robust adaptive fault-tolerant tracking control of multiple...

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt. J. Robust Nonlinear Control 2015; 25:2922–2938Published online 8 September 2014 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/rnc.3241

Robust adaptive fault-tolerant tracking control of multipletime-delays systems with mismatched parameter

uncertainties and actuator failures

Li-Bing Wu1,2 and Guang-Hong Yang1,3,*,†

1College of Information Science and Engineering, Northeastern University, Shenyang 110819, China2College of Sciences, University of Science and Technology Liaoning, Anshan 114051, China

3State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University,Shenyang 110819, China

SUMMARY

This study deals with the problem of robust adaptive fault-tolerant tracking for uncertain systems withmultiple delayed state perturbations, mismatched parameter uncertainties, external disturbances, and actu-ator faults including loss of effectiveness, outage, and stuck. It is assumed that the upper bounds of thedelayed state perturbations, the external disturbances and the unparameterizable time-varying stuck faultsare unknown. Then, by estimating online such unknown bounds and on the basis of the updated values ofthese unknown bounds from the adaptive mechanism, a class of memoryless state feedback fault-tolerantcontroller with switching signal function is constructed for robust tracking of dynamical signals. Further-more, by making use of the proposed adaptive robust tracking controller, the tracking error can be guaranteedto be asymptotically zero in spite of multiple delayed state perturbations, mismatched parameter uncertain-ties, external disturbances, and actuator faults. In addition, it is also proved that the solutions with trackingerror of resulting adaptive closed-loop system are uniformly bounded. Finally, a simulation example forB747-100/200 aircraft system is provided to illustrate the efficiency of the proposed fault-tolerant designapproach. Copyright © 2014 John Wiley & Sons, Ltd.

Received 19 December 2013; Revised 21 May 2014; Accepted 2 August 2014

KEY WORDS: robust adaptive control; mismatched parameter uncertainties; fault-tolerant tracking scheme;multiple time-delays systems; asymptotically stable

1. INTRODUCTION

In many practical control problems, actuator failures can cause severe performance deteriorationof control systems, or even system instability leading to catastrophic events. So the research oncontrolling systems with actuator faults is a challenging issue, which is attracting more and moreattention in the field of fault-tolerant control (FTC) systems [1, 2]. The methods and techniquesthat are developed for FTC systems can be broadly classified into two types: passive approach andactive approach. The typical passive approach consists in designing the same controller throughoutfaulty-free as well as fault cases, for example, [3] and [4]. It is simple to design and implement,but the system stability and satisfactory performance cannot be guaranteed if any fault outside thepredefined faulty set occurs. In contrast to the passive solution, an FTC system based on activeapproaches is aimed at selecting a precomputed control law or synthesizing a new control strategy

*Correspondence to: Guang-Hong Yang, College of Information Science and Engineering, Northeastern University,Shenyang, Liaoning, 110819, China.

†E-mail: [email protected]

Copyright © 2014 John Wiley & Sons, Ltd.

ROBUST ADAPTIVE FAULT-TOLERANT TRACKING CONTROL 2923

online. Then the stability as well as the acceptable performance of the system can be maintained(see, e.g., [5, 6] and the references therein). In particular, it should be noted that the adaptive actu-ator failure compensation control scheme, as one of the effective active approach, has been widelyapplied to compensation of unknown actuator failures [7–18].

On the other hand, the robust tracking control problem for systems with significant uncertaintieshas been well developed over the last few decades. Some approaches and results to track dynami-cal signals in such uncertain systems have been investigated [17–19]. As is well-known, except forsignificant uncertainties and disturbances, the time delays are often encountered in various engi-neering systems to be controlled, and the existence of the delays is frequently a source of instability.Furthermore, for systems with matched parameter uncertainties, external disturbances, and delayedstate perturbations, the problems of robust tracking and model following have been proposed in[20] and [21]. It should be pointed out, by using an improved adaptation law with � -modificationto estimate the unknown upper bounds of uncertain systems, that the literature [21] that proposedadaptive robust tracking controller can guarantee that the tracking error decreases asymptoticallyto zero. Unfortunately, the previous research does not study with mismatched parameter uncertain-ties, as well as FTC problem. Because of mismatched parameter uncertainties and actuator faulteffects, the traditional robust adaptive techniques and approaches cannot deal with the FTC prob-lems. For this purpose, to the author’s best knowledge, few results are well addressed to considerthe problem of robust tracking control for uncertain systems with the unknown bounds of mul-tiple delayed state perturbations, mismatched parameter uncertainties, external disturbances, andactuator faults.

The aforementioned considerations motivate the study in this paper. In the present work,the problem of robust adaptive fault-tolerant tracking control has been investigated for a classof uncertain systems with multiple delayed state perturbations, mismatched parameter uncer-tainties, external disturbances, and actuator faults including loss of effectiveness, outage, andstuck. It is assumed that the upper bounds of the delayed state perturbations, the unparameter-izable time-varying stuck faults, and the external disturbances are unknown. Under the specialactuator redundancy assumptions, and by introducing a norm error estimate between the mis-matched parameter matrix and the input matrix, the novel adaptive laws based on switchingsignal functions with the proposed norm error estimate are given. Then, by estimating onlinethe unknown bounds and on the basis of the updated values of these unknown bounds fromthe adaptive mechanism, a class of memoryless state feedback fault-tolerant controllers is con-structed for robust tracking of dynamical signals. Moreover, by making use of the proposedadaptive robust tracking fault-tolerant controller, the tracking error can be guaranteed to beasymptotically zero in the presence of multiple delayed state perturbations, mismatched parame-ter uncertainties, external disturbances, and actuator faults. A B747-100/200 aircraft system fromthe references [9] and [24] is used to investigate the feasibility of the proposed fault-tolerantdesign scheme.

The remaining part of the paper is organized as follows. In Section 2, the fault-tolerant trackingcontrol problem to be tackled is presented, and some standard assumptions are introduced. A robustadaptive fault-tolerant tracking controller is proposed in Section 3. In Section 4, one simulationexample taken from the existing literature is given to illustrate the use of our results, and someconclusions end this paper in Section 5.

Notation: Throughout the paper, the following notations are used, that is, Rn denotes the n-dimensional Euclidean space and k � k represents Euclidean norm of vectors or matrices. In�nstands for the n � n identity matrix with appropriate dimensions. AT represents the transpose ofmatrix A. A block diagonal matrix with matrices X1; X2; � � � ; Xn on its main diagonal is denoted asdiag.X1; X2; � � � ; Xn/.

Copyright © 2014 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2015; 25:2922–2938DOI: 10.1002/rnc

2924 L.-B. WU AND G.-H. YANG

2. PROBLEM STATEMENT AND PRELIMINARIES

2.1. System model

Consider a linear uncertain multiple time-delay systems with disturbances via the following state-space representation [20] and [21]

Px.t/ D .AC�A.t//x.t/C

rXlD1

�El.t/x.t � hl/C Bu.t/C B1d.t/

y.t/ D Cx.t/

(1)

where x.t/ 2 Rn is the state vector, u.t/ 2 Rm is the control input, d.t/ 2 Rq is the externaldisturbance, and y.t/ 2 Rp represents the output vector which is to track the reference outputym.t/. A;B;C , and B1 are known real constant matrices with appropriate dimensions. As in [9,20] and [21],�A.t/ and�El.t/ represent the time-varying parameter uncertainties, and B1; �A.t/and �El.t/ are assumed to satisfy the following conditions:

B1 D BF1; �A.t/ D FN0.t/;�El.t/ D BNl.t/; l D 1; 2; � � � ; r (2)

where F and F1 are known real constant matrices with appropriate dimensions, N0.t/ and Nl.t/are continuous and bounded matrix functions, which satisfy kN0.t/k 6 ��0 , kNl.t/k 6 ��

l, and

��0 , ��l.l D 1; 2; � � � ; r/ are unknown positive constants. In addition, for each l 2 ¹1; 2; � � � ; rº, the

time delay hl is assumed to be unknown constant, and is not required to be known for the controlsystem designer. For system (1), the initial condition is given by x.t/ D �.t/; t 2 Œ� Nh; 0� whereNh D max16l6r¹hlº and �.t/ is a continuous vector function on Œ� Nh; 0�.

Remark 1In [20] and [21], the external disturbances, multiple delayed state perturbations, and parameteruncertainties of system (1) are considered simultaneously in terms of matched condition, that is,B1 D BF1; �A.t/ D BN0.t/;�El.t/ D BNl.t/; l D 1; 2; � � � ; r . However, from a practi-cal point of view, the mismatched parameter uncertainty is a generalized scenario. In this paper,a robust adaptive fault-tolerant tracking control scheme is proposed to deal with the effect onmultiple time-delays state perturbations, actuator faults, mismatched parameter uncertainties (i.e.,�A.t/ D FN0.t/), and external disturbances for system (1) (see Sections 2 and 3 for details).

To make use of our robust adaptive tracking method, the reference output ym.t/ is assumed tosatisfy the reference model described by the following form:

Pxm.t/ D Amxm.t/; ym.t/ D Cmxm.t/ (3)

where xm 2 Rn is the state vector of the reference model and ym has the same dimensions as y.t/.Furthermore, it is required that the model state must be bounded, that is, there exists a positiveconstant M such that kxm.t/k 6 M; 8t > 0. As discussed in [20] and [21], the design for thedeveloped controller gains to track model described by system (3) is subject to such matrix-matchedequations, that is, there exist the matrices G 2 Rn�n and H 2 Rm�n such that the followingexpression

AG C BH D GAm; CG D Cm (4)

If a solution cannot be found to satisfy this algebraic equation, a different model or output matrix Cmust be chosen. More specially, the approach to find the solution to the algebraic matrix equation isalso discussed in detail in [21] and the references therein.

2.2. Fault model

To formulate the fault-tolerant tracking control problem, the fault model must be established first. Inthis paper, the actuator faults including outage, loss of effectiveness, and stuck are considered. Let



uFij .t/ represent the output signal from the i th actuator that has failed in the j th faulty mode, andui .t/ represent the input signal of the i th actuator. Then, a general actuator fault model proposed in[8–10, 14] is described as follows:

uFij .t/ D �ji ui .t/C �

ji usi .t/; �

ji �

ji D 0; i D 1; 2; � � � ; m; j D 1; 2; � � � ; L (5)

where �ji and �ji are the unknown time-varying actuator efficiency factors, the index j denotesthe j th faulty mode, and L is the number of total faulty modes. For every faulty mode, �j

iand �ji

represent the known lower and upper bounds of �ji , respectively. usi .t/ is the unparameterizablebounded time-varying stuck fault in the i th actuator. Following the practical case, we have 0 6 �j

i6

�ji 6 �

ji 6 1, and �ji is defined as

�ji D

´0; 0 < �

ji 6 1

0 or 1; �ji D 0

(6)

Equation (5) implies the following three cases:

1. �ji ¤ 0 and �ji D 0: in this case, uFij D �ji ui .t/, where 0 < �j

i6 �

ji 6 �

ji < 1. This

indicates partial loss of effectiveness in [18].2. �ji D 0 and �ji ¤ 0: the case of �j

iD �

ji D �

ji D 0 indicates that uFij can no longer

be influence by the control inputs ui .t/. This means that ui .t/ is stuck at unparameterizablebounded time-varying function usi .t/ in [17].

3. �ji D 0 and �ji D 0: this case corresponds to outage type as discussed in [16].

Remark 2Note that actuators operating in the failure-free case can also be represented as (5) with �j

iD �

ji D

�ji D 1 and �ji D 0 for t > 0.

Denote uFj .t/ D �ju.t/ C �jus.t/, where �j D diag��j1 ; � � � ; �

jm

�, �ji 2

h�ji; �ji

i, �j D

diag��j1 ; � � � ; �

jm

�, i D 1; 2; � � � ; m; j D 1; 2; � � � ; L, and uFj .t/ D

huF1j .t/; � � � ; u

Fmj .t/

iT,

us.t/ D Œus1.t/; � � � ; usm.t/�T . Then, the sets with the previous structure are defined by

��j D®�j j �j D diag

°�j1 ; � � � ; �

jm

±; �

ji 2

h�ji; �ji

i±��j D

®�j j �j D diag

°�j1 ; � � � ; �

jm

±; �

ji D 0 or 1

±For the sake of convenience, for all possible faulty modes L, the uniform actuator fault model isexploited:

uF .t/ D �u.t/C �us.t/ (7)

where � 2 ��j and � 2 ��j ; j D 1; 2; � � � ; L.Hence, the dynamics of system (1) with actuator faults (7) is written by the following form:

Px.t/ D.AC�A.t//x.t/C

rXlD1

�El.t/x.t � hl/C B.�u.t/C �us.t//C B1d.t/

y.t/ D Cx.t/

(8)

Remark 3For technical reasons, most of the known adaptive FTC approaches, [7, 8] and [16–18], are out-lined in a framework of disturbance-free .d.t/ D 0/ or without incorporating parameter uncertainty.�A.t/ D 0/, which might be very restrictive in reality. It should be pointed out that the disturbanceand matched parameter uncertainty scenario .�A.t/ D BN.t// are considered in [9] and a directadaptive control method is presented to investigate the FTC problem for the linear system with themismatched parameter uncertain .�A.t/ D FN.t// in [10]. Unfortunately, the controller (13) with



the variable gains (14), (19)–(20) and the corresponding adaptive control laws (14), (22)–(23) in [10]cannot be directly extended to (1) in this paper, because the time delay hl is assumed to be unknownconstant and the variable gain counterpart of x.t � hl/ with the appropriate adaptive control laware not constructed effectively. In this paper, we simultaneously consider the effects of external dis-turbances, mismatched parameter uncertainties (i.e., �A.t/ D FN.t/) and multiple delayed stateperturbations, where the situation turned out to be more complex than the existing works. More-over, being different [10], in the present work, the control problem of robust adaptive fault-toleranttracking is studied, and the parameter-updated laws proposed in this paper are simpler than in [10].

To ensure the achievement of the fault-tolerant objective, the following assumptions in the FTCdesign are also made to be valid.

Assumption 1All the states of system are available at every instant.

Assumption 2The pair ¹A;Bº given in (1) is completely controllable.

Assumption 3The unparameterizable stuck-actuator fault and external disturbance are piece-wise continuousbounded functions, that is, there exist unknown positive constants Nd and Nus such that kus.t/k 6Nus; kd.t/k 6 Nd , respectively.

Assumption 4rankŒB��=rankŒB� for all � 2 ��j ; j D 1; 2; � � � ; L.

Assumption 5In the presence of up to any m � s .1 6 s 6 m � 1/ actuators that undergo stuck or outage fault,the remaining actuators can still be used to implement control signals to achieve a desired controlobjective.

Remark 4Assumption 1 is a standard for state feedback system design. Assumption 2 denotes the internalstabilizability of the normal system, and Assumption 3 is quite natural and is common in the robustFTC references [8] and [9]. As discussed in [7], Assumption 4 introduces a condition of actuatorredundancy of the faulty system, and is necessary for completely compensating the stuck-actuatorfaults and disturbances. From [7–9, 15] and [17], Assumption 5 is a basic assumption to ensurethe controllability of the system and the existence of a nominal solution for the actuator failurecompensation problem.

According to the analysis of the literature [9] and Remark 4 in [10], there exist aK� 2 Rm�n anda positive constant � such that the following matrix inequality:

1

�.1C r/I C P.AC B�K�/C .AC B�K�/

TP < 0 (9)

where r is the number of multiple delayed state perturbations with system (1). On the other hand,from Assumption 4 and (4), there exists a H� 2 Rm�n satisfying matrix equations as follows:

AG C B�H� D GAm; CG D Cm (10)

2.3. Control objective

In the following discussion, based on Assumptions 1–5 and the previous analysis, the main controlobjective is to construct a robust adaptive memoryless state feedback fault-tolerant controller foruncertain system (1) in the presence of actuator failures (5) with unknown failure pattern j , failureparameters �j ; �j , and us.t/, to guarantee that the remaining actuators can still ensure that allthe closed-loop signals are bounded and the tracking error between the plant output y.t/ and thereference output signal ym.t/ decreases asymptotically to zero.



3. ADAPTIVE FAULT-TOLERANT CONTROLLER DESIGN AND STABILITY ANALYSIS

In this section, to achieve the desired robust tracking and fault compensation control goal given inSection 2, the adaptive memoryless state feedback fault-tolerant controller model is constructed asthe form

u.t/ D�OK�.t/CK1.t/CK2.t/CK3.t/CK4.t/

�´.t/C OH�xm.t/ (11)

Here, the auxiliary state vector ´.t/ is defined as follows:

´.t/ D x.t/ �Gxm.t/ (12)

where G 2 Rn�n is assumed to satisfy (4). OK�.t/ and OH�.t/ are the estimates of unknown matrices

K� in (9) andH� in (10), respectively, where OK�.t/ DhOK�1.t/; OK�2.t/; � � � OK�m.t/

iTand OH�.t/ Dh

OH�1.t/; OH�2.t/; � � � ; OH�m.t/iT2 Rm�n.

Then, OK�i .t/ and OH�i .t/.i D 1; 2; � � � ; m/ are updated by the following adaptive laws:

d OK�i .t/

dtD �1i´´

TPbi (13a)

d OH�i .t/

dtD �2ixm´

TPbi (13b)

where 1i and 2i are any positive constant, OK�i .t0/ and OH�i .t0/ are finite, respectively, bi is thei th column of B , and P is a positive-definite matrix satisfying (9). Also, it is worth noting thatK1.t/;K2.t/;K3.t/, and K4.t/ are four auxiliary control functions, which will be given later.

Consequently, applying the auxiliary control function given in (11)–(8), and combining (10) and(12), yields the following closed-loop auxiliary system:

P.t/ D .AC�A.t/C B�. OK�.t/CK1.t/CK2.t/CK3.t/CK4.t///´.t/

C B��OH�.t/ �H�.t/

�xm C

rXlD1

�El.t/´.t � hl/C B�us.t/C BFd.t/

C�A.t/Gxm C

rXlD1

�El.t/Gxm.t � hl/

(14)

In what follows, the following lemmas will be used in our main results.

Lemma 1([10]) For the diagonal matrix � in (7), there exists a positive constant ı satisfying the followinginequality:

xTPB�BTPx > ıkxTPBk2 (15)

Lemma 2([22]) For any matrices X and Y with appropriate dimensions, the following inequality holds:

XT Y C Y TX 6 ˛XTX C 1

˛Y T Y; 8˛ > 0 (16)

Denote H.t/ WDPrlD1Nl.t/Gxm.t � hl/ C �us.t/ C Fd.t/. From (3) and based on

Assumption 3, we have

kH.t/k 6 kGkMrXlD1

��l C k�k Nus C kF kNd (17)



It is clear from (17) that H.t/ is a continuous bounded function. Moreover, there also existsa positive constant k� satisfying kH.t/k 6 k�. Without loss of generality, we introduce thefollowing notion:

k1 D

PrlD1 �

�l2

ı; k2 D

��02

ı; k3 D

k�

ı; k4 D

��0MkGk

ı(18)

where ı is chosen as the inequality (15), as well as ��0 and ��l

are the upper bounds of N0.t/ andNl.t/ defined in (2), respectively. Here, it is worth pointing out that the constants k�; ��0 ; �

�l.l D

1; 2; � � � ; r/ are unknown; thus, k1; k2; k3, and k4 are all unknown.Now we give the auxiliary control functions K1.t/;K2.t/;K3.t/, and K4.t/ as follows:

K1.t/ D �1

2� Ok1B

TP (19a)

K2.t/ D �1

2� Ok2B

TP �1

2� Ok2#1B

TP (19b)

K3.t/ D �Ok23B

TP

kBTP´k Ok3 C �.t/(19c)

K4.t/ D �Ok24B

TP

kF TP´k Ok4 C �.t/�

Ok24#2BTP

kF TP´k Ok4 C �.t/(19d)

where � is given in (9), #1 and #2 are the switching signals as in (22), and �.t/ 2 RC is any positiveuniform continuous and bounded function, which satisfies

limt!1

Z t

t0

�./d 6 N� <1 (20)

where N� is a positive constant. Ok1; Ok2; Ok3, and Ok4 are the estimates of k1; k2; k3, and k4, which are,respectively, updated by the following adaptive laws:

d Ok1.t/

dtD ��1�.t/ Ok1 C �1�kB

TP´k2 (21a)

d Ok2.t/

dtD ��2�.t/ Ok2 C �2�kF

TP´k2 ��2 Ok2�eF�Ok2 � �2

�C #1

(21b)

d Ok3.t/

dtD ��3�.t/ Ok3 C 2�3kB

TP´k (21c)

d Ok4.t/

dtD ��4�.t/ Ok4 C 2�4kF

TP´k �2�4 Ok

24eF�

Ok4 � �4

� �kF TP´k Ok4 C �.t/

�C #2

(21d)

where eF D kF TP´k2�kBTP´k2 is the error estimate between the mismatched matrix F and theinput matrix BI �2 and �4 belong to any small neighborhood of Ok2 and Ok4, respectively; �1; �2; �3,and �4 are any positive constants; Ok1.t0/; Ok2.t0/; Ok3.t0/, and Ok4.t0/ are finite; and #1 and #2 are theswitching signal functions, and designed by the following switching laws:

#1 D

´0; Ok2 ¤ �2;eF sign.eF /

kBTP´k2C�; Ok2 D �2;

#2 D

´0; Ok4 ¤ �4eF sign.eF /

kBTP´k2C�; Ok4 D �4

(22)



with as an arbitrarily small positive scalar. Denote

QK�.t/ D OK�.t/ �K�; QH�.t/ D OH�.t/ �H�; Qk1.t/ D Ok1.t/ � k1; Qk2.t/ D Ok2.t/ � k2 (23)

We can obtain the following error systems:

d QK�i .t/

dtD �1i´´

TPbi ;d QH�i .t/

dtD �2ixm´

TPbi ; i D 1; 2; � � � ; m (24a)

d Qk1.t/

dtD ��1�.t/ Qk1 C �1�kB

TP´k2 � �1�.t/k1 (24b)

d Qk2.t/

dtD ��2�.t/ Qk2 C �2�kF

TP´k2 � �2�.t/k2 ��2

�Qk2 C k2

��eF�

Qk2 C k2 � �2

�C #1

(24c)

d Qk3.t/

dtD ��3�.t/ Qk3 C 2�3kB

TP´k � �3�.t/k3 (24d)

d Qk4.t/

dtD� �4�.t/ Qk4 C 2�4kF

TP´k � �4�.t/k4

�2�4

�Qk4 C k4

�2eF�

Qk4 C k4 � �4

� �kF TP´k

�Qk4 C k4

�C �.t/

�C #2

(24e)

Remark 5The previously proposed adaptive laws with � modification are capable of effectively avoiding highgain; this conclusion has been drawn in [9, 21] and [23]. Meanwhile, the error estimate eF DkF TPxk2�kBTPxk2 between the mismatched parameter matrix F and the input matrixB is usedto weaken the effect on mismatched parameter uncertainties, and it has been introduced in [10].

Remark 6As mentioned in Remark 3, the control approach presented in [10] is invalid for the system (8) inthis paper. To deal with the multiple time-delays state perturbations and achieve to the fault-toleranttracking objective, in this paper, we employ the auxiliary functions (19a) and (19c) with the novelparameter-updated laws (21a) and (21c) to eliminate the effect of the multiple time delays. Besides,the novel adaptive FTC law (13b) is also designed to implement the tracking control. Moreover, theauxiliary gain function and the corresponding adaptive control law proposed in [10] are as follows:

K3.t/ D �1

2� Ok4B

TP �1

2

�2 Ok24sign.eF /e2FB

TP

� Ok4sign.eF /eF kBTPxk2 C �.t/(25a)

d Ok4.t/

dtD ��1�.t/ Ok4 C �1�kF

TPxk2 C �1�.t/� Ok4

�sign.eF /eF kB

TPxk2 � eF�

� Ok4sign.eF /eF kBTPxk2 C �.t/(25b)

Therefore, comparing (25a) and (25b) with (19d) and (21d), respectively, it can be seen that theparameter update laws proposed in this paper are simpler than in [10]. In addition, it is worthpointing out that the robust adaptive FTC scheme in [10] only is able to deal with the case in the non-negative error estimate between the mismatched parameter matrix F and the input matrix B , that is,eF D kF

TPxk2 � kBTPxk2 > 0. In the following discussion, we should show that the proposedrobust adaptive fault-tolerant controller (11) can guarantee that the tracking error converges to zeroasymptotically both the cases with eF � 0 and eF 6 0.



Next, from (11), (23), and by virtue of condition (2), one can rewrite (14) as the following closed-loop auxiliary system:

P.t/ D�AC�A.t/C B�

�OK�.t/CK1.t/CK2.t/CK3.t/CK4.t/

��´.t/

C B� QH�.t/xm C

rXlD1

�El.t/´.t � hl/C BH.t/C�A.t/Gxm(26)

Moreover, by Q.t/ Dh´T ; QKT� ;

QHT� ;Qk1; Qk2; Qk3; Qk4

iT.t/, we denote a solution of the closed-loop

auxiliary system and the error system. Then, we can obtain the following theorem.

Theorem 1Consider the adaptive closed-loop auxiliary system described by (26) and error systems describedby (24). Suppose that Assumptions 1–5 are satisfied. Then, the solutions Q.t I t0; Q.t0// to theclosed-loop auxiliary system and error systems are uniformly bounded, and the states convergeasymptotically to zero, that is, limt!1 ´.t I t0; ´.t0// D 0.

ProofFor the adaptive closed-loop system described by (26), we first choose a Lyapunov–Krasovskiifunction candidate as

V. Q/ D ´T .t/P ´.t/C

rXlD1

��1Z t

t�hl

´T .s/´.s/ds C

mXiD1

�i QKT�i�11iQK�i

C

mXiD1

�i QHT�i�12iQH�i C

1

2ı��11Qk21 C �

�12Qk22 C �

�13Qk23 C �

�14Qk24

� (27)

Then, according to (19), by taking the time derivative of V.�/ along the trajectories of (24), and from(15) of Lemma 1 and (17), it is obtained that for any t > 0

PV . Q/ 6 ´T�P.AC B� OK�/C .AC B� OK�

�TP /´C 2��0k´

TPF kk´k C 2��0Mk´TPF kkGk

C 2k´TPBk

rXlD1

��l k´.t � hl/k C 2´TPB� QH�.t/xm � � Ok1ık´TPBk2

� � Ok2ık´TPBk2 � � Ok2ı#1k´

TPBk2 �2ı Ok23k´

TPBk2

k´TPBk Ok3 C ��2ı Ok24k´

TPBk2

k´TPF k Ok4 C �

�2ı#2 Ok

24k´

TPBk2

k´TPF k Ok4 C �C 2k�k´TPBk C ��1

rXlD1

�k´.t/k2 � k´.t � hl/k

2�

C 2

mXiD1

�i QKT�i�11iPQK�i C 2

mXiD1

�i QHT�i�12iPQH�i

C ı��11Qk1PQk1 C �

�12Qk2PQk2 C �

�13Qk3PQk3 C �

�14Qk4PQk4

�(28)

Now according to (16) of Lemma 2, for any positive constant > 0, one obtains

2xy 6 1

x2 C y2 8x; y > 0 (29)



Then combining (28) with (29), one has

PV . Q/ 6´T .P.AC B�K�/C .AC B�K�/TP /Ć 2´TPB� QK�Ć �"�02k´TPF k2

C ��1k´.t/k2 C

rXlD1

��l

2k´TPBk2 C ��1k´.t � hl/k

2�C 2��0Mk´

TPF kkGk

C 2´TPB� QH�.t/xm � � Ok1ık´TPBk2 � � Ok2ık´

TPBk2 � � Ok2ı#1k´TPBk2

�2ı Ok23k´

TPBk2

k´TPBk Ok3 C ��2ı Ok24k´

TPBk2

k´TPF k Ok4 C ��2ı#2 Ok

24k´

TPBk2

k´TPF k Ok4 C �C 2k�k´TPBk

C ��1rXlD1

�k´.t/k2 � k´.t � hl/k

2�C 2

mXiD1

�i QKT�i�11iPQK�i C 2

mXiD1

�i QHT�i�12iPQH�i

C ı��11Qk1PQk1 C �

�12Qk2PQk2 C �

�13Qk3PQk3 C �

�14Qk4PQk4

�(30)

For inequality (9), denote �Q D ��1.1C r/I C P.AC B�K�/C .AC B�K�/TP , taking theadaptive control laws (24) into (30), and with the switching laws (22), one can obtain the followinginequality:

PV . Q/ 6 � ´TQĆ ı�k1k´TPBk2 C ı�k2k´TPF k2 C 2ık3k´TPBk C 2ık4k´TPF k

� � Ok1ık´TPBk2 � � Ok2ık´

TPBk2 � � Ok2ı#1k´TPBk2 �

2ı Ok23k´TPBk2

k´TPBk Ok3 C �

�2ı Ok24k´

TPBk2

k´TPF k Ok4 C ��2ı#2 Ok

24k´

TPBk2

k´TPF k Ok4 C �C ı� Qk1k´

TPBk2 C ı� Qk2k´TPF k2

�Qk2 Ok2ı�eFQk2 C #1

C 2ı Qk3k´TPBk C 2ı Qk4k´

TPF k �2 Qk4 Ok

24ıeF

Qk4

�kF TP´k Ok4 C �.t/

�C #2

� ı��Qk21 C

Qk1k1 C Qk22 CQk2k2 C Qk

23 CQk3k3 C Qk

24 CQk4k4

�6 � �min.Q/k´k2 C ��

(31)

where �min.Q/ denotes the minimum eigenvalue of the matrix Q and � D

ı

�k21

4C

k22

4C

k23

4C

k24

4C 2

�. Moreover, according to the definition of Lyapunov–Krasovskii

given in (27), there exists a positive constant ımin such that ımink Q.t/k2 6 V. Q.t//. Then, from(20) and (31), for any t > t0, we have

0 6 ımink Q.t/k2 6 V. Q.t// D V. Q.t0//CZ t

t0

PV . Q.//d

6V. Q.t0// �Z t

t0

�min.Q/k´./k2d C � N�

(32)

which means that the solutions of the adaptive closed-loop auxiliary system described by (26) anderror systems described by (24) are uniformly bounded. Also, (32) implies that

limt!1

Z t

t0

�min.Q/k´./k2d 6 V . Q.t0//C � N� (33)

Because Q.t/ is uniformly bounded, and it follows from (24) and (26) that Q.t/ is continu-ous, which implies Q.t/ is uniformly continuous, �min.Q/k´.t/k2 is also uniformly continuous.Applying the Barbalat Lemma [22] to (33) yields limt!1 �min.Q/k´.t/k

2 D 0, which implieslimt!1 k´.t/k D 0. Namely, limt!1 ´.t I t0; ´.t0// D 0.



Furthermore, by utilizing Theorem 1, we have the following robust adaptive fault-toleranttracking theorem. �

Theorem 2Consider the tracking control problem of uncertain multiple time-delays system (1) with actuatorfaults (5) satisfying Assumptions 1–5. Then, by making use of the robust adaptive memoryless statefeedback fault-tolerant controller described in (11) with error systems (24), we can show that thetracking error e.t/ decreases uniformly asymptotically to zero.

ProofBased on Theorem 1, it is shown that the adaptive closed-loop auxiliary system (26) with (24) isuniformly bounded and their auxiliary states can converge uniformly asymptotically to zero. Thatis, for the auxiliary state ´.t/, we have limt!1 k´.t/k D 0. Then, it is clear from (4) and (12) thatthe tracking error e.t/ D y � ym D C´.t/ also decreases asymptotically to zero. �

Remark 7In the literature [21], a class of improved adaptation laws with ��modification is developed for therobust tracking problem (1)–(4), which can guarantee that the tracking error decreases asymptoti-cally to zero. However, for mismatched parameter uncertainties (2) as well as the general actuatorfaults model (5), the robust tracking approach provided in [21] is invalid. In other words, whenthe more serious robustness and the actuator fault situation occur, most of the traditional robusttechniques and approaches cannot deal with the fault-tolerant tracking problems effectively. In thispaper, by making use of the proposed robust adaptive fault-tolerant tracking controller with switch-ing signal function, the tracking error can be guaranteed to be asymptotically zero in the presence ofmultiple delayed state perturbations, mismatched parameter uncertainties, disturbances, and actua-tor faults model (5). That is, we can make sure that the outputs of the controlled uncertain time-delaysystem track exactly the outputs of the reference model without time-delay.

4. SIMULATION STUDIES

In this section, an example of tracking for a linear B747-100/200 aircraft model is given to demon-strate the proposed method, where the technical data and the underlying differential equations havebeen obtained from NASA [9] and [24]. For design purposes, only the first four states have been

0 5 10 15 20 25 30 35 40−4

−2

0

2

4

6

time(s)

the

trac

king

err

or e

(t)

0 5 10 15 20 25 30 35 40−5

0

5

10

15

20

time(s)

the

trac

king

err

or e

(t)

e1(t)

e2(t)

e1(t)

e2(t)

Figure 1. Response curves of the tracking error e.t/ using proposed adaptive fault-tolerant control method(above) and the method in [21] without considering actuator faults (below).



retained, and they are pitch rate (radians per second), true airspeed (meters per second), angle ofattack (radians), and pitch angle (radians), respectively. The inputs are elevator deflection, totalthrust, and horizontal stabilizer. The system, input, and output distribution matrices are

A D

264�0:6803 0:0002 �0:1049 0

�0:1463 �0:0062 �4:6726 �9:79421:0050 �0:0006 �0:5717 0

1 0 0 0

375 ; C D � 1 0 1 0

0 1 0 1

B D

264�1:5539 0:0154 �0:1556

0 1:3287 0:2

0 0 0

0 0 0

375 ; F D

264�1 0 0

0 1 0

0 0 0

�1 0 0:5

375

0 5 10 15 20 25 30 35 40

0

1

2

3

4

5

6

7

8

time(s)

the estimate of k1

the estimate of k2

the estimate of k3

the estimate of k4

Figure 2. Online estimates of k1, k2, k3, and k4.

0 5 10 15 20 25 30 35 40−12

−10

−8

−6

−4

−2

0

2

4

6

time(s)

the estimate of the first row of Kρ

Figure 3. Online estimates of K�.



In addition, it is assumed that

F1 D

24 100

35 ;H D

24 1 0 0 �10 1 0 0

0 0 1 2

35 ; G D

264

2:7515 1:6250 0:0659 �0:3515�57:4471 �53:5676 �0:1780 0:1557

�6:5369 �5:7426 �0:0273 0:0951

�2:7515 �2:1882 �0:0220 0:0824

375

and

N� .t/ D

24 n.�/11 .t/ 0 0 0

0 n.�/22 .t/ 0 0

0 0 0 0

35 ; � D 0; 1; 2; 3:

the estimate of the second row of Kρ

0 5 10 15 20 25 30 35 40

−15

−10

−5

0

5

10

time(s)


the estimate of the third row of Kρ

0 5 10 15 20 25 30 35 40−3

−2

−1

0

1

2

3

time(s)




where n.0/11 .t/ D 0:5sin.t/; n.0/22 .t/ D sin.t/; n.1/11 .t/ D n.3/11 .t/ D n

.2/22 .t/ D n

.3/22 .t/ D 1C0:5sin.t/,

and n.1/22 .t/ D n.2/11 .t/ D 0. Accordingly, the reference model matrices are given in the form of

Am D

264�1 1 0 0

0 �2 0 0

0 0 �3 1

0 0 0 �4

375 ; Cm D

��3:7854 �4:1176 0:0386 �0:2564�60:1987 �55:7558 �0:2000 0:2381

Furthermore, by selecting Q D 1Cr�D 0:2I from (9) and solving matrix inequality Q C .A C

BK/TP C P.AC BK/ < 0, one obtains that

P D

2640:0299 �0:0384 �0:0013 0:0489

�0:0384 0:0569 0:0013 �0:0669�0:0013 0:0013 0:0021 �0:00220:0489 �2:1882 �0:0022 0:0857

375

0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

1.2

1.4the estimate of the first row of Hρ

time(s)

Figure 6. Online estimates of H�.

0 5 10 15 20 25 30 35 40−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

time(s)

the estimate of the second row of Hρ




To verify the effectiveness of the presented adaptive approach, simulation parameters are selectedas �.t/ D 2e�0:145t ; Ok1.0/ D 8; Ok2.0/ D 5; Ok3.0/ D 8; Ok4.0/ D 5; �2 D �4 D 800; 1i D 2i D55; i D 1; 2; 3; � D 20; �1 D �2 D �3 D �4 D 500, and the initial values are chosen as x.0/ DŒ0:5;�0:45;�0:35; 2:65�T , xm.0/ D Œ1:5;�1:5; 0:245; 0:45�T , �.t/ D Œ0:2;�0:15; 0:25; 0:5�T ,h1 D 1; h2 D 2; h3 D 3; Nh D max16l63¹hlº D 3, OK�i .0/ D OH�i .0/ D Œ1; 1; 1; 1�T ; i D 1; 2; 3.

Next, the following fault model is considered. Before 6 s, the system operates in normal case.After 6 s, the third actuator is stuck at 20 C cos.t/, and the first actuator loses 20% of its effec-tiveness simultaneously. Suppose the disturbance d.t/ D sin.t/ enters the system at the beginning.t > 0/. With the previously chosen parameter settings, the simulation results of the tracking errore.t/ is given in Figure 1. The response curves in the above figures are asymptotically stable by utiliz-ing proposed adaptive FTC method, while the corresponding curves in the below ones are divergentusing the method in [21] without considering actuator faults. In addition, the respond curves of con-troller parameters Ok1; Ok2; Ok3; Ok4; OK�; OH� and the control input curve u.t/ are shown in Figures 2–9.It is also seen that the estimations can converge and all signals are uniformly bounded.

0 5 10 15 20 25 30 35 400.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

time(s)

the estimate of the third row of Hρ


0 5 10 15 20 25 30 35 40−35

−30

−25

−20

−15

−10

−5

0

5

10

15

time(s)

the

cont

rol i

nput

sig

nal u

(t)

the first control input signal u1(t)

the second control input signal u2(t)

the third control input signal u3(t)

Figure 9. Response curves of the control input signal u.t/.



5. CONCLUSIONS

The problem of robust adaptive fault-tolerant tracking for a class of linear systems with mismatchedparameter uncertainties, external disturbances, multiple time-delays state perturbations, and actuatorfaults (5) has been investigated. Here, the upper bounds of the delayed state perturbations, mis-matched parameter uncertainties, and external disturbances are assumed to be unknown. Under thespecial actuator redundancy assumptions, a novel adaptive update law is given. Then, by estimatingonline such unknown bounds and on the basis of the updated values of these unknown bounds fromthe adaptive mechanism, a class of memoryless state feedback fault-tolerant controller with switch-ing signal function is constructed for robust tracking of dynamical signals. It is also shown that byemploying the proposed robust adaptive fault-tolerant tracking controller, all signals of closed-loopsystem are bounded and the tracking error can be guaranteed to converge asymptotically to zero.

ACKNOWLEDGEMENTS

This work was supported in part by the Funds of National Science of China (Grant No. 61273148), theFunds for the Author of National Excellent Doctoral Dissertation of Peoples Republic of China (Grant No.201157), the Fundamental Research Funds for the Central Universities (Grant No. N110804001), and theIAPI Fundamental Research Funds (Grant No. 2013ZCX01-01).

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