iexclVshellipxhelliptdaggerdaggert 0 and centVshellipxhelliptkdaggerdagger micro yT

d helliptkdaggerQdydhelliptkdagger micro 0 k 2 N Hence the undisturbed non-linear impulsive dynamicalsystem (10)plusmn(13) is asymptotically stable If in additionthere exist constants not shy gt 0 and p 1 such that

notkxkp micro Vshellipxdagger micro shy kxkp x 2 n then the undisturbednon-linear impulsive dynamical system (10)plusmn(13) is ex-ponentially stable

Next we provide necessary and su cient conditionsfor the case where G given by (10)plusmn(13) is lossless withrespect to a quadratic supply rate helliprc rddagger

Theorem 10 Let Qc 2 lc Sc 2 lc mc Rc 2 mc Qd 2 ld Sd 2 ld md and Rd 2 md Then the non-linear impulsive system G given by hellip10daggerplusmnhellip13dagger is losslesswith respect to the quadratic supply rate


c Scuc Dagger uTc Rcuc

yTd Qdyd Dagger 2yT

d Sdud Dagger uTd Rduddagger

if and only if there exist functions Vsn P1ud

n 1 md and P2ud

n md such that Vshellip dagger is con-tinuously di erentiable and positive deregnite Vshellip0dagger ˆ 0and for all x 2 n hellip79dagger holds and


c hellipxdaggerQchchellipxdagger hellip99dagger

0 ˆ 12V 0

s hellipxdaggerGchellipxdagger iexcl hTc hellipxdaggerhellipQcJchellipxdagger Dagger Scdagger hellip100dagger


c hellipxdaggerSc Dagger JTc hellipxdaggerQcJchellipxdagger hellip101dagger

0 ˆ Vshellipx Dagger fdhellipxdaggerdagger iexcl Vshellipxdagger iexcl hTd hellipxdaggerQdhdhellipxdagger hellip102dagger

0 ˆ 12P1ud

hellipxdagger iexcl hTd hellipxdaggerhellipQdJdhellipxdagger Dagger Sddagger hellip103dagger


d hellipxdaggerSd Dagger JTd hellipxdaggerQdJdhellipxdagger iexcl P2ud

hellipxdagger

hellip104dagger

If in addition Vshellip dagger is two-times continuously di erenti-able then

P1udhellipxdagger ˆ V 0

s hellipx Dagger fdhellipxdaggerdaggerGdhellipxdagger

P2udhellipxdagger ˆ 1

2GT

d hellipxdaggerVshellipx Dagger fdhellipxdaggerdaggerGdhellipxdagger

Proof Su ciency follows as in the proof of Theorem9 To show necessity suppose that the non-linear im-pulsive dynamical system G is lossless with respect tothe quadratic supply rate helliprc rddagger Then it follows fromTheorem 6 that for all k 2 N


t



and

Vshellipxhelliptkdagger Dagger fdhellipxhelliptkdaggerdagger Dagger Gdhellipxhelliptkdaggerdaggerudhelliptkdaggerdagger

ˆ Vshellipxhelliptkdaggerdagger Dagger rdhellipudhelliptkdagger ydhelliptkdaggerdagger hellip106dagger

Now dividing (105) by tt iexcl tDagger and letting tt tDagger (105) isequivalent to

_VsVshellipxhelliptdaggerdagger ˆ V 0s hellipxhelliptdaggerdaggerpermilfchellipxhelliptdaggerdagger Dagger GchellipxhelliptdaggerdaggeruchelliptdaggerŠ

ˆ rchellipuchelliptdagger ychelliptdaggerdagger tk lt t micro tkDagger1 hellip107dagger

Next with t ˆ 0 it follows from (107) that

V 0s hellipx0daggerpermilfchellipx0dagger Dagger Gchellipx0daggeruchellip0daggerŠ ˆ rchellipuchellip0dagger ychellip0daggerdagger

x0 2 n uchellip0dagger 2 mc hellip108dagger

Since x0 2 n is arbitrary it follows that

V 0s hellipxdaggerpermilfchellipxdagger Dagger GchellipxdaggerucŠ ˆ yT


c Rcuc

ˆ hTc hellipxdaggerQchchellipxdagger Dagger 2hT

c hellipxdaggerhellipQcJchellipxdagger

Dagger Scdaggeruc Dagger uTc hellipRc Dagger ST

c Jchellipxdagger

Dagger JTc hellipxdaggerSc Dagger JT

c hellipxdaggerQcJchellipxdaggerdaggeruc

x 2 n uc 2 mc

Now equating coe cients of equal powers yields (99)plusmn(101) Next it follows from (106) with k ˆ 1 that

Vshellipxhellipt1dagger Dagger fdhellipxhellipt1daggerdagger Dagger Gdhellipxhellipt1daggerdaggerudhellipt1daggerdagger

ˆ Vshellipxhellipt1daggerdagger Dagger rdhellipudhellipt1dagger ydhellipt1daggerdagger hellip109dagger

Now since the continuous-time dynamics (10) areLipschitz it follows that for arbitrary x 2 n there existsx0 2 n such that xhellipt1dagger ˆ x Hence it follows from (109)that

VshellipxDaggerfdhellipxdagger Dagger Gdhellipxdaggeruddagger

ˆ Vshellipxdagger Dagger yTd Qdyd Dagger 2yT

d Sdud Dagger uTd Rdud

ˆ Vshellipxdagger Dagger hTd hellipxdaggerQdhdhellipxdagger Dagger 2hT

d hellipxdaggerhellipQdJdhellipxdagger

Dagger Sddaggerud Dagger uTd hellipRd Dagger ST

d Jdhellipxdagger Dagger JTd hellipxdaggerSd Dagger JT

d hellipxdagger

QdJdhellipxdaggerdaggerud x 2 n ud 2 md hellip110dagger

Since the right-hand-side of (110) is quadratic in ud itfollows that Vshellipx Dagger fdhellipxdagger Dagger Gdhellipxdaggeruddagger is quadratic in ud

and hence there exists P1ud n 1 md and P2ud

n md such that



d P2udhellipxdaggerud hellip111dagger

Now using (111) and equating coe cients of equalpowers in (110) yields (102)plusmn(104) Finally if Vshellip dagger is


two-times continuously di erentiable applying a Taylorseries expansion on (111) about ud ˆ 0 yields

P1udhellipxdagger ˆ Vshellipx Dagger fdhellipxdagger Dagger Gdhellipxdaggeruddagger

ud udˆ0

ˆ V 0s hellipx Dagger fdhellipxdaggerdaggerGdhellipxdagger hellip112dagger

P2udhellipxdagger ˆ 2Vshellipx Dagger fdhellipxdagger Dagger Gdhellipxdaggeruddagger

ud2

udˆ0

ˆ 12GT

d hellipxdaggerVshellipx Dagger fdhellipxdaggerdaggerGdhellipxdagger hellip113dagger

amp

Next we provide two deregnitions of non-linearimpulsive dynamical systems which are dissipative(resp exponentially dissipative) with respect to supplyrates of a speciregc form

Deregnition 7 A system G of the form (1)plusmn(4) withmc ˆ lc and md ˆ ld is passive (resp exponentially pas-sive) if G is dissipative (resp exponentially dissipative)with respect to the supply rate helliprchellipuc ycdagger rdhellipud yddaggerdagger ˆhellip2uT

c yc 2uTd yddagger

Deregnition 8 A system G of the form (1)plusmn(4) is non-expansive (resp exponentially non-expansive) if G isdissipative (resp exponentially dissipative) with re-spect to the supply rate helliprchellipuc ycdagger rdhellipud yddaggerdagger ˆhellipreg2

c uTc uc iexcl yT

c yc reg2duT

d ud iexcl yTd yddagger where regc regd gt 0 are

given

Remark 17 Note that a mixed passive-non-expansiveformulation of G can also be considered Speciregcallyone can consider impulsive dynamical systems G whichare dissipative with respect to supply rates of the formhelliprchellipuc ycdagger rdhellipud yddaggerdagger ˆ hellip2uT

c yc reg2duT

d ud iexcl yTd yddagger where

regd gt 0 and vice versa Furthermore supply rates forinput strict passivity (Hill and Moylan 1977) outputstrict passivity (Hill and Moylan 1977) and inputplusmnout-put strict passivity (Hill and Moylan 1977) can also beconsidered However for simplicity of exposition wedo not do so here

The following results present the non-linear versionsof the KalmanplusmnYakubovichplusmnPopov positive real lemmaand the bounded real lemma for non-linear impulsivesystems G of the form (10)plusmn(13)

Corollary 3 Consider the non-linear impulsive systemG given by hellip10daggerplusmnhellip13dagger If there exist functionsVs

n `cn pc `d n pd Wc

n pc mc Wd n pd md P1ud

n 1 md and P2ud

n md such that Vshellip dagger is continuously di erentiableand positive deregnite Vshellip0dagger ˆ 0





and for all x 2 n

0 ˆ V 0s hellipxdaggerfchellipxdagger Dagger `T

c hellipxdagger`chellipxdagger hellip115dagger

0 ˆ 12V 0

s hellipxdaggerGchellipxdagger iexcl hTc hellipxdagger Dagger `T

c hellipxdaggerWchellipxdagger hellip116dagger

0 ˆ Jchellipxdagger Dagger JTc hellipxdagger iexcl WT


0 ˆ Vshellipx Dagger fdhellipxdaggerdagger iexcl Vshellipxdagger Dagger `Td hellipxdagger`dhellipxdagger hellip118dagger

0 ˆ 12P1ud

hellipxdagger iexcl hTd hellipxdagger Dagger `T


0 ˆ Jdhellipxdagger Dagger JTd hellipxdagger iexcl P2ud

hellipxdagger iexcl WTd hellipxdaggerWdhellipxdagger hellip120dagger

then G is passive If alternatively Jchellipxdagger Dagger JTc hellipxdagger gt 0

x 2 n and there exist a continuously di erentiable func-tion Vs

n and matrix functions P1ud n 1 md

and P2ud n md such that Vshellip dagger is positive deregnite

Vshellip0dagger ˆ 0 (114) holds and for all x 2 n

0 lt Jdhellipxdagger Dagger JTd hellipxdagger iexcl P2ud

hellipxdagger hellip121dagger

0 V 0s hellipxdaggerfchellipxdagger Dagger permil1

2V 0

s hellipxdaggerGchellipxdagger iexcl hTc hellipxdaggerŠ

permilJchellipxdagger Dagger JTc hellipxdaggerŠiexcl1permil1

2V 0

s hellipxdaggerGchellipxdagger iexcl hTc hellipxdaggerŠT hellip122dagger

0 Vshellipx Dagger fdhellipxdaggerdagger iexcl Vshellipxdagger Dagger permil12P1ud

hellipxdagger iexcl hTd hellipxdaggerŠ

permilJdhellipxdagger Dagger JTd hellipxdagger iexcl P2ud

hellipxdaggerŠiexcl1permil12P1udhellipxdagger iexcl hT

d hellipxdaggerŠT

hellip123dagger

then G is passive

Proof The result is a direct consequence of Theorem9 with lc ˆ mc ld ˆ md Qc ˆ 0 Qd ˆ 0 Sc ˆ Imc

Sd ˆ Imd

Rc ˆ 0 and Rd ˆ 0 Speciregcally withmicrochellipycdagger ˆ iexclyc and microdhellipyddagger ˆ iexclyd it follows thatrchellipmicrochellipycdagger ycdagger ˆ iexcl2yT

c yc lt 0 yc 6ˆ 0 and rdhellipmicrodhellipyddagger yddagger ˆiexcl2yT

d yd lt 0 yd 6ˆ 0 so that all of the assumptions ofTheorem 9 are satisreged amp


n `cn pc `d n pd Wc


n 1 md and P2ud n md

such that Vshellip dagger is continuously di erentiable and positivederegnite Vshellip0dagger ˆ 0





and for all x 2 n


0 ˆ V 0s hellipxdaggerfchellipxdagger Dagger hT

c hellipxdaggerhchellipxdagger Dagger `Tc hellipxdagger`chellipxdagger hellip125dagger

0 ˆ 12V 0

s hellipxdaggerGchellipxdagger Dagger hTc hellipxdaggerJchellipxdagger Dagger `T


0 ˆ reg2c Imc

iexcl JTc hellipxdaggerJchellipxdagger iexcl WT


0 ˆ Vshellipx Dagger fdhellipxdaggerdagger iexcl Vshellipxdagger Dagger hTd hellipxdaggerhdhellipxdagger Dagger `T


hellip128dagger

0 ˆ 12P1ud

hellipxdagger Dagger hTd hellipxdaggerJdhellipxdagger Dagger `T


0 ˆ reg2dImd

iexcl JTd hellipxdaggerJdhellipxdagger iexcl P2ud


then G is non-expansive If alternatively reg2c Imc

iexclJT

c hellipxdaggerJchellipxdagger gt 0 x 2 n and there exist a continuouslydi erentiable function Vs

n and matrix functionsP1ud

n 1 md and P2ud n md such that Vshellip dagger is

positive deregnite Vshellip0dagger ˆ 0 hellip124dagger holds and for all x 2 n

0 lt reg2dImd



0 V 0s hellipxdaggerfchellipxdagger Dagger hT

c hellipxdaggerhchellipxdagger

Dagger permil12V 0

s hellipxdaggerGchellipxdagger Dagger hTc hellipxdaggerJchellipxdaggerŠ

permilreg2c Imc

iexcl JTc hellipxdaggerJchellipxdaggerŠiexcl1

permil12V 0

s hellipxdaggerGchellipxdagger Dagger hTc hellipxdaggerJchellipxdaggerŠT hellip132dagger

0 Vshellipx Dagger fdhellipxdaggerdagger iexcl Vshellipxdagger Dagger hTd hellipxdaggerhdhellipxdagger

Dagger permil12P1ud

hellipxdagger Dagger hTd hellipxdaggerJdhellipxdaggerŠ

permilreg2dImd


hellipxdaggerŠiexcl1

permil12P1ud

hellipxdagger Dagger hTd hellipxdaggerJdhellipxdaggerŠT hellip133dagger

then G is non-expansive

Proof The result is a direct consequence of Theorem9 with Qc ˆ iexclIlc Qd ˆ iexclIld Sc ˆ 0 Sd ˆ 0 Rc ˆreg2

c Imcand Rd ˆ reg2

dImd Speciregcally with microchellipycdagger ˆ

iexclhellip1=2regcdaggeryc and microdhellipyddagger ˆ iexclhellip1=2regddaggeryd it followsthat rchellipmicrochellipycdagger ycdagger ˆ iexcl 3

4yT

c yc lt 0 yc 6ˆ 0 andrdhellipmicrodhellipyddagger yddagger ˆ iexcl 3

4yT

d yd lt 0 yd 6ˆ 0 so that all of theassumptions of Theorem 9 are satisreged amp

Next we provide necessary and su cient conditionsfor dissipativity of a non-linear impulsive dynamicalsystem G of the form (10)plusmn(13) in the case whererdhellipud yddagger sup2 0 and Gdhellipxdagger sup2 0

Theorem 11 Let Qc 2 lc Sc 2 lc mc and Rc 2 mc Then the non-linear impulsive system G given by hellip10daggerplusmn

hellip13dagger with Gdhellipxdagger sup2 0 is dissipative with respect to the

supply rate helliprchellipuc ycdagger rdhellipud yddaggerdagger ˆ hellipyTc Qcyc Dagger 2yT

c ScucDaggeruT

c Rcuc 0dagger if and only if there exist functions Vsn `c

n pc `d n pd and Wcn

pc mc such that Vshellip dagger is continuously di erentiable and

positive deregnite Vshellip0dagger ˆ 0 and for all x 2 n



0 ˆ 12V 0


Dagger `Tc hellipxdaggerWchellipxdagger hellip135dagger



iexcl WTc hellipxdaggerWchellipxdagger hellip136dagger


Proof Su ciency follows from Theorem 9 with

Qd ˆ 0 Sd ˆ 0 Rd ˆ 0 Gdhellipxdagger ˆ 0 P1udhellipxdagger ˆ 0 and

P2udhellipxdagger ˆ 0 Necessity follows from Theorem 6 using a

similar construction as in the proof of Theorem 10 amp

Remark 18 Note that in the case where

rdhellipud yddagger sup2 0 and Gdhellipxdagger sup2 0 it follows from Theorem11 that the non-linear impulsive system G given by(10)plusmn(13) is passive (resp non-expansive) if and onlyif there exist functions Vs

n `cn pc

`d n pd and Wcn pc mc such that Vshellip dagger is

continuously di erentiable and positive deregniteVshellip0dagger ˆ 0 and (115)plusmn(117) and (137) (resp (125)plusmn(127)and (137)) are satisreged

Finally we present two key results on linearizationof impulsive dynamical systems For these resultswe assume that there exist functions microc

lc mc

and microd ld md such that microchellip0dagger ˆ 0 microdhellip0dagger ˆ 0

rchellipmicrochellipycdagger ycdagger lt 0 yc 6ˆ 0 rdhellipmicrodhellipyddagger yddagger lt 0 yd 6ˆ 0 andthe available storage Vahellipxdagger x 2 n is a three-times con-tinuously di erentiable function

Theorem 12 Let Qc 2 lc Sc 2 lc mc Rc 2 mc Qd 2 ld Sd 2 ld md and Rd 2 md and suppose thatthe non-linear impulsive system G given by hellip10daggerplusmnhellip13dagger is

dissipative with respect to the quadratic supply rate



yTd Qdyd Dagger 2yT


Then there exists matrices P 2 n n Lc 2 pc nWc 2 pc mc Ld 2 pd n and Wd 2 pd md with P non-

negative deregnite such that


0 ˆ ATc P Dagger PAc iexcl CT

c QcCc Dagger LTc Lc hellip138dagger

0 ˆ PBc iexcl CTc hellipQcDc Dagger Scdagger Dagger LT

c Wc hellip139dagger

0 ˆ Rc Dagger STc Dc Dagger DT

c Sc Dagger DTc QcDc iexcl WT


0 ˆ ATd PAd iexcl P iexcl CT

d QdCd Dagger LTd Ld hellip141dagger

0 ˆ ATd PBd iexcl CT

d hellipQdDd Dagger Sddagger Dagger LTd Wd hellip142dagger

0 ˆ Rd Dagger STd Dd Dagger DT

d Sd Dagger DTd QdDd iexcl BT

d PBd iexcl WTd Wd

hellip143dagger

where

Ac ˆ fc

x xˆ0

Bc ˆ Gchellip0dagger

Cc ˆ hc

x xˆ0

Dc ˆ Jchellip0dagger

9gtgtgt=

gtgtgthellip144dagger

Ad ˆ fd

x xˆ0

DaggerIn Bd ˆ Gdhellip0dagger

Cd ˆ hd

x xˆ0

Dd ˆ Jdhellip0dagger

9gtgtgt=


If in addition hellipAc Ccdagger and hellipAd Cddagger are observable thenP gt 0

Proof First note that since G is dissipative with re-spect to the supply rate helliprc rddagger it follows fromTheorem 6 that there exists a storage functionVs

n such that for all k 2 N

Vshellipxhellipttdaggerdagger iexcl Vshellipxhelliptdaggerdagger microhellip tt

t



and


micro Vshellipxhelliptkdaggerdagger Dagger rdhellipudhelliptkdagger ydhelliptkdaggerdagger hellip147dagger



micro rchellipuchelliptdagger ychelliptdaggerdagger tk lt t micro tkDagger1 hellip148dagger

Next with t ˆ 0 it follows that

V 0s hellipx0daggerpermilfchellipx0dagger Dagger Gchellipx0daggeruchellip0daggerŠ

micro rchellipuchellip0dagger ychellip0daggerdagger x0 2 n uchellip0dagger 2 mc hellip149dagger


V 0s hellipxdaggerpermilfchellipxdagger Dagger GchellipxdaggerucŠ micro rchellipuc hchellipxdagger Dagger Jchellipxdaggerucdagger

x 2 n uc 2 mc hellip150dagger

Furthermore it follows from (147) with k ˆ 1 that


micro Vshellipxhellipt1daggerdagger Dagger rdhellipudhellipt1dagger ydhellipt1daggerdagger hellip151dagger



micro Vshellipxdagger Dagger rdhellipud hdhellipxdagger Dagger Jdhellipxdaggeruddagger


Next it follows from (150) and (152) that there existssmooth functions dc

n mc and dd n md such that dchellipx ucdagger 0 dchellip0 0dagger ˆ 0 ddhellipx uddagger 0ddhellip0 0dagger ˆ 0 and

0 ˆ V 0s hellipxdaggerpermilfchellipxdagger Dagger GchellipxdaggerucŠ iexcl rchellipuc hchellipxdagger

Dagger Jchellipxdaggerucdagger Dagger dchellipx ucdagger x 2 n uc 2 mc hellip153dagger

0 ˆ Vshellipx Dagger fdhellipxdagger Dagger Gdhellipxdaggeruddagger iexcl Vshellipxdagger

iexcl rdhellipud hdhellipxdagger Dagger Jdhellipxdaggeruddagger Dagger ddhellipx uddagger


Now expanding Vshellip dagger dchellip dagger and ddhellip dagger via a Taylorseries expansion about x ˆ 0 uc ˆ 0 ud ˆ 0 and usingthe fact that Vshellip dagger dchellip dagger and ddhellip dagger are non-negativederegnite and Vshellip0dagger ˆ 0 dchellip0 0dagger ˆ 0 and ddhellip0 0dagger ˆ 0 itfollows that there exist matrices P 2 n n Lc 2 pc nWc 2 pc mc Ld 2 pd n and Wd 2 pd md with P non-negative deregnite such that

Vshellipxdagger ˆ xTPx Dagger Vrhellipxdagger hellip155dagger

dchellipx ucdagger ˆ hellipLcx Dagger WcucdaggerThellipLcx Dagger Wcucdagger Dagger dcrhellipx ucdagger

hellip156dagger

ddhellipx uddagger ˆ hellipLdx Dagger WduddaggerThellipLdx Dagger Wduddagger Dagger ddrhellipx uddagger

hellip157dagger

where Vrn dcr

n mc and ddrn

md contain the higher-order terms of Vshellip daggerdchellip dagger and ddhellip dagger respectively Next let fchellipxdagger ˆ AcxDaggerfcrhellipxdagger hchellipxdagger ˆ Ccx Dagger hcrhellipxdagger fdhellipxdagger ˆ hellipAd iexcl Indaggerx Dagger fdrhellipxdaggerand hdhellipxdagger ˆ Cdx Dagger hdrhellipxdagger where fcrhellip dagger hcrhellip dagger fdrhellip dagger andhdrhellip dagger contain the non-linear terms of fchellipxdagger hchellipxdagger fdhellipxdaggerand hdhellipxdagger respectively and let Gchellipxdagger ˆ Bc Dagger GcrhellipxdaggerJchellipxdagger ˆ Dc Dagger Jcrhellipxdagger Gdhellipxdagger ˆ Bd Dagger Gdrhellipxdagger Jdhellipxdagger ˆ DdDaggerJdrhellipxdagger where Gcrhellipxdagger Jcrhellipxdagger Gdrhellipxdagger and Jdrhellipxdagger containthe nonplusmnconstant terms of Gchellipxdagger Jchellipxdagger Gdhellipxdagger and


Jdhellipxdagger respectively Using the above expressions (153)and (154) can be written as

0 ˆ 2xTPhellipAcx Dagger Bcucdagger iexcl hellipxTCTc QcCcx Dagger 2xTCT

c QcDcuc

Dagger uTc DT

c QcDcuc Dagger 2xTCTc Scuc Dagger 2uT

c DTc Scuc Dagger uT

c Rcucdagger

Dagger hellipLcx Dagger WcucdaggerThellipLcx Dagger Wcucdagger Dagger macrchellipx ucdagger hellip158dagger

0 ˆ hellipAdx Dagger BduddaggerTPhellipAdx Dagger Bduddagger iexcl xTPx

iexcl hellipxTCTd QdCdx Dagger 2xTCT

d QdDdud Dagger uTd DT

d QdDdud

Dagger 2xTCTd Sdud Dagger 2uT

d DTd Sdud Dagger uT

d Rduddagger

Dagger hellipLdx Dagger WduddaggerThellipLdx Dagger Wduddagger Dagger macrdhellipx uddagger hellip159dagger

where macrchellipx ucdagger and macrdhellipx uddagger are such that

limkxk2Daggerkuck20

j macrchellipx ucdagger jkxk2 Dagger kuck2

ˆ 0

limkxk2Daggerkudk20

j macrdhellipx uddagger jkxk2 Dagger kudk2

ˆ 0

Now viewing (158) and (159) as the Taylor series expan-sion of (153) and (154) respectively about x ˆ 0 uc ˆ 0and ud ˆ 0 it follows that

0 ˆ xThellipATc P Dagger PAc iexcl CT

c QcCc Dagger LTc Lcdaggerx

Dagger 2xThellipPBc iexcl CTc Sc iexcl CT

c QcDc Dagger LTc Wcdaggeruc

Dagger uTc hellipWT

c Wc iexcl DTc QcDc iexcl DT

c Sc

iexcl STc Dc iexcl Rcdaggeruc x 2 n uc 2 mc hellip160dagger

0 ˆ xThellipATd PAd iexcl P iexcl CT

d QdCd Dagger LTd Lddaggerx

Dagger 2xThellipATd PBd iexcl CT

d Sd iexcl CTd QdDd Dagger LT

d Wddaggerud

Dagger uTd hellipWT

d Wd iexcl DTd QdDd iexcl DT

d Sd iexcl STd Dd

iexcl Rd Dagger BTd PBddaggerud x 2 n ud 2 md hellip161dagger

Next equating coe cients of equal powers in (160) and(161) yields (138)plusmn(143)

Finally to show that P gt 0 in the case wherehellipAc Ccdagger and hellipAd Cddagger are observable note that it followsfrom Theorem 9 and (138)plusmn(143) that the linearizedsystem G with storage function Vshellipxdagger ˆ xTPx is dissipa-tive with respect to the quadratic supply rate helliprchellipuc ycdaggerrdhellipud yddaggerdagger Now the positive deregniteness of P followsfrom Theorem 7 amp

It is important to note that Theorem 12 does nothold for state-dependent impulsive dynamical systemsTo see this note that (138)plusmn(143) follow from (160) and(161) if and only if x 2 n For state-dependent impul-sive dynamical systems x 62 Zx in (160) and x 2 Zx in(161) For state-dependent impulsive dynamical systemswe have the following linearization result

Theorem 13 Let Qc 2 lc Sc 2 lc mc Rc 2 mc Qd 2 ld Sd 2 ld md and Rd 2 md and suppose thatthe non-linear impulsive system G given by hellip23daggerplusmnhellip26dagger isdissipative with respect to the quadratic supply rate



yTd Qdyd Dagger 2yT


Then there exists matrices P 2 n n Lc 2 pc nWc 2 pc mc Ld 2 pd n and Wd 2 pd md with P non-negative deregnite such that


c QcCc Dagger LTc Lcdaggerx x 62 Zx

hellip162dagger

0 ˆ xThellipPBc iexcl CTc hellipQcDc Dagger Scdagger Dagger LT

c Wcdagger x 62 Zx

hellip163dagger





d QdCd Dagger LTd Lddaggerx x 2 Zx

hellip165dagger

0 ˆ xThellipATd PBd iexcl CT

d hellipQdDd Dagger Sddagger Dagger LTd Wddagger x 2 Zx

hellip166dagger



d PBd iexcl WTd Wd

hellip167dagger

where Ac Bc Cc Dc Ad Bd Cd and Dd are given byhellip144dagger and hellip145dagger If in addition hellipAc Ccdagger and hellipAd Cddagger areobservable then P gt 0

Proof The proof is identical to the proof of Theorem12 with (160) and (161) replaced by





Dagger uTc hellipWT

c Wc iexcl DTc QcDc

iexcl DTc Sc iexcl ST

c Dc iexcl Rcdaggeruc x 62 Zx uc 2 mc

hellip168dagger





d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd

iexcl Rd Dagger BTd PBddaggerud x 2 Zx ud 2 md hellip169dagger

Now setting uc ˆ 0 and ud ˆ 0 in (168) and (169) re-spectively yields (162) and (165) In this case (168) and(169) become


0 ˆ 2xTRcxuuc Dagger uTc Rcuuuc x 62 Zx uc 2 mc hellip170dagger

0 ˆ 2xTR dxuud Dagger uTd R duuud x 2 Zx ud 2 md hellip171dagger

where

Rcxu 7 PBc iexcl CTc Sc iexcl CT

c QcDc Dagger LTc Wc

Rcuu 7 WTc Wc iexcl DT

c QcDc iexcl DTc Sc iexcl ST

c Dc iexcl Rc

Rdxu 7 ATd PBd iexcl CT


d Wd

R duu 7 WTd Wd iexcl DT

d QdDd iexcl DTd Sd

iexcl STd Dd iexcl Rd Dagger BT

d PBd

Next let x 62 Zx and uuc 2 mc so that with uc ˆ 2uuc(170) implies

0 ˆ 4xTRcxuuuc Dagger 4uuTc Rcuuuuc hellip172dagger

Now forming 12(172)plusmn(171) yields uuT

c Rcuuuuc ˆ 0uuc 2 mc Hence Rcuu ˆ 0 or equivalently (164)holds Furthermore uuT

c Rcuuuuc ˆ 0 uuc 2 mc implies2xTRcxuuuc ˆ 0 uuc 2 mc Hence 2xTRcxu ˆ 0 x 62 Zxwhich implies (163) Using similar arguments (166)and (167) hold Finally the positive deregniteness of Pin the case where hellipAc Ccdagger and hellipAd Cddagger are observablefollows as in the proof of Theorem 12 using Theorem 9Remark 13 and Theorem 7 amp

6 Specialization to linear impulsive dynamical systems

In this section we specialize the results of 5 to thecase of linear impulsive dynamical systems Speciregcallysetting fchellipxdagger ˆ Acx Gchellipxdagger ˆ Bc hchellipxdagger ˆ Ccx Jchellipxdagger ˆ Dcfdhellipxdagger ˆ hellipAd iexcl Indaggerx Gdhellipxdagger ˆ Bd hdhellipxdagger ˆ Cdx andJdhellipxdagger ˆ Dd the non-linear time-dependent impulsivedynamical system given by (10)plusmn(13) specializes to

_xxhelliptdagger ˆ Acxhelliptdagger Dagger Bcuchelliptdagger xhellip0dagger ˆ x0 t 6ˆ tk hellip173dagger

centxhelliptdagger ˆ hellipAd iexcl Indaggerxhelliptdagger Dagger Bdudhelliptdagger t ˆ tk hellip174dagger

ychelliptdagger ˆ Ccxhelliptdagger Dagger Dcuchelliptdagger t 6ˆ tk hellip175dagger

ydhelliptdagger ˆ Cdxhelliptdagger Dagger Ddudhelliptdagger t ˆ tk hellip176dagger

where Ac 2 n n Bc 2 n mc Cc 2 lc n Dc 2 lc mc Ad 2 n n Bd 2 n md Cd 2 ld n and Dd 2 ld md

Theorem 14 Let Qc 2 lc Sc 2 lc mc Rc 2 mc Qd 2 ld Sd 2 ld md Rd 2 md consider the linear im-pulsive dynamical system G given by hellip173daggerplusmnhellip176dagger and as-sume G is minimal Then the following statements areequivalent

(i) G is dissipative with respect to the quadratic supplyrate helliprchellipuc ycdagger rdhellipud yddaggerdagger ˆ hellipyT

c Qcyc Dagger 2yTc Scuc Dagger

uTc Rcuc yT

d Qdyd Dagger 2yTd Sdud Dagger uT

d Rduddagger(ii) There exist matrices P 2 n n Lc 2 pc n

Wc 2 pc mc Ld 2 pd n and Wd 2 pd md with

P positive deregnite such that hellip138daggerplusmnhellip143dagger aresatisreged

If alternatively Rc Dagger STc Dc Dagger DT

c Sc Dagger DTc QcDc gt 0 then

G is dissipative with respect to the quadratic supply rate



yTd Qdyd Dagger 2yT


if and only if there exists an n n positive-deregnite matrix

P such that

0 lt Rd Dagger STd Dd Dagger DT


d PBd hellip177dagger

0 ATc P Dagger PAc iexcl CT

c QcCc Dagger permilPBc iexcl CTc hellipQcDc Dagger ScdaggerŠ

permilRc Dagger STc Dc Dagger DT

c Sc Dagger DTc QcDcŠ

iexcl1

permilPBc iexcl CTc hellipQcDc Dagger ScdaggerŠT hellip178dagger

0 ATd PAd iexcl P iexcl CT

d QdCd

Dagger permilATd PBd iexcl CT

d hellipQdDd Dagger SddaggerŠ

permilRd Dagger STd Dd Dagger DT

d Sd Dagger DTd QdDd

iexcl BTd PBdŠiexcl1permilAT

d PBd iexcl CTd hellipQdDd Dagger SddaggerŠT hellip179dagger

Proof The fact that (ii) implies (i) follows from

Theorem 9 with fchellipxdagger ˆ Acx Gchellipxdagger ˆ Bc hchellipxdagger ˆ CcxJchellipxdagger ˆ Dc fdhellipxdagger ˆ hellipAd iexcl Indaggerx Gdhellipxdagger ˆ Bd hdhellipxdagger ˆCdx Jdhellipxdagger ˆ Dd Vshellipxdagger ˆ xTPx `chellipxdagger ˆ Lcx `dhellipxdagger ˆLdx Wchellipxdagger ˆ Wc and Wdhellipxdagger ˆ Wd To show that hellipidaggerimplies hellipiidagger note that if the linear impulsive dynamical

system given by (173)plusmn(176) is dissipative then it fol-

lows from Theorem 12 with fchellipxdagger ˆ Acx Gchellipxdagger ˆ Bc

hchellipxdagger ˆ Ccx Jchellipxdagger ˆ Dc fdhellipxdagger ˆ hellipAd iexcl Indaggerx Gdhellipxdagger ˆ Bdhdhellipxdagger ˆ Cdx and Jdhellipxdagger ˆ Dd that there exists matrices

P 2 n n Lc 2 pc n Wc 2 pc mc Ld 2 pd n and

Wd 2 pd md with P positive deregnite such that (138)plusmn

(143) are satisreged Finally (177)plusmn(179) follow from

(87)plusmn(89) and Theorem 12 with the linearization given

above amp

Remark 19 Note that the proof of Theorem 14 relies

on Theorem 12 which a priori assumes that the storage

function Vshellipxdagger x 2 n is three-times continuously dif-ferentiable Unlike linear time-invariant dissipative dy-

namical systems with continuous macrows (Willems

1972 b) there does not always exists a smooth (ie

C1) storage function Vshellipxdagger x 2 n for linear dissipa-

tive impulsive dynamical systems

Remark 20 Note that (138)plusmn(143) are equivalent to


Ac Bc

BTc Dc

ˆ

LTc

WTc

Lc Wcpermil Š 0 hellip180dagger

Ad Bd

BTd Dd

ˆ

LTd

WTd

Ld Wdpermil Š 0 hellip181dagger

where

Ac ˆ iexclATc P iexcl PAc Dagger CT

c QcCc

Bc ˆ iexclPBc Dagger CTc hellipQcDc Dagger Scdagger

Dc ˆ Rc Dagger STc Dc Dagger DT

c Sc Dagger DTc QcDc

Ad ˆ P iexcl ATd PAd Dagger CT

d QdCd

Bd ˆ iexclATd PBd Dagger CT

d hellipQdDd Dagger Sddagger

and

Dd ˆ Rd Dagger STd Dd Dagger DT


d PBd

Hence dissipativity of linear impulsive dynamicalsystems with respect to quadratic supply rates can be

characterized via linear matrix inequalities (LMIs)(Boyd et al 1994) Similar remarks hold for the passivity

and non-expansivity results given in Corollaries 5 and 6

respectively

Remark 21 It follows from Theorem 13 that theequivalence between (138)plusmn(143) and dissipativity of a

linear state-dependent impulsive dynamical system doesnot hold In particular for linear state-dependent im-pulsive dynamical systems (138)plusmn(143) are only su -

cient conditions for dissipativity However under theassumptions of Theorem 14 the equivalence betweenthe more involved conditions (162)plusmn(167) and dissipa-

tivity of a linear state-dependent impulsive dynamicalsystem does hold The proof of this fact is identical tothe proof of Theorem 14 using Theorem 13 in place of

Theorem 12 Similar remarks hold for the passivityand non-expansivity results given in Corollaries 5 and

6 respectivelyThe following results present generalizations of the

positive real lemma and the bounded real lemma forlinear impulsive systems respectively

Corollary 5 Consider the linear impulsive dynamicalsystem G given by hellip173daggerplusmnhellip176dagger with mc ˆ lc and md ˆ ldand assume G is minimal Then the following statements

are equivalent

(i) G is passive

(ii) There exist matrices P 2 n n Lc 2 pc nWc 2 pc mc Ld 2 pd n and Wd 2 pd md withP positive deregnite such that

0 ˆ ATc P Dagger PAc Dagger LT

c Lc hellip182dagger

0 ˆ PBc iexcl CTc Dagger LT


0 ˆ Dc Dagger DTc iexcl W T


0 ˆ ATd PAd iexcl P Dagger LT

d Ld hellip185dagger


d Dagger LTd Wd hellip186dagger

0 ˆ Dd Dagger DTd iexcl BT

d PBd iexcl WTd Wd hellip187dagger

If alternatively Dc Dagger DTc gt 0 then G is passive if and

only if there exists an n n positive-deregnite matrix Psuch that

0 lt Dd Dagger DTd iexcl BT


0 ATc P Dagger PAc Dagger hellipPBc iexcl CT

c dagger

hellipDc Dagger DTc daggeriexcl1hellipPBc iexcl CT

c daggerT hellip189dagger

0 ATd PAd iexcl P Dagger hellipAT

d PBd iexcl CTd dagger

hellipDd Dagger DTd iexcl BT

d PBddaggeriexcl1hellipATd PBd iexcl CT

d daggerT hellip190dagger

Proof The result is a direct consequence of Theorem14 with mc ˆ lc md ˆ ld Qc ˆ 0 Sc ˆ Imc

Rc ˆ 0Qd ˆ 0 Sd ˆ Imd

and Rd ˆ 0 amp

Remark 22 Equations (182)plusmn(184) are identical inform to the equations appearing in the continuous-time positive real lemma (Anderson 1967) used tocharacterize positive realness for continuous-timelinear systems in the state-space while (185)plusmn(187) areidentical in form to the equations appearing in thediscrete-time positive real lemma (Hitz and Anderson1969) This is not surprising since as noted in Remark14 impulsive dynamical systems involve a hybridformulation of continuous-time and discrete-time dy-namics A key di erence however is the fact that inthe impulsive case a single positive-deregnite matrix P isrequired to satisfy all six equations Similar remarkshold for Corollary 6

Corollary 6 Consider the linear impulsive dynamicalsystem G given by hellip173daggerplusmnhellip176dagger and assume G is minimalThen the following statements are equivalent

(i) G is non-expansive


0 ˆ ATc P Dagger PAc Dagger CT

c Cc Dagger LTc Lc hellip191dagger

0 ˆ PBc Dagger CTc Dc Dagger LT


0 ˆ reg2c Imc

iexcl DTc Dc iexcl WT



0 ˆ ATd PAd iexcl P Dagger CT

d Cd Dagger LTd Ld hellip194dagger

0 ˆ ATd PBd Dagger CT

d Dd Dagger LTd Wd hellip195dagger

0 ˆ reg2dId iexcl DT

d Dd iexcl BTd PBd iexcl WT

d Wd hellip196dagger

If alternatively reg2c Imc

iexcl DTc Dc gt 0 then G is non-expan-

sive if and only if there exists an n n positive-deregnitematrix P such that

0 lt reg2dImd

iexcl DTd Dd iexcl BT


0 ATc P Dagger PAc Dagger hellipPBc Dagger CT

c Dcdaggerhellipreg2c Imc

iexcl DTc Dcdagger

iexcl1

hellipPBc Dagger CTc DcdaggerT Dagger CT

c Cc hellip198dagger


d PBd Dagger CTd Dddaggerhellipreg2

dImdiexcl DT

d Dd

iexcl BTd PBddaggeriexcl1hellipAT

d PBd Dagger CTd DddaggerT Dagger CT

d Cd hellip199dagger

Proof The result is a direct consequence of Theorem14 with Qc ˆ iexclIlc Sc ˆ 0 Rc ˆ reg2

c Imc Qd ˆ iexclIld

Sd ˆ 0 and Rd ˆ reg2dImd

amp

Remark 23 It follows from Remark 15 that if (182)and (191) are replaced respectively by

0 ˆ ATc P Dagger PAc Dagger P Dagger LT


0 ˆ ATc P Dagger PAc Dagger P Dagger CT


where gt 0 then (200) (183)plusmn(187) provide necessaryand su cient conditions for exponential passivity while(201) (192)plusmn(196) provide necessary and su cient con-ditions for exponential non-expansivity These con-ditions present generalizations of the strict positivereal lemma and the strict bounded real lemma for linearimpulsive systems respectively

7 Conclusion

In this paper we developed new invariant set stabilitytheorems for non-linear impulsive dynamical systemsFurthermore we extended the classical notions of dis-sipativity theory to non-linear dynamical systems withimpulsive e ects Speciregcally the concepts of storagefunctions and supply rates are extended to impulsivedynamical systems providing a generalized hybridsystem energy interpretation in terms of stored energydissipated energy over the continuous-time dynamicsand dissipated energy at the resetting instantsFurthermore extended KalmanplusmnYakubovichplusmnPopovalgebraic conditions in terms of the impulsive systemdynamics for characterizing dissipativeness via systemstorage functions are also derived In the case of quad-ratic supply rates involving net system power and inputplusmnoutput energy these results provide generalizations ofthe classical notions of passivity and non-expansivity In

addition for linear impulsive systems the proposedresults provide a generalization of the positive reallemma and the bounded real lemma In Part II of thispaper (Haddad et al 2001) we develop general stabilitycriteria for feedback interconnections of non-linearimpulsive systems as well as a unireged framework forhybrid feedback optimal and inverse optimal control

Acknowledgements

This research was supported in part by the NationalScience Foundation under Grant ECS-9496249 the AirForce O ce of Scientiregc Research under Grant F49620-00-1-0095 and the Army Research O ce under GrantDAAH04-96-1-0008 The authors would like to thankProfessor S P Bhat for several helpful discussionsregarding Assumption 1 and Theorem 3

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Popov V M 1973 Hyperstability of Control Systems (NewYork Springer-Verlag)

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Multivariable Feedback Systems (Cambridge MIT Press)Samoilenko A M and Perestyuk N A 1995 Impulsive

Di erential Equations (Singapore World Scientiregc)Simeonov P S and Bainov D D 1985 The second method

of Lyapunov for systems with an impulse e ect TamkangJournal of Mathematics 16 19plusmn40

Simeonov P S and Bainov D D 1987 Stability withrespect to part of the variables in systems with impulsee ect Journal of Mathematics Analysis and Applications124 547plusmn560

Tomlin C Pappas G J and Sastry S 1998 Conmacrictresolution for air tra c management a study in multiagenthybrid systems IEEE Transactions on Automatic Control43 509plusmn521

Vidyasagar M 1993 Nonlinear Systems Analysis(Englewood Cli s NJ Prentice-Hall)

Willems J C 1972a Dissipative dynamical systems PartI General theory Arch Rational Mech Anal 45 321plusmn351


Willems J C 1972b Dissipative dynamical systems Part IIQuadratic supply rates Arch Rational Mech Anal 45 359plusmn393

Yang T 1999 Impulsive control IEEE Transactions onAutomatic Control 44 1081plusmn1083

Ye H Michel A N and Hou L 1998a Stability analysisof systems with impulsive e ects IEEE Transactions onAutomatic Control 43 1719plusmn1723

Ye H Michel A N and Hou L 1998b Stability theoryfor hybrid dynamical systems IEEE Transactions onAutomatic Control 43 461plusmn474

Zames G 1966 On the inputplusmnoutput stability of time-varyingnonlinear feedback systems Part I Conditions derived usingconcepts of loop gain conicity and positivity IEEE Trans-actions on Automatic Control 11 228plusmn238


using exponentially weighted system storage functionswith appropriate exponentially weighted supply ratesthe concept of exponential dissipativity was introducedin Chellaboina and Haddad (2000)

Dissipativity theory along with its connections toLyapunov stability theory has been extensively devel-oped for dynamical systems possessing continuousmacrows However in light of the increasingly complex nat-ure of the dynamical systems discussed above discontin-uous system macrows arise naturally Alternatively withinthe context of feedback control active energy macrow reset-ting control for interconnected subsystems also gives riseto discontinuous closed-loop system macrows Speciregcallyif a dissipative or lossless plant is at a high energy leveland a dissipative feedback controller at a low energylevel is attached to it then energy will generally tendto macrow from the plant into the controller decreasingthe plant energy and increasing the controller energy(Kishimoto et al 1995) Of course emulated energyand not physical energy is accumulated by the control-ler Conversely if the attached controller is at a highenergy level and a plant is at a low energy level thenenergy can macrow from the controller to the plant sincea controller can generate real physical energy to e ectthe required energy macrow Hence if and when the con-troller states coincide with a high emulated energy levelthen we can reset these states to remove the emulatedenergy so that the emulated energy is not returned to theplant In this case the overall closed-loop system con-sisting of the plant and the controller possesses discon-tinuous macrows characterized by impulsive di erentialequations (Lakshmikantham et al 1989) Within thecontext of vibration control using resetting virtualabsorbers these ideas were regrst presented in Bupp etal (2000)

Motivated by complex hybrid dynamical systemspossessing discontinuous macrows in this paper we developstability dissipativity and exponential dissipativityconcepts for non-linear impulsive dynamical systemsSpeciregcally we develop an invariance principle forimpulsive dynamical systems wherein system trajectoriesconverge to a largest invariant set contained in a hybridlevel surface composed of a union involving vanishingLyapunov derivatives and di erences of the continuous-time trajectories and resetting instants respectivelyFurthermore we extend the notions of classical dissipa-tivity theory using generalized storage functions andsupply rates for impulsive dynamical systems The over-all approach provides an interpretation of a generalizedhybrid energy balance for an impulsive dynamicalsystem in terms of the stored or accumulated general-ized energy dissipated energy over the continuous-timedynamics and dissipated energy at the resetting instantsFurthermore as in the case of dynamical systems pos-sessing continuous macrows (Willems 1972 a) we show that

the set of all possible storage functions of an impulsive

dynamical system forms a convex set and is bounded

from below by the systemrsquos available stored generalized

energy which can be recovered from the system and

bounded from above by the systemrsquos required general-ized energy supply needed to transfer the system from an

initial state of minimum generalized energy to a given

state In addition for two kinds of non-linear impulsive

dynamical systems namely time-dependent and state-dependent impulsive systems we develop extended

KalmanplusmnYakubovichplusmnPopov algebraic conditions in

terms of the system dynamics for characterizing dissipa-

tiveness via system storage functions for impulsive dyna-mical systems

Although the results of this paper are conregned to

analysis stability and optimality results of feedback

non-linear impulsive systems are discussed in the secondpart of this paper (Haddad et al 2001) The main con-

tribution of this two-part paper is to develop a unireged

framework for the analysis and control synthesis of

non-linear impulsive systems However since impulsive

dynamical systems involve a hybrid formulation of con-tinuous-time and discrete-time dynamics these papers

also provide a tutorial for stability dissipativity feed-

back interconnections and optimality of continuous-time and discrete-time dynamical systems which can be

viewed as a specialization of impulsive systems

The contents of the paper are as follows In 2 we

establish deregnitions notation and review some basic

results on impulsive dynamical systems In 3 we presentLyapunov asymptotic and exponential stability results

for impulsive dynamical systems Furthermore new

invariant set theorems are derived wherein system tra-

jectories converge to a largest invariant set contained ina hybrid Lyapunov level surface composed of a union

involving vanishing Lyapunov derivatives and di er-

ences of the hybrid system dynamics Then in 4 we

extend the notion of dissipative dynamical systems todevelop the concept of dissipativity for impulsive dyna-

mical systems In 5 we develop extended Kalmanplusmn

YakubovichplusmnPopov algebraic conditions in terms of

the hybrid system dynamics for characterizing dissipa-tiveness via system storage functions for impulsive

systems Furthermore a generalized hybrid energy bal-

ance interpretation involving the systemrsquos stored or

accumulated energy dissipated energy over the contin-

uous-time dynamics and dissipated energy at the reset-ting instants is given Specialization of these results to

passive and non-expansive impulsive systems is also pro-

vided In 6 we specialize the results of 5 to linearimpulsive systems to obtain extended hybrid Kalmanplusmn

YakubovichplusmnPopov equations for positive real and

bounded real impulsive systems Finally we draw con-

clusions in 7

























ting events





















9=

hellip1dagger


hellip2dagger


hellip3dagger


hellip4dagger














ˆ lim0Dagger

















interior of S







S ˆ[

k






where






















0 iexclI


0

I




where


0






































0





Daggerhellipt

tk




Daggerhellipt

tk





tk




0


DaggerXk

iˆ1






0


t 0 hellip37dagger






s





































ˆ T x0ˆ
























1iˆ1 converging


















infp2hellipx0dagger














0


DaggerXk

iˆ1








ˆ limn1









where Mregx0
















bdquo tt






0 microhellipT

t0




T t0 hellip48dagger




hellipT

t0


DaggerX



hellip49dagger




hellipT

t0


DaggerX



hellip50dagger




hellipT

t0


DaggerX




hellipT

0


DaggerX

k2N permil0T dagger



where


hellipT

0


DaggerX

k2N permil0T dagger


hellip52dagger





t0


DaggerX





DaggerhellipT

t0


DaggerX













microhellipT

t0




ˆhellipt

t0




DaggerhellipT

t


k2N permiltT dagger


which implies


t0


iexclX



microhellipT

t


k2N permiltT dagger


Hence


t0


DaggerX




hellipT

t


DaggerX

k2N permiltT dagger





0 microhellipT

tt




ˆhellipt0

tt




DaggerhellipT

t0






t0


DaggerX





hellipT

t0


DaggerX







t0


iexclX



iexcl hellipT

t0


DaggerX



which implies



hellipT

t0


DaggerX










t







t









microhelliptDagger

k

tk


















microhellip tt

tDaggerj


Daggerhelliptj

tDaggerjiexcl1


Daggerhellipti

t


ˆhellip tt

t


k2N permiltttdagger



















hellipT

t0


X



ˆ 0 hellip67dagger





hellipt0

T


DaggerX



hellip68dagger



hellipt0

T


DaggerX



hellip69dagger









T


DaggerX



ˆhellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX



and hence


hellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX




t


DaggerX





0 microhellipt0

T




and hence


hellipt0

T


DaggerX



hellip73dagger

which implies that






T


DaggerX



and hence


hellipt0

T


DaggerX






























c Scuc DaggeruT
















functions Vsn `c


n 1 md and







and for all x 2 n



0 ˆ 12V 0




c hellipxdaggerSc






0 ˆ 12P1ud










c Rcuc yTd Qdyd


d Rduddagger

If alternatively









d hellipxdaggerSd





Dagger permil12V 0



2V 0




Dagger permil12P1ud



2P1ud





c Scuc

Dagger uTc Rcuc yT

d Qdyd


d Rduddagger



t


microhellip tt

t




ˆhellip tt

t






ˆhellip tt

t








ˆhellip tt

t




ˆhellip tt

t








Dagger uTd P2ud

hellipxdaggerud
















ˆ iexcl

`Tc hellipxdagger

WTc hellipxdagger





ˆ iexcl

`Td hellipxdagger

WTd hellipxdagger



where










and


d hellipxdaggerSd


hellipxdaggerdagger






T c 71 0


c hellipxdagger Imc


T d 71 0


d hellipxdagger Imd

amp





section































hellip tt

t


k2N permiltttdagger



Daggerhellip tt

t



DaggerX

k2N permiltttdagger

























yTd Qdyd Dagger 2yT



n 1 md and P2ud




0 ˆ 12V 0





0 ˆ 12P1ud




hellipxdagger

hellip104dagger





2GT




t



and












c Rcuc




c Jchellipxdagger



x 2 n uc 2 mc












d hellipxdagger




n md such that








ud udˆ0



ud2

udˆ0

ˆ 12GT


amp



c yc 2uTd yddagger


c uTc uc iexcl yT

c yc reg2duT


given


c yc reg2duT





n `cn pc `d n pd Wc


n 1 md and P2ud






and for all x 2 n



0 ˆ 12V 0






0 ˆ 12P1ud













2V 0



2V 0






d hellipxdaggerŠT

hellip123dagger

then G is passive


Sd ˆ Imd





n `cn pc `d n pd Wc








and for all x 2 n




0 ˆ 12V 0



0 ˆ reg2c Imc





hellip128dagger

0 ˆ 12P1ud



0 ˆ reg2dImd




iexclJT





0 lt reg2dImd





Dagger permil12V 0


permilreg2c Imc


permil12V 0



Dagger permil12P1ud


permilreg2dImd



permil12P1ud




c Imcand Rd ˆ reg2



4yT


4yT






c ScucDaggeruT







0 ˆ 12V 0













n `cn pc




lc mc







yTd Qdyd Dagger 2yT


















d PBd iexcl WTd Wd

hellip143dagger

where

Ac ˆ fc

x xˆ0


Cc ˆ hc

x xˆ0


9gtgtgt=


Ad ˆ fd

x xˆ0


Cd ˆ hd

x xˆ0


9gtgtgt=






t



and





























hellip156dagger


hellip157dagger

where Vrn dcr

n mc and ddrn





c QcDcuc

Dagger uTc DT



c Rcucdagger





d QdDdud



d Rduddagger



limkxk2Daggerkuck20


ˆ 0

limkxk2Daggerkudk20


ˆ 0






Dagger uTc hellipWT


c Sc






d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd








yTd Qdyd Dagger 2yT





hellip162dagger


c Wcdagger x 62 Zx

hellip163dagger






hellip165dagger



hellip166dagger



d PBd iexcl WTd Wd

hellip167dagger







Dagger uTc hellipWT

c Wc iexcl DTc QcDc



hellip168dagger





d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd






where





c Dc iexcl Rc



d Wd


d QdDd iexcl DTd Sd


d PBd
















uTc Rcuc yT










yTd Qdyd Dagger 2yT



P such that








iexcl1



d QdCd
















above amp










Ac Bc

BTc Dc

ˆ

LTc

WTc


Ad Bd

BTd Dd

ˆ

LTd

WTd


where


c QcCc





d QdCd



and



d PBd




respectively









are equivalent

(i) G is passive



















c dagger










and Rd ˆ 0 amp









0 ˆ reg2c Imc














0 lt reg2dImd





iexcl DTc Dcdagger

iexcl1





dImdiexcl DT

d Dd







amp







7 Conclusion



Acknowledgements


References





















































































ting events





















9=

hellip1dagger


hellip2dagger


hellip3dagger


hellip4dagger














ˆ lim0Dagger

















interior of S







S ˆ[

k






where






















0 iexclI


0

I




where


0






































0





Daggerhellipt

tk




Daggerhellipt

tk





tk




0


DaggerXk

iˆ1






0


t 0 hellip37dagger






s





































ˆ T x0ˆ
























1iˆ1 converging


















infp2hellipx0dagger














0


DaggerXk

iˆ1








ˆ limn1









where Mregx0
















bdquo tt






0 microhellipT

t0




T t0 hellip48dagger




hellipT

t0


DaggerX



hellip49dagger




hellipT

t0


DaggerX



hellip50dagger




hellipT

t0


DaggerX




hellipT

0


DaggerX

k2N permil0T dagger



where


hellipT

0


DaggerX

k2N permil0T dagger


hellip52dagger





t0


DaggerX





DaggerhellipT

t0


DaggerX













microhellipT

t0




ˆhellipt

t0




DaggerhellipT

t


k2N permiltT dagger


which implies


t0


iexclX



microhellipT

t


k2N permiltT dagger


Hence


t0


DaggerX




hellipT

t


DaggerX

k2N permiltT dagger





0 microhellipT

tt




ˆhellipt0

tt




DaggerhellipT

t0






t0


DaggerX





hellipT

t0


DaggerX







t0


iexclX



iexcl hellipT

t0


DaggerX



which implies



hellipT

t0


DaggerX










t







t









microhelliptDagger

k

tk


















microhellip tt

tDaggerj


Daggerhelliptj

tDaggerjiexcl1


Daggerhellipti

t


ˆhellip tt

t


k2N permiltttdagger



















hellipT

t0


X



ˆ 0 hellip67dagger





hellipt0

T


DaggerX



hellip68dagger



hellipt0

T


DaggerX



hellip69dagger









T


DaggerX



ˆhellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX



and hence


hellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX




t


DaggerX





0 microhellipt0

T




and hence


hellipt0

T


DaggerX



hellip73dagger

which implies that






T


DaggerX



and hence


hellipt0

T


DaggerX






























c Scuc DaggeruT
















functions Vsn `c


n 1 md and







and for all x 2 n



0 ˆ 12V 0




c hellipxdaggerSc






0 ˆ 12P1ud










c Rcuc yTd Qdyd


d Rduddagger

If alternatively









d hellipxdaggerSd





Dagger permil12V 0



2V 0




Dagger permil12P1ud



2P1ud





c Scuc

Dagger uTc Rcuc yT

d Qdyd


d Rduddagger



t


microhellip tt

t




ˆhellip tt

t






ˆhellip tt

t








ˆhellip tt

t




ˆhellip tt

t








Dagger uTd P2ud

hellipxdaggerud
















ˆ iexcl

`Tc hellipxdagger

WTc hellipxdagger





ˆ iexcl

`Td hellipxdagger

WTd hellipxdagger



where










and


d hellipxdaggerSd


hellipxdaggerdagger






T c 71 0


c hellipxdagger Imc


T d 71 0


d hellipxdagger Imd

amp





section































hellip tt

t


k2N permiltttdagger



Daggerhellip tt

t



DaggerX

k2N permiltttdagger

























yTd Qdyd Dagger 2yT



n 1 md and P2ud




0 ˆ 12V 0





0 ˆ 12P1ud




hellipxdagger

hellip104dagger





2GT




t



and












c Rcuc




c Jchellipxdagger



x 2 n uc 2 mc












d hellipxdagger




n md such that








ud udˆ0



ud2

udˆ0

ˆ 12GT


amp



c yc 2uTd yddagger


c uTc uc iexcl yT

c yc reg2duT


given


c yc reg2duT





n `cn pc `d n pd Wc


n 1 md and P2ud






and for all x 2 n



0 ˆ 12V 0






0 ˆ 12P1ud













2V 0



2V 0






d hellipxdaggerŠT

hellip123dagger

then G is passive


Sd ˆ Imd





n `cn pc `d n pd Wc








and for all x 2 n




0 ˆ 12V 0



0 ˆ reg2c Imc





hellip128dagger

0 ˆ 12P1ud



0 ˆ reg2dImd




iexclJT





0 lt reg2dImd





Dagger permil12V 0


permilreg2c Imc


permil12V 0



Dagger permil12P1ud


permilreg2dImd



permil12P1ud




c Imcand Rd ˆ reg2



4yT


4yT






c ScucDaggeruT







0 ˆ 12V 0













n `cn pc




lc mc







yTd Qdyd Dagger 2yT


















d PBd iexcl WTd Wd

hellip143dagger

where

Ac ˆ fc

x xˆ0


Cc ˆ hc

x xˆ0


9gtgtgt=


Ad ˆ fd

x xˆ0


Cd ˆ hd

x xˆ0


9gtgtgt=






t



and





























hellip156dagger


hellip157dagger

where Vrn dcr

n mc and ddrn





c QcDcuc

Dagger uTc DT



c Rcucdagger





d QdDdud



d Rduddagger



limkxk2Daggerkuck20


ˆ 0

limkxk2Daggerkudk20


ˆ 0






Dagger uTc hellipWT


c Sc






d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd








yTd Qdyd Dagger 2yT





hellip162dagger


c Wcdagger x 62 Zx

hellip163dagger






hellip165dagger



hellip166dagger



d PBd iexcl WTd Wd

hellip167dagger







Dagger uTc hellipWT

c Wc iexcl DTc QcDc



hellip168dagger





d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd






where





c Dc iexcl Rc



d Wd


d QdDd iexcl DTd Sd


d PBd
















uTc Rcuc yT










yTd Qdyd Dagger 2yT



P such that








iexcl1



d QdCd
















above amp










Ac Bc

BTc Dc

ˆ

LTc

WTc


Ad Bd

BTd Dd

ˆ

LTd

WTd


where


c QcCc





d QdCd



and



d PBd




respectively









are equivalent

(i) G is passive



















c dagger










and Rd ˆ 0 amp









0 ˆ reg2c Imc














0 lt reg2dImd





iexcl DTc Dcdagger

iexcl1





dImdiexcl DT

d Dd







amp







7 Conclusion



Acknowledgements


References









































































interior of S







S ˆ[

k






where






















0 iexclI


0

I




where


0






































0





Daggerhellipt

tk




Daggerhellipt

tk





tk




0


DaggerXk

iˆ1






0


t 0 hellip37dagger






s





































ˆ T x0ˆ
























1iˆ1 converging


















infp2hellipx0dagger














0


DaggerXk

iˆ1








ˆ limn1









where Mregx0
















bdquo tt






0 microhellipT

t0




T t0 hellip48dagger




hellipT

t0


DaggerX



hellip49dagger




hellipT

t0


DaggerX



hellip50dagger




hellipT

t0


DaggerX




hellipT

0


DaggerX

k2N permil0T dagger



where


hellipT

0


DaggerX

k2N permil0T dagger


hellip52dagger





t0


DaggerX





DaggerhellipT

t0


DaggerX













microhellipT

t0




ˆhellipt

t0




DaggerhellipT

t


k2N permiltT dagger


which implies


t0


iexclX



microhellipT

t


k2N permiltT dagger


Hence


t0


DaggerX




hellipT

t


DaggerX

k2N permiltT dagger





0 microhellipT

tt




ˆhellipt0

tt




DaggerhellipT

t0






t0


DaggerX





hellipT

t0


DaggerX







t0


iexclX



iexcl hellipT

t0


DaggerX



which implies



hellipT

t0


DaggerX










t







t









microhelliptDagger

k

tk


















microhellip tt

tDaggerj


Daggerhelliptj

tDaggerjiexcl1


Daggerhellipti

t


ˆhellip tt

t


k2N permiltttdagger



















hellipT

t0


X



ˆ 0 hellip67dagger





hellipt0

T


DaggerX



hellip68dagger



hellipt0

T


DaggerX



hellip69dagger









T


DaggerX



ˆhellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX



and hence


hellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX




t


DaggerX





0 microhellipt0

T




and hence


hellipt0

T


DaggerX



hellip73dagger

which implies that






T


DaggerX



and hence


hellipt0

T


DaggerX






























c Scuc DaggeruT
















functions Vsn `c


n 1 md and







and for all x 2 n



0 ˆ 12V 0




c hellipxdaggerSc






0 ˆ 12P1ud










c Rcuc yTd Qdyd


d Rduddagger

If alternatively









d hellipxdaggerSd





Dagger permil12V 0



2V 0




Dagger permil12P1ud



2P1ud





c Scuc

Dagger uTc Rcuc yT

d Qdyd


d Rduddagger



t


microhellip tt

t




ˆhellip tt

t






ˆhellip tt

t








ˆhellip tt

t




ˆhellip tt

t








Dagger uTd P2ud

hellipxdaggerud
















ˆ iexcl

`Tc hellipxdagger

WTc hellipxdagger





ˆ iexcl

`Td hellipxdagger

WTd hellipxdagger



where










and


d hellipxdaggerSd


hellipxdaggerdagger






T c 71 0


c hellipxdagger Imc


T d 71 0


d hellipxdagger Imd

amp





section































hellip tt

t


k2N permiltttdagger



Daggerhellip tt

t



DaggerX

k2N permiltttdagger

























yTd Qdyd Dagger 2yT



n 1 md and P2ud




0 ˆ 12V 0





0 ˆ 12P1ud




hellipxdagger

hellip104dagger





2GT




t



and












c Rcuc




c Jchellipxdagger



x 2 n uc 2 mc












d hellipxdagger




n md such that








ud udˆ0



ud2

udˆ0

ˆ 12GT


amp



c yc 2uTd yddagger


c uTc uc iexcl yT

c yc reg2duT


given


c yc reg2duT





n `cn pc `d n pd Wc


n 1 md and P2ud






and for all x 2 n



0 ˆ 12V 0






0 ˆ 12P1ud













2V 0



2V 0






d hellipxdaggerŠT

hellip123dagger

then G is passive


Sd ˆ Imd





n `cn pc `d n pd Wc








and for all x 2 n




0 ˆ 12V 0



0 ˆ reg2c Imc





hellip128dagger

0 ˆ 12P1ud



0 ˆ reg2dImd




iexclJT





0 lt reg2dImd





Dagger permil12V 0


permilreg2c Imc


permil12V 0



Dagger permil12P1ud


permilreg2dImd



permil12P1ud




c Imcand Rd ˆ reg2



4yT


4yT






c ScucDaggeruT







0 ˆ 12V 0













n `cn pc




lc mc







yTd Qdyd Dagger 2yT


















d PBd iexcl WTd Wd

hellip143dagger

where

Ac ˆ fc

x xˆ0


Cc ˆ hc

x xˆ0


9gtgtgt=


Ad ˆ fd

x xˆ0


Cd ˆ hd

x xˆ0


9gtgtgt=






t



and





























hellip156dagger


hellip157dagger

where Vrn dcr

n mc and ddrn





c QcDcuc

Dagger uTc DT



c Rcucdagger





d QdDdud



d Rduddagger



limkxk2Daggerkuck20


ˆ 0

limkxk2Daggerkudk20


ˆ 0






Dagger uTc hellipWT


c Sc






d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd








yTd Qdyd Dagger 2yT





hellip162dagger


c Wcdagger x 62 Zx

hellip163dagger






hellip165dagger



hellip166dagger



d PBd iexcl WTd Wd

hellip167dagger







Dagger uTc hellipWT

c Wc iexcl DTc QcDc



hellip168dagger





d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd






where





c Dc iexcl Rc



d Wd


d QdDd iexcl DTd Sd


d PBd
















uTc Rcuc yT










yTd Qdyd Dagger 2yT



P such that








iexcl1



d QdCd
















above amp










Ac Bc

BTc Dc

ˆ

LTc

WTc


Ad Bd

BTd Dd

ˆ

LTd

WTd


where


c QcCc





d QdCd



and



d PBd




respectively









are equivalent

(i) G is passive



















c dagger










and Rd ˆ 0 amp









0 ˆ reg2c Imc














0 lt reg2dImd





iexcl DTc Dcdagger

iexcl1





dImdiexcl DT

d Dd







amp







7 Conclusion



Acknowledgements


References

















































































0 iexclI


0

I




where


0






































0





Daggerhellipt

tk




Daggerhellipt

tk





tk




0


DaggerXk

iˆ1






0


t 0 hellip37dagger






s





































ˆ T x0ˆ
























1iˆ1 converging


















infp2hellipx0dagger














0


DaggerXk

iˆ1








ˆ limn1









where Mregx0
















bdquo tt






0 microhellipT

t0




T t0 hellip48dagger




hellipT

t0


DaggerX



hellip49dagger




hellipT

t0


DaggerX



hellip50dagger




hellipT

t0


DaggerX




hellipT

0


DaggerX

k2N permil0T dagger



where


hellipT

0


DaggerX

k2N permil0T dagger


hellip52dagger





t0


DaggerX





DaggerhellipT

t0


DaggerX













microhellipT

t0




ˆhellipt

t0




DaggerhellipT

t


k2N permiltT dagger


which implies


t0


iexclX



microhellipT

t


k2N permiltT dagger


Hence


t0


DaggerX




hellipT

t


DaggerX

k2N permiltT dagger





0 microhellipT

tt




ˆhellipt0

tt




DaggerhellipT

t0






t0


DaggerX





hellipT

t0


DaggerX







t0


iexclX



iexcl hellipT

t0


DaggerX



which implies



hellipT

t0


DaggerX










t







t









microhelliptDagger

k

tk


















microhellip tt

tDaggerj


Daggerhelliptj

tDaggerjiexcl1


Daggerhellipti

t


ˆhellip tt

t


k2N permiltttdagger



















hellipT

t0


X



ˆ 0 hellip67dagger





hellipt0

T


DaggerX



hellip68dagger



hellipt0

T


DaggerX



hellip69dagger









T


DaggerX



ˆhellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX



and hence


hellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX




t


DaggerX





0 microhellipt0

T




and hence


hellipt0

T


DaggerX



hellip73dagger

which implies that






T


DaggerX



and hence


hellipt0

T


DaggerX






























c Scuc DaggeruT
















functions Vsn `c


n 1 md and







and for all x 2 n



0 ˆ 12V 0




c hellipxdaggerSc






0 ˆ 12P1ud










c Rcuc yTd Qdyd


d Rduddagger

If alternatively









d hellipxdaggerSd





Dagger permil12V 0



2V 0




Dagger permil12P1ud



2P1ud





c Scuc

Dagger uTc Rcuc yT

d Qdyd


d Rduddagger



t


microhellip tt

t




ˆhellip tt

t






ˆhellip tt

t








ˆhellip tt

t




ˆhellip tt

t








Dagger uTd P2ud

hellipxdaggerud
















ˆ iexcl

`Tc hellipxdagger

WTc hellipxdagger





ˆ iexcl

`Td hellipxdagger

WTd hellipxdagger



where










and


d hellipxdaggerSd


hellipxdaggerdagger






T c 71 0


c hellipxdagger Imc


T d 71 0


d hellipxdagger Imd

amp





section































hellip tt

t


k2N permiltttdagger



Daggerhellip tt

t



DaggerX

k2N permiltttdagger

























yTd Qdyd Dagger 2yT



n 1 md and P2ud




0 ˆ 12V 0





0 ˆ 12P1ud




hellipxdagger

hellip104dagger





2GT




t



and












c Rcuc




c Jchellipxdagger



x 2 n uc 2 mc












d hellipxdagger




n md such that








ud udˆ0



ud2

udˆ0

ˆ 12GT


amp



c yc 2uTd yddagger


c uTc uc iexcl yT

c yc reg2duT


given


c yc reg2duT





n `cn pc `d n pd Wc


n 1 md and P2ud






and for all x 2 n



0 ˆ 12V 0






0 ˆ 12P1ud













2V 0



2V 0






d hellipxdaggerŠT

hellip123dagger

then G is passive


Sd ˆ Imd





n `cn pc `d n pd Wc








and for all x 2 n




0 ˆ 12V 0



0 ˆ reg2c Imc





hellip128dagger

0 ˆ 12P1ud



0 ˆ reg2dImd




iexclJT





0 lt reg2dImd





Dagger permil12V 0


permilreg2c Imc


permil12V 0



Dagger permil12P1ud


permilreg2dImd



permil12P1ud




c Imcand Rd ˆ reg2



4yT


4yT






c ScucDaggeruT







0 ˆ 12V 0













n `cn pc




lc mc







yTd Qdyd Dagger 2yT


















d PBd iexcl WTd Wd

hellip143dagger

where

Ac ˆ fc

x xˆ0


Cc ˆ hc

x xˆ0


9gtgtgt=


Ad ˆ fd

x xˆ0


Cd ˆ hd

x xˆ0


9gtgtgt=






t



and





























hellip156dagger


hellip157dagger

where Vrn dcr

n mc and ddrn





c QcDcuc

Dagger uTc DT



c Rcucdagger





d QdDdud



d Rduddagger



limkxk2Daggerkuck20


ˆ 0

limkxk2Daggerkudk20


ˆ 0






Dagger uTc hellipWT


c Sc






d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd








yTd Qdyd Dagger 2yT





hellip162dagger


c Wcdagger x 62 Zx

hellip163dagger






hellip165dagger



hellip166dagger



d PBd iexcl WTd Wd

hellip167dagger







Dagger uTc hellipWT

c Wc iexcl DTc QcDc



hellip168dagger





d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd






where





c Dc iexcl Rc



d Wd


d QdDd iexcl DTd Sd


d PBd
















uTc Rcuc yT










yTd Qdyd Dagger 2yT



P such that








iexcl1



d QdCd
















above amp










Ac Bc

BTc Dc

ˆ

LTc

WTc


Ad Bd

BTd Dd

ˆ

LTd

WTd


where


c QcCc





d QdCd



and



d PBd




respectively









are equivalent

(i) G is passive



















c dagger










and Rd ˆ 0 amp









0 ˆ reg2c Imc














0 lt reg2dImd





iexcl DTc Dcdagger

iexcl1





dImdiexcl DT

d Dd







amp







7 Conclusion



Acknowledgements


References














































































0





Daggerhellipt

tk




Daggerhellipt

tk





tk




0


DaggerXk

iˆ1






0


t 0 hellip37dagger






s





































ˆ T x0ˆ
























1iˆ1 converging


















infp2hellipx0dagger














0


DaggerXk

iˆ1








ˆ limn1









where Mregx0
















bdquo tt






0 microhellipT

t0




T t0 hellip48dagger




hellipT

t0


DaggerX



hellip49dagger




hellipT

t0


DaggerX



hellip50dagger




hellipT

t0


DaggerX




hellipT

0


DaggerX

k2N permil0T dagger



where


hellipT

0


DaggerX

k2N permil0T dagger


hellip52dagger





t0


DaggerX





DaggerhellipT

t0


DaggerX













microhellipT

t0




ˆhellipt

t0




DaggerhellipT

t


k2N permiltT dagger


which implies


t0


iexclX



microhellipT

t


k2N permiltT dagger


Hence


t0


DaggerX




hellipT

t


DaggerX

k2N permiltT dagger





0 microhellipT

tt




ˆhellipt0

tt




DaggerhellipT

t0






t0


DaggerX





hellipT

t0


DaggerX







t0


iexclX



iexcl hellipT

t0


DaggerX



which implies



hellipT

t0


DaggerX










t







t









microhelliptDagger

k

tk


















microhellip tt

tDaggerj


Daggerhelliptj

tDaggerjiexcl1


Daggerhellipti

t


ˆhellip tt

t


k2N permiltttdagger



















hellipT

t0


X



ˆ 0 hellip67dagger





hellipt0

T


DaggerX



hellip68dagger



hellipt0

T


DaggerX



hellip69dagger









T


DaggerX



ˆhellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX



and hence


hellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX




t


DaggerX





0 microhellipt0

T




and hence


hellipt0

T


DaggerX



hellip73dagger

which implies that






T


DaggerX



and hence


hellipt0

T


DaggerX






























c Scuc DaggeruT
















functions Vsn `c


n 1 md and







and for all x 2 n



0 ˆ 12V 0




c hellipxdaggerSc






0 ˆ 12P1ud










c Rcuc yTd Qdyd


d Rduddagger

If alternatively









d hellipxdaggerSd





Dagger permil12V 0



2V 0




Dagger permil12P1ud



2P1ud





c Scuc

Dagger uTc Rcuc yT

d Qdyd


d Rduddagger



t


microhellip tt

t




ˆhellip tt

t






ˆhellip tt

t








ˆhellip tt

t




ˆhellip tt

t








Dagger uTd P2ud

hellipxdaggerud
















ˆ iexcl

`Tc hellipxdagger

WTc hellipxdagger





ˆ iexcl

`Td hellipxdagger

WTd hellipxdagger



where










and


d hellipxdaggerSd


hellipxdaggerdagger






T c 71 0


c hellipxdagger Imc


T d 71 0


d hellipxdagger Imd

amp





section































hellip tt

t


k2N permiltttdagger



Daggerhellip tt

t



DaggerX

k2N permiltttdagger

























yTd Qdyd Dagger 2yT



n 1 md and P2ud




0 ˆ 12V 0





0 ˆ 12P1ud




hellipxdagger

hellip104dagger





2GT




t



and












c Rcuc




c Jchellipxdagger



x 2 n uc 2 mc












d hellipxdagger




n md such that








ud udˆ0



ud2

udˆ0

ˆ 12GT


amp



c yc 2uTd yddagger


c uTc uc iexcl yT

c yc reg2duT


given


c yc reg2duT





n `cn pc `d n pd Wc


n 1 md and P2ud






and for all x 2 n



0 ˆ 12V 0






0 ˆ 12P1ud













2V 0



2V 0






d hellipxdaggerŠT

hellip123dagger

then G is passive


Sd ˆ Imd





n `cn pc `d n pd Wc








and for all x 2 n




0 ˆ 12V 0



0 ˆ reg2c Imc





hellip128dagger

0 ˆ 12P1ud



0 ˆ reg2dImd




iexclJT





0 lt reg2dImd





Dagger permil12V 0


permilreg2c Imc


permil12V 0



Dagger permil12P1ud


permilreg2dImd



permil12P1ud




c Imcand Rd ˆ reg2



4yT


4yT






c ScucDaggeruT







0 ˆ 12V 0













n `cn pc




lc mc







yTd Qdyd Dagger 2yT


















d PBd iexcl WTd Wd

hellip143dagger

where

Ac ˆ fc

x xˆ0


Cc ˆ hc

x xˆ0


9gtgtgt=


Ad ˆ fd

x xˆ0


Cd ˆ hd

x xˆ0


9gtgtgt=






t



and





























hellip156dagger


hellip157dagger

where Vrn dcr

n mc and ddrn





c QcDcuc

Dagger uTc DT



c Rcucdagger





d QdDdud



d Rduddagger



limkxk2Daggerkuck20


ˆ 0

limkxk2Daggerkudk20


ˆ 0






Dagger uTc hellipWT


c Sc






d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd








yTd Qdyd Dagger 2yT





hellip162dagger


c Wcdagger x 62 Zx

hellip163dagger






hellip165dagger



hellip166dagger



d PBd iexcl WTd Wd

hellip167dagger







Dagger uTc hellipWT

c Wc iexcl DTc QcDc



hellip168dagger





d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd






where





c Dc iexcl Rc



d Wd


d QdDd iexcl DTd Sd


d PBd
















uTc Rcuc yT










yTd Qdyd Dagger 2yT



P such that








iexcl1



d QdCd
















above amp










Ac Bc

BTc Dc

ˆ

LTc

WTc


Ad Bd

BTd Dd

ˆ

LTd

WTd


where


c QcCc





d QdCd



and



d PBd




respectively









are equivalent

(i) G is passive



















c dagger










and Rd ˆ 0 amp









0 ˆ reg2c Imc














0 lt reg2dImd





iexcl DTc Dcdagger

iexcl1





dImdiexcl DT

d Dd







amp







7 Conclusion



Acknowledgements


References






























































































ˆ T x0ˆ
























1iˆ1 converging


















infp2hellipx0dagger














0


DaggerXk

iˆ1








ˆ limn1









where Mregx0
















bdquo tt






0 microhellipT

t0




T t0 hellip48dagger




hellipT

t0


DaggerX



hellip49dagger




hellipT

t0


DaggerX



hellip50dagger




hellipT

t0


DaggerX




hellipT

0


DaggerX

k2N permil0T dagger



where


hellipT

0


DaggerX

k2N permil0T dagger


hellip52dagger





t0


DaggerX





DaggerhellipT

t0


DaggerX













microhellipT

t0




ˆhellipt

t0




DaggerhellipT

t


k2N permiltT dagger


which implies


t0


iexclX



microhellipT

t


k2N permiltT dagger


Hence


t0


DaggerX




hellipT

t


DaggerX

k2N permiltT dagger





0 microhellipT

tt




ˆhellipt0

tt




DaggerhellipT

t0






t0


DaggerX





hellipT

t0


DaggerX







t0


iexclX



iexcl hellipT

t0


DaggerX



which implies



hellipT

t0


DaggerX










t







t









microhelliptDagger

k

tk


















microhellip tt

tDaggerj


Daggerhelliptj

tDaggerjiexcl1


Daggerhellipti

t


ˆhellip tt

t


k2N permiltttdagger



















hellipT

t0


X



ˆ 0 hellip67dagger





hellipt0

T


DaggerX



hellip68dagger



hellipt0

T


DaggerX



hellip69dagger









T


DaggerX



ˆhellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX



and hence


hellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX




t


DaggerX





0 microhellipt0

T




and hence


hellipt0

T


DaggerX



hellip73dagger

which implies that






T


DaggerX



and hence


hellipt0

T


DaggerX






























c Scuc DaggeruT
















functions Vsn `c


n 1 md and







and for all x 2 n



0 ˆ 12V 0




c hellipxdaggerSc






0 ˆ 12P1ud










c Rcuc yTd Qdyd


d Rduddagger

If alternatively









d hellipxdaggerSd





Dagger permil12V 0



2V 0




Dagger permil12P1ud



2P1ud





c Scuc

Dagger uTc Rcuc yT

d Qdyd


d Rduddagger



t


microhellip tt

t




ˆhellip tt

t






ˆhellip tt

t








ˆhellip tt

t




ˆhellip tt

t








Dagger uTd P2ud

hellipxdaggerud
















ˆ iexcl

`Tc hellipxdagger

WTc hellipxdagger





ˆ iexcl

`Td hellipxdagger

WTd hellipxdagger



where










and


d hellipxdaggerSd


hellipxdaggerdagger






T c 71 0


c hellipxdagger Imc


T d 71 0


d hellipxdagger Imd

amp





section































hellip tt

t


k2N permiltttdagger



Daggerhellip tt

t



DaggerX

k2N permiltttdagger

























yTd Qdyd Dagger 2yT



n 1 md and P2ud




0 ˆ 12V 0





0 ˆ 12P1ud




hellipxdagger

hellip104dagger





2GT




t



and












c Rcuc




c Jchellipxdagger



x 2 n uc 2 mc












d hellipxdagger




n md such that








ud udˆ0



ud2

udˆ0

ˆ 12GT


amp



c yc 2uTd yddagger


c uTc uc iexcl yT

c yc reg2duT


given


c yc reg2duT





n `cn pc `d n pd Wc


n 1 md and P2ud






and for all x 2 n



0 ˆ 12V 0






0 ˆ 12P1ud













2V 0



2V 0






d hellipxdaggerŠT

hellip123dagger

then G is passive


Sd ˆ Imd





n `cn pc `d n pd Wc








and for all x 2 n




0 ˆ 12V 0



0 ˆ reg2c Imc





hellip128dagger

0 ˆ 12P1ud



0 ˆ reg2dImd




iexclJT





0 lt reg2dImd





Dagger permil12V 0


permilreg2c Imc


permil12V 0



Dagger permil12P1ud


permilreg2dImd



permil12P1ud




c Imcand Rd ˆ reg2



4yT


4yT






c ScucDaggeruT







0 ˆ 12V 0













n `cn pc




lc mc







yTd Qdyd Dagger 2yT


















d PBd iexcl WTd Wd

hellip143dagger

where

Ac ˆ fc

x xˆ0


Cc ˆ hc

x xˆ0


9gtgtgt=


Ad ˆ fd

x xˆ0


Cd ˆ hd

x xˆ0


9gtgtgt=






t



and





























hellip156dagger


hellip157dagger

where Vrn dcr

n mc and ddrn





c QcDcuc

Dagger uTc DT



c Rcucdagger





d QdDdud



d Rduddagger



limkxk2Daggerkuck20


ˆ 0

limkxk2Daggerkudk20


ˆ 0






Dagger uTc hellipWT


c Sc






d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd








yTd Qdyd Dagger 2yT





hellip162dagger


c Wcdagger x 62 Zx

hellip163dagger






hellip165dagger



hellip166dagger



d PBd iexcl WTd Wd

hellip167dagger







Dagger uTc hellipWT

c Wc iexcl DTc QcDc



hellip168dagger





d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd






where





c Dc iexcl Rc



d Wd


d QdDd iexcl DTd Sd


d PBd
















uTc Rcuc yT










yTd Qdyd Dagger 2yT



P such that








iexcl1



d QdCd
















above amp










Ac Bc

BTc Dc

ˆ

LTc

WTc


Ad Bd

BTd Dd

ˆ

LTd

WTd


where


c QcCc





d QdCd



and



d PBd




respectively









are equivalent

(i) G is passive



















c dagger










and Rd ˆ 0 amp









0 ˆ reg2c Imc














0 lt reg2dImd





iexcl DTc Dcdagger

iexcl1





dImdiexcl DT

d Dd







amp







7 Conclusion



Acknowledgements


References




































































ˆ T x0ˆ
























1iˆ1 converging


















infp2hellipx0dagger














0


DaggerXk

iˆ1








ˆ limn1









where Mregx0
















bdquo tt






0 microhellipT

t0




T t0 hellip48dagger




hellipT

t0


DaggerX



hellip49dagger




hellipT

t0


DaggerX



hellip50dagger




hellipT

t0


DaggerX




hellipT

0


DaggerX

k2N permil0T dagger



where


hellipT

0


DaggerX

k2N permil0T dagger


hellip52dagger





t0


DaggerX





DaggerhellipT

t0


DaggerX













microhellipT

t0




ˆhellipt

t0




DaggerhellipT

t


k2N permiltT dagger


which implies


t0


iexclX



microhellipT

t


k2N permiltT dagger


Hence


t0


DaggerX




hellipT

t


DaggerX

k2N permiltT dagger





0 microhellipT

tt




ˆhellipt0

tt




DaggerhellipT

t0






t0


DaggerX





hellipT

t0


DaggerX







t0


iexclX



iexcl hellipT

t0


DaggerX



which implies



hellipT

t0


DaggerX










t







t









microhelliptDagger

k

tk


















microhellip tt

tDaggerj


Daggerhelliptj

tDaggerjiexcl1


Daggerhellipti

t


ˆhellip tt

t


k2N permiltttdagger



















hellipT

t0


X



ˆ 0 hellip67dagger





hellipt0

T


DaggerX



hellip68dagger



hellipt0

T


DaggerX



hellip69dagger









T


DaggerX



ˆhellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX



and hence


hellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX




t


DaggerX





0 microhellipt0

T




and hence


hellipt0

T


DaggerX



hellip73dagger

which implies that






T


DaggerX



and hence


hellipt0

T


DaggerX






























c Scuc DaggeruT
















functions Vsn `c


n 1 md and







and for all x 2 n



0 ˆ 12V 0




c hellipxdaggerSc






0 ˆ 12P1ud










c Rcuc yTd Qdyd


d Rduddagger

If alternatively









d hellipxdaggerSd





Dagger permil12V 0



2V 0




Dagger permil12P1ud



2P1ud





c Scuc

Dagger uTc Rcuc yT

d Qdyd


d Rduddagger



t


microhellip tt

t




ˆhellip tt

t






ˆhellip tt

t








ˆhellip tt

t




ˆhellip tt

t








Dagger uTd P2ud

hellipxdaggerud
















ˆ iexcl

`Tc hellipxdagger

WTc hellipxdagger





ˆ iexcl

`Td hellipxdagger

WTd hellipxdagger



where










and


d hellipxdaggerSd


hellipxdaggerdagger






T c 71 0


c hellipxdagger Imc


T d 71 0


d hellipxdagger Imd

amp





section































hellip tt

t


k2N permiltttdagger



Daggerhellip tt

t



DaggerX

k2N permiltttdagger

























yTd Qdyd Dagger 2yT



n 1 md and P2ud




0 ˆ 12V 0





0 ˆ 12P1ud




hellipxdagger

hellip104dagger





2GT




t



and












c Rcuc




c Jchellipxdagger



x 2 n uc 2 mc












d hellipxdagger




n md such that








ud udˆ0



ud2

udˆ0

ˆ 12GT


amp



c yc 2uTd yddagger


c uTc uc iexcl yT

c yc reg2duT


given


c yc reg2duT





n `cn pc `d n pd Wc


n 1 md and P2ud






and for all x 2 n



0 ˆ 12V 0






0 ˆ 12P1ud













2V 0



2V 0






d hellipxdaggerŠT

hellip123dagger

then G is passive


Sd ˆ Imd





n `cn pc `d n pd Wc








and for all x 2 n




0 ˆ 12V 0



0 ˆ reg2c Imc





hellip128dagger

0 ˆ 12P1ud



0 ˆ reg2dImd




iexclJT





0 lt reg2dImd





Dagger permil12V 0


permilreg2c Imc


permil12V 0



Dagger permil12P1ud


permilreg2dImd



permil12P1ud




c Imcand Rd ˆ reg2



4yT


4yT






c ScucDaggeruT







0 ˆ 12V 0













n `cn pc




lc mc







yTd Qdyd Dagger 2yT


















d PBd iexcl WTd Wd

hellip143dagger

where

Ac ˆ fc

x xˆ0


Cc ˆ hc

x xˆ0


9gtgtgt=


Ad ˆ fd

x xˆ0


Cd ˆ hd

x xˆ0


9gtgtgt=






t



and





























hellip156dagger


hellip157dagger

where Vrn dcr

n mc and ddrn





c QcDcuc

Dagger uTc DT



c Rcucdagger





d QdDdud



d Rduddagger



limkxk2Daggerkuck20


ˆ 0

limkxk2Daggerkudk20


ˆ 0






Dagger uTc hellipWT


c Sc






d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd








yTd Qdyd Dagger 2yT





hellip162dagger


c Wcdagger x 62 Zx

hellip163dagger






hellip165dagger



hellip166dagger



d PBd iexcl WTd Wd

hellip167dagger







Dagger uTc hellipWT

c Wc iexcl DTc QcDc



hellip168dagger





d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd






where





c Dc iexcl Rc



d Wd


d QdDd iexcl DTd Sd


d PBd
















uTc Rcuc yT










yTd Qdyd Dagger 2yT



P such that








iexcl1



d QdCd
















above amp










Ac Bc

BTc Dc

ˆ

LTc

WTc


Ad Bd

BTd Dd

ˆ

LTd

WTd


where


c QcCc





d QdCd



and



d PBd




respectively









are equivalent

(i) G is passive



















c dagger










and Rd ˆ 0 amp









0 ˆ reg2c Imc














0 lt reg2dImd





iexcl DTc Dcdagger

iexcl1





dImdiexcl DT

d Dd







amp







7 Conclusion



Acknowledgements


References









































































infp2hellipx0dagger














0


DaggerXk

iˆ1








ˆ limn1









where Mregx0
















bdquo tt






0 microhellipT

t0




T t0 hellip48dagger




hellipT

t0


DaggerX



hellip49dagger




hellipT

t0


DaggerX



hellip50dagger




hellipT

t0


DaggerX




hellipT

0


DaggerX

k2N permil0T dagger



where


hellipT

0


DaggerX

k2N permil0T dagger


hellip52dagger





t0


DaggerX





DaggerhellipT

t0


DaggerX













microhellipT

t0




ˆhellipt

t0




DaggerhellipT

t


k2N permiltT dagger


which implies


t0


iexclX



microhellipT

t


k2N permiltT dagger


Hence


t0


DaggerX




hellipT

t


DaggerX

k2N permiltT dagger





0 microhellipT

tt




ˆhellipt0

tt




DaggerhellipT

t0






t0


DaggerX





hellipT

t0


DaggerX







t0


iexclX



iexcl hellipT

t0


DaggerX



which implies



hellipT

t0


DaggerX










t







t









microhelliptDagger

k

tk


















microhellip tt

tDaggerj


Daggerhelliptj

tDaggerjiexcl1


Daggerhellipti

t


ˆhellip tt

t


k2N permiltttdagger



















hellipT

t0


X



ˆ 0 hellip67dagger





hellipt0

T


DaggerX



hellip68dagger



hellipt0

T


DaggerX



hellip69dagger









T


DaggerX



ˆhellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX



and hence


hellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX




t


DaggerX





0 microhellipt0

T




and hence


hellipt0

T


DaggerX



hellip73dagger

which implies that






T


DaggerX



and hence


hellipt0

T


DaggerX






























c Scuc DaggeruT
















functions Vsn `c


n 1 md and







and for all x 2 n



0 ˆ 12V 0




c hellipxdaggerSc






0 ˆ 12P1ud










c Rcuc yTd Qdyd


d Rduddagger

If alternatively









d hellipxdaggerSd





Dagger permil12V 0



2V 0




Dagger permil12P1ud



2P1ud





c Scuc

Dagger uTc Rcuc yT

d Qdyd


d Rduddagger



t


microhellip tt

t




ˆhellip tt

t






ˆhellip tt

t








ˆhellip tt

t




ˆhellip tt

t








Dagger uTd P2ud

hellipxdaggerud
















ˆ iexcl

`Tc hellipxdagger

WTc hellipxdagger





ˆ iexcl

`Td hellipxdagger

WTd hellipxdagger



where










and


d hellipxdaggerSd


hellipxdaggerdagger






T c 71 0


c hellipxdagger Imc


T d 71 0


d hellipxdagger Imd

amp





section































hellip tt

t


k2N permiltttdagger



Daggerhellip tt

t



DaggerX

k2N permiltttdagger

























yTd Qdyd Dagger 2yT



n 1 md and P2ud




0 ˆ 12V 0





0 ˆ 12P1ud




hellipxdagger

hellip104dagger





2GT




t



and












c Rcuc




c Jchellipxdagger



x 2 n uc 2 mc












d hellipxdagger




n md such that








ud udˆ0



ud2

udˆ0

ˆ 12GT


amp



c yc 2uTd yddagger


c uTc uc iexcl yT

c yc reg2duT


given


c yc reg2duT





n `cn pc `d n pd Wc


n 1 md and P2ud






and for all x 2 n



0 ˆ 12V 0






0 ˆ 12P1ud













2V 0



2V 0






d hellipxdaggerŠT

hellip123dagger

then G is passive


Sd ˆ Imd





n `cn pc `d n pd Wc








and for all x 2 n




0 ˆ 12V 0



0 ˆ reg2c Imc





hellip128dagger

0 ˆ 12P1ud



0 ˆ reg2dImd




iexclJT





0 lt reg2dImd





Dagger permil12V 0


permilreg2c Imc


permil12V 0



Dagger permil12P1ud


permilreg2dImd



permil12P1ud




c Imcand Rd ˆ reg2



4yT


4yT






c ScucDaggeruT







0 ˆ 12V 0













n `cn pc




lc mc







yTd Qdyd Dagger 2yT


















d PBd iexcl WTd Wd

hellip143dagger

where

Ac ˆ fc

x xˆ0


Cc ˆ hc

x xˆ0


9gtgtgt=


Ad ˆ fd

x xˆ0


Cd ˆ hd

x xˆ0


9gtgtgt=






t



and





























hellip156dagger


hellip157dagger

where Vrn dcr

n mc and ddrn





c QcDcuc

Dagger uTc DT



c Rcucdagger





d QdDdud



d Rduddagger



limkxk2Daggerkuck20


ˆ 0

limkxk2Daggerkudk20


ˆ 0






Dagger uTc hellipWT


c Sc






d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd








yTd Qdyd Dagger 2yT





hellip162dagger


c Wcdagger x 62 Zx

hellip163dagger






hellip165dagger



hellip166dagger



d PBd iexcl WTd Wd

hellip167dagger







Dagger uTc hellipWT

c Wc iexcl DTc QcDc



hellip168dagger





d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd






where





c Dc iexcl Rc



d Wd


d QdDd iexcl DTd Sd


d PBd
















uTc Rcuc yT










yTd Qdyd Dagger 2yT



P such that








iexcl1



d QdCd
















above amp










Ac Bc

BTc Dc

ˆ

LTc

WTc


Ad Bd

BTd Dd

ˆ

LTd

WTd


where


c QcCc





d QdCd



and



d PBd




respectively









are equivalent

(i) G is passive



















c dagger










and Rd ˆ 0 amp









0 ˆ reg2c Imc














0 lt reg2dImd





iexcl DTc Dcdagger

iexcl1





dImdiexcl DT

d Dd







amp







7 Conclusion



Acknowledgements


References





































































bdquo tt






0 microhellipT

t0




T t0 hellip48dagger




hellipT

t0


DaggerX



hellip49dagger




hellipT

t0


DaggerX



hellip50dagger




hellipT

t0


DaggerX




hellipT

0


DaggerX

k2N permil0T dagger



where


hellipT

0


DaggerX

k2N permil0T dagger


hellip52dagger





t0


DaggerX





DaggerhellipT

t0


DaggerX













microhellipT

t0




ˆhellipt

t0




DaggerhellipT

t


k2N permiltT dagger


which implies


t0


iexclX



microhellipT

t


k2N permiltT dagger


Hence


t0


DaggerX




hellipT

t


DaggerX

k2N permiltT dagger





0 microhellipT

tt




ˆhellipt0

tt




DaggerhellipT

t0






t0


DaggerX





hellipT

t0


DaggerX







t0


iexclX



iexcl hellipT

t0


DaggerX



which implies



hellipT

t0


DaggerX










t







t









microhelliptDagger

k

tk


















microhellip tt

tDaggerj


Daggerhelliptj

tDaggerjiexcl1


Daggerhellipti

t


ˆhellip tt

t


k2N permiltttdagger



















hellipT

t0


X



ˆ 0 hellip67dagger





hellipt0

T


DaggerX



hellip68dagger



hellipt0

T


DaggerX



hellip69dagger









T


DaggerX



ˆhellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX



and hence


hellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX




t


DaggerX





0 microhellipt0

T




and hence


hellipt0

T


DaggerX



hellip73dagger

which implies that






T


DaggerX



and hence


hellipt0

T


DaggerX






























c Scuc DaggeruT
















functions Vsn `c


n 1 md and







and for all x 2 n



0 ˆ 12V 0




c hellipxdaggerSc






0 ˆ 12P1ud










c Rcuc yTd Qdyd


d Rduddagger

If alternatively









d hellipxdaggerSd





Dagger permil12V 0



2V 0




Dagger permil12P1ud



2P1ud





c Scuc

Dagger uTc Rcuc yT

d Qdyd


d Rduddagger



t


microhellip tt

t




ˆhellip tt

t






ˆhellip tt

t








ˆhellip tt

t




ˆhellip tt

t








Dagger uTd P2ud

hellipxdaggerud
















ˆ iexcl

`Tc hellipxdagger

WTc hellipxdagger





ˆ iexcl

`Td hellipxdagger

WTd hellipxdagger



where










and


d hellipxdaggerSd


hellipxdaggerdagger






T c 71 0


c hellipxdagger Imc


T d 71 0


d hellipxdagger Imd

amp





section































hellip tt

t


k2N permiltttdagger



Daggerhellip tt

t



DaggerX

k2N permiltttdagger

























yTd Qdyd Dagger 2yT



n 1 md and P2ud




0 ˆ 12V 0





0 ˆ 12P1ud




hellipxdagger

hellip104dagger





2GT




t



and












c Rcuc




c Jchellipxdagger



x 2 n uc 2 mc












d hellipxdagger




n md such that








ud udˆ0



ud2

udˆ0

ˆ 12GT


amp



c yc 2uTd yddagger


c uTc uc iexcl yT

c yc reg2duT


given


c yc reg2duT





n `cn pc `d n pd Wc


n 1 md and P2ud






and for all x 2 n



0 ˆ 12V 0






0 ˆ 12P1ud













2V 0



2V 0






d hellipxdaggerŠT

hellip123dagger

then G is passive


Sd ˆ Imd





n `cn pc `d n pd Wc








and for all x 2 n




0 ˆ 12V 0



0 ˆ reg2c Imc





hellip128dagger

0 ˆ 12P1ud



0 ˆ reg2dImd




iexclJT





0 lt reg2dImd





Dagger permil12V 0


permilreg2c Imc


permil12V 0



Dagger permil12P1ud


permilreg2dImd



permil12P1ud




c Imcand Rd ˆ reg2



4yT


4yT






c ScucDaggeruT







0 ˆ 12V 0













n `cn pc




lc mc







yTd Qdyd Dagger 2yT


















d PBd iexcl WTd Wd

hellip143dagger

where

Ac ˆ fc

x xˆ0


Cc ˆ hc

x xˆ0


9gtgtgt=


Ad ˆ fd

x xˆ0


Cd ˆ hd

x xˆ0


9gtgtgt=






t



and





























hellip156dagger


hellip157dagger

where Vrn dcr

n mc and ddrn





c QcDcuc

Dagger uTc DT



c Rcucdagger





d QdDdud



d Rduddagger



limkxk2Daggerkuck20


ˆ 0

limkxk2Daggerkudk20


ˆ 0






Dagger uTc hellipWT


c Sc






d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd








yTd Qdyd Dagger 2yT





hellip162dagger


c Wcdagger x 62 Zx

hellip163dagger






hellip165dagger



hellip166dagger



d PBd iexcl WTd Wd

hellip167dagger







Dagger uTc hellipWT

c Wc iexcl DTc QcDc



hellip168dagger





d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd






where





c Dc iexcl Rc



d Wd


d QdDd iexcl DTd Sd


d PBd
















uTc Rcuc yT










yTd Qdyd Dagger 2yT



P such that








iexcl1



d QdCd
















above amp










Ac Bc

BTc Dc

ˆ

LTc

WTc


Ad Bd

BTd Dd

ˆ

LTd

WTd


where


c QcCc





d QdCd



and



d PBd




respectively









are equivalent

(i) G is passive



















c dagger










and Rd ˆ 0 amp









0 ˆ reg2c Imc














0 lt reg2dImd





iexcl DTc Dcdagger

iexcl1





dImdiexcl DT

d Dd







amp







7 Conclusion



Acknowledgements


References
































































hellipT

t0


DaggerX



hellip50dagger




hellipT

t0


DaggerX




hellipT

0


DaggerX

k2N permil0T dagger



where


hellipT

0


DaggerX

k2N permil0T dagger


hellip52dagger





t0


DaggerX





DaggerhellipT

t0


DaggerX













microhellipT

t0




ˆhellipt

t0




DaggerhellipT

t


k2N permiltT dagger


which implies


t0


iexclX



microhellipT

t


k2N permiltT dagger


Hence


t0


DaggerX




hellipT

t


DaggerX

k2N permiltT dagger





0 microhellipT

tt




ˆhellipt0

tt




DaggerhellipT

t0






t0


DaggerX





hellipT

t0


DaggerX







t0


iexclX



iexcl hellipT

t0


DaggerX



which implies



hellipT

t0


DaggerX










t







t









microhelliptDagger

k

tk


















microhellip tt

tDaggerj


Daggerhelliptj

tDaggerjiexcl1


Daggerhellipti

t


ˆhellip tt

t


k2N permiltttdagger



















hellipT

t0


X



ˆ 0 hellip67dagger





hellipt0

T


DaggerX



hellip68dagger



hellipt0

T


DaggerX



hellip69dagger









T


DaggerX



ˆhellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX



and hence


hellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX




t


DaggerX





0 microhellipt0

T




and hence


hellipt0

T


DaggerX



hellip73dagger

which implies that






T


DaggerX



and hence


hellipt0

T


DaggerX






























c Scuc DaggeruT
















functions Vsn `c


n 1 md and







and for all x 2 n



0 ˆ 12V 0




c hellipxdaggerSc






0 ˆ 12P1ud










c Rcuc yTd Qdyd


d Rduddagger

If alternatively









d hellipxdaggerSd





Dagger permil12V 0



2V 0




Dagger permil12P1ud



2P1ud





c Scuc

Dagger uTc Rcuc yT

d Qdyd


d Rduddagger



t


microhellip tt

t




ˆhellip tt

t






ˆhellip tt

t








ˆhellip tt

t




ˆhellip tt

t








Dagger uTd P2ud

hellipxdaggerud
















ˆ iexcl

`Tc hellipxdagger

WTc hellipxdagger





ˆ iexcl

`Td hellipxdagger

WTd hellipxdagger



where










and


d hellipxdaggerSd


hellipxdaggerdagger






T c 71 0


c hellipxdagger Imc


T d 71 0


d hellipxdagger Imd

amp





section































hellip tt

t


k2N permiltttdagger



Daggerhellip tt

t



DaggerX

k2N permiltttdagger

























yTd Qdyd Dagger 2yT



n 1 md and P2ud




0 ˆ 12V 0





0 ˆ 12P1ud




hellipxdagger

hellip104dagger





2GT




t



and












c Rcuc




c Jchellipxdagger



x 2 n uc 2 mc












d hellipxdagger




n md such that








ud udˆ0



ud2

udˆ0

ˆ 12GT


amp



c yc 2uTd yddagger


c uTc uc iexcl yT

c yc reg2duT


given


c yc reg2duT





n `cn pc `d n pd Wc


n 1 md and P2ud






and for all x 2 n



0 ˆ 12V 0






0 ˆ 12P1ud













2V 0



2V 0






d hellipxdaggerŠT

hellip123dagger

then G is passive


Sd ˆ Imd





n `cn pc `d n pd Wc








and for all x 2 n




0 ˆ 12V 0



0 ˆ reg2c Imc





hellip128dagger

0 ˆ 12P1ud



0 ˆ reg2dImd




iexclJT





0 lt reg2dImd





Dagger permil12V 0


permilreg2c Imc


permil12V 0



Dagger permil12P1ud


permilreg2dImd



permil12P1ud




c Imcand Rd ˆ reg2



4yT


4yT






c ScucDaggeruT







0 ˆ 12V 0













n `cn pc




lc mc







yTd Qdyd Dagger 2yT


















d PBd iexcl WTd Wd

hellip143dagger

where

Ac ˆ fc

x xˆ0


Cc ˆ hc

x xˆ0


9gtgtgt=


Ad ˆ fd

x xˆ0


Cd ˆ hd

x xˆ0


9gtgtgt=






t



and





























hellip156dagger


hellip157dagger

where Vrn dcr

n mc and ddrn





c QcDcuc

Dagger uTc DT



c Rcucdagger





d QdDdud



d Rduddagger



limkxk2Daggerkuck20


ˆ 0

limkxk2Daggerkudk20


ˆ 0






Dagger uTc hellipWT


c Sc






d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd








yTd Qdyd Dagger 2yT





hellip162dagger


c Wcdagger x 62 Zx

hellip163dagger






hellip165dagger



hellip166dagger



d PBd iexcl WTd Wd

hellip167dagger







Dagger uTc hellipWT

c Wc iexcl DTc QcDc



hellip168dagger





d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd






where





c Dc iexcl Rc



d Wd


d QdDd iexcl DTd Sd


d PBd
















uTc Rcuc yT










yTd Qdyd Dagger 2yT



P such that








iexcl1



d QdCd
















above amp










Ac Bc

BTc Dc

ˆ

LTc

WTc


Ad Bd

BTd Dd

ˆ

LTd

WTd


where


c QcCc





d QdCd



and



d PBd




respectively









are equivalent

(i) G is passive



















c dagger










and Rd ˆ 0 amp









0 ˆ reg2c Imc














0 lt reg2dImd





iexcl DTc Dcdagger

iexcl1





dImdiexcl DT

d Dd







amp







7 Conclusion



Acknowledgements


References


































































microhellipT

t0




ˆhellipt

t0




DaggerhellipT

t


k2N permiltT dagger


which implies


t0


iexclX



microhellipT

t


k2N permiltT dagger


Hence


t0


DaggerX




hellipT

t


DaggerX

k2N permiltT dagger





0 microhellipT

tt




ˆhellipt0

tt




DaggerhellipT

t0






t0


DaggerX





hellipT

t0


DaggerX







t0


iexclX



iexcl hellipT

t0


DaggerX



which implies



hellipT

t0


DaggerX










t







t









microhelliptDagger

k

tk


















microhellip tt

tDaggerj


Daggerhelliptj

tDaggerjiexcl1


Daggerhellipti

t


ˆhellip tt

t


k2N permiltttdagger



















hellipT

t0


X



ˆ 0 hellip67dagger





hellipt0

T


DaggerX



hellip68dagger



hellipt0

T


DaggerX



hellip69dagger









T


DaggerX



ˆhellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX



and hence


hellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX




t


DaggerX





0 microhellipt0

T




and hence


hellipt0

T


DaggerX



hellip73dagger

which implies that






T


DaggerX



and hence


hellipt0

T


DaggerX






























c Scuc DaggeruT
















functions Vsn `c


n 1 md and







and for all x 2 n



0 ˆ 12V 0




c hellipxdaggerSc






0 ˆ 12P1ud










c Rcuc yTd Qdyd


d Rduddagger

If alternatively









d hellipxdaggerSd





Dagger permil12V 0



2V 0




Dagger permil12P1ud



2P1ud





c Scuc

Dagger uTc Rcuc yT

d Qdyd


d Rduddagger



t


microhellip tt

t




ˆhellip tt

t






ˆhellip tt

t








ˆhellip tt

t




ˆhellip tt

t








Dagger uTd P2ud

hellipxdaggerud
















ˆ iexcl

`Tc hellipxdagger

WTc hellipxdagger





ˆ iexcl

`Td hellipxdagger

WTd hellipxdagger



where










and


d hellipxdaggerSd


hellipxdaggerdagger






T c 71 0


c hellipxdagger Imc


T d 71 0


d hellipxdagger Imd

amp





section































hellip tt

t


k2N permiltttdagger



Daggerhellip tt

t



DaggerX

k2N permiltttdagger

























yTd Qdyd Dagger 2yT



n 1 md and P2ud




0 ˆ 12V 0





0 ˆ 12P1ud




hellipxdagger

hellip104dagger





2GT




t



and












c Rcuc




c Jchellipxdagger



x 2 n uc 2 mc












d hellipxdagger




n md such that








ud udˆ0



ud2

udˆ0

ˆ 12GT


amp



c yc 2uTd yddagger


c uTc uc iexcl yT

c yc reg2duT


given


c yc reg2duT





n `cn pc `d n pd Wc


n 1 md and P2ud






and for all x 2 n



0 ˆ 12V 0






0 ˆ 12P1ud













2V 0



2V 0






d hellipxdaggerŠT

hellip123dagger

then G is passive


Sd ˆ Imd





n `cn pc `d n pd Wc








and for all x 2 n




0 ˆ 12V 0



0 ˆ reg2c Imc





hellip128dagger

0 ˆ 12P1ud



0 ˆ reg2dImd




iexclJT





0 lt reg2dImd





Dagger permil12V 0


permilreg2c Imc


permil12V 0



Dagger permil12P1ud


permilreg2dImd



permil12P1ud




c Imcand Rd ˆ reg2



4yT


4yT






c ScucDaggeruT







0 ˆ 12V 0













n `cn pc




lc mc







yTd Qdyd Dagger 2yT


















d PBd iexcl WTd Wd

hellip143dagger

where

Ac ˆ fc

x xˆ0


Cc ˆ hc

x xˆ0


9gtgtgt=


Ad ˆ fd

x xˆ0


Cd ˆ hd

x xˆ0


9gtgtgt=






t



and





























hellip156dagger


hellip157dagger

where Vrn dcr

n mc and ddrn





c QcDcuc

Dagger uTc DT



c Rcucdagger





d QdDdud



d Rduddagger



limkxk2Daggerkuck20


ˆ 0

limkxk2Daggerkudk20


ˆ 0






Dagger uTc hellipWT


c Sc






d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd








yTd Qdyd Dagger 2yT





hellip162dagger


c Wcdagger x 62 Zx

hellip163dagger






hellip165dagger



hellip166dagger



d PBd iexcl WTd Wd

hellip167dagger







Dagger uTc hellipWT

c Wc iexcl DTc QcDc



hellip168dagger





d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd






where





c Dc iexcl Rc



d Wd


d QdDd iexcl DTd Sd


d PBd
















uTc Rcuc yT










yTd Qdyd Dagger 2yT



P such that








iexcl1



d QdCd
















above amp










Ac Bc

BTc Dc

ˆ

LTc

WTc


Ad Bd

BTd Dd

ˆ

LTd

WTd


where


c QcCc





d QdCd



and



d PBd




respectively









are equivalent

(i) G is passive



















c dagger










and Rd ˆ 0 amp









0 ˆ reg2c Imc














0 lt reg2dImd





iexcl DTc Dcdagger

iexcl1





dImdiexcl DT

d Dd







amp







7 Conclusion



Acknowledgements


References































































hellipT

t0


DaggerX










t







t









microhelliptDagger

k

tk


















microhellip tt

tDaggerj


Daggerhelliptj

tDaggerjiexcl1


Daggerhellipti

t


ˆhellip tt

t


k2N permiltttdagger



















hellipT

t0


X



ˆ 0 hellip67dagger





hellipt0

T


DaggerX



hellip68dagger



hellipt0

T


DaggerX



hellip69dagger









T


DaggerX



ˆhellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX



and hence


hellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX




t


DaggerX





0 microhellipt0

T




and hence


hellipt0

T


DaggerX



hellip73dagger

which implies that






T


DaggerX



and hence


hellipt0

T


DaggerX






























c Scuc DaggeruT
















functions Vsn `c


n 1 md and







and for all x 2 n



0 ˆ 12V 0




c hellipxdaggerSc






0 ˆ 12P1ud










c Rcuc yTd Qdyd


d Rduddagger

If alternatively









d hellipxdaggerSd





Dagger permil12V 0



2V 0




Dagger permil12P1ud



2P1ud





c Scuc

Dagger uTc Rcuc yT

d Qdyd


d Rduddagger



t


microhellip tt

t




ˆhellip tt

t






ˆhellip tt

t








ˆhellip tt

t




ˆhellip tt

t








Dagger uTd P2ud

hellipxdaggerud
















ˆ iexcl

`Tc hellipxdagger

WTc hellipxdagger





ˆ iexcl

`Td hellipxdagger

WTd hellipxdagger



where










and


d hellipxdaggerSd


hellipxdaggerdagger






T c 71 0


c hellipxdagger Imc


T d 71 0


d hellipxdagger Imd

amp





section































hellip tt

t


k2N permiltttdagger



Daggerhellip tt

t



DaggerX

k2N permiltttdagger

























yTd Qdyd Dagger 2yT



n 1 md and P2ud




0 ˆ 12V 0





0 ˆ 12P1ud




hellipxdagger

hellip104dagger





2GT




t



and












c Rcuc




c Jchellipxdagger



x 2 n uc 2 mc












d hellipxdagger




n md such that








ud udˆ0



ud2

udˆ0

ˆ 12GT


amp



c yc 2uTd yddagger


c uTc uc iexcl yT

c yc reg2duT


given


c yc reg2duT





n `cn pc `d n pd Wc


n 1 md and P2ud






and for all x 2 n



0 ˆ 12V 0






0 ˆ 12P1ud













2V 0



2V 0






d hellipxdaggerŠT

hellip123dagger

then G is passive


Sd ˆ Imd





n `cn pc `d n pd Wc








and for all x 2 n




0 ˆ 12V 0



0 ˆ reg2c Imc





hellip128dagger

0 ˆ 12P1ud



0 ˆ reg2dImd




iexclJT





0 lt reg2dImd





Dagger permil12V 0


permilreg2c Imc


permil12V 0



Dagger permil12P1ud


permilreg2dImd



permil12P1ud




c Imcand Rd ˆ reg2



4yT


4yT






c ScucDaggeruT







0 ˆ 12V 0













n `cn pc




lc mc







yTd Qdyd Dagger 2yT


















d PBd iexcl WTd Wd

hellip143dagger

where

Ac ˆ fc

x xˆ0


Cc ˆ hc

x xˆ0


9gtgtgt=


Ad ˆ fd

x xˆ0


Cd ˆ hd

x xˆ0


9gtgtgt=






t



and





























hellip156dagger


hellip157dagger

where Vrn dcr

n mc and ddrn





c QcDcuc

Dagger uTc DT



c Rcucdagger





d QdDdud



d Rduddagger



limkxk2Daggerkuck20


ˆ 0

limkxk2Daggerkudk20


ˆ 0






Dagger uTc hellipWT


c Sc






d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd








yTd Qdyd Dagger 2yT





hellip162dagger


c Wcdagger x 62 Zx

hellip163dagger






hellip165dagger



hellip166dagger



d PBd iexcl WTd Wd

hellip167dagger







Dagger uTc hellipWT

c Wc iexcl DTc QcDc



hellip168dagger





d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd






where





c Dc iexcl Rc



d Wd


d QdDd iexcl DTd Sd


d PBd
















uTc Rcuc yT










yTd Qdyd Dagger 2yT



P such that








iexcl1



d QdCd
















above amp










Ac Bc

BTc Dc

ˆ

LTc

WTc


Ad Bd

BTd Dd

ˆ

LTd

WTd


where


c QcCc





d QdCd



and



d PBd




respectively









are equivalent

(i) G is passive



















c dagger










and Rd ˆ 0 amp









0 ˆ reg2c Imc














0 lt reg2dImd





iexcl DTc Dcdagger

iexcl1





dImdiexcl DT

d Dd







amp







7 Conclusion



Acknowledgements


References











































































hellipT

t0


X



ˆ 0 hellip67dagger





hellipt0

T


DaggerX



hellip68dagger



hellipt0

T


DaggerX



hellip69dagger









T


DaggerX



ˆhellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX



and hence


hellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX




t


DaggerX





0 microhellipt0

T




and hence


hellipt0

T


DaggerX



hellip73dagger

which implies that






T


DaggerX



and hence


hellipt0

T


DaggerX






























c Scuc DaggeruT
















functions Vsn `c


n 1 md and







and for all x 2 n



0 ˆ 12V 0




c hellipxdaggerSc






0 ˆ 12P1ud










c Rcuc yTd Qdyd


d Rduddagger

If alternatively









d hellipxdaggerSd





Dagger permil12V 0



2V 0




Dagger permil12P1ud



2P1ud





c Scuc

Dagger uTc Rcuc yT

d Qdyd


d Rduddagger



t


microhellip tt

t




ˆhellip tt

t






ˆhellip tt

t








ˆhellip tt

t




ˆhellip tt

t








Dagger uTd P2ud

hellipxdaggerud
















ˆ iexcl

`Tc hellipxdagger

WTc hellipxdagger





ˆ iexcl

`Td hellipxdagger

WTd hellipxdagger



where










and


d hellipxdaggerSd


hellipxdaggerdagger






T c 71 0


c hellipxdagger Imc


T d 71 0


d hellipxdagger Imd

amp





section































hellip tt

t


k2N permiltttdagger



Daggerhellip tt

t



DaggerX

k2N permiltttdagger

























yTd Qdyd Dagger 2yT



n 1 md and P2ud




0 ˆ 12V 0





0 ˆ 12P1ud




hellipxdagger

hellip104dagger





2GT




t



and












c Rcuc




c Jchellipxdagger



x 2 n uc 2 mc












d hellipxdagger




n md such that








ud udˆ0



ud2

udˆ0

ˆ 12GT


amp



c yc 2uTd yddagger


c uTc uc iexcl yT

c yc reg2duT


given


c yc reg2duT





n `cn pc `d n pd Wc


n 1 md and P2ud






and for all x 2 n



0 ˆ 12V 0






0 ˆ 12P1ud













2V 0



2V 0






d hellipxdaggerŠT

hellip123dagger

then G is passive


Sd ˆ Imd





n `cn pc `d n pd Wc








and for all x 2 n




0 ˆ 12V 0



0 ˆ reg2c Imc





hellip128dagger

0 ˆ 12P1ud



0 ˆ reg2dImd




iexclJT





0 lt reg2dImd





Dagger permil12V 0


permilreg2c Imc


permil12V 0



Dagger permil12P1ud


permilreg2dImd



permil12P1ud




c Imcand Rd ˆ reg2



4yT


4yT






c ScucDaggeruT







0 ˆ 12V 0













n `cn pc




lc mc







yTd Qdyd Dagger 2yT


















d PBd iexcl WTd Wd

hellip143dagger

where

Ac ˆ fc

x xˆ0


Cc ˆ hc

x xˆ0


9gtgtgt=


Ad ˆ fd

x xˆ0


Cd ˆ hd

x xˆ0


9gtgtgt=






t



and





























hellip156dagger


hellip157dagger

where Vrn dcr

n mc and ddrn





c QcDcuc

Dagger uTc DT



c Rcucdagger





d QdDdud



d Rduddagger



limkxk2Daggerkuck20


ˆ 0

limkxk2Daggerkudk20


ˆ 0






Dagger uTc hellipWT


c Sc






d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd








yTd Qdyd Dagger 2yT





hellip162dagger


c Wcdagger x 62 Zx

hellip163dagger






hellip165dagger



hellip166dagger



d PBd iexcl WTd Wd

hellip167dagger







Dagger uTc hellipWT

c Wc iexcl DTc QcDc



hellip168dagger





d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd






where





c Dc iexcl Rc



d Wd


d QdDd iexcl DTd Sd


d PBd
















uTc Rcuc yT










yTd Qdyd Dagger 2yT



P such that








iexcl1



d QdCd
















above amp










Ac Bc

BTc Dc

ˆ

LTc

WTc


Ad Bd

BTd Dd

ˆ

LTd

WTd


where


c QcCc





d QdCd



and



d PBd




respectively









are equivalent

(i) G is passive



















c dagger










and Rd ˆ 0 amp









0 ˆ reg2c Imc














0 lt reg2dImd





iexcl DTc Dcdagger

iexcl1





dImdiexcl DT

d Dd







amp







7 Conclusion



Acknowledgements


References


































































T


DaggerX



ˆhellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX



and hence


hellipt

T


DaggerX

k2N permilT tdagger


Daggerhellipt0

t


DaggerX




t


DaggerX





0 microhellipt0

T




and hence


hellipt0

T


DaggerX



hellip73dagger

which implies that






T


DaggerX



and hence


hellipt0

T


DaggerX






























c Scuc DaggeruT
















functions Vsn `c


n 1 md and







and for all x 2 n



0 ˆ 12V 0




c hellipxdaggerSc






0 ˆ 12P1ud










c Rcuc yTd Qdyd


d Rduddagger

If alternatively









d hellipxdaggerSd





Dagger permil12V 0



2V 0




Dagger permil12P1ud



2P1ud





c Scuc

Dagger uTc Rcuc yT

d Qdyd


d Rduddagger



t


microhellip tt

t




ˆhellip tt

t






ˆhellip tt

t








ˆhellip tt

t




ˆhellip tt

t








Dagger uTd P2ud

hellipxdaggerud
















ˆ iexcl

`Tc hellipxdagger

WTc hellipxdagger





ˆ iexcl

`Td hellipxdagger

WTd hellipxdagger



where










and


d hellipxdaggerSd


hellipxdaggerdagger






T c 71 0


c hellipxdagger Imc


T d 71 0


d hellipxdagger Imd

amp





section































hellip tt

t


k2N permiltttdagger



Daggerhellip tt

t



DaggerX

k2N permiltttdagger

























yTd Qdyd Dagger 2yT



n 1 md and P2ud




0 ˆ 12V 0





0 ˆ 12P1ud




hellipxdagger

hellip104dagger





2GT




t



and












c Rcuc




c Jchellipxdagger



x 2 n uc 2 mc












d hellipxdagger




n md such that








ud udˆ0



ud2

udˆ0

ˆ 12GT


amp



c yc 2uTd yddagger


c uTc uc iexcl yT

c yc reg2duT


given


c yc reg2duT





n `cn pc `d n pd Wc


n 1 md and P2ud






and for all x 2 n



0 ˆ 12V 0






0 ˆ 12P1ud













2V 0



2V 0






d hellipxdaggerŠT

hellip123dagger

then G is passive


Sd ˆ Imd





n `cn pc `d n pd Wc








and for all x 2 n




0 ˆ 12V 0



0 ˆ reg2c Imc





hellip128dagger

0 ˆ 12P1ud



0 ˆ reg2dImd




iexclJT





0 lt reg2dImd





Dagger permil12V 0


permilreg2c Imc


permil12V 0



Dagger permil12P1ud


permilreg2dImd



permil12P1ud




c Imcand Rd ˆ reg2



4yT


4yT






c ScucDaggeruT







0 ˆ 12V 0













n `cn pc




lc mc







yTd Qdyd Dagger 2yT


















d PBd iexcl WTd Wd

hellip143dagger

where

Ac ˆ fc

x xˆ0


Cc ˆ hc

x xˆ0


9gtgtgt=


Ad ˆ fd

x xˆ0


Cd ˆ hd

x xˆ0


9gtgtgt=






t



and





























hellip156dagger


hellip157dagger

where Vrn dcr

n mc and ddrn





c QcDcuc

Dagger uTc DT



c Rcucdagger





d QdDdud



d Rduddagger



limkxk2Daggerkuck20


ˆ 0

limkxk2Daggerkudk20


ˆ 0






Dagger uTc hellipWT


c Sc






d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd








yTd Qdyd Dagger 2yT





hellip162dagger


c Wcdagger x 62 Zx

hellip163dagger






hellip165dagger



hellip166dagger



d PBd iexcl WTd Wd

hellip167dagger







Dagger uTc hellipWT

c Wc iexcl DTc QcDc



hellip168dagger





d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd






where





c Dc iexcl Rc



d Wd


d QdDd iexcl DTd Sd


d PBd
















uTc Rcuc yT










yTd Qdyd Dagger 2yT



P such that








iexcl1



d QdCd
















above amp










Ac Bc

BTc Dc

ˆ

LTc

WTc


Ad Bd

BTd Dd

ˆ

LTd

WTd


where


c QcCc





d QdCd



and



d PBd




respectively









are equivalent

(i) G is passive



















c dagger










and Rd ˆ 0 amp









0 ˆ reg2c Imc














0 lt reg2dImd





iexcl DTc Dcdagger

iexcl1





dImdiexcl DT

d Dd







amp







7 Conclusion



Acknowledgements


References

















































































c Scuc DaggeruT
















functions Vsn `c


n 1 md and







and for all x 2 n



0 ˆ 12V 0




c hellipxdaggerSc






0 ˆ 12P1ud










c Rcuc yTd Qdyd


d Rduddagger

If alternatively









d hellipxdaggerSd





Dagger permil12V 0



2V 0




Dagger permil12P1ud



2P1ud





c Scuc

Dagger uTc Rcuc yT

d Qdyd


d Rduddagger



t


microhellip tt

t




ˆhellip tt

t






ˆhellip tt

t








ˆhellip tt

t




ˆhellip tt

t








Dagger uTd P2ud

hellipxdaggerud
















ˆ iexcl

`Tc hellipxdagger

WTc hellipxdagger





ˆ iexcl

`Td hellipxdagger

WTd hellipxdagger



where










and


d hellipxdaggerSd


hellipxdaggerdagger






T c 71 0


c hellipxdagger Imc


T d 71 0


d hellipxdagger Imd

amp





section































hellip tt

t


k2N permiltttdagger



Daggerhellip tt

t



DaggerX

k2N permiltttdagger

























yTd Qdyd Dagger 2yT



n 1 md and P2ud




0 ˆ 12V 0





0 ˆ 12P1ud




hellipxdagger

hellip104dagger





2GT




t



and












c Rcuc




c Jchellipxdagger



x 2 n uc 2 mc












d hellipxdagger




n md such that








ud udˆ0



ud2

udˆ0

ˆ 12GT


amp



c yc 2uTd yddagger


c uTc uc iexcl yT

c yc reg2duT


given


c yc reg2duT





n `cn pc `d n pd Wc


n 1 md and P2ud






and for all x 2 n



0 ˆ 12V 0






0 ˆ 12P1ud













2V 0



2V 0






d hellipxdaggerŠT

hellip123dagger

then G is passive


Sd ˆ Imd





n `cn pc `d n pd Wc








and for all x 2 n




0 ˆ 12V 0



0 ˆ reg2c Imc





hellip128dagger

0 ˆ 12P1ud



0 ˆ reg2dImd




iexclJT





0 lt reg2dImd





Dagger permil12V 0


permilreg2c Imc


permil12V 0



Dagger permil12P1ud


permilreg2dImd



permil12P1ud




c Imcand Rd ˆ reg2



4yT


4yT






c ScucDaggeruT







0 ˆ 12V 0













n `cn pc




lc mc







yTd Qdyd Dagger 2yT


















d PBd iexcl WTd Wd

hellip143dagger

where

Ac ˆ fc

x xˆ0


Cc ˆ hc

x xˆ0


9gtgtgt=


Ad ˆ fd

x xˆ0


Cd ˆ hd

x xˆ0


9gtgtgt=






t



and





























hellip156dagger


hellip157dagger

where Vrn dcr

n mc and ddrn





c QcDcuc

Dagger uTc DT



c Rcucdagger





d QdDdud



d Rduddagger



limkxk2Daggerkuck20


ˆ 0

limkxk2Daggerkudk20


ˆ 0






Dagger uTc hellipWT


c Sc






d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd








yTd Qdyd Dagger 2yT





hellip162dagger


c Wcdagger x 62 Zx

hellip163dagger






hellip165dagger



hellip166dagger



d PBd iexcl WTd Wd

hellip167dagger







Dagger uTc hellipWT

c Wc iexcl DTc QcDc



hellip168dagger





d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd






where





c Dc iexcl Rc



d Wd


d QdDd iexcl DTd Sd


d PBd
















uTc Rcuc yT










yTd Qdyd Dagger 2yT



P such that








iexcl1



d QdCd
















above amp










Ac Bc

BTc Dc

ˆ

LTc

WTc


Ad Bd

BTd Dd

ˆ

LTd

WTd


where


c QcCc





d QdCd



and



d PBd




respectively









are equivalent

(i) G is passive



















c dagger










and Rd ˆ 0 amp









0 ˆ reg2c Imc














0 lt reg2dImd





iexcl DTc Dcdagger

iexcl1





dImdiexcl DT

d Dd







amp







7 Conclusion



Acknowledgements


References































































c Scuc

Dagger uTc Rcuc yT

d Qdyd


d Rduddagger



t


microhellip tt

t




ˆhellip tt

t






ˆhellip tt

t








ˆhellip tt

t




ˆhellip tt

t








Dagger uTd P2ud

hellipxdaggerud
















ˆ iexcl

`Tc hellipxdagger

WTc hellipxdagger





ˆ iexcl

`Td hellipxdagger

WTd hellipxdagger



where










and


d hellipxdaggerSd


hellipxdaggerdagger






T c 71 0


c hellipxdagger Imc


T d 71 0


d hellipxdagger Imd

amp





section































hellip tt

t


k2N permiltttdagger



Daggerhellip tt

t



DaggerX

k2N permiltttdagger

























yTd Qdyd Dagger 2yT



n 1 md and P2ud




0 ˆ 12V 0





0 ˆ 12P1ud




hellipxdagger

hellip104dagger





2GT




t



and












c Rcuc




c Jchellipxdagger



x 2 n uc 2 mc












d hellipxdagger




n md such that








ud udˆ0



ud2

udˆ0

ˆ 12GT


amp



c yc 2uTd yddagger


c uTc uc iexcl yT

c yc reg2duT


given


c yc reg2duT





n `cn pc `d n pd Wc


n 1 md and P2ud






and for all x 2 n



0 ˆ 12V 0






0 ˆ 12P1ud













2V 0



2V 0






d hellipxdaggerŠT

hellip123dagger

then G is passive


Sd ˆ Imd





n `cn pc `d n pd Wc








and for all x 2 n




0 ˆ 12V 0



0 ˆ reg2c Imc





hellip128dagger

0 ˆ 12P1ud



0 ˆ reg2dImd




iexclJT





0 lt reg2dImd





Dagger permil12V 0


permilreg2c Imc


permil12V 0



Dagger permil12P1ud


permilreg2dImd



permil12P1ud




c Imcand Rd ˆ reg2



4yT


4yT






c ScucDaggeruT







0 ˆ 12V 0













n `cn pc




lc mc







yTd Qdyd Dagger 2yT


















d PBd iexcl WTd Wd

hellip143dagger

where

Ac ˆ fc

x xˆ0


Cc ˆ hc

x xˆ0


9gtgtgt=


Ad ˆ fd

x xˆ0


Cd ˆ hd

x xˆ0


9gtgtgt=






t



and





























hellip156dagger


hellip157dagger

where Vrn dcr

n mc and ddrn





c QcDcuc

Dagger uTc DT



c Rcucdagger





d QdDdud



d Rduddagger



limkxk2Daggerkuck20


ˆ 0

limkxk2Daggerkudk20


ˆ 0






Dagger uTc hellipWT


c Sc






d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd








yTd Qdyd Dagger 2yT





hellip162dagger


c Wcdagger x 62 Zx

hellip163dagger






hellip165dagger



hellip166dagger



d PBd iexcl WTd Wd

hellip167dagger







Dagger uTc hellipWT

c Wc iexcl DTc QcDc



hellip168dagger





d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd






where





c Dc iexcl Rc



d Wd


d QdDd iexcl DTd Sd


d PBd
















uTc Rcuc yT










yTd Qdyd Dagger 2yT



P such that








iexcl1



d QdCd
















above amp










Ac Bc

BTc Dc

ˆ

LTc

WTc


Ad Bd

BTd Dd

ˆ

LTd

WTd


where


c QcCc





d QdCd



and



d PBd




respectively









are equivalent

(i) G is passive



















c dagger










and Rd ˆ 0 amp









0 ˆ reg2c Imc














0 lt reg2dImd





iexcl DTc Dcdagger

iexcl1





dImdiexcl DT

d Dd







amp







7 Conclusion



Acknowledgements


References






























































T c 71 0


c hellipxdagger Imc


T d 71 0


d hellipxdagger Imd

amp





section































hellip tt

t


k2N permiltttdagger



Daggerhellip tt

t



DaggerX

k2N permiltttdagger

























yTd Qdyd Dagger 2yT



n 1 md and P2ud




0 ˆ 12V 0





0 ˆ 12P1ud




hellipxdagger

hellip104dagger





2GT




t



and












c Rcuc




c Jchellipxdagger



x 2 n uc 2 mc












d hellipxdagger




n md such that








ud udˆ0



ud2

udˆ0

ˆ 12GT


amp



c yc 2uTd yddagger


c uTc uc iexcl yT

c yc reg2duT


given


c yc reg2duT





n `cn pc `d n pd Wc


n 1 md and P2ud






and for all x 2 n



0 ˆ 12V 0






0 ˆ 12P1ud













2V 0



2V 0






d hellipxdaggerŠT

hellip123dagger

then G is passive


Sd ˆ Imd





n `cn pc `d n pd Wc








and for all x 2 n




0 ˆ 12V 0



0 ˆ reg2c Imc





hellip128dagger

0 ˆ 12P1ud



0 ˆ reg2dImd




iexclJT





0 lt reg2dImd





Dagger permil12V 0


permilreg2c Imc


permil12V 0



Dagger permil12P1ud


permilreg2dImd



permil12P1ud




c Imcand Rd ˆ reg2



4yT


4yT






c ScucDaggeruT







0 ˆ 12V 0













n `cn pc




lc mc







yTd Qdyd Dagger 2yT


















d PBd iexcl WTd Wd

hellip143dagger

where

Ac ˆ fc

x xˆ0


Cc ˆ hc

x xˆ0


9gtgtgt=


Ad ˆ fd

x xˆ0


Cd ˆ hd

x xˆ0


9gtgtgt=






t



and





























hellip156dagger


hellip157dagger

where Vrn dcr

n mc and ddrn





c QcDcuc

Dagger uTc DT



c Rcucdagger





d QdDdud



d Rduddagger



limkxk2Daggerkuck20


ˆ 0

limkxk2Daggerkudk20


ˆ 0






Dagger uTc hellipWT


c Sc






d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd








yTd Qdyd Dagger 2yT





hellip162dagger


c Wcdagger x 62 Zx

hellip163dagger






hellip165dagger



hellip166dagger



d PBd iexcl WTd Wd

hellip167dagger







Dagger uTc hellipWT

c Wc iexcl DTc QcDc



hellip168dagger





d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd






where





c Dc iexcl Rc



d Wd


d QdDd iexcl DTd Sd


d PBd
















uTc Rcuc yT










yTd Qdyd Dagger 2yT



P such that








iexcl1



d QdCd
















above amp










Ac Bc

BTc Dc

ˆ

LTc

WTc


Ad Bd

BTd Dd

ˆ

LTd

WTd


where


c QcCc





d QdCd



and



d PBd




respectively









are equivalent

(i) G is passive



















c dagger










and Rd ˆ 0 amp









0 ˆ reg2c Imc














0 lt reg2dImd





iexcl DTc Dcdagger

iexcl1





dImdiexcl DT

d Dd







amp







7 Conclusion



Acknowledgements


References






































































yTd Qdyd Dagger 2yT



n 1 md and P2ud




0 ˆ 12V 0





0 ˆ 12P1ud




hellipxdagger

hellip104dagger





2GT




t



and












c Rcuc




c Jchellipxdagger



x 2 n uc 2 mc












d hellipxdagger




n md such that








ud udˆ0



ud2

udˆ0

ˆ 12GT


amp



c yc 2uTd yddagger


c uTc uc iexcl yT

c yc reg2duT


given


c yc reg2duT





n `cn pc `d n pd Wc


n 1 md and P2ud






and for all x 2 n



0 ˆ 12V 0






0 ˆ 12P1ud













2V 0



2V 0






d hellipxdaggerŠT

hellip123dagger

then G is passive


Sd ˆ Imd





n `cn pc `d n pd Wc








and for all x 2 n




0 ˆ 12V 0



0 ˆ reg2c Imc





hellip128dagger

0 ˆ 12P1ud



0 ˆ reg2dImd




iexclJT





0 lt reg2dImd





Dagger permil12V 0


permilreg2c Imc


permil12V 0



Dagger permil12P1ud


permilreg2dImd



permil12P1ud




c Imcand Rd ˆ reg2



4yT


4yT






c ScucDaggeruT







0 ˆ 12V 0













n `cn pc




lc mc







yTd Qdyd Dagger 2yT


















d PBd iexcl WTd Wd

hellip143dagger

where

Ac ˆ fc

x xˆ0


Cc ˆ hc

x xˆ0


9gtgtgt=


Ad ˆ fd

x xˆ0


Cd ˆ hd

x xˆ0


9gtgtgt=






t



and





























hellip156dagger


hellip157dagger

where Vrn dcr

n mc and ddrn





c QcDcuc

Dagger uTc DT



c Rcucdagger





d QdDdud



d Rduddagger



limkxk2Daggerkuck20


ˆ 0

limkxk2Daggerkudk20


ˆ 0






Dagger uTc hellipWT


c Sc






d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd








yTd Qdyd Dagger 2yT





hellip162dagger


c Wcdagger x 62 Zx

hellip163dagger






hellip165dagger



hellip166dagger



d PBd iexcl WTd Wd

hellip167dagger







Dagger uTc hellipWT

c Wc iexcl DTc QcDc



hellip168dagger





d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd






where





c Dc iexcl Rc



d Wd


d QdDd iexcl DTd Sd


d PBd
















uTc Rcuc yT










yTd Qdyd Dagger 2yT



P such that








iexcl1



d QdCd
















above amp










Ac Bc

BTc Dc

ˆ

LTc

WTc


Ad Bd

BTd Dd

ˆ

LTd

WTd


where


c QcCc





d QdCd



and



d PBd




respectively









are equivalent

(i) G is passive



















c dagger










and Rd ˆ 0 amp









0 ˆ reg2c Imc














0 lt reg2dImd





iexcl DTc Dcdagger

iexcl1





dImdiexcl DT

d Dd







amp







7 Conclusion



Acknowledgements


References
































































ud udˆ0



ud2

udˆ0

ˆ 12GT


amp



c yc 2uTd yddagger


c uTc uc iexcl yT

c yc reg2duT


given


c yc reg2duT





n `cn pc `d n pd Wc


n 1 md and P2ud






and for all x 2 n



0 ˆ 12V 0






0 ˆ 12P1ud













2V 0



2V 0






d hellipxdaggerŠT

hellip123dagger

then G is passive


Sd ˆ Imd





n `cn pc `d n pd Wc








and for all x 2 n




0 ˆ 12V 0



0 ˆ reg2c Imc





hellip128dagger

0 ˆ 12P1ud



0 ˆ reg2dImd




iexclJT





0 lt reg2dImd





Dagger permil12V 0


permilreg2c Imc


permil12V 0



Dagger permil12P1ud


permilreg2dImd



permil12P1ud




c Imcand Rd ˆ reg2



4yT


4yT






c ScucDaggeruT







0 ˆ 12V 0













n `cn pc




lc mc







yTd Qdyd Dagger 2yT


















d PBd iexcl WTd Wd

hellip143dagger

where

Ac ˆ fc

x xˆ0


Cc ˆ hc

x xˆ0


9gtgtgt=


Ad ˆ fd

x xˆ0


Cd ˆ hd

x xˆ0


9gtgtgt=






t



and





























hellip156dagger


hellip157dagger

where Vrn dcr

n mc and ddrn





c QcDcuc

Dagger uTc DT



c Rcucdagger





d QdDdud



d Rduddagger



limkxk2Daggerkuck20


ˆ 0

limkxk2Daggerkudk20


ˆ 0






Dagger uTc hellipWT


c Sc






d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd








yTd Qdyd Dagger 2yT





hellip162dagger


c Wcdagger x 62 Zx

hellip163dagger






hellip165dagger



hellip166dagger



d PBd iexcl WTd Wd

hellip167dagger







Dagger uTc hellipWT

c Wc iexcl DTc QcDc



hellip168dagger





d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd






where





c Dc iexcl Rc



d Wd


d QdDd iexcl DTd Sd


d PBd
















uTc Rcuc yT










yTd Qdyd Dagger 2yT



P such that








iexcl1



d QdCd
















above amp










Ac Bc

BTc Dc

ˆ

LTc

WTc


Ad Bd

BTd Dd

ˆ

LTd

WTd


where


c QcCc





d QdCd



and



d PBd




respectively









are equivalent

(i) G is passive



















c dagger










and Rd ˆ 0 amp









0 ˆ reg2c Imc














0 lt reg2dImd





iexcl DTc Dcdagger

iexcl1





dImdiexcl DT

d Dd







amp







7 Conclusion



Acknowledgements


References
































































0 ˆ 12V 0



0 ˆ reg2c Imc





hellip128dagger

0 ˆ 12P1ud



0 ˆ reg2dImd




iexclJT





0 lt reg2dImd





Dagger permil12V 0


permilreg2c Imc


permil12V 0



Dagger permil12P1ud


permilreg2dImd



permil12P1ud




c Imcand Rd ˆ reg2



4yT


4yT






c ScucDaggeruT







0 ˆ 12V 0













n `cn pc




lc mc







yTd Qdyd Dagger 2yT


















d PBd iexcl WTd Wd

hellip143dagger

where

Ac ˆ fc

x xˆ0


Cc ˆ hc

x xˆ0


9gtgtgt=


Ad ˆ fd

x xˆ0


Cd ˆ hd

x xˆ0


9gtgtgt=






t



and





























hellip156dagger


hellip157dagger

where Vrn dcr

n mc and ddrn





c QcDcuc

Dagger uTc DT



c Rcucdagger





d QdDdud



d Rduddagger



limkxk2Daggerkuck20


ˆ 0

limkxk2Daggerkudk20


ˆ 0






Dagger uTc hellipWT


c Sc






d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd








yTd Qdyd Dagger 2yT





hellip162dagger


c Wcdagger x 62 Zx

hellip163dagger






hellip165dagger



hellip166dagger



d PBd iexcl WTd Wd

hellip167dagger







Dagger uTc hellipWT

c Wc iexcl DTc QcDc



hellip168dagger





d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd






where





c Dc iexcl Rc



d Wd


d QdDd iexcl DTd Sd


d PBd
















uTc Rcuc yT










yTd Qdyd Dagger 2yT



P such that








iexcl1



d QdCd
















above amp










Ac Bc

BTc Dc

ˆ

LTc

WTc


Ad Bd

BTd Dd

ˆ

LTd

WTd


where


c QcCc





d QdCd



and



d PBd




respectively









are equivalent

(i) G is passive



















c dagger










and Rd ˆ 0 amp









0 ˆ reg2c Imc














0 lt reg2dImd





iexcl DTc Dcdagger

iexcl1





dImdiexcl DT

d Dd







amp







7 Conclusion



Acknowledgements


References











































































d PBd iexcl WTd Wd

hellip143dagger

where

Ac ˆ fc

x xˆ0


Cc ˆ hc

x xˆ0


9gtgtgt=


Ad ˆ fd

x xˆ0


Cd ˆ hd

x xˆ0


9gtgtgt=






t



and





























hellip156dagger


hellip157dagger

where Vrn dcr

n mc and ddrn





c QcDcuc

Dagger uTc DT



c Rcucdagger





d QdDdud



d Rduddagger



limkxk2Daggerkuck20


ˆ 0

limkxk2Daggerkudk20


ˆ 0






Dagger uTc hellipWT


c Sc






d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd








yTd Qdyd Dagger 2yT





hellip162dagger


c Wcdagger x 62 Zx

hellip163dagger






hellip165dagger



hellip166dagger



d PBd iexcl WTd Wd

hellip167dagger







Dagger uTc hellipWT

c Wc iexcl DTc QcDc



hellip168dagger





d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd






where





c Dc iexcl Rc



d Wd


d QdDd iexcl DTd Sd


d PBd
















uTc Rcuc yT










yTd Qdyd Dagger 2yT



P such that








iexcl1



d QdCd
















above amp










Ac Bc

BTc Dc

ˆ

LTc

WTc


Ad Bd

BTd Dd

ˆ

LTd

WTd


where


c QcCc





d QdCd



and



d PBd




respectively









are equivalent

(i) G is passive



















c dagger










and Rd ˆ 0 amp









0 ˆ reg2c Imc














0 lt reg2dImd





iexcl DTc Dcdagger

iexcl1





dImdiexcl DT

d Dd







amp







7 Conclusion



Acknowledgements


References
































































c QcDcuc

Dagger uTc DT



c Rcucdagger





d QdDdud



d Rduddagger



limkxk2Daggerkuck20


ˆ 0

limkxk2Daggerkudk20


ˆ 0






Dagger uTc hellipWT


c Sc






d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd








yTd Qdyd Dagger 2yT





hellip162dagger


c Wcdagger x 62 Zx

hellip163dagger






hellip165dagger



hellip166dagger



d PBd iexcl WTd Wd

hellip167dagger







Dagger uTc hellipWT

c Wc iexcl DTc QcDc



hellip168dagger





d Wddaggerud

Dagger uTd hellipWT


d Sd iexcl STd Dd






where





c Dc iexcl Rc



d Wd


d QdDd iexcl DTd Sd


d PBd
















uTc Rcuc yT










yTd Qdyd Dagger 2yT



P such that








iexcl1



d QdCd
















above amp










Ac Bc

BTc Dc

ˆ

LTc

WTc


Ad Bd

BTd Dd

ˆ

LTd

WTd


where


c QcCc





d QdCd



and



d PBd




respectively









are equivalent

(i) G is passive



















c dagger










and Rd ˆ 0 amp









0 ˆ reg2c Imc














0 lt reg2dImd





iexcl DTc Dcdagger

iexcl1





dImdiexcl DT

d Dd







amp







7 Conclusion



Acknowledgements


References
































































where





c Dc iexcl Rc



d Wd


d QdDd iexcl DTd Sd


d PBd
















uTc Rcuc yT










yTd Qdyd Dagger 2yT



P such that








iexcl1



d QdCd
















above amp










Ac Bc

BTc Dc

ˆ

LTc

WTc


Ad Bd

BTd Dd

ˆ

LTd

WTd


where


c QcCc





d QdCd



and



d PBd




respectively









are equivalent

(i) G is passive



















c dagger










and Rd ˆ 0 amp









0 ˆ reg2c Imc














0 lt reg2dImd





iexcl DTc Dcdagger

iexcl1





dImdiexcl DT

d Dd







amp







7 Conclusion



Acknowledgements


References






























































Ac Bc

BTc Dc

ˆ

LTc

WTc


Ad Bd

BTd Dd

ˆ

LTd

WTd


where


c QcCc





d QdCd



and



d PBd




respectively









are equivalent

(i) G is passive



















c dagger










and Rd ˆ 0 amp









0 ˆ reg2c Imc














0 lt reg2dImd





iexcl DTc Dcdagger

iexcl1





dImdiexcl DT

d Dd







amp







7 Conclusion



Acknowledgements


References








































































0 lt reg2dImd





iexcl DTc Dcdagger

iexcl1





dImdiexcl DT

d Dd







amp







7 Conclusion



Acknowledgements


References



















































































































Non-linear impulsive dynamical systems. Part I: Stability ... · In this paper we develop Lyapunov and invariant set stability theorems for non-linear impulsive dynamical systems.

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