Output feedback control of saturated discrete-time linear systems using parameter-dependent Lyapunov functions

Systems & Control Letters 57 (2008) 896–903

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Systems & Control Letters

journal homepage: www.elsevier.com/locate/sysconle

Output feedback control of saturated discrete-time linear systems usingparameter-dependent Lyapunov functionsI

Qian Zheng, Fen Wu ∗

Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695, United States

a r t i c l e i n f o

Article history:Received 21 July 2005Received in revised form23 November 2007Accepted 17 December 2007Available online 4 June 2008

Keywords:Actuator saturationParameter-dependent Lyapunov functionOutput feedback controlDomain of attractionRestricted `2 gain

a b s t r a c t

In this paper, we present a newoutput feedback control approach for discrete-time linear systems subjectto actuator saturations using parameter-dependent Lyapunov functions. The saturation level indicatorserves as a scheduling parameter. The resulting nonlinear controller is expressed in a quasi-LPV (linearparameter-varying) form, and the stabilization and disturbance attenuation problems are formulatedand solved as finite-dimensional linear matrix inequality (LMI) optimization problems. Our approachis less conservative than a single quadratic Lyapunov function method. Specifically, the proposed outputfeedback control law asymptotically stabilizes the open loop system with a larger domain of attractionand achieves better disturbance attenuation under energy and magnitude bounded disturbances.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

Saturation is a widely encountered and most dangerousnonlinearity in control systems. It is well recognized that actuatorsaturation degrades the performance of the control system andmay even lead to instability. As a result, the problem of actuatorsaturation has received increasing attention from the researchcommunity and industry [1,21,11,8].

The works on this topic can be divided into two categories,those that deal with open loop systems that are not exponentiallyunstable (or simply called semi-stable) and those that areexponentially unstable. The stabilization of semi-stable systems isnowwell-understood [22,13,20,14,19]. For exponentially unstableopen loop systems, the objectives are to characterize the nullcontrollable region [8] and to design feedback laws that work onthe entire null controllable region or a large portion of it (see,for example, [7,15,8,16,12,9,5]). However, most research work onexponentially unstable systems pertains to state feedback. A fewexceptions where output feedback are used include [15,12,9].

Recently, we have developed a method for the design of outputfeedback laws that result in large domains of attraction [23]. Thismethod is based on constant Lyapunov functions, and the resultapplies to general linear systems including strictly unstable onesand enlarges the estimated domain of attraction. By utilizing the

I The work was supported in part by NSF Grant CMS-0324397.∗ Corresponding author. Tel.: +1 (919) 515 5268; fax: +1 (919) 515 7968.

E-mail address: [email protected] (F. Wu).

0167-6911/$ – see front matter© 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.sysconle.2007.12.011

convex hull expression of a saturating linear feedback law [10],a nonlinear output feedback controller is first expressed in theform of a quasi-LPV (linear parameter-varying) system. Conditionsunder which the closed loop system is locally asymptoticallystable is then established as a finite-dimensional optimizationproblem with LMI constraints. In the same framework, we havealso developed output feedback laws to attenuate the disturbanceeffect on the system output in [24]. The level of disturbanceattenuation is measured in terms of the restricted L2 gain and L2toL∞ gain over a bounded disturbance set. Numerical results havedemonstrated the effectiveness of the resulting output feedbacklaws.

In this paper, we will extend the saturation control approachin [23,24] to design output feedback laws using parameter-dependent Lyapunov method. Specifically, we consider a class ofparameter-dependent Lyapunov matrix

P(ρ) =

2m−1∑j=0

ρjPj, Pj > 0,2m−1∑j=0

ρj = 1

where Pj’s are constant matrices involved in the control synthesisprocess. ρj’s are saturation level indicator and measurable inreal time. We will first parameterize the nonlinear outputfeedback law in the form of quasi-LPV systems. By introducing anintermediate matrix variable [3], the control synthesis conditionsfor stabilization and disturbance attenuation problems will beestablished in terms of convex optimization problems involvingonly finite number of LMIs, which can be solved using efficientinterior-point algorithms. It is noted that most of previous workin saturation control were based on a single quadratic Lyapunov

Q. Zheng, F. Wu / Systems & Control Letters 57 (2008) 896–903 897

function approach, which is potentially conservative in dealingwith saturated control systems. With the help of parameter-dependent Lyapunov functions, the proposed controller willprovide less conservative results such as larger domain ofattraction and better disturbance attenuation capabilities. Similarwork employing parameter-dependent Lyapunov functions havebeen reported in [6,18,2] for state feedback problems. Moreover,an output feedback control problem was addressed in [12] usingparameter-dependent Lyapunov functions. However, the resultingsynthesis condition was formulated as an infinite-dimensionaloptimization problem and is difficult to solve.

The notation used in this paper is fairly standard. R stands forthe set of real numbers and R+ for the non-negative real numbers.Rm×n is the set of real m × n matrices. We use Sn×n to denotereal, symmetric n × n matrices, and Sn×n

+for positive definite

matrices. A block diagonal matrix with matrices X1, X2, . . . , Xp

on its main diagonal is denoted by diagX1, X2, . . . , Xp

. In large

symmetric matrix expressions, terms denoted ?will be induced bysymmetry. For two integers k1, k2, k1 < k2, we denote I[k1, k2] =

k1, k1 + 1, . . . , k2. An infinite sequence x := x1, x2, . . . is said tobe in `2 if

∑∞

i=1 xTi xi < ∞.

2. Problem statement

Consider a discrete-time linear system subject to inputsaturation

xp(k + 1) = Apxp(k) + Bp1w(k) + Bp2σ(u(k))z(k) = Cp1xp(k) + Dp11w(k) + Dp12σ(u(k))y(k) = Cp2xp(k) + Dp21w(k),

(1)

where xp ∈ Rn, u ∈ Rm is the control input and y ∈ Rp is themeasurement output, w ∈ Rnw is the disturbance and z ∈ Rnz isthe controlled output. The matrix triple (Ap, Bp2, Cp2) is assumedstabilizable and detectable. Also, the function σ : Rm

→ Rm is avector valued standard saturation function, i.e.,

σ(u) =[σ(u1) σ(u2) · · · σ(um)

]with σ(ui) = sgn(ui) min 1, |ui|.

We will consider a dynamic output feedback control law of theform,xc(k + 1) = Ac(xc(k), y(k))xc(k) + Bc(xc(k), y(k))y(k)u(k) = Ccxc(k) + Dcy(k),

(2)

where xc(k) ∈ Rn is the controller state. Ac, Bc are Lipschitzfunctions of xc and y, and Cc and Dc are constant matrices ofappropriate dimensions.

For matrices HC ∈ Rm×n and HD ∈ Rm×p, we define

L(HC,HD) =(xc, y) ∈ Rn+p

: |HCixc + HDiy| ≤ 1, i ∈ I[1,m],

where HCi and HDi represent the ith row of matrices HC and HD

respectively. We note that L(HC,HD) represents the region in Rn+p

where the auxiliary feedback HCxc + HDy does not saturate.From [8], if (xc, y) ∈ L(HC,HD), the saturated input σ(u) can

be expressed on a convex hull of control command Ccxc + Dcy andauxiliary control input HCxc + DCy as

σ(Ccxc + Dcy) =

2m−1∑j=0

ρj

[Ej(Ccxc + Dcy) + E−

j (HCxc + HDy)], (3)

for some scalars 0 ≤ ρj ≤ 1, j ∈ I[0, 2m−1], such that

∑2m−1j=0 ρj = 1.

Ej = diag1 − z1, 1 − z2, . . . , 1 − zm, j ∈ I[0, 2m− 1] are m × m

diagonalmatriceswith the value of zi’s either 1 or 0, and E−

j = I−Ej.

In general, there are many choices of ρj’s satisfying Eq. (3).However, one choice of such ρj’s of particularly interest for thisstudy will be [23]

ρj(xc, y) =

m∏i=1

[zi(1 − λi(xc, y)) + (1 − zi)λi(xc, y)] ,

j ∈ I[0, 2m− 1],

where j = z12m−1+z22m−2

+· · ·+zm with zi ∈ 0, 1.λi’s are definedas

λi(xc, y) =

1, if Ccixc + Dciy = HCixc + HDiyσ(Ccixc + Dciy) − (HCixc + HDiy)

(Cci − HCi)xc + (Dci − HDi)y, otherwise

for each i ∈ I[1,m]. As shown in [23], ρj(xc, y)’s are Lipschitzfunctions in xc and y, such that,

∑2m−1j=0 ρj = 1, 0 ≤ ρj ≤ 1, j ∈

I[0, 2m− 1]. We note that the values of ρj’s reflect in a way the

severity of control saturation and are available for real-time use ingain-scheduling control.

We will use the functions ρj(xc, y)’s to parameterize the outputfeedback controller (2) into the following quasi-LPV systemxc(k + 1) =

(2m−1∑j=0

ρjAcj

)xc(k) +

(2m−1∑j=0

ρjBcj

)y(k)

u(k) = Ccxc(k) + Dcy(k)

, (4)

The coefficient matrices Acj’s, Bcj’s, Cc and Dc are to be designed.Since ρj is a time-varying parameter which is used to schedule thecontroller gain, the resulting controller is called a nonlinear gain-scheduled controller.

One of our design objectives is to construct a dynamic outputfeedback law (2) that asymptotically stabilizes the plant (1) atthe origin with a domain of attraction as large as possible. This isachieved by parameterizing the controller in a quasi-LPV form (4)and then establish stabilization conditions within an ellipsoid ofthe form

Ω(P(ρ),β) =

x : xTP(ρ)x ≤ β, 0 ≤ ρj ≤ 1,

2m−1∑j=0

ρj = 1

,

P(ρ) =

2m−1∑j=0

ρjPj, Pj > 0, β > 0,

which is contained in the domain of attraction. Secondly, wewill develop output feedback laws of the form (4) that establishinvariance property in an ellipsoid for bounded disturbances, andminimize the `2 gain from disturbance to the controlled outputwithin the ellipsoid. For this purpose, we will introduce the set ofenergy and magnitude bounded disturbances

Wβ,w =

w : R+ → Rnw ,

∞∑k=0

wT(k)w(k) ≤ β, wTw ≤ w2

, (5)

where β is some positive number and w is the magnitude boundof disturbance. In both stabilization and disturbance attenuationproblems, the determination of the controller coefficient matriceswill be formulated as optimization problemswith finite number ofLMI constraints, which can be solved using efficient interior-pointalgorithms.

Using the scheduling parameter ρ, the nonlinear plant (1) canalso be rewritten in a quasi-LPV form if (xc, y) ∈ L(HC,HD)

xp(k + 1) = Apxp(k) + Bp1w(k)

+ Bp2

2m−1∑j=0

ρj

[Ej(Ccxc(k) + Dcy(k)) + E−

j (HCxc(k) + HDy(k))]

(6)

z(k) = Cp1xp(k) + Dp11w(k)

+Dp12

2m−1∑j=0

ρj

[Ej(Ccxc(k) + Dcy(k)) + E−

j (HCxc(k) + HDy(k))](7)

y(k) = Cp2x(k) + Dp21w(k). (8)

898 Q. Zheng, F. Wu / Systems & Control Letters 57 (2008) 896–903

Motivated by the quasi-LPV structure of both the plant andthe controller, we consider the following auxiliary LPV system, ofwhich the nonlinear closed loop system comprising of (1) and (4)is a special case,[x(k + 1)

z(k)

]=

[A(ρ) B(ρ)C(ρ) D(ρ)

] [x(k)w(k)

], (9)

where xT(k) =

[xTp(k) xTc(k)

]and ρ ∈ Γ with

Γ :=

ρ ∈ R2m

:

2m−1∑j=0

ρj = 1, 0 ≤ ρj ≤ 1, j ∈ I[0, 2m− 1]

.

The state-space data of the closed loop LPV system (9) is given by[A(ρ) B(ρ)C(ρ) D(ρ)

]=

Ap 0 Bp10 0 0Cp1 0 Dp11

+

0 Bp2I 00 Dp12

×

(2m−1∑j=0

ρj

[Acj Bcj

EjCc + E−

j HC EjDc + E−

j HD

]) [0 I 0Cp2 0 Dp21

]

:=

2m−1∑j=0

ρj

[Aj Bj

Cj Dj

],

which are linear functions of scheduling parameter ρ.

3. Output feedback stabilization

For the stabilization problem, we assume the disturbancew = 0. The following theorem establishes conditions on thecontroller coefficient matrices under which the LPV system (9)is asymptotically stable with a quadratic parameter-dependentLyapunov function.

Theorem 1. Consider the plant (1). If there exist positive definitematrices Pj ∈ S2n×2n

+, j ∈ I[0, 2m

− 1] and matrices X, Y,W ∈

Rn×n, (Acj, Bcj) ∈ Rn×n×Rn×p, j ∈ I[0, 2m

−1], (Cc, Dc) ∈ Rm×n×Rm×p

and (HC, HD) ∈ Rm×n× Rm×p such that −Pj ?[

YTAp + BcjCp2 Acj

Ap + Bp2VcjCp2 ApXT+ Bp2Ucj

]P` −

[Y + YT ?

I + W X + XT

] < 0, j, ` ∈ I[0, 2m

− 1], (10)1β

?[CTp2H

TDi

HTCi

]Pj

≥ 0, i ∈ I[1,m], j ∈ I[0, 2m− 1], (11)

where Ucj = EjCc + E−

j HC and Vcj = EjDc + E−

j HD. Then, with thecontroller coefficient matrices for j ∈ I[0, 2m

− 1],Acj BcjCc DcHC HD

=

N YTBp2Ej YTBp2E−

j

0 Im 00 0 Im

−1

×

Acj − YTApXT Bcj

Cc Dc

HC HD

[ MT 0Cp2X

T Ip

]−1

, (12)

where M,N ∈ Rn×n and MNT= W − XY , the output feedback

controller (2) asymptotically stabilizes the plant (1) at the origin withthe ellipsoid Ω(P(ρ),β) contained in the domain of attraction. Theparameter-dependent Lyapunov matrix function P(ρ) is given by

P(ρ) =

2m−1∑j=0

ρjPj =

2m−1∑j=0

ρj

[I 0X M

]−1

Pj

[I XT

0 MT

]−1

. (13)

Proof. We will show that the saturated linear system (1) isasymptotically stabilized by the output feedback controller (2)using the parameter-dependent Lyapunov function

V(x) = xTP(ρ)x, P(ρ) =

2m−1∑j=0

ρjPj, Pj > 0.

Since P(ρ) is defined over the compact set Γ , its minimumeigenvalue will be strictly larger than 0, i.e., α1 ≤ λmin(P(ρ)) ≤ α2for some positive scalars α1,α2.

Introducing two non-singular matrices

T1 =

[I XT

0 MT

], T2 =

[Y INT 0

]

withMNT= W − XY and specifying G = T2T

−11 , it can be verified by

a generalized congruent transformation [17,4] that

TT1GTT1 = TT2T1 =

[YT WT

I XT

],

TT1(GTAj

)T1 =

[YTAp + BcjCp2 Acj

Ap + Bp2VcjCp2 ApXT+ Bp2Ucj

],

where the transformed controller data relates to the originalcontroller data (Acj, Bcj, Cc,Dc,HC,HD) through Eq. (12). Alsodefining TT1PjT1 = Pj and multiplying diag

T−11 , T−1

1

from the right

hand side and its transpose from left to Eq. (10), we obtain[−Pj AT

j GGTAj P` − G − GT

]< 0, j, ` ∈ I[0, 2m

− 1].

Taking linear combinations over j and `, then[−P(ρ) AT(ρ)GGTA(ρ) P(ρ) − G − GT

]< 0 (14)

for any ρ, ρ ∈ Γ . Since −P−1(ρ) ≤ −G−T(G+ GT− P(ρ))G−1 for any

non-singular G, it is clear that Eq. (14) guarantees[−P(ρ) AT(ρ)

A(ρ) −P−1(ρ)

]< 0.

By Schur complement, the above inequality is equivalent to

AT(ρ)P(ρ)A(ρ) − P(ρ) < 0, ∀ρ, ρ ∈ Γ (15)

inwhich the symbol ρ stands for the parameter value at step (k+1)and ρ represents its value at step k. Therefore,

V(k + 1) − V(k) = xT(k)[AT(ρ(k))P(ρ(k + 1))A(ρ(k)) − P(ρ(k))

]× x(k) < 0. (16)

When there is no disturbance w, we have

L(HC,HD) =(xc, y) : |HCixc + HDiy| ≤ 1, i ∈ I[1,m]

=(xp, xc) : |HDiCp2xp + HCixc| ≤ 1, i ∈ I[1,m]

= L(HDCp2,HC).

Then multiplying diag1, T−1

1

from the right hand side and its

transpose from left to (11), and noting that[HDCp2 HC

]T−11 =

[HDCp2 HC

],

we get an equivalent condition of (11) as1β

[HDiCp2 HCi

][CTp2H

TDi

HTCi

]Pj

≥ 0, i ∈ I[1,m], j ∈ I[0, 2m− 1].

Q. Zheng, F. Wu / Systems & Control Letters 57 (2008) 896–903 899

By Schur complement and linear combination arguments, theabove equation can be rewritten as

β[HDiCp2 HCi

]P−1(ρ)

[CTp2H

TDi

HTCi

]≤ 1, i ∈ I[1,m].

Therefore,

Ω(P(ρ),β) ⊂ L(HDCp2,HC) (17)

for all ρ ∈ Γ . Then we conclude the stability of the quasi-LPVsystem (9) from condition (15), while condition (17) validatesthe LPV representation of nonlinear saturated control systems inΩ(P(ρ),β).

In Theorem 1, a parameter-dependent Lyapunov function wasused to expand the stability region in construction of a stabilizingoutput feedback controller. Specifically, condition (11) validatesthe convex hull relation (3) by enforcing (xc, y) ∈ L(HC,HD);while condition (10) provides closed-loop stability guarantee usingoutput feedback control and parameter-dependent Lyapunovfunctions. The estimated domain of attraction is thus given by

D =

x ∈ R2n

: xTP(ρ)x < β, ∀ρ ∈ Γ.

To maximize the cross-section of the ellipsoid in the plant state-space, one can solve the following LMI optimization problem:

maxη,

s.t.1η2 XR −

[I 0

] Pjβ

[I0

]≥ 0, j ∈ I[0, 2m

− 1],

(10) and (11),

(18)

where Ω(XR, 1) is a reference ellipsoid with a given XR ∈ Sn×n+

.By setting Pj = P, j ∈ I[0, 2m

−1] and G = P, then conditions (10)and (11) recover the results of a single quadratic Lyapunov functionin [23]. as a special case. Therefore, it will be less conservative inestimating the domain of attraction. However, the conditions (10)and (11) involve 2m+1 and 2m numbers of LMIs respectively. So thecomputational cost will increase rapidly as the number of controlinput becomes large.

4. `2 gain control

Now, we will focus on the performance control problems fordiscrete-time saturated systems under energy and magnitudebounded disturbances.

4.1. Disturbance tolerance

For energy bounded disturbances, it was shown in [5] thatthe trajectories of saturated systems can not be invariant withinan ellipsoid. Nevertheless, one can find a slightly larger ellipsoidto accommodate the disturbance effect. By introducing an over-bounding set of Ω(P(ρ),β)

Ω(P(ρ),Λ,β + αw2

)=

(x,w) : xTP(ρ)x + wTΛw ≤ β + αw2, σ(Λ) ≤ α

,

the following set inclusion relation will hold.

Lemma 2 ([23]). If x ∈ Ω(P(ρ),β), then we have (x,w) ∈

Ω(P(ρ),Λ,β + αw2) for any w ∈ Wβ,w.

Also we note that

L(HC,HD) =(xc, y) : |HCixc + HDiy| ≤ 1, i ∈ I[1,m]

= (xp, xc,w) : |HDiCp2xp + HCixc + HDiDp21w| ≤ 1,

i ∈ I[1,m]

= L(HDCp2,HC,HDDp21).

Using Lemma 2, we will be able to replace the following relation

(xc, y) ∈ L(HC,HD), ∀(x,w) ∈ Ω(P(ρ),β) × Wβ,w

with a more stringent condition

Ω(P(ρ),Λ,β + αw2

)⊂ L(HDCp2,HC,HDDp21).

Although this will introduce some degree of conservatism, thelatter condition can be readily converted into an LMI constraint.

The following lemma provides the disturbance tolerancecondition for the saturated control system.

Lemma 3. If there exist parameter-dependent positive definitematrix function P(ρ) ∈ S2n×2n

+, positive definite matrix Λ ∈ Snw×nw

+,

and a scalar alpha such that−P(ρ) 0 AT(ρ)

0 −I BT(ρ)

A(ρ) B(ρ) −P−1(ρ)

< 0, (19)

Ω(P(ρ),Λ,β + αw2

)⊂ L(HDCp2,HC,HDDp21), (20)

Λ ≤ αI, (21)

where ρ, ρ ∈ Γ , then all trajectories of the saturated control systemthat starts from the origin will remain inside the ellipsoid Ω(P(ρ),β)for every w ∈ Wβ,w.

Proof. Due to the set inclusion relation in Lemma 2, we have thatconditions (20) and (21) imply

(xc, y) ∈ L(HC,HD), ∀(x,w) ∈ Ω(P(ρ),β) × Wβ,w.

This validates the LPV representation of the closed loop systemwithin Ω(P(ρ),β).

Using a parameter-dependent Lyapunov function V(x) =

xTP(ρ)x for the closed loop system (9),wehave the equation in Box Iwhere the last inequality comes from (19) by Schur complement.Summing up from 0 to N and assuming x(0) = 0, it can be seen that

V(x(N + 1)) <N∑

k=0wT(k)w(k) < β.

Therefore x(N + 1) ∈ Ω(P(ρ),β) for any integer N. This concludesthe proof.

For any disturbance w ∈ Wβ,w, Lemma 3 provides a boundedstate region in which the saturated system can be stabilized byan output feedback controller in the form of (2). Furthermore,one can find the largest disturbance level βmax by solving an LMIoptimization problem subject to constraints (19)–(21).

4.2. Restricted `2 gain control synthesis

The `2 gain of a saturated control system may not well definedfor a sufficiently large disturbance. For this reason, we will onlyconsider energy and magnitude bounded disturbances (5) andminimize the energy amplification from this class of disturbancew to controlled output z. Specifically, the restricted `2 gain underconsideration is given by

maxx(0)=0,w∈Wβ,w

‖z‖2

‖w‖2.

900 Q. Zheng, F. Wu / Systems & Control Letters 57 (2008) 896–903

V(x(k + 1)) − V(x(k)) − wT(k)w(k)

= [A(ρ(k))x(k) + B(ρ(k))w(k)]T P(ρ(k + 1)) [A(ρ(k))x(k) + B(ρ(k))w(k)] − xT(k)P(ρ(k))x(k) − wT(k)w(k)

=

[x(k)w(k)

]T [AT(ρ(k))P(ρ(k + 1))A(ρ(k)) − P(ρ(k)) AT(ρ(k))P(ρ(k + 1))B(ρ(k))

BT(ρ(k))P(ρ(k + 1))A(ρ(k)) BT(ρ(k))P(ρ(k + 1))B(ρ(k)) − I

] [x(k)w(k)

]< 0

Box I.

2)

3)

4)

−Pj ? ? ?0 −I ? ?[

YTAp + BcjCp2 Acj

Ap + Bp2VcjCp2 ApXT+ Bp2Ucj

] [YTBp1 + BcjDp21

Bp1 + Bp2VcjDp21

]P` −

[Y + YT ?

I + W X + XT

]?[

Cp1 + Dp12VcjCp2 Cp1XT+ Dp12Ucj

]Dp11 + Dp12VcjDp21 0 −γ2I

< 0, j, ` ∈ I[0, 2m− 1], (2

1

β + αw2 ? ?[CTp2H

TDi

HTCi

]Pj ?

DTp21H

TDi

0 Λ

≥ 0, i ∈ I[1,m], j ∈ I[0, 2m− 1], (2

Λ ≤ αI, (2

where Ucj = EjCc + E−

j HC, Vcj = EjDc + E−

j HD

Box II.

Theorem 4. Given scalars β < βmax and α, there exists a stabilizingoutput feedback controller (2) rendering `2 gain of the saturatedcontrol system (1) less than γ for all w ∈ Wβ,w if there exist positivedefinite matrices Pj ∈ Sn×n

+, j ∈ I[0, 2m

− 1], and matrices X, Y,W ∈

Rn×n, Λ ∈ Snw×nw+

, (Acj, Bcj) ∈ Rn×n× Rn×p, j ∈ I[0, 2m

− 1],(Cc, Dc) ∈ Rm×n

× Rm×p and (HC, HD) ∈ Rm×n× Rm×p such that the

equations (22)–(24) in Box II are satisfied. The coefficient matrices ofsuch an nth-order output feedback controller and its associated linearsubspace are given by (12). Moreover, all trajectories of the closedloop system that starts from the origin will remain inside the ellipsoidΩ(P(ρ),β) for every w ∈ Wβ,w, where P(ρ) is defined by (13).

Proof. Form Lemma 3, conditions (22)–(24) in Box II guaranteethat the trajectories of saturated control system contain inΩ(P(ρ),β) for bounded disturbance w ∈ Wβ,w. Within Ω(P(ρ),β),the `2 gain of the closed loop system is no more than γ if thereexists a parameter-dependent Lyapunov function as defined in(13), such that

V(x(k + 1)) − V(x(k)) +1γ2 z

T(k)z(k) − wT(k)w(k) < 0, (25)

Ω(P(ρ),Λ,β + αw2

)⊂ L(HDCp2,HC,HDDp21). (26)

To prove the claim, we apply the generalized congruenttransformation [4] as in the proof of Theorem 1 and obtain

TT1GTT1 =

[XT WT

I YT

],[

TT1(GTAj

)T1 TT1

(GTBj

)CjT1 Dj

]

=

YTAp + BcjCp2 Acj YTBp1 + BcjDp21

Ap + Bp2VcjCp2 ApXT+ Bp2Ucj Bp1 + Bp2VcjDp21

Cp1 + Dp12VcjCp2 Cp1XT+ Dp12Ucj Dp11 + Dp12VcjDp21

.

Also, the transformed controller data is related to the original onesby (12). Similar to the proof of Theorem 1, it can be shown that

condition (22) in Box II leads to−P(ρ) 0 AT(ρ) CT(ρ)

0 −I BT(ρ) DT(ρ)

A(ρ) B(ρ) −P−1(ρ) 0C(ρ) D(ρ) 0 −γ2I

< 0

for anyρ, ρ ∈ Γ . Therefore, the equation in Box III proves condition(25).

Next, we note that[HDCp2 HC HDDp21

] [T−11 00 I

]=[HDCp2 HC HDDp21

],

and multiply diag1, T−1

1 , I

from the right hand side and itstranspose from left to (23) in Box II, then

1β + αw2

[HDiCp2 HCi HDiDp21

] CT

p2HTDi

HTCi

DTp21H

TDi

diagPj,Λ

≥ 0,

i ∈ I[1,m], j ∈ I[0, 2m− 1]. (27)

Following standard procedure, it can be shown that condition (27)is equivalent to

(β + αw2)[HDiCp2 HCi HDiDp21

]diag

P−1(ρ),Λ−1

CTp2H

TDi

HTCi

DTp21H

TDi

≤ 1, i ∈ I[1,m].

The above inequality combined with (24) in Box II validatescondition (26). Therefore the saturated control system has itsrestricted `2 gain less than γ.

Theorem 4 provides an upper bound on the restricted `2 gainfor saturated control system subject to the class of disturbance(5). Condition (22) in Box II guarantees stability and performanceproperties of the closed-loop system, and conditions (23)–(24) inBox II are equivalent to the set inclusion relation in Lemma 2. Note

Q. Zheng, F. Wu / Systems & Control Letters 57 (2008) 896–903 901

V(x(k + 1)) − V(x(k)) +1γ2 z

T(k)z(k) − wT(k)w(k)

=

[x(k)w(k)

]T [AT(ρ(k))P(ρ(k + 1))A(ρ(k)) − P(ρ(k)) AT(ρ(k))P(ρ(k + 1))B(ρ(k))

BT(ρ(k))P(ρ(k + 1))A(ρ(k)) BT(ρ(k))P(ρ(k + 1))B(ρ(k))

]

+

1γ2 C

T(ρ(k))C(ρ(k))1γ2 C

T(ρ(k))D(ρ(k))

1γ2 D

T(ρ(k))C(ρ(k))1γ2 D

T(ρ(k))D(ρ(k)) − I

[x(k)w(k)

]

< 0

Box III.

Fig. 1. Comparison of domain of attraction using constant and parameter-dependent Lyapunov functions.

that the synthesis conditions (22)–(24) in Box II become a set ofLMIs for each fixed values of β and α. In order to obtain the globalminimum γ, we will conduct linear search over scalar variablesβ,α. Similar to the stabilization result, it can be shown thatTheorem4 always provides a smaller γ value (or better disturbanceattenuation) than constant Lyapunov function approach in [24].

5. Example

To demonstrate the advantages of the proposed controlapproach, we consider a discrete-time linear system (1) subject toinput saturation with

Ap =

[0.8 −0.90.8 0.9

], Bp1 =

[0.51.5

], Bp2 =

[3.8

−0.9

],

Cp1 =[0.1 0.2

], Dp11 = −0.3, Dp12 = −0.8,

Cp2 =[−0.5 2

], Dp21 = 1.

We first compare the domain of attractions predicted by constantand parameter-dependent Lyapunov functions using Theorem 1.In Fig. 1, the larger ellipsoids are the cross-section of the domain ofattraction derived from parameter-dependent Lyapunov functionwhich clearly contains the one achieved by a single quadraticLyapunov function.

Next, we set the magnitude bound of disturbance as 1.186, andrestrict the singular value of Λ no more than 0.1. Using Lemma 3,we obtained the maximal disturbance energy level βPD

max = 5.638.Therefore, any trajectory starting from the origin will stay withina bounded state region for the disturbances of magnitude lessthan 1.186 and energy less than 5.638. On the other hand, the

Table 1`2 gain comparison using constant and parameter-dependent Lyapunov functions

Method β `2 gain CPU time (s)

Constant LF 5.4969 0.8257 1.052Parameter-dependent LF 5.4969 0.7427 4.196

maximal disturbance energy calculated using a constant Lyapunovfunction [24] is βC

max = 5.607 < βPDmax.

Finally, we applied Theorem 4 to solve the disturbanceattenuation problem. Table 1 shows the comparison of restricted`2 gains using constant and parameter-dependent Lyapunovfunctions under the same disturbance energy level β =

5.4969 < βPDmax. It can be seen that parameter-dependent

Lyapunov functions render better disturbance attenuation (10%)than constant Lyapunov functions with a reasonable increase ofcomputational cost.

Moreover, the coefficient matrices of output feedback con-troller for parameter-dependent Lyapunov function case are

Ac0 =

[0.0026 −0.11371.7825 −2.0535

], Ac1 =

[0.6536 −1.10341.7006 −2.2212

],

Bc0 =

[−0.08371.4072

], Bc1 =

[0.36241.4948

],

Cc =[−0.2744 0.3765

], Dc = −0.0855,

and the auxiliary matrices HC and HD are given byHC =

[−0.1220 0.1734

], HD = −0.0176.

In the simulation, a disturbance trajectory with its energy of5.4969 is chosen as

w(k) =

1.17, 1 ≤ k < 40.0, k ≥ 4.

902 Q. Zheng, F. Wu / Systems & Control Letters 57 (2008) 896–903

(a) State trajectory. (b) Control input.

Fig. 2. State trajectory and control input profile.

(a) Bounded state region. (b) Truncated `2 gain.

Fig. 3. Bounded state region and truncated `2 gain.

Fig. 2 provides the state trajectory and control input under thisbounded disturbance. As can be seen, the control input saturates atstep 5 and the disturbance effect is quickly attenuated after step 15.

We have also plotted the Lyapunov function values andtruncated `2 gain over finite horizon in Fig. 3. Note that the value ofV(x(k)) is always less than the energy bound β, which implies thetrajectory of the saturated system starting from the origin remainswithin the bounded state regionD . The subplot 3(b) shows that thetruncated `2 gain from w to z over any finite time interval [0,N] isless than γ, as expected.

6. Conclusion

Using parameter-dependent Lyapunov functions, we havesolved output feedback stabilization and disturbance attenuationproblems for saturated discrete-time linear systems. The resultingoutput feedback controller is nonlinear in nature, and wasparameterized in quasi-LPV form. By introducing an intermediatematrix, we formulated the control synthesis conditions as LMI

optimization problems with finite number of LMI constraints. Theparameter-dependent Lyapunov approach improved our previouswork in [23,24] by enlarging the domain of attraction and reducingthe disturbance effect on the output.

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