1410 IEEE TRANSACTIONS ON AUTOMATIC … Collection/TAC/2006...1410 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 9, SEPTEMBER 2006 Generalized Sliding-Mode Control for Multi-Input

1410 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 9, SEPTEMBER 2006

Generalized Sliding-Mode Control for Multi-InputNonlinear Systems

Ronald M. Hirschorn

Abstract—The purpose of this paper is to present a frameworkfor the design of sliding-mode based controllers for multi-inputaffine nonlinear systems. In this framework the control effort goesto zero as the state approaches the sliding surface, a key feature forpractical implementation. The sliding surface is a submanifold withcodimension equal to the dimension of the involutive distributiongenerated by the controlled vector fields rather than the dimensionof their linear span. We also allow nonzero vectors in to be tangentto the sliding surface, the singular case in sliding-mode control.

Index Terms—Nonlinear systems, sliding-mode control, stabi-lization.

I. INTRODUCTION

THERE are a number of more or less standard assumptionsmade in the sliding control literature, either explicitly or

implicitly (cf. [24], [26], and the references therein). The slidingsurface is assumed to have a codimension equal to the number ofindependent controlled vector fields. Due to the use of full con-trol authority state trajectories under sliding-mode controllersare solutions to differential inclusions rather than differentialequations. Finally, to ensure that state trajectory can be forcedto evolve on the sliding surface, at each point in the sliding sur-face the span of the controlled vector fields are assumed to havea trivial intersection with the tangent space to the sliding sur-face. The design of such controllers can be problematic for sys-tems whose linearization is not controllable. The purpose of thispaper is to present a framework for the design of multi-inputnonlinear controllers which act, in some sense, as a practicalimplementation of sliding-mode control and do not require anyof the above assumptions.

Our generalized sliding-mode controller is modelled after“sample and hold” controllers used in digital control, hence ourstate trajectories are continuous piecewise smooth curves. Thishas some connections with the work of Clarke, Ledyaev, andSontag on asymptotic controllability [7]. By allowing the stateto evolve close to the sliding surface we can accommodatethe singular situation in sliding-mode control where the inputvector fields become tangent to on some submanifold [13].In particular we show that for a class of homogeneous sys-tems with this feature our generalized sliding-mode controllerachieves global asymptotic stabilization. We note that finite

Manuscript received September 13, 2004; revised August 11, 2005,November 30, 2005, and December 14, 2005. Recommended by AssociateEditor L. Glielmo. This work was supported by the Natural Sciences andEngineering Research Council of Canada.

The author is with the Department of Mathematics and Statistics, Queen’sUniversity, Kingston, Ontario K7L 3N6, Canada (e-mail: [email protected]).

Digital Object Identifier 10.1109/TAC.2006.880959

time stability for homogeneous switched systems has beenstudied in [17] in the case where the controllers cause the stateto evolve on the sliding surface after a finite time interval.

One great advantage of sliding-mode control is the reductionof order which comes from keeping the state on or close to thesliding surface . Typically (cf. [13]–[26]) the dimension ofis where is the dimension of the state manifold andthe number of (independent) input vector fields. For the gener-alized sliding-mode controllers introduced here the dimensionof is where is the dimension of the involutive dis-tribution generated by the controlled vector fields. This cangreatly simplify the problem of designing the sliding surface atthe price of a more involved controller to carry the state trajec-tory to , since we use Lie brackets of the input vector fields tohelp move the state to the sliding surface. That the state trajec-tory can be made to closely follow trajectories of an extendedsystem (whose extra input vector fields are Lie brackets of inputvector fields) has been noted by Brockett, Pomet, and others [3],[18]. We note that in [10] a quite different way of increasing thecodimension of the sliding surface is presented. Here the con-troller causes the state to approach a submanifold which maydepend explicitly on the control and its derivatives. The require-ments on the controllers take the form of ordinary differentialequations. The assumptions needed are related to those whichallow for dynamic feedback linearization. In contrast to the re-sults presented here the controller design and dimension of thesliding surface are independent of the structure of the distribu-tion generated by the controlled vector fields. In the frameworkdeveloped here the codimension of the sliding surface is not therelative degree of some output function but is determined by thedimension of the involutive distribution generated by the con-trolled vector fields.

Under conventional sliding-mode control the state is movedto the sliding surface in finite time. Practical considerations dic-tate that the control effort be attenuated as the state trajectorynears the sliding surface and this requires a controller designwhich accounts for the contribution of the drift vector field tothe state trajectory (cf. [24]). Here we do not insist that that theactual trajectory reach the sliding surface in finite time but de-sign our controller to move the state of the corresponding drift-less system to the sliding surface in finite time. This can be arelatively straightforward task, especially in the case where thecontrolled vector fields span an involutive distribution.

The paper is organized as follows: in Section II we present ourdefinitions for practical and asymptotic stability for families ofcontrollers. In Section III we define sliding surfaces, our gen-eralized sliding-mode controller, and the -Lyapunov function.Section IV contains Theorem IV.1, our main result on practical

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HIRSCHORN: GENERALIZED SLIDING-MODE CONTROL FOR MULTI-INPUT NONLINEAR SYSTEMS 1411

stability for a family of generalized sliding-mode controllersand an illustrative example. In Section V contains results

on asymptotic stability for a class of homogeneous systems.

II. PRELIMINARIES

Suppose that is a smooth manifold, a smoothfunction and a vector field on . Thendenotes the Lie Derivative of along and the integralcurve of passing through at , so that

. If is a smooth vector field on thendenotes the Lie Bracket of and . A dimensional distri-bution on is the choice of a dimensional subspaceof the tangent space to at for each . is smooth iffor each there exist a neighbourhood of and smoothvector fields which span at each point of . Avector field belongs to if . If is asmooth function on then belongs to .is involutive if the Lie Bracket whenever aresmooth vector fields which belong to . A continuously differ-entiable function is called positive definite at

if iff and proper if the preimageof every compact set in is compact in . We will consideraffine system models of the form

(1)

where is a smooth -dimensional manifold,, the drift vector field and the con-

trolled vector fields are smooth, and is anequilibrium point where vanishes. We will assume that thevector fields with constant are complete.This simplifies the exposition.

Definition II.1: Suppose that is an equilibrium state forsystem (1), a family of (possibly time dependent) feed-back controllers indexed by . Then is practicallystabilized by the controllers if

i) the state trajectories corresponding toexist for all ;

ii) given any a compact neighbourhood of and opensubset of containing there exists suchthat (cf. [25]).

If instead of ii) we have the stronger condition ii)’ given anya compact neighbourhood of there exists suchthat , we say that issemiglobally asymptotically stabilized by . If ii)’ holdsfor all we say that is globally asymptotically sta-bilized by .

III. A FRAMEWORK FOR SLIDING-MODE CONTROL

A. Sliding Surfaces

A sliding surface for system (1) is a subset of containingthe equilibrium and such that is a smooth subman-ifold of the state manifold and states in can be movedto along piecewise smooth integral curves of the controlledvector fields . We let denote the smooth involutivedistribution on generated by . To save accounting

and to present semi-global results we assume that the integralcurves for the controlled and drift vector fields are defined forall , that is -dimensional , and is thedisjoint union of the leaves of .

Definition III.1: A connected subset of containing iscalled a sliding surface for the system (1) if

a) is an embedded codimension submanifold of;

b) .Our notion of the sliding surface differs from the usual

one (cf. [13]–[26]) in significant ways. The dimension ofour sliding surface is . Typically is taken to be

dimensional. The drop in dimension betweenand is one of the attractions of sliding-mode control andthe dimension of is often (generically in the multi-inputcase) greater than . This can simplify the problem of de-signing an appropriate sliding surface (see Example IV.2).Another standard assumption in sliding-mode control is that

span . This ensuressufficient direct control to keep the state trajectory on thesliding surface (cf. [24][26])). For a significant class of controlaffine systems this requirement is too restrictive. DefinitionIII.1 allows for states in such that .This lets us use sliding-mode control to generate new robustlow-gain controllers to globally asymptotically stabilize awell known class of homogeneous systems (see Section V).We note that the high-gain continuous stabilizer introducedin [19] for these systems causes a globally defined Lyapunovfunction to decrease along the state trajectory. Since we doimpose this requirement we can stabilize these systems usinga comparatively low-gain discontinuous feedback controller.We note that for the linear system evolving on

, the controlled vector fields are the constant vector fieldswhere are the columns of the matrix . Becausewe have span Range .

The leaves of this involutive distribution are the affinesubsets Range . A standard sliding surface is a subspace

such that Range . This choice of satisfiesDefinition III.1 but it is clear that in our framework a greatmany submanifolds of can also act as sliding surfaces forthese linear systems.

B. Generalized Sliding-Mode Controllers

Two issues arise naturally in sliding-mode control. The firstis finding a feedback controller which causes the state trajectoryto reach the sliding surface in finite time and thereafter remainon . The second is to guarantee that the resulting trajectory on

, the “ sliding motion” (cf. [26]), is stable. Our issues are some-what different. We wish to allow for the possibility of “singularstates” where the controlled vector fields are tangent to . Wealso want our controller to tends to zero as the state trajectoryapproaches while being sufficiently forceful so as to keep thestate close to in the presence of outside disturbances and un-modelled dynamics. Our philosophy is to design our controllerindependent of the drift vector field. This implies a certain ro-bustness in the face of unmodelled dynamics. Thus we defineour generalized sliding-mode controller so that the state of the


driftless system reaches the sliding surfacein finite time. The drift vector field only plays a direct role in

the choice of the sliding surface.We first consider steering the state of the driftless system toin finite time. Suppose that , where is a leaf of

the distribution . Then Chow’s Theorem (cf. [6][12]) impliesthat every element of can be expressed as

where forand . Thus the leaf is traced out by piecewisesmooth curves which are the concatenations of smooth integralcurves of the controlled vector fields emanating from . Fur-thermore, since we are considering a driftless system evolvingon , we have small time local controllability (cf. [23]). Sup-pose that an open set in containing and . Wecan find an open neighbourhood of contained in suchthat every element can be reached from in time

along a concatenation of integral curves of controlledvector fields which evolve in . Since our system is driftlesswe can go “backward” along trajectories, so that we can selecttimes , vector fields ,and points defined by

such that and . Of courseare all functions of but, if we are given the point

, we can deduce , etc. We can therefore treat as a functionof . In particular set for .Then . If we rescale by the rescaledcurve has . Heremay be thought of as a “hybrid control” which is held constantfor a time interval of length 1 i.e., sample and hold with sampleinterval 1 in the language of digital control theory. Thus, for

for

with we have .This reachability property for driftless systems can be re-

stated in terms the existence of “hybrid controllers” for thedriftless system

(2)

such that states are transfered to in time andthe controls are constant on each subinterval of .Furthermore state trajectories under this controller have thelocal controllability properties discussed above, namely for anyneighbourhood of there exists a neighbourhood

such that is transfered to intime along trajectories contained in . In light of the abovediscussion we have if we define thecontrols appropriately. In particular, since forsome , we can set

for (3)

and for for to . We note that thefunctions tend to zero as approaches (as )hence the controllers also have this property. Also has a“memory” of length 1 as depends only on over thetime interval .

The above discussion serves to motivate our definition of ageneralized sliding-mode controller. To simplify notation we(implicitly) assume a somewhat stronger version of controlla-bility for the driftless system (2), namely that one can select afixed such that transfers between points in the leaves of

can be achieved using piecewise smooth trajectories with atmost switches. We retain the feature that the the controllerstend to zero as approaches . As we shorten the “samplingtime” our controller will act more forcefully to keep the statetrajectory close to the sliding surface and hence retain some ro-bustness. We will allow our controllers to have a memory oflength as opposed to the memory of length 1 above. In partic-ular, over time intervals , the controls will befunctions of if

. Here , is a sample historyof the state trajectory with memory at most .

Definition III.2: Let be a sliding surface for system (1) and. A control is called a generalized sliding-mode

controller for with memory ifi) on the time interval

if ;ii) over the time interval

for functions if ;iii) trajectories of the driftless system (2) with

have and for;

iv) if is an open neighbourhood of a point andthere exists an open neighbourhood of such thattrajectories of the driftless system (2) withand have andfor .

We will show that controllable linear systems satisfy Defini-tion III.2 with and a stronger version of Definition III.2iv)holds, namely if is an open neighbourhood of a pointthere exists an open neighbourhood of which is in-variant under trajectories of the corresponding driftless system.For example consider with

. Then, for the corresponding driftless system, we can steer to

using . Then the openset where con-tains and is forward time invariant under trajectories ofthe driftless system with . That is implies

. This motivates the following definition.Definition III.3: Let be a sliding surface for system (1).

A generalized sliding-mode controller for with memoryis called a direct generalized sliding-mode controller

for if Definition III.2-iv) holds for an open set forwardinvariant under trajectories of the driftless system with .

In most applications of sampled-data control a highsample rate is desirable. In the discussion so far we haveassumed a sample period of , the controller memory. Todefine a controller with sample interval where


we use the modified state sample history. If is a gener-

alized sliding-mode controller with memory , over the timeinterval we set

(4)

We note that the trajectory of the driftless system (2) withis a reparametrization of the trajectory corresponding

to . Thus is constant over intervals of lengthwhereas is constant over intervals of length 1. Since ourgeneralized sliding-mode controller is piecewise constant and

with constant is complete by assumption,solutions to the resulting state differential equation exist for

. Finally, decreasing increases both and the controller’sability to reject external disturbances.

C. S-Lyapunov Functions

In sliding-mode control, the sliding surface is designed toensure the stability of the equivalent dynamics, the dynamics ofthe state evolution (sliding motion) on the sliding surface. Wewill show that for any controllable time-invariant linear systemon there exists a subspace , a quadratic function

with on for some, such that the restriction of to is proper and positive

definite at the origin. This motivates the following definition.Definition III.4: Suppose that is a sliding surface for system

(1). A continuously differentiable function iscalled an -Lyapunov function for system (1) at if

i) ;ii) on for some ;

iii) the restriction is a proper positive definitefunction at .

IV. PRACTICAL STABILIZATION

When the distribution generated by the input vector fieldshas a higher dimension than the number of independent inputs

the motion in the state space required to generatethe higher dimensional Lie brackets of the input vector fieldscan make asymptotic stabilization problematic. In this sectiontherefore we consider practical stabilization and establish thefollowing theorem.

Theorem IV.1: Suppose that is a sliding surface for system(1), a generalized sliding-mode controller for , and an

-Lyapunov function at . Then the equilibrium state ispractically stabilized by .

Before proving this theorem we present an example and thenestablish some technical lemmas.

Example IV.2: Consider the system model

(5)

where and . A lower dimensionalversion of this system is discussed in the study of local small

time controllability in [14]. It is a system which does not satisfythe necessary conditions for the construction of a continuousstabilizing feedback controller [3], [22]. Our system model is

, withand

for . It follows that the involutive distributionhas dimension and is the linear span of .Thus, with the usual identification of with itself,

and the leaf of through a pointis . Since it is natural

to select a codimension 3 siding surface , that is

On we have . Since the leaf of throughis it follows that

and is a sliding surface for system(5). The function , when restricted to is positivedefinite and proper at . Furthermore , so that on

we have . Since it followsthat is an Lyapunov function for system (5) at . TheoremIV.1 asserts that if is a generalized sliding-mode controllerfor then system (5) is practically stabilized by .

For this example finding a generalized sliding-mode con-troller is fairly straightforward. If we use piecewise constantcontrols the integral curves for and are easy to com-pute. There are an infinite number of possible generalizedsliding-mode controls for this example. We will construct arather simple controller based on the following idea. We firstmake nonzero using constant and zero. Next we use

to move to . Then is used to set back to zero.Finally is used to set to zero. This results in a general-ized sliding-mode controller with memory . To satisfyDefinition III.2 iv) we need to ensure that when the state isclose to we only move a small amount in step one above.To measure the “distance” of a state from we introduce thecontinuous function and define

as follows. If then on ; ifthen the following hold.

1) Over the time interval we set. Then

and the state trajectory over is the integral curve ofthrough . Thus

and


2) Over the time interval we set.

Then

and the state trajectory over is the integral curve ofthrough . Thus

and

3) Over the time interval we set. Then

the state trajectory over is the integral curve ofthrough , and

4) Over the time interval we set .Then and

This defines over the time interval . The above stepsare repeated to define over , etc. The controller

defined previously satisfies Definition III.2. Conditions i)and ii) of Definition III.2 hold by our construction of . Sim-ilarly we have demonstrated above that trajectories of thedriftless system (2) with haveand for so that Definition III.2 iii) holds.To show that Definition III.2 iv) holds we suppose that is anyopen set containing and . From our constructionof we see that both and tend tozero as approaches . Furthermore we can easily writedown explicit formulae for for (we have done sofor above) and verify that the distance from to

can be made arbitrarily small by choosing sufficientlyclose to . In particular we can choose an open neighbourhood

of with such that trajectories of the driftlesssystem (2) with and have and

for so that Definition III.2iv) holds. Thus is a generalized sliding-mode controller andTheorem IV.1 asserts that system (5) is practically stabilizableby for sufficiently small.

Fig. 1. Practical stabilization using generalized sliding-mode control.

Fig. 2 shows the results of a simulation performed usingSimnon/PCW for Windows Version 2.01 (SSPA MaritimeConsulting AB, Sweden). The simulation results werefor initial condition and using aRunge-Kutta-Fehlberg 4/5 scheme. For the system (5) isnot clear how one would go about designing a standardsliding-mode controller which requires the state to slide on asliding submanifold of codimension 2.

For the linear system a natural slidingsurface is and acts as a

-Lyapunov function. For let denote the region incontaining and bounded by the lines and

. Then, on , we have . Thuswe have is proper andpositive definite at .More generally, suppose that is a sliding surface for system(1), is an -Lyapunov function, and .

Definition IV.3: A closed subset containing iscalled a -extension of if

a) ;b) the restriction of to is proper and positive definite at

;c) .

Lemma IV.4 shows that -extensions of exist.Lemma IV.4: Suppose that is a sliding surface for system

(1) and is an -Lyapunov function for system (1) at . Thenthere exists a -extension of .

Proof: Here is an -Lyapunov function hence there ex-ists such that for all . Choose

and . Then there exists an open neighbour-hood of on which and

. This ensures that does not vanish on .Since is metrizable we can further shrink if necessary toensure that the distance from each point in to is at most

. Let be the closure of the union of and all such


Fig. 2. Global stabilization of a mechanical system using generalized sliding-mode control.

sets. Then , the restriction of to is pos-itive definite at (because restricted to is proper anddoes not vanish on the closure of the sets ), and

. Thus i) and iii) of Definition III.4 are satisfiedand is positive definite. To show that this map isalso proper we note that is proper, so for any closedbounded interval , we know thatis closed and bounded. Since is continuous

is closed in hence in . By construction thedistance from any point in the bounded setto a point is at most . Since is the closure of theunion of and the sets it followsdirectly that is bounded as well. In particular

is compact hence a proper map. Finally, byconstruction, is nonzero on hence is positive defi-nite at and ii) of Definition III.4 is satisfied as well.

Lemma IV.5 shows that, as the sample rate increases (i.e.,decreases), the state trajectory for the system (1) withapproaches the sliding surface .

Lemma IV.5: Suppose that is a sliding surface, acompact neighbourhood of any open set containing , and

a generalized sliding-mode controller with memory forsystem (1). Then there exists such that for alland the sufficiently small, the state trajectoryfor system (1) with and reaches in time .

Proof: Let be a compact neighbourhood of anopen set containing , and . Since is a general-ized sliding-mode controller with memory the state trajec-tory for the driftless system (2) with initial condition

has . By definition the -th com-

ponent of is over the time interval , hencethe trajectory for is the integral curve ofthe smooth vector field . Thus,

. Similarly, for willbe the integral curve of the smooth vector field where

. It follows that. We now consider the state trajectory

for the system (1) with andover . As above, for , the state tra-

jectory is the integral curve of so thathence

approaches as tends to0. A similar reasoning applies on each interval ,hence we conclude that approaches

as . Thus there existsand an open neighbourhood such that for

and . We repeat this construction at eachpoint in get an open covering by sets of the form andcorresponding constants such that for

and . Since is compact and is anopen covering of , we can find a finite subcovering and a finiteset of constants . We set .Then for the state trajectory for system (1) withand reaches in time .

Suppose that is the -extension of for the linearsystem considered above. It is straightforwardto verify that, for sufficiently small, is invariant understate trajectories corresponding to . A more generalversion of this invariance in the linear case is established inCorollary V.3. The following Lemma shows that a local version


of this invariance holds for driftless control affine systemsunder generalized sliding-mode controllers.

Lemma IV.7: Suppose that is a sliding surface for system(1), an -Lyapunov function at a -extensionof , and a generalized sliding-mode controllerwith memory . Let denote the trajectory of thedriftless system (2) with andand let denote the trajectory for system (1) with

. Then there exists a -extensionof such that

a) ;b) if then for and

for ;c) if such that, for

and .Proof: Let be a generalized sliding-

mode controller with memory , and a -extension of .Then Definition III.2 iv) implies that there exists an open setsuch that and such that for all we have

and as a consequenceof Definition III.2 iii). Let be the closure of the set whichis the union of and the subsets for all .Then the set satisfies i) and ii) above. To show that isa -extension we note that, by construction, is a closedsubset of containing in its interior. Since is a

-extension of Definition IV.3 hold for , hence holds forthe subset of . In particular is a -extension of .To establish iii) we note that from ii) above if then

for and . But in the proofof Lemma IV.5 we showed that, for eachapproaches as tends to zero. Thus for sufficientlysmall and , it follows that .Furthermore, since , we can ensure that

by further shrinking .We know from Lemma IV.5 that a generalized sliding-mode

controller will move states of system (1) into any open neigh-bourhood of a sliding surface , but a -extension ofneed not be an open neighbourhood of (only of ).The following Lemma provides a way to enlarge using anopen set so that contains in its interior. This willbe needed in the proof of Theorem IV.1.

Lemma IV.7: Suppose that is a sliding surface for system(1), a set containing in its interior, a gener-alized sliding-mode controller with memory , and an openneighbourhood of with compact closure. Let de-note the trajectory for system (1) withfor . Then there exists an open set and such that

a) ;b) if and then for

and .Proof: From our definition of a generalized sliding-mode

controller with memory [Definition III.2 iii) and iv)] there ex-ists an open neighbourhood of in such that for allwe have and , where

denotes the trajectory of the driftless system (2) withand . By definition is an open

neighbourhood of . In the proof of Lemma IV.5 we showedthat approaches as . Choose .

Then, for sufficiently small, we have forand . Thus, we can find

an open neighbourhood of such that forand for .

Shrinking if necessary the above holds for all ,the closure of . Since and is compact the set

is also compact and we can find a finite covering by sets. Set . Thus i) and ii) hold

using and .Proof (Theorem IV.1): Let be a compact a compact

neighbourhood of and an open subset of containing .We must show that there exist such that for anywe have for .

Suppose that is a sliding surface for system (1), a gen-eralized sliding-mode controller for , and an -Lyapunovfunction at . Let be a -extension of (whose existenceis guaranteed by Lemma IV.4). Let denote the trajec-tory of system (1) with and .Let be a -extension of satisfying the hypotheses ofLemma IV.6.

Our first task is to enlarge by adjoining an open neigh-bourhood of to create a set which contains in its interior.Since and is compact we know that is acompact neighbourhood of . Lemma IV.7 with and

implies that there exists an open set andsuch that and for all and if then

for and .Set . We have established that, for all

for and .We now show that all states in will be steered to in fi-

nite time, so that, without loss of generality, we can assume thatour initial state is in a compact subset of . Since is con-tained in the interior of Lemma IV.5 implies that, for suf-ficiently small, for all . Furthermore theimage of under our state trajectory mapis compact. Thus, after one “sample interval” of length , thestate is transfered from to .Thus at time the state will be in . This is a boundedsubset of . Since the restriction of to the -ex-tension is proper and positive definite, we can assume thatthe initial state for system (1) lies in forsome . Thus any bounded subset of is contained in abounded set for some . This means that

. Thus we can assumethat the state of system (1) starts off from

. Furthermore, since the restriction of to the -exten-sion is proper and for if and only if ,there exists such that and

. Since we have two casesto consider, or where is the compact set

We note that the intersection of and need not be empty.In the case we have shown that for

and . To complete our proof it sufficesto show that for an initial state the trajectory of system(1) reaches in finite time and then stays in thereafter. From


Lemma IV.6iii) we know that if there existssuch that, for all ,

and . In fact we can choose an openneighbourhood of such that for we have

and . The setsform an open cover of the compact set . Choosing a finite

subcover we can find such that, for all ,and for any .

To complete our proof we use that fact that the -Lyapunovfunction is decreasing along the system trajectories whichcauses the state trajectory to reach . From the definition of an

Lyapunov function, we haveand . Thus, while the state trajectory evolves in ,the -Lyapunov function decreases. In particular

(6)

for . Thus, after the “initial sample period” ofduration we have

. If thenand we can repeat the above argument over the time in-terval to show that

and .Continuing in this manner we see that eventually we have

for some positive integer . Thus aftersome time we have hence

. As we notedabove if we have for and

. Of course can increase on .Set . If then and over thenext time interval of length this cycle repeats and the statestays in . If, on the other hand, ( ) thenwe must ensure that over the following time interval of length

the state does not leave . Recall that was picked so thatand and

we have established that for initial states inthe function decreases along the state trajectory over a timeinterval of length . Suppose that . Then over thenext sample period (of duration ) decreases along the statetrajectory so stays less than and the state stays in . It fol-lows that the state stays in for all if we can ensurethat whenever the state trajectoryhas the property that ).This is easy to do by decreasing . The rate of change ofalong the state trajectory was shown to be .But and the closure of is compact. Thus wehave an upper bound for the rate at which can increasealong state trajectories in . This means that is an upperbound on the increase of over a sample interval. If we choose

then we will have .We have established that, for sufficiently small, for all

there exists such that. This implies that for in

some open neighbourhood of . These neighbourhoods

are an open covering of . Since is compact we canfind a finite subcover . Let be the max-imum of . Then for any we have

. This completes the proof.

V. ASYMPTOTIC STABILIZATION

If is a direct generalized sliding-mode controller one canshow that there exists a -extension of such that, givenany open neighbourhood of and initial state ,there exists such that forif for . Assumption A1i) belowis a stronger version of this forward invariance which hasimplications for asymptotic stability. Assumption A1ii) belowis a somewhat stronger version of the controllability result ofLemma IV.5 and is needed to establish global results.

A1. is a sliding surface, a direct generalizedsliding-mode controller for an -Lyapunov functionfor system (1) at and a -extension of such that,for sufficiently small

a) is forward invariant under trajectories of thesystem (1) for ;

b) the state trajectory of system (1) with carriesany initial state to in finite time.

The following local version of A1 suffices for semiglobalstabilization.A2. is a sliding surface, a compact neighbourhood of

a direct generalized sliding-mode con-troller for an -Lyapunov function for system (1) at

and a -extension of such that, for sufficientlysmall

i) trajectories of the system (1) withstay in for all ;

ii) the state trajectory of system (1) with carriesany initial state in to in time .

Theorem V.1: Suppose that is a sliding surface for system(1), a direct generalized sliding-mode controller foran -Lyapunov function at , and a -extension ofsuch that A1 (A2) holds. Then is globally (semi-globally)asymptotically stabilized by .

Before proving this theorem we introduce two corollarieswhich show that Theorem V.1 can be used to stabilize a class ofsingle-input homogeneous systems and controllable multi-inputlinear systems. Consider the single-input affine system model

on in triangular form described by

...

(7)

whereare odd positive integers,is , and is homogeneous of degree 0 with respectto the dilation , where . Avector field on is said to be homogeneous


of degree with respect to the dilation (cf [4][8])if, for all ,

A function on is said to be homogeneous of degreewith respect to the dilation if, for all ,

. The state equiv-alence of a nonlinear system to a triangular form is consideredin [5]. It is well known that for many systems of the form (7)(for example systems whose linearization is not stabilizable vialinear state feedback) which cannot be stabilized by means ofsmooth state feedback (cf [1], [2], [15], [22], and the referencestherein).

Corollary V.2: There exists a sliding surface for system(7), an -Lyapunov function at , a direct generalizedsliding-mode controller , and -extension of suchthat A1 holds. In particular system (7) is globally asymptoticallystabilized by .

Corollary V.3: Suppose that is a controllablelinear system with and rank . Then there existsan dimensional subspace which is a sliding surfacefor this system, a quadratic -Lyapunov function at ,a direct generalized sliding-mode controller , and a -ex-tension of such that A1 holds. In particular the controllablelinear system can be asymptotically stabilizedby .

We note that in the proof of Corollary V.2 we obtain a simpleformula (13) for a stabilizing direct generalized sliding-modecontroller. Before proving the above we present the followingexample studied in [21].

Example V.4: Consider the system model

(8)

introduced in [21]. These equations model an under-actuatedtwo degree of freedom mechanical system composed of amass on a horizontal surface connected on one side to a wallby a linear spring and on the opposite side to an invertedpendulum by a nonlinear spring (spring force proportional tothe cube of the displacement). The control is a force actingon the mass. Here is the angular displacement of theinverted pendulum, its angular velocity, the displace-ment of the mass and its velocity, is the length of thependulum, the acceleration due to gravity. After a nonlinearfeedback to cancel terms in the above equations of motionresult (cf. [21]). As in [21] we take . Here

. This homogeneous system has, and thus is homogeneous with respect

to the dilation . Here the func-tions in system (7) are . Itis straightforward to verify from the proof of Corollary V.2 that

is a sliding surface for this system wherefor constants

arising from the backstepping construction and estimatesin [19]. For example . These are veryconservative estimates and smaller values of can usuallybe used. In any event here ( holds for anysingle input system) and, from the proof of Corollary V.2

is an -Lyapunov function for this example, where. The direct generalized sliding-mode

controller from the proof of Corollary V.2 ((13)) is

for .CorollaryV.2asserts that, for sufficientlysmall, the generalized controller achievesglobal asymptotic stability for system (8). Fig. 2 shows the resultsof a simulation performed using Simnon/PCW for WindowsVersion 2.01 (SSPA Maritime Consulting AB, Sweden), wherethe simulation has .

Proof: (Theorem V.1) Suppose that is a sliding surfacefor system (1), a direct generalized sliding-mode controllerfor an -Lyapunov function at , and is a -ex-tension of such that A1 holds. Then, for sufficiently small,A1ii) asserts that the state trajectory of system (1)with carries any initial state to in fi-nite time. A1i) asserts that is invariant under trajectoriesfor system (1) where . This implies that

such that . Becauseis a -extension of we have proper and pos-itive definite at and . It followsthat for . In par-ticular hence

. Because restricted to is pos-itive definite at it follows that .

In the case where A2 holds we proceed in the same manneras above. Suppose that is a compact neighbourhood ofand . For sufficiently small A2ii) asserts that the statetrajectory of system (1) with carries anyinitial state to a point in time less than or equalto . Since is compact the set of all such points is containedin a compact neighbourhood of . Shrinking if necessaryA2i) asserts that is invariant under trajectories for system (1)where . We can repeat the aboveargument to conclude that .

Before proving Corollary V.2 we present the followinglemma.

Lemma V.5: Let denote the affine system(7). Then there exist a function and smoothfunctions on such that the following hold.

i) The restriction of to is proper and positivedefinite at 0.


ii)

forsome , where .

iii) for iff .Proof: (Lemma V.5) It is straightforward to verify (cf. [4])

that the homogeneity of the functions in (7) implies that,after a simple homogeneous feedback, the system (7) can beexpressed as

(9)

where

ifif

and

We only need a slight modification of the proof of Theorem 3.1in [19]. Here we have the inequality

(10)

where the functions can be chosen to beconstants, namely . Nowwe repeat the backstepping constructions of [19] which result ina sequence of proper positive definite functions . Beginwith the proper positive definite function of ,

. As in the proof of [19, Th. 3.1]where for

. Now suppose that we have found constantsfunctions and defined by

(11)

whereand proper positive definite func-

tions such that

for . Then

is a proper positive definite function and

(12)

where for some constant . Thus, by in-duction, and setting

and wesatisfy i) and ii). Conclusion iii) follows from the definition of

. One can incorporate the change in coor-dinates and homogeneous feedback to show that this result alsoholds for system (7).

Proof: (Corollary V.2) From Lemma V.5 there exist func-tions defined on such that Lemma V.5 i)-iii)hold. In light Lemma V.5ii) we define our sliding surface forsystem (7) to be where

and . From (12) wehave and

on . System (7) has hence the integral curve. By definition is the graph

of the function

and thus, given , there exists a unique such that. In particular, where is the

leaf of containing . By construction isa smooth function and is a nonzero constant. Thus0 is a regular value for and also for

. This implies that is a codi-mension one embedded submanifold of containing

, and thus is a sliding surface for system (7).To show that is an Lyapunov function for system (7) at

it suffices to verify that for someand that the restriction of to is positive definite at 0. But

is the graph of , hence therestriction of to is positive definite as a consequence ofLemma V.5 i). To see that for some weuse homogeneity. First, we note that on we have establishedthat and

is positive definite at 0. Thus, given any , there existssuch that .

Shrinking of necessary we can choose an open neighbour-hood of in such that

. Consider the compact set whereis the “homogeneous -sphere” corresponding to the di-

lation . Here is a “di-lated norm” with respect to and the homogeneous“ -sphere” is . Then, corre-sponding to to each , there exists an open set

such that. This gives rise to an open covering of

. Thus there exists a finite subcover of by opensets . Setting we have

on . Inparticular on . In light of (9) itstraightforward to verify that .

We note that, for a given dilation , a vectorfield is homogeneous of degree iff is homogeneous


of degree for all homogeneous of degree . Thusis homogeneous of degree 0 with respect to the dilation

. Sim-ilarly the functions are ho-mogeneous of degree 1 with respect to the dilation .Now, by construction, is homogeneous of degreewith respect to the dilation and, since is homo-geneous of degree 0 with respect to willbe homogeneous of degree . Thus, forand , we havewhich implies that

This implies that holds on all of , henceis an -Lyapunov function.To define a direct generalized sliding-mode controller we

first note that we defined to beFor we set

and define our direct generalized sliding-modecontroller to be

(13)

We note that is a function ofonly and if and only if . It

is straightforward to verify that is homogeneous of degreewith respect to the dilation . Over the time interval

the trajectory of the driftless system (2) withisand . Inparticular

hence . Furthermore, if , thenand for . It is straightforward to check

that satisfies all of the remaining conditions necessary toqualify as a direct generalized sliding-mode controller.

We now define a -extension for which A1 holds. Letand denote the homogeneous sphere in . Set

so that. Since and is homogeneous of

degree 1 with respect to the dilation we know thatif then for all

. We set

so that . Then iff forsome (that is and

). We now show that, for sufficiently small,on . Let . Then for

. We set. Then Lemma V.5ii) implies that the rate of change

of along the system trajectories is where.

Since we haveBut are defined to be even

powers of hence . Sup-pose that we choose . Then .To see that this is the case we suppose that

. From our definition of and Lemma V.5iii) itfollows that . Since wehave and . This contradicts

. It follow that by further shrinking we can ensure that. Thus given any we can choose

and an open neighbourhood of in such that. Since with the relative topology is com-

pact and an open covering ofthere exists such that . Forwe saw above that for

and . It is easy to verify that the functionsare homogeneous of de-

gree with respect to the dilation . Thus

if . In particular on . With this fact it isstraightforward to verify that is a -extension of .

To complete the proof we must show that with the abovechoices for , and that assumption A1 holds. That isfor sufficiently small we have i) is invariant under tra-jectories for system (7) with , and ii)the statetrajectory of system (1) with carries any initial state to

in finite time.i) Let . We first consider the case where .

Then over the first sample period andwe must show that for . Wefirst suppose that (so that the dilatednorm and ) and examine ,the integral curve for the homogeneous drift vector fieldfor the system (7) through . Since

there exists such that for all. From the compactness of it follows

that there exists such that for alland . More generally suppose that

. Then for some. Because is homogeneous

of degree 0 with respect to the dilation wehave where

. But we showed above that for alland . From our definition of as

a “homogeneous extension” it follows that forall . In particular for . Now


consider the case where is arbitrary. Letdenote the trajectory of system (7) with

and . The controlled vector field for the system(7) can be represented as . In lightof the above discussion we can choose sufficiently smallto ensure that . Set

, a compact set. If thenand for . In

particular for . Sincetends to zero as as approaches and is a constantvector field it follows that for and

in some open neighbourhood of in . Ifthen trajectories of the driftless system with and

stay in for as a straightforwardconsequence of our constructions. Thus, as in the proof ofLemma IV.7, we can choose sufficiently small and anopen neighbourhood of in such that

for and . Since is compact wecan find such that for and

. More precisely we have shown that

for (14)

Now is a homogeneous vector field of de-gree 0 with respect to the dilation .In particular we have

for . Here, and ,

a homogeneous function of degree with respect to thedilation . It follows that is homoge-neous of degree 0 with respect to hence, for

Thus if and it follows that

From (14) we have where. This implies

hence . Sincethe set of points for isall of we have established i).

ii) To show that the state trajectory of system (7) withcarries any initial state to in finite time we first

note that the state trajectory of the driftlesssystem with carries any initial stateto in time . Of course initial states of the form

have hence the trajectory ofthe system (7) will have where we set

. Thus is suffices to show that trajectoryof the system (7) with initial states in reach in finitetime. We first suppose that . For the drift-less system with we have .Thus for sufficiently small it follows that the trajectoryof the system (7) will have for all insome open neighbourhood of in . From thecompactness of we can conclude that, forsufficiently small, for all .As above we can use the homogeneity of withrespect to the initial condition and the homogeneityused to generate to conclude that forall . Since we indicated above that all states aresteered to in finite time it follows that A1 holds. Thatsystem (7) is asymptotically stabilized by fol-lows from Theorem V.1.

Proof: (Corollary V.3) We begin by considering a singleinput system in Brunovsky form, that is

(15)

This is a homogeneous system of the form (7) with. The relevant di-

lation is thus . Corollary V.2 yields a sliding sur-face , a direct generalized sliding-mode controller, an -Lya-punov function , and -extension such that A1 is satisfied.Furthermore the sliding surface is a subspace of codimen-sion 1. We then apply the linear static state feedback and linearchange of coordinates in the state and input spaces to convertthe Brunovsky form back to the original system. The resulting

satisfy A1 and our linear system is asymptoti-cally stabilized via the direct sliding-mode controller . Forthe multi-input case with independent inputs in Brunovskyform we repeat the above on each single-input single-outputsub-block and let the sliding surface be the intersection ofthe hyperspaces which act as sliding surface for each of thesingle-input subsystems. Similarly is the sum of the -Lya-punov functions created for each of the single-input sub-sys-tems, is the intersection of the -extensions and

the vector controller whose components are thedirect generalized sliding-mode controllers for the single-inputsubsystems. We then apply the linear static state feedback andlinear change of coordinates in the state and input spaces to con-vert back to the original system from the Brunovsky form. It isstraightforward to show that the resulting is a direct gener-alized sliding-mode controller, is a codimension subspaceof and satisfy A1. Theorem V.1 implies thatour linear system is asymptotically stabilized by .

ACKNOWLEDGMENT

The author would like to thank the anonymous referees fortheir very careful reading of this paper and their constructivecomments which guided the significant revisions to the man-uscript. Thanks also to N.H. McClamroch for suggesting thephysical system considered in Example 2.


REFERENCES

[1] D. Aeyels, “Stabilization of a class of nonlinear systems by smoothfeedback control,” Syst. Control Lett., vol. 5, pp. 289–294, 1985.

[2] R. W. Brockett, , R. W. Brockett, R. S. Millman, and H. J. Sussmann,Eds., “Asymptotic stability and feedback stabilization,” in Differen-tial Geometric Control Theory. Boston, MA: Birkhäuser, 1983, pp.181–191.

[3] R. W. Brockett, “Pattern generation and the control of nonlinear sys-tems,” IEEE Trans. Autom. Control, vol. 48, no. 10, pp. 1699–1711,Oct. 2003.

[4] S. Celikovsky and E. A. Bricaire, “Constructive nonsmooth stabiliza-tion of triangular systems,” Syst. Control Lett., vol. 36, pp. 21–37, 1999.

[5] S. Celikovsky and H. Nijmeijer, “Equivalence of nonlinear systems totriangular form: The singular case,” Syst. Control Lett., vol. 27, pp.135–144, 1996.

[6] W. L. Chow, “Uber systeme von linearen partiellen differentialgle-ichungen erster ordnung,” Math. Ann., vol. 117, pp. 98–105, 1939.

[7] F. Clarke, Y. Ledyaev, E. Sontag, and A. I. Subbotin, “Asymptotic con-trollability implies feedback stabilization,” IEEE Trans. Autom. Con-trol, vol. 42, no. 10, pp. 1394–1407, Oct. 1997.

[8] J. M. Coron and L. Praly, “Adding an integrator for the stabilizationproblem,” Syst. Control Lett., vol. 17, pp. 89–104, 1991.

[9] A. F. Filippov, Differential Equations with Discontinuous RighthandSides. Dordrecht, The Netherlands: Kluwer, 1988.

[10] L. Fridman and A. Levant, , W. Perruquetti and J.-P. Barbout, Eds.,“Higher order sliding-modes,” in Sliding Mode Control in Engi-neering. New York: Marcel Dekker, 2002, pp. 53–102.

[11] L. Grune, “Homogeneous state feedback stabilization of homogeneoussystems,” SIAM J. Control Optim., vol. 38, no. 4, pp. 1288–1308, 2000.

[12] R. Hermann, “On the accessibility problem in control theory,” in Inter-national Symposium Nonlinear Differential Equations and NonlinearMechanics. New York: Academic, 1963, pp. 325–332.

[13] R. Hirschorn, “Singular sliding-mode control,” IEEE Trans. Autom.Control, vol. 46, no. 2, pp. 276–286, Feb. 2001.

[14] R. Hirschorn and A. Lewis, Geometric local controllability: second-order conditions 2006.

[15] M. Kawski, “Homogeneous stabilizing feedback laws,” in ControlTheory Adv. Technol., Tokyo, Japan, 1990, vol. 6, pp. 497–516.

[16] H. Nijmeijer and A. van der Schaft, Nonlinear Dynamical Control Sys-tems. New York: Springer-Verlag, 1990.

[17] Y. Orlov, “Finite-time stability and robust control synthesis of uncer-tain switched systems,” SIAM J. Control Optim., vol. 43, no. 4, pp.1253–1271, 2005.

[18] J.-B. Pomet, “On the curves that may be approached by trajectories of asmooth affine system,” Syst. Control Lett., vol. 36, pp. 143–149, 1999.

[19] C. Qian and W. Lin, “Non-Lipschitz continuous stabilizers for non-linear systems with uncontrollable unstable linearization,” Syst. Con-trol Lett., vol. 42, pp. 185–200, 2001.

[20] C. Qian and W. Lin, “A continuous feedback approach to global strongstabilization of nonlinear systems,” IEEE Trans. Autom. Control, vol.46, no. 7, pp. 1061–1079, Jul. 2001.

[21] C. Rui, M. Reyhanoglu, I. Kolmanovsky, S. Cho, and N. H. McClam-roch, “Nonsmooth stabilization of an under-actuated two degree offreedom mechanical system,” in Proc. 36th IEEE Conf. Decision andControl. New York: IEEE Publications, 1997, pp. 3998–4003.

[22] E. P. Ryan, “On Brockett’s condition for smooth stabilizability andits necessity in a context of nonsmooth feedback,” SIAM J. ControlOptim., vol. 32, pp. 1597–1604, 1994.

[23] S. Sastry, “Nonlinear systems, analysis stability and control,” inInterdisciplinary Applied Mathematics. New York: Springer-Verlag,1999.

[24] J.-J. E. Slotine, Applied Nonlinear Control. Upper Saddle River, NJ:Prentice-Hall, 1991.

[25] J. Tsinias, “Planar nonlinear systems: Practical stabilization andHermes controllability condition,” Syst. Control Lett., vol. 17, pp.291–296, 1991.

[26] V. I. Utkin, Sliding Modes in Control Optimization. Berlin, Germany:Springer-Verlag, 1992, CCES.

Ronald M. Hirschorn received the B.A.Sc. degreein engineering from the University of Toronto,Toronto, ON, Canada, the S.M. degree in aeronauticsand astronautics from the Massachusetts Instituteof Technology, Cambridge, and the Ph.D. degree inapplied mathematics from Harvard University, Cam-bridge, MA, in 1968, 1970, and 1973, respectively.

Since 1973, he has been a Faculty Member with theDepartment of Mathematics and Statistics, Queen’sUniversity, Kingston, ON, Canada. His research in-terests have included controllability and disturbance

rejection for nonlinear systems, output tracking, model-based friction compen-sation, and sliding-mode control.

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