Applied Mathematics and Computation 220 (2013) 382–393

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Applied Mathematics and Computation

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

Anti-windup-based dynamic controller synthesis for nonlinearsystems under input saturation

0096-3003/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.amc.2013.05.080

⇑ Corresponding author.E-mail addresses: rehanqau@gmail.com (M. Rehan), aqkhan@pieas.edu.pk (A.Q. Khan), mabid@pieas.edu.pk (M. Abid), naeem@pieas.edu.pk (N

bhussai1@uncc.edu (B. Hussain).

Muhammad Rehan a,⇑, Abdul Qayyum Khan a, Muhammad Abid a, Naeem Iqbal a, Babar Hussain b

a Department of Electrical Engineering, Pakistan Institute of Engineering and Applied Sciences (PIEAS), P.O. Box 45650, Islamabad, Pakistanb Department of Electrical and Computer Engineering, EPIC Building, University of North Carolina (UNC) at Charlotte, 9201 University City Blvd., Charlotte,NC 28223-0001, USA

a r t i c l e i n f o

Keywords:Input saturationH1 controllerAnti-windup compensatorLipschitz nonlinearitySector condition

a b s t r a c t

This paper describes the design of dynamic controller and static anti-windup compensator(AWC) for Lipschitz nonlinear systems under input saturation. Global and local AWC-basedcontrol schemes for stabilization of the nonlinear systems are proposed, and necessaryconditions for feasibility of the control approaches are investigated. A one-step approachfor simultaneous design of H1 controller and AWC by means of linear matrix inequalities(LMIs) is presented herein, which supports multi-objective synthesis to attain stabilizationor tracking, robustness against disturbance and noise, and penalization of large and highfrequency control signals. This multi-objective synthesis can be accomplished by incorpo-rating design weights, as commonly used in the standard H1 control theory, to design aperformance-oriented anti-windup-based control scheme. LMIs for the global control ofthe nonlinear systems subject to input saturation are derived by application of a quadraticLyapunov function, the Lipschitz condition, the global sector condition, L2 gain reduction,substantial matrix algebra and variable transformation. In order to cope with unstableand oscillatory nonlinear systems, LMI-based local results are established using a local sec-tor condition. Additional conditions are derived, by incorporating properties of the satura-tion function, to ensure well-posedness of the controller. Two simulation examples areprovided to show the effectiveness of the proposed control schemes for control of stableand chaotic nonlinear systems under input saturation.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

Every input of a linear or a nonlinear system is bounded by an upper and a lower limit owing to physical restriction ofactuators, which causes saturation of the control signal applied to achieve desired control objectives like stabilization, track-ing and disturbance rejection. Actuator saturation, usually ignored to simplify the design of a control system, causes perfor-mance degradation, lag, overshoot, undershoot as well as instability in the closed-loop response of practical systems due to awell-known phenomenon called windup effect [1–5]. Due to inherent complexity of nonlinear systems, actuator saturationis particularly neglected in a control law derivation, which can cause serious troubles to physical systems and their sur-roundings, such as damage, plant failure and accidents [6–7]. In order to overcome the effects of actuator saturation, ananti-windup compensator (AWC) is applied in addition to an output feedback controller [3–10].

. Iqbal),

M. Rehan et al. / Applied Mathematics and Computation 220 (2013) 382–393 383

One-step control approaches to deal with the windup effect and to ensure the closed-loop performance such asstability, tracking and disturbance rejection have been considered in the literature [11–15]. These schemes are utilizingthe knowledge of actuator nonlinearity, for output or state feedback controller synthesis, to achieve global or local stabilityand robust performance. Recently, a one-step approach for simultaneous linear controller and static AWC design of linearsystems, by extending the standard results of H1 control theory [16], has been proposed in the work [17] (see also Refs.[18–22]). Since both linear controller and AWC are simultaneously involved, in the design process, by incorporating theknowledge of actuator saturation, such one-step approaches for control of input constrained systems offer an attractivefeature of more clear multi-objective synthesis. However, the one-step synthesis approach requires rigorous efforts forcontroller design of different types of stable, unstable and chaotic nonlinear systems under input saturation owing todifficulties in mathematical derivations for simultaneous controller and AWC synthesis and due to in-built complexityof nonlinear systems.

Some remarkable exceptions on AWC synthesis for different types of nonlinear systems are available in the literature.A full order AWC design methodology, based on parametric tuning, is proposed in [23] for feedback linearizable Euler–Lagrange systems. This nonlinear AWC scheme guarantees stabilization of the overall closed-loop system; though, itlacks in attaining the performance objectives in the presence of input saturation. In order to keep the state of a feedbackcontroller in the presence of input saturation same as for the case of nominal closed-loop system (without inputsaturation), a dynamic linear AWC design scheme has been proposed in the work [24] for a specific class of feedbacklinearizable nonlinear systems. A preliminary analytical study on nonlinear decoupled-architecture-based AWC designhas been carried out in [25]. This contribution addressed the aforesaid problem using nonlinear matrix inequalitiesfor feedback linearizable asymptotically stable Lipschitz nonlinear systems with invertible dynamics. It is worthmentioning that the aforementioned studies on AWC design had proposed two-step control approaches for nonlinearsystems under actuator saturation, in which the design of a nominal feedback controller (using existing techniques)by ignoring input saturation is followed by incorporation of an AWC in the closed-loop system to compensate the effectsof saturation. Moreover, control of nonlinear systems under input saturation is an important and appealing researchissue owing to the bounded-input limitation for every system, which unfortunately could not get the desired researchattention due to complexity of the problem.

In this paper, simultaneous synthesis of dynamic controller and AWC, motivated by the results of multi-objective controlmethodology [16], linear AWC scheme [26] and linear one-step control approach [17], is studied for Lipschitz nonlinear sys-tems under input saturation. The proposed one-step techniques can be applied to a more general class of nonlinear systems,which are not necessarily stable, containing both actuator saturation and Lipschitz nonlinearities in contrast to the recentwork [17]. Design schemes for global and local output feedback stabilizing controllers and necessary conditions for feasibil-ity of the design constraints are provided. Further, a global anti-windup-based H1 controller design approach utilizing linearmatrix inequalities (LMIs) is developed by application of a quadratic Lyapunov function, the L2 gain reduction, the global sec-tor condition, considerable matrix algebra and enormous variable transformation. Furthermore, a one-step local H1 LMI-based control approach, to deal with nonlinear systems for which global design is not feasible, is derived by applicationof the local sector condition. With this local control approach, the acceptable bound on L2 norm of exogenous input canbe enlarged by means of an LMI-based optimization algorithm. Additional conditions are derived by employing propertiesof the saturation function to obtain a well-posed controller avoiding algebraic loops for numerical simulation and practicalimplementation. Numerical examples are provided to demonstrate the effectiveness of the proposed global and local anti-windup-based one-step control approaches.

In contrast to the aforementioned nonlinear AWC design schemes, the present work proposes a one-step approach fordesigning output feedback controller and AWC simultaneously. The key feature of the proposed one-step designapproach, over the two-step schemes for nonlinear systems under actuator saturation, is involvement of the bothcontroller and AWC, simultaneously, in synthesis process to achieve desired closed-loop performance objectives as well asprevention against windup effects. Therefore, the proposed one-step anti-windup-based control approach can assemblea better combination of feedback controller and AWC in terms of performance, robustness, stability, disturbancerejection and region of convergence. Consequently, the one-step approach for control of input constrained nonlinearsystems can offer an attractive feature of more clear multi-objective synthesis. Performance weights for control error,noise, disturbance, control signal, input and output, as in the standard H1 multi-objective synthesis, can be incorporatedto design a higher order performance-oriented anti-windup-based controller. For the local synthesis, this multi-objectivesynthesis can be advantageous in enlarging the tolerate-able bound on L2 norm of exogenous input because a higherorder controller, due to incorporation of proper design weights in the synthesis process, can have more degree offreedom to achieve the desired performance objectives. In addition, the present work deals with the design of acomputationally simple static AWC rather than a dynamic AWC. The proposed controller and AWC can be designedand implemented without any requirement of the exact information of nonlinearity in a plant in contrast to the tradi-tional nonlinear AWC methodologies. Further, our control schemes do not require measurement or estimation of thestate of a system in addition to measured output.

This paper is organized as follows. Section 2 presents the system description. Section 3 derives LMI-based conditions fordesigning global and local stabilizing controllers. Section 4 presents the global and local conditions for simultaneous designof H1 controller and static AWC. Section 5 provides the simulation results for the proposed control schemes. Section 6 drawsconclusions of the study.

384 M. Rehan et al. / Applied Mathematics and Computation 220 (2013) 382–393

Standard notation is used in this paper. The L2 gain from a vector d to another z is represented by supkdk2–0ðkzk2=kdk2Þ,

where k � k2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR1

0 k � k2dt

qand ||�|| denote the L2 norm and the Euclidian norm, respectively, for a vector. A symmetric po-

sitive-definite (or semi-positive-definite) matrix X is denoted by X > 0 (or X P 0). For a symmetric matrix X, He(X) = X + XT.The ith row of a matrix A is represented by A(i). For control signal u 2 Rmu , the saturated control input is defined asusat ¼ signðuðiÞÞminð�uðiÞ; juðiÞjÞ, where �uðiÞ > 0 denotes the ith bound on the saturation. Bold symbols are used to representthe decision variables.

2. System description

Consider a nonlinear system

dxp

dt¼ f ðt; xpÞ þ Apxp þ Bpusat þ Bww;

z ¼ Czxp þ Dzwwþ Dzusat

y ¼ Cyxp þ Dyww; ð1Þ

where xp e Rn, usat ¼ satðuÞ 2 Rmu , u 2 Rmu , w 2 Rmw y 2 Rpy , and z 2 Rpz represent the state, the saturated control input, thecontrol input, the exogenous input (reference r, disturbance d, and noise N), the output, and the exogenous output vectors,respectively. The initial condition is assumed to be xp(0) = 0. The function f(t,xp) e Rn represents a time-varying nonlinearvector satisfying f(0,0) = 0. Note that the condition f(0,0) = 0 is usually required to develop LMI-based results. For a specificcase, this condition can be relaxed (to be discussed later).

Assumption 1. The nonlinearity f(t,x), for all x; �x 2 Rn, satisfies the Lipschitz condition given by

kf ðt; xÞ � f ðt; �xÞk 6 kLðx� �xÞk ð2Þ

where L represents the Lipschitz constant matrix of appropriate dimensions.The output feedback controller, along with the static AWC, is given by

_xc ¼ Acxc þ Bcyþ h1ðusat � uÞ;u ¼ Ccxc þ Dcyþ h2ðusat � uÞ; ð3Þ

where xc 2 Rq is the controller state vector and h1 2 Rq�mu and h2 2 Rmu�mu are components of the static AWC. The initial con-dition of the controller can be taken as xc(0) = 0. Throughout the paper, we assume that q = n. Defining v ¼ u� usat ,n ¼ �hv ¼ ½hT

1 hT2�

Tv and xcl ¼ ½xTp xT

c �T for xclð0Þ ¼ 0, the overall-closed loop system can be written as

dxcl

dt¼ �f ðt; xclÞ þ Axcl þ Bwwþ ðBv � BnhÞv ;

u ¼ Cuxcl þ Duww� Dunhv ;z ¼ Czxcl þ Dzwwþ ðDzv � DznhÞv ; ð4Þ

�f ðt; xclÞ ¼f ðt; xpÞ

0

� �; A ¼

Ap þ BpDcCy BpCc

BcCy Ac

� �; ð5Þ

Bw ¼Bw þ BpDcDyw

BcDyw

� �; Bv ¼

�Bp

0

� �; Bn ¼

0 Bp

I 0

� �ð6Þ

Cu ¼ ½DcCy Cc�; Duw ¼ ½DcDyw�; Dun ¼ ½0 I�; Cz ¼ ½Cz þ DzDcCy DzCc� ð7Þ

Dzw ¼ ½Dzw þ DzDcDyw�; Dzv ¼ ½�Dz�; Dzn ¼ ½0 Dz�; h ¼ ½hT1 hT

2�T ð8Þ

For a diagonal positive-definite matrix W 2 Rmu�mu , saturation nonlinearity satisfies the classical global sector condition[4,8] given by

vT W½u� v �P 0 ð9Þ

This sector condition can be used to obtain global results for asymptotically stable systems. If global results cannot beobtained, a local sector condition such as [6,8,27,28] can be used to establish local results. For a region

Sð�uÞ ¼ x 2 Rmu ;��u 6 u�x 6 �u� �

; ð10Þ

where �u 2 Rmu represents the saturation bound, the local sector condition

vT W½x� v �P 0 ð11Þ

remains valid (see [6,27]).

M. Rehan et al. / Applied Mathematics and Computation 220 (2013) 382–393 385

3. Stabilizing controller and AWC design

This section addresses the synthesis of AWC-based global and local stabilizing controllers and derives necessary condi-tions for existence of the controllers. First, we provide the design of a global controller (3) for a nonlinear system (1) underinput saturation.

Theorem 1. Consider the plant (1) under w = 0 and let Assumption 1 be satisfied.

(a) Suppose there exist symmetric matrices Q 11 2 Rn�n, P11 2 Rn�n, and P12 2 Rn�n and matrices A 2 Rn�n, B 2 Rn�py , C 2 Rmu�n,D 2 Rmu�py , h1 2 Rn�mu , and h2 2 Rmu�mu , such that the LMIs

Q 11 I

� P11

� �> 0; P12 > 0; ð12Þ

He

ApQ 11 þ BpC Ap þ BpDCy I 0 �Bph2 0

A P11Ap þ BCy PT11 I �h1 0

0 0 � 12 I 0 0 0

0 0 0 � 12 P12 0 0

C DCy 0 0 �h2 0LQ 11 L 0 0 0 � 1

2 I

266666666664

377777777775< 0 ð13Þ

are satisfied, then there exists a controller of the form (3) with q = n such that the closed-loop system is globally asymptoticallystable. A controller satisfying the above can be constructed by

P12 ¼ ðP�112 Þ

2;

Q 12PT12 ¼ I � Q 11P11;

D ¼ Dc;

C ¼ CcQ T12 þ DcCyQ11;

B ¼ P12Bc þ P11BpDc;

A ¼ P12AcQT12 þ P12BcCyQ 11

þP11BpCcQ T12 þ P11ðAp þ BpDcCyÞQ 11;

h1 ¼ P12h1U þ P11Bpð1þ h2ÞU;h2 ¼ ð1þ h2ÞU;

9>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>;

ð14Þ

for any positive-definite diagonal matrix U 2 Rnu�nu .

(b) A necessary condition for existence of solution in Theorem 1(a) is that there exists a symmetric positive-definite matrix Q11,such that the LMI

Q 11ATp þ ApQ 11 I Q 11LT

� �I 0� � �I

264

375 < 0 ð15Þ

is satisfied.

Proof. Consider a quadratic Lyapunov function given by

Vðxcl; tÞ ¼ xTclPxcl; ð16Þ

with P ¼ PT > 0 and Q = P�1. Partitioning matrix P according to xcl ¼ ½xTp xT

c �T such that

P ¼P11 P12

PT12 P22

� �; Q ¼

Q 11 Q12

Q T12 Q22

" #; ð17Þ

P1 ¼Q 11 I

Q T12 0

" #; P2 ¼

I P11

0 PT12

� �ð18Þ

386 M. Rehan et al. / Applied Mathematics and Computation 220 (2013) 382–393

Note that PP1 = P2. For stability, _Vðxcl; tÞ < 0 is required. Using Assumption 1 and (9), it can be verified that _Vðxcl; tÞ < 0remains valid if

_V þ vT Wðu� vÞ þ ðu� vÞT Wv � �f T�f þ xTclM

T Mxcl < 0; ð19Þ

where M ¼L 00 0

� �ð20Þ

Taking the derivative of (16) along (4) and, further, using the resultant, w = 0 and (4) into (19) yields ZT1U1Z1 < 0, where

Z1 ¼ xTcl

�f Tðt; xclÞ vT� �T ð21Þ

U1 ¼AT P þ PAþMT M P PðBv � BnhÞ þ CT

uW

� �I 0� � �2W �WDunh� ðWDunhÞ

T

264

375 < 0 ð22Þ

Applying Schur complement and congruence transform with diagðP1; I; I; I;U; IÞ to (22), where U = W�1, and putting var-ious matrices from (5)–(8), (14), (17), and (18), the following LMI is obtained:

He

ApQ 11 þ BpC Ap þ BpDCy I 0 �Bph2 0

A P11Ap þ BCy PT11 P12 �h1 0

0 0 � 12 I 0 0 0

0 0 0 � 12 I 0 0

C DCy 0 0 �h2 0LQ 11 L 0 0 0 � 1

2 I

26666666664

37777777775< 0 ð23Þ

Applying congruence transform, diagðI; I; P�112 ; I; I; IÞ, to LMI (23) and using P12 ¼ ðP�1

12 Þ2

into the resultant, we obtain LMI(13). LMI (12) is obtained by applying congruence transform, P1, to P > 0 and using (17), (18) into the resultant. It completesthe proof of statement (a) in Theorem 1.

Applying the congruence transform [29–30], diag(Q, I,U), to (22) and using the same steps as in [17] for matrixelimination, we obtain

wTU2w < 0; fTU2f < 0 ð24Þ

U2 ¼QAT þ AQ þ QMT MQ I BvU þ QCT

u

� �I 0� � �2U

264

375; w ¼

w1

w2

0

264

375; f ¼

f11

0f2

�Bpf11

26664

37775 ð25Þ

Using wTU2w < 0 and (25), we obtain" #

QAT þ AQ þ QMT MQ I

� �I< 0 ð26Þ

Using fTU2f < 0, (5)–(8), (14), (17), (20), and (25), we obtain

Q 11ATp þ ApQ 11 þ Q 11LT LQ 11 � BpC � CT BT

p I

� �I

" #< 0 ð27Þ

It can be easily verified by using matrix algebra and Eqs. (5), (14), (20), (26), and (27) that the condition" #

Q 11AT

p þ ApQ 11 þ Q 11LT LQ 11 I

� �I< 0 ð28Þ

is necessary for validation of (26), (27). Applying Schur complement to (28) gives (15), which completes the proof of state-ment (b) in Theorem 1. h

The LMIs in Theorem 1(b) offer a sufficient condition for asymptotic stability of a Lipschitz nonlinear system (1) (see, forexample, [31]). Note that the LMI-based condition in Theorem 1(b) is necessary to design a controller through Theorem 1(a).An asymptotically stable plant (1) may not satisfy the condition in Theorem 1(b); therefore, global results are not guaranteedby Theorem 1(a). Moreover, global results for unstable plants are not guaranteed through Theorem 1(a) due to infeasibility ofLMIs in Theorem 1(b). Consequently, the global controller and anti-windup compensator design schemes are not applicableto unstable, oscillatory and chaotic systems due to the infeasibility problem. Therefore, local anti-windup and controller de-sign scheme is addressed by application of the local sector condition (10), (11) by selecting x as

M. Rehan et al. / Applied Mathematics and Computation 220 (2013) 382–393 387

x ¼ ~Cuxcl þ Duww� Dunhv; ð29Þ

where ~Cu ¼ ½~DcCy~Cc� ð30Þ

Using (4), (10), (11), and (29), we obtain

Sð�uÞ ¼ w 2 Rmu ;��u 6 Cu � ~Cu

xcl 6 �u

n oð31Þ

vT W ~Cuxcl þ Duww� Dunhv � vh i

P 0 ð32Þ

Now we provide an LMI-based design condition, using sector condition (31), (32), for local stabilization of plant (1)through the following theorem.

Theorem 2. Consider the plant (1) and let Assumption 1 be satisfied.

(a) Suppose there exist symmetric matrices Q 11 2 Rn�n, P11 2 Rn�n, and P12 2 Rn�n and matrices A 2 Rn�n, B 2 Rn�py , C 2 Rmu�n,~C 2 Rmu�n, D 2 Rmu�py , ~D 2 Rmu�py , h1 2 Rn�mu , and h2 2 Rmu�mu , such that the LMIs (12),

Q 11 I CTðiÞ � ~CT

ðiÞ

� P11 ðCTyÞðiÞðD

T � ~DT� � �u2

ðiÞ

2664

3775P 0; i ¼ 1; . . . ;m ð33Þ

He

ApQ 11 þ BpC Ap þ BpDCy I 0 �Bph2 0A P11Ap þ BCy PT

11 I �h1 00 0 � 1

2 I 0 0 00 0 0 � 1

2 P12 0 0~C ~DCy 0 0 �h2 0

LQ 11 L 0 0 0 � 12 I

266666664

377777775< 0 ð34Þ

are satisfied, then there exists a controller of the form (3) with q = n such that the closed-loop system is locally asymptotically stable

for all initial conditions belonging to xTpð0ÞQ

�1xpð0Þ 6 1. A controller satisfying the above and the matrices ~Cc , ~Dc , and Q for can beconstructed by (14) and

~D ¼ ~Dc;~C ¼ ~CcQ T

12 þ ~DcCyQ 11;Q 22 > 0:

9=; ð35Þ

for any positive-definite diagonal matrix U 2 Rnu�nu .

(b) A necessary condition for existence of solution in Theorem 2(a) is that there exist a symmetric positive-definite matrix Q 11

and a matrix, c 2 Rmu�n such that the LMI

Q 11ATp þ ApQ 11 � Bpc� cT BT

p I Q 11LT

� �I 0� � �I

24

35 < 0 ð36Þ

is satisfied.

Proof. Consider a quadratic Lyapunov function (16). By ensuring _Vðxcl; tÞ < 0, asymptotic stability of the closed-loop system(4) can be achieved. It implies Vðxcl; tÞ 6 Vðxcl;0Þ for any time t P 0, which further implies that xT

clðtÞQ�1xclðtÞ 6

xTclð0ÞQ

�1xclð0Þ 6 1, where Q ¼ P�1. By including xTclðtÞQ

�1xclðtÞ 6 1 into Sð�uÞ, the region (31) and the condition (32) are satis-fied. It reveals that

P CTu � ~CT

u

ðiÞ

� �u2ðiÞ

24

35P 0; i ¼ 1; . . . ;m ð37Þ

The LMI (33) is derived by applying congruence transform, diagðP1; IÞ, to (37) and, further, using (7), (17), (18), and (30).The LMI (34) is obtained, using (14), (32), and (35), by applying the same steps as for the proof of Theorem 1(a). The LMI (36)is obtained by applying transform c ¼ C � ~C and using same steps as for the proof of Theorem 1(b). h

388 M. Rehan et al. / Applied Mathematics and Computation 220 (2013) 382–393

Remark 1. The traditional control approaches for nonlinear systems subject to input saturation such as [23–25] are based ondesigning a dynamic AWC, which may utilize the dynamics of plant for implementation. However, the present scenario sup-ports the idea of static AWC synthesis, which reduces the computational complexity and does not require plant dynamics forimplementation.

Remark 2. Because of the additional term �Bpc� cT BTp , the condition (36) can be satisfied either by asymptotically stable,

oscillatory or unstable systems (not validating Theorem 1(b)); therefore, the proposed local stabilization methodologycan be applied to stable as well as oscillatory, chaotic and unstable nonlinear systems.

Remark 3. The traditional two-step control approaches (for example, [8,10,23]) for unstable systems under actuator satu-ration incorporate a local AWC into the closed-loop system containing a global output feedback controller. However, the pro-posed technique by Theorems 2 guarantees both the controller and the anti-windup for a desired locality, which can besuitable for closed-loop performance.

4. H‘ controller and AWC design

This section derives LMI-based global and local conditions for designing H1 controller and AWC, simultaneously, for anonlinear system (1). The global design, by means of the classical sector condition (9), is provided in Theorem 3.

Theorem 3. Consider the plant (1) and let Assumption 1 be satisfied. Suppose there exist a real scalar r > 0, symmetric matrices

Q 11 2 Rn�n, P11 2 Rn�n and P12 2 Rn�n, and matrices A 2 Rn�n, B 2 Rn�py , C 2 Rmu�n, D 2 Rmu�py , h1 2 Rn�mu and h2 2 Rmu�mu , suchthat the LMIs (12) and

He

ApQ 11 þ BpC Ap þ BpDCy I 0 Bw þ BpDDyw 0 �Bph2 0

A P11Ap þ BCy PT11 I P11Bw þ BDyw 0 �h1 0

0 0 � 12 I 0 0 0 0 0

0 0 0 � 12 P12 0 0 0 0

0 0 0 0 � 12 I 0 DT

ywDT 0

CzQ 11 þ DzC Cz þ DzDCy 0 0 Dzw þ DzDDyw � 12 rI �Dzh2 0

C DCy 0 0 0 0 �h2 0LQ 11 L 0 0 0 0 0 � 1

2 I

26666666666666664

37777777777777775

< 0 ð38Þ

are satisfied, then there exists a controller of the form (3) with q = n such that

(i) the closed-loop system is globally asymptotically stable, if w = 0,(ii) the L2 gain from w to z is less than c ¼

ffiffiffiffirp

.

A controller satisfying the above can be constructed by (14) for any positive-definite diagonal matrix U 2 Rnu�nu .

Proof. Consider a quadratic Lyapunov function given by (16). Defining

J1 ¼ _Vðxcl; tÞ þ r�1zT z�wT w < 0; ð39Þ

for a scalar r > 0. Integrating (39) from 0 to T !1 reveals

Z T

0J1dt ¼ ðVðxcl; TÞ � Vðxcl;0ÞÞ þ r�1

Z T

0zT zdt � c2

Z T

0wT wdt < 0; ð40Þ

which implies

(a) if w = 0, (39) implies that _V < 0, i.e., the nonlinear closed-loop system is asymptotically stable;(b) if xclð0Þ ¼ 0, then Vðxcl; 0Þ ¼ 0, which along with Vðxcl; TÞ > 0 and (40) implies kzk2

2 < c2kwk22 (because r ¼ c2).

By applying the global sector condition (9) and Assumption 1, it can be easily verified that the inequality J1 < 0 remainsvalid if

J2 ¼ _V þ r�1zT z�wT wþ vT Wðu� vÞ þ ðu� vÞT Wv � �f T�f þ xTclM

T Mxcl < 0 ð41Þ

M. Rehan et al. / Applied Mathematics and Computation 220 (2013) 382–393 389

Taking the derivative of (16) along (4) and, further, using the resultant and (4) into (41) yields J2 ¼ ZT2U3Z2 < 0, where

Z2 ¼ xTcl

�f Tðt; xclÞ wT vT� �T

; ð42Þ

U3 ¼

AT P þ PAþMT M þ r�1CTz Cz P PBw þ r�1CT

z DTzw PðBv � BnhÞ þ CT

uW þ r�1CTz ðDzv � DznhÞ

� �I 0 0� � �I þ r�1DT

zwDzw r�1ðDzv � DznhÞTDzw þ DT

uwW

� � �r�1ðDzv � DznhÞ

TðDzv � DznhÞn�2W �WDunh� ðWDunhÞ

To

26666664

37777775< 0: ð43Þ

Applying Schur complement and conguerence transform, diag I; I; I; 0 II 0

� �� �, to the matrix inequality (43) reveals

AT P þ PAþMT M P PBw CTz PðBv � BnhÞ þ CT

uW� �I 0 0 0� � �I DT

zw DTuwW

� � � �rI ðDzv � DznhÞ� � � � �2W �WDunh� ðWDunhÞ

T

2666664

3777775 < 0 ð44Þ

Again applying Schur complement and congruence transform with diagðP1; I; I; I;U; IÞ to (44) and using (5)–(8) and (17),(18), followed by a successive congruence transform with diagðI; I; I; P�1

12 ; I; I; I; I; IÞ, the LMI (38) is obtained, which completesthe proof of Theorem 3. h

Now an LMI-based sufficient condition for designing a local anti-windup-based H1 output feedback controller isprovided.

Theorem 4. Consider the plant (1) and let Assumption 1 be satisfied. Suppose there exist a real scalar r > 0, symmetric matricesQ 11 2 Rn�n, P11 2 Rn�n, P12 2 Rn�n and Q22 2 Rn�n, and matrices A 2 Rn�n, B 2 Rn�py , C 2 Rmu�n, ~C 2 Rmu�n, D 2 Rmu�py ,~D 2 Rmu�py , h1 2 Rn�mu and h2 2 Rmu�mu , such that the LMIs (12), Q22 > 0,

Q 11 I CTðiÞ � ~CT

ðiÞ

� P11 CTy

ðiÞ

DT � ~DT

� � d�u2ðiÞ

2664

3775P 0; i ¼ 1; . . . ;m ð45Þ

He

ApQ 11 þ BpC Ap þ BpDCy I 0 Bw þ BpDDyw 0 �Bph2 0

A P11Ap þ BCy PT11 I P11Bw þ BDyw 0 �h1 0

0 0 � 12 I 0 0 0 0 0

0 0 0 � 12 P12 0 0 0 0

0 0 0 0 � 12 I 0 DT

ywDT 0

CzQ 11 þ DzC Cz þ DzDCy 0 0 Dzw þ DzDDyw � 12 rI �Dzh2 0

~C ~DCy 0 0 0 0 �h2 0LQ 11 L 0 0 0 0 0 � 1

2 I

26666666666666664

37777777777777775

< 0 ð46Þ

are satisfied, where d�1 is the acceptable bound on the L2 norm of w, then there exists a controller of the form (3) with q = n suchthat

(i). the nonlinear closed-loop is locally asymptotically stable with region of convergence xTclðtÞdQ�1xclðtÞ < 1;8t > 0, if w = 0,

(ii). the state of the closed-loop system remains bounded by xTclðtÞdQ�1xclðtÞ < 1;8t > 0, and the L2 gain from w to z is less than

c ¼ffiffiffiffirp

.

A controller satisfying the above and the matrices ~Cc ,~Dc and Q can be constructed by (14) and (35) for any positive-definitediagonal matrix U 2 Rnu�nu .

Proof. It has been already seen in the proof of Theorem 1 that the inequality (39) ensures asymptotic stability of the closed-loop system under w = 0 and the L2 gain from w to z less than c ¼

ffiffiffiffirp

for all time. Further, it can be derived from (40) that

Vðxcl; TÞ < kwk22 6 d�1; 8T > 0; because V(xcl,0) = 0 for xcl(0) = 0 and r�1

R T0 zT zdt P 0. Therefore, the state xcl of the nonlinear

closed-loop system remains bounded by xTcldPxcl < 1 for all time. By including the region xT

clðtÞdPxclðtÞ < 1 into Sð�uÞ, applyingcongruence transform with diagðP1; IÞ to the resultant and, further, using (7), (16), (17), and (27), the LMI (45) is obtained. It

390 M. Rehan et al. / Applied Mathematics and Computation 220 (2013) 382–393

implies that (31), (32) remains valid. Hence, by virtue of the local sector condition, (11), (32), under Assumption 1, theinequality J1 < 0 is satisfied if

J3 ¼ _V þ r�1zT z�wT wþ vT Wðw� vÞ þ ðw� vÞT Wv � �f T�f þ xTclM

T Mxcl < 0 ð47Þ

By applying the same steps as for the proof of Theorem 3, the LMIs (46) is obtained, which completes the proof ofTheorem 4. h

Remark 4. In the traditional two-step approaches [23–25], an unconstrained optimal controller, designed in the absence of satu-ration, does not remain optimal when control signal is saturated. In contrast, an optimal combination of global controller and AWC,to ensure closed-loop performance as well as to deal with saturated control signal, can be obtained through Theorem 3 by minimiz-ing the parameter c. To obtain semi-optimal results through LMIs in Theorem 4, either the L2 gainc from w to z can be minimized fora fixed value of d or the tolerate-able bound on L2 norm of w can be enlarged by minimizing d for a fixed value of c. In contrast to theconventional two-step control approaches, the proposed less conservative one-step approach applies optimization of performanceobjectives to determine both controller and AWC parameterizations at the same time.

Remark 5. By means of multi-objective synthesis, the proposed one-step schemes by Theorems 3–4 can be used to design arobust stabilization or tracking controller against noise and/or disturbance in contrast to the conventional two-stepapproaches for nonlinear systems under input saturation. Similar to the standard multi-objective techniques for H1 control-ler synthesis, frequency-dependent design weights for control error, control signal, exogenous input, and exogenous output,as seen in the literature [16,19], can be used to attain desired performance.

Remark 6. It is interesting to note that the simultaneous H1 controller and anti-windup design scheme utilizing LMIs, for-merly developed in the literature [17], can be expanded to nonlinear systems by applying the properties of Lipschitz non-linearity [31]. In contrast to [17], the local LMI-based results, for control of unstable, oscillatory and chaotic systemsunder input saturation, are developed by application of the local sector condition in [6,27]. The traditional controlapproaches for nonlinear systems subject to input saturation (such as [23–25]) are based on designing a dynamic AWC,which may utilize the dynamics of a plant for implementation. However, the present scenario supports the idea of staticAWC synthesis, which reduces the computational complexity and does not require plant dynamics.

In addition to stability and performance of the closed-loop system, another important issue is the well-posedness of thecontroller (3). This issue can be addressed by considering the fact that every entry of the saturated control signal usat, at anyinstant of time, is equal to the corresponding entry of either control signal u or saturation bound ��u. Consequently, satura-tion function satisfies

usat ¼ gjuþ ðI � gjÞXk�u 8j ¼ 1;2; . . . ;2mu ; k ¼ 1;2; . . . ;2mu ; ð48Þ

where gj and Xk are the entries of the sets

g ¼ fji ¼ 0 or ji ¼ 1 8i ¼ 1;2; . . . ;mu; diagðj1;j2; . . . ;jmuÞg 8j ¼ 1;2; . . . ;2mu ; ð49Þ

X ¼ fmi ¼ �1 or mi ¼ 1 8i ¼ 1;2; . . . ;mu; diagðm1; m2; . . . ; mmuÞg 8k ¼ 1;2; . . . ;2mu ð50Þ

For example, if mu = 1 then g ¼ f0;1g and X ¼ f�1;1g. Correspondingly, usat is equal to either u or ��u at any instant oftime. Using (3) and (48), we get

ðI þ h2 � h2gjÞu ¼ Ccxc þ Dcyþ h2ðI � gjÞXk�u 8j ¼ 1;2; . . . ;2mu ; k ¼ 1;2; . . . ;2mu ð51Þ

It is clear from (51) that a unique solution to control signal u always exists, if the matrix ðI þ h2 � h2gjÞ 8j ¼ 1;2; . . . ;2mu isinvertible. One method to ensure well-posedness of the controller is to verify this condition from the solution of (14). Thecomputation of h2 depends on U (because h2 ¼ ð1þ h2ÞU), the selection of U can play an important role to ensureðI þ h2 � h2gjÞ; 8j ¼ 1;2; . . . ;2mu ; invertible. It is notable that the matrix ðI þ h2 � h2gjÞ is always invertible if

ðI þ h2 � h2gjÞ > 0 ð52Þ

Using (52) and h2 ¼ ðI þ h2ÞU, we obtain

h2U�1 � ðh2U�1 � IÞgj > 0; 8j ¼ 1;2; . . . ;2mu ð53Þ

The second method to guarantee well-posedness of the controller can be to introduce the additional inequality (53) inTheorems 1–4. The matrix inequality (53) can be treated as an LMI for a selection of U.

Remark 7. It was assumed in H1 controller synthesis that the nonlinearity in system (1) satisfies f(0,0) = 0. If this conditionis not verified, an H1 can be designed for a specific case. Suppose there exists a parameter b of appropriate dimensionssatisfying Bpbf ð0;0Þ ¼ f ð0;0Þ and bf ð0;0Þ 6 �u, then by changing the control law (3) to

_xc ¼ Acxc þ Bcyþ h1ðusat � uÞ;u ¼ �bf ð0;0Þ þ Ccxc þ Dcyþ h2ðusat � uÞ; ð54Þ

M. Rehan et al. / Applied Mathematics and Computation 220 (2013) 382–393 391

and by replacing LMI (45) in Theorem 4 with

Q 11 I CTðiÞ � ~CT

ðiÞ

� P11 ðCTyÞðiÞðD

T � ~DTÞ

� � d min �uðiÞ � bf ð0;0ÞðiÞ 2

; �uðiÞ þ bf ð0;0ÞðiÞ 2

� � �26664

37775P 0; i ¼ 1; . . . ;m ð55Þ

a local H1 controller with AWC can be designed. The LMI (55) can be derived by changing the region (31) to��u 6 ðCu � ~CuÞxcl � bf ð0;0Þ 6 �u and using the same procedure as for the proof of Theorem 4.

5. Simulation results

Controller synthesis for complex nonlinear networks and chaotic systems is an appealing research area owing to its appli-cations in science and engineering and due to intricate nature of the problem [31–36]. We consider two simulation examplesto verify the proposed methodologies for control of nonlinear and chaotic systems subject to input saturation.

Example 1. Consider a nonlinear system (1) having dynamics given by

Ap ¼�5:3 0:1 0:2�0:12 �5:4 0:2

0 0:15 �5:2

264

375; Bp ¼

100

264

375; Bw ¼

000

264

375; f ðt; xÞ ¼

0:1xp1 sin 5t

0:1 tanh xp2

0:1jxp3j

264

375;

Cy ¼ ½�5 0 0�; Cz ¼ 0:01Cy; Dz ¼ 0; Dyw ¼ 1; Dzw ¼ 0:01Dyw

The objective is to design a tracking controller such that the actual output yo = xp1 tracks the reference signal w = r. Bysolving the LMIs in Theorem 3 for feasibility, the parameters of controller and AWC are obtained as

Ac ¼�0:0063 0:208 0:111�0:0754 �9:764 0:767

0:038 0:719 �8:912

264

375; Bc ¼

�2:261�0:0140:108

264

375; h1 ¼

�8:81�0:0310:098

264

375;

Cc ¼ ½�0:647 � 0:018 0:043�; Dc ¼ �0:043; h2 ¼ 0:427;

U = 10, c = 1.36, and L = 0.1. Fig. 1 shows the closed-loop responses of the nonlinear system for different scenarios. Theclosed-loop response without saturation is following the reference with a reasonable performance. The closed-loop responseunder input saturation by using the proposed static anti-windup-based dynamic controller is tracking the reference signal;however, the tracking error is higher than the unconstrained closed-loop system due to saturation. The closed-loop responseunder input saturation by using the proposed controller without anti-windup (by taking h1 = h2 = 0) is unable to track thereference signal due to lag. In the presence of saturation, the controller without AWC is not sufficient for reference tracking.Hence, the proposed combination of H1 controller and static AWC offers closed-loop performance and reference trackingunder both saturated and unsaturated control inputs.

0 5 10 15 20 25 30

-1

-0.5

0

0.5

1

1.5

2

Time (s)

Out

put

Reference signalNominal closed-loop responseClosed-loop response under input saturation (without AWC)Closed-loop response with the proposed controller and AWC

Fig. 1. Closed-loop response of the proposed control methodology for the input-constrained system in Example 1.

0 5 10 15 20 25 30 35 40-15

-10

-5

0

5

10

15

20

Time (s)

Out

put y

o (V)

Reference signalClosed-loop response under input saturation (without AWC)Closed-loop response with the proposed controller and AWC

Fig. 2. Control of chaotic Chua’s circuit under input saturation using the proposed anti-windup-based control scheme.

392 M. Rehan et al. / Applied Mathematics and Computation 220 (2013) 382–393

Example 2. Consider a third-order chaotic Chua’s circuit [31] with performance weights, given by

Ap ¼

�2:548 9:1 0 01 �1 1 00 �14:2 0 0�1 0 0 �100

26664

37775; Bp ¼

1000

26664

37775; Bw ¼

0 10 00 01 0

26664

37775;

f ðt; xÞ ¼ 9:11

jxpð1Þ þ 1j � jxpð1Þ � 1j000

26664

37775; Cy ¼

�1000

26664

37775

T

; Cz ¼

�1001

26664

37775

T

;

Dz ¼ 0; Dzw ¼ Dyw ¼ ½1 0�

The aim of the present study is to design a multi-objective robust controller and AWC for reference tracking and distur-

bance rejection. Therefore, the exogenous input can be written as w ¼ ½r d�T , where r is a desired reference for tracking ofthe output yo = xp1 and d is a time-varying disturbance. Note that the first three states represent the dynamics of the third-order Chua’s circuit, while the remaining state is used for a first order performance weight for tracking error. The otherweights for reference and disturbance are selected as unity for the sake of simplicity. By solving the LMIs in Theorem 4,the controller parameters are obtained as

Ac ¼

�597:98 �516:34 �168:34 0:174237:56 207:81 95:49 �0:079�10:12 �10:12 �4:74 0:0040:467 0:149 0:392 �299:78

26664

37775;

Bc ¼ 106

6:3� 106

13:022:07

�3:17� 106

26664

37775; Cc ¼

0:00180:00160:0005

0

26664

37775

T

; h1 ¼

�7:0853:267�0:126�0:002

26664

37775

Dc ¼ 0; h2 ¼ �1; c ¼ 17:32:

The disturbance was selected as d = 2 sin 20t. The closed-loop response of the Chua’s circuit with the proposed controllerand AWC under saturation and disturbance is tracking the reference signal as shown in Fig. 2. In contrast, the constrainedclosed-loop response with controller without AWC is not following the reference. Hence, the proposed anti-windup-basedH1 control methodologies can be used to acquire multiple control objectives, like stability, tracking, robustness, disturbancerejection and noise handling, with the help of design weights for nonlinear systems under input saturation.

M. Rehan et al. / Applied Mathematics and Computation 220 (2013) 382–393 393

6. Conclusions

In this paper, simultaneous dynamic controller and anti-windup compensator design methodologies have been devel-oped for a class of nonlinear systems with Lipschitz nonlinearities under actuator saturation. Separate LMI-based sufficientconditions were derived to design a combination of controller and AWC, for local and global stability, by utilizing a quadraticLyapunov function, Lipschitz condition, L2 gain reduction, and global and local sector conditions. The proposed one-step ap-proaches can be used for a more clear multi-objective synthesis, through which a set of controller and AWC can be designedto ensure performance objectives and to penalize windup effects. To overcome the infeasibility problems, a higher order con-troller with large size AWC can be designed, for a desired closed-loop performance, by incorporating the design weights forexogenous inputs (reference, disturbance and/or noise etc.) and exogenous outputs (tracking error, control signal and/or out-put etc.). Simulation results have been provided for a stable nonlinear system and a chaotic Chua’s circuit subject to actuatorsaturation, which shows that the proposed anti-windup-based H1 control strategy has satisfactory performance.

References

[1] C. Edwards, I. Postlethwaite, An anti-windup scheme with closed-loop stability considerations, Automatica 35 (1999) 761–765.[2] P.F. Weston, I. Postlethwaite, Linear conditioning for systems containing saturating actuators, Automatica 36 (2000) 1347–1354.[3] A. Zheng, M.V. Kothare, M. Morari, Anti-windup design for internal model control, Int. J. Control 60 (1994) 1015–1024.[4] M. Rehan, A. Ahmed, N. Iqbal, Static and low order anti-windup synthesis for cascade control systems with actuator saturation: an application to

temperature-based process control, ISA Trans. 49 (2010) 293–301.[5] S. Tarbouriech, T. Loquen, C. Prieur, Anti-windup strategy for reset control systems, Int. J. Robust Nonlinear Control 21 (2011) 1159–1177.[6] S. Tarbouriech, M.C. Turner, Anti-windup design: an overview of some recent advances and open problems, IET Control Theory A. 3 (2009) 1–19.[7] T.A. Kendi, F.J. Doyle III, An anti-windup scheme for multivariable nonlinear systems, J. Process Control 7 (1997) 329–343.[8] M.C. Turner, G. Hermann, I. Postlethwaite, Incorporating robustness requirements into anti-windup design, IEEE Trans. Automat. Control 52 (2007)

1842–1854.[9] C. Roos, J.M. Biannic, A convex characterization of dynamically-constrained anti-windup controllers, Automatica 44 (2008) 2449–2452.

[10] S. Tarbouriech, J.M. Gomes-Da-Silva Jr, G. Garcia, Delay-dependent anti-windup loops for enlarging the stability region of time-delay systems withsaturating inputs, J. Dyn. Syst-T. ASME 125 (2003) 265–267.

[11] D. Henrion, S. Tarbouriech, G. Garcia, Output feedback robust stabilization of uncertain linear systems with saturating controls: an LMI approach, IEEETrans. Automat. Control 44 (1999) 2230–2237.

[12] D. Henrion, S. Tarbouriech, LMI relaxations for robust stability of linear systems with saturating controls, Automatica 35 (1999) 1599–1604.[13] T. Nguyen, F. Jabbari, Output feedback controller for disturbance attenuation with actuator amplitude and rate saturation, Automatica 36 (2000) 1339–

1346.[14] I.E. Kose, F. Jabbari, Scheduled controllers for linear systems with saturating actuator, Automatica 39 (2003) 1377–1387.[15] S.M. Lee, Augmented Lyapunov function approach to gain analysis for discrete-time systems with saturation nonlinearities, Appl. Math. Comput. 217

(2011) 10205–10212.[16] C. Scherer, P. Gahinet, M. Chilali, Multiobjective output-feedback control via LMI optimization, IEEE Trans. Automat. Control 42 (1997) 896–911.[17] E.F. Mulder, P.Y. Tiwari, M.V. Kothare, Simultaneous linear and anti-windup controller synthesis using multiobjective convex optimization, Automatica

45 (2009) 805–811.[18] C. Burgat, S. Tarbouriech, Intelligent anti-windup for systems with input magnitude saturation, Int. J. Robust Nonlinear Control 8 (1998) 1085–1100.[19] F. Wu, K.M. Grigoriadis, A. Packard, Anti-windup controller design using linear parameter-varying control methods, Int. J. Control 73 (2000) 1104–

1114.[20] T. Kiyama, T. Iwasaki, On the use of multi-loop circle criterion for saturating control synthesis, Syst. Control Lett. 41 (2000) 105–114.[21] T. Nguyen, F. Jabbari, Output feedback controllers for disturbance attenuation with actuator amplitude and rate saturation, Automatica 36 (2000)

1339–1346.[22] T. Kiyama, S. Hara, T. Iwasaki, Effectiveness and limitation of circle criterion for LTI robust control systems with control input nonlinearities of sector

type, Int. J. Robust Nonlinear Control 15 (2005) 873–901.[23] F. Morabito, A.R. Teel, L. Zaccarian, Nonlinear antiwindup applied to Euler–Lagrange systems, IEEE Trans. Rob. Autom. 20 (2004) 526–537.[24] S.S. Yoon, J.-K. Park, T.-W. Yoon, Dynamic anti-windup scheme for feedback linearizable nonlinear control systems with saturating inputs, Automatica

44 (2008) 3176–3180.[25] G. Herrmann, P.P. Menon, M.C. Turner, D.G. Bates, I. Postlethwaite, Anti-windup synthesis for nonlinear dynamic inversion control schemes, Int. J.

Robust Nonlinear Control 20 (2008) 1465–1482.[26] E.F. Mulder, M.V. Kothare, M. Morari, Multivariable anti-windup controller synthesis using linear matrix inequalities, Automatica 37 (2001) 1407–

1416.[27] S. Tarbouriech, C. Prieur, J.M. Gomes-da-Silva Jr, Stability analysis and stabilization of systems presenting nested saturations, IEEE Trans. Automat.

Control 51 (2006) 1364–1371.[28] S.M. Lee, O.M. Kwon, J.H. Park, Regional asymptotic stability analysis for discrete-time delayed systems with saturation nonlinearity, Nonlinear Dyn. 67

(2012) 885–892.[29] S.M. Lee, J.H. Park, Robust model predictive control for norm-bounded uncertain systems using new parameter dependent terminal weighting matrix,

Chaos Solitons Fractals 38 (2008) 199–208.[30] S.M. Lee, O.M. Kwon, J.H. Park, Predictive control for sector bounded nonlinear model and its application to solid oxide fuel cell systems, Appl. Math.

Comput. 218 (2012) 9296–9304.[31] M. Rehan, K.-S. Hong, S.S. Ge, Stabilization and tracking control for a class of nonlinear systems, Nonlinear Anal. Real World Appl. 12 (2011) 1786–

1796.[32] T.H. Lee, D.H. Ji, J.H. Park, H.Y. Jung, Decentralized guaranteed cost dynamic control for synchronization of a complex dynamical network with

randomly switching topology, Appl. Math. Comput. 219 (2012) 996–1010.[33] T.H. Lee, J.H. Park, Z.-G. Wu, S.-C. Lee, D.H. Lee, Robust H1 decentralized dynamic control for synchronization of a complex dynamical network with

randomly occurring uncertainties, Nonlinear Dyn. 70 (2012) 559–570.[34] J.H. Park, Synchronization of cellular neural networks of neutral type via dynamic feedback controller, Chaos Solitons Fractals 42 (2009) 1299–1304.[35] M. Rehan, K.-S. Hong, M. Aqil, Synchronization of multiple chaotic FitzHugh–Nagumo neurons with gap junctions under external electrical stimulation,

Neurocomputing 74 (2011) 3296–3304.[36] M. Rehan, K.-S. Hong, LMI-based robust adaptive synchronization of FitzHugh–Nagumo neurons with unknown parameters under uncertain external

electrical stimulation, Phys. Lett. A 375 (2011) 1666–1670.