Output Feedback Based Adaptive Robust Fault-tolerant Control For A .

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Journal of Systems Engineering and Electronics Vol. 22, No. 1, February 2011, pp.38–51Available online at www.jseepub.comOutput feedback based adaptive robust fault-tolerant controlfor a class of uncertain nonlinear systemsShreekant Gayaka and Bin Yao*Ray W. Herrick Labs, School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USAAbstract: An adaptive robust approach for actuator fault-tolerantcontrol of a class of uncertain nonlinear systems is proposed.The two chief ways in which the system performance can degrade following an actuator-fault are undesirable transients andunacceptably large steady-state tracking errors. Adaptive controlbased schemes can achieve good final tracking accuracy in spiteof change in system parameters following an actuator fault, androbust control based designs can achieve guaranteed transientresponse. However, neither adaptive control nor robust controlbased fault-tolerant designs can address both the issues associated with actuator faults. In the present work, an adaptive robustfault-tolerant control scheme is claimed to solve both the problems,as it seamlessly integrates adaptive and robust control designtechniques. Comparative simulation studies are performed usinga nonlinear hypersonic aircraft model to show the effectiveness ofthe proposed scheme over a robust adaptive control based faulttolerant scheme.Keywords: fault-tolerant system, actuator fault, adaptive control,robust control.DOI: 10.3969/j.issn.1004-4132.2011.01.0051. IntroductionIn complex systems like chemical plants, nuclear reactors, flight control systems etc., reliability is as importantas performance. Conventional feedback design for suchcomplex systems may result in unacceptable degradationin performance or even instability in the event of an actuator, sensor or component failure. Hence, it is desirable to have a certain degree of fault tolerance with respect to various faults. In the present work, we focus onthe problem of fault accommodation for unknown actuator failures for a class of nonlinear systems in presenceof parametric and non-parametric uncertainties like disturbances and uncertain nonlinearities. The faults considered here comprise of stuck actuators, loss in actuatorManuscript received November 18, 2010.*Corresponding author.The work was supported by the US National Science Foundation(CMMI-1052872) and the Ministry of Education of China.efficiency or a combination of the two faults. We do notassume the knowledge of failed actuators, instant of failure or the type of failure in the present work. The effectof such actuator faults on the system dynamics can be captured as unknown, sudden change in system parameters,and it can degrade the system performance in two chiefways: (i) it can cause large transients, which may eventually cause instability and, (ii) it may result in unacceptablylarge steady-state tracking errors.The adaptive scheme is a promising approach to dealwith such failures as it can learn the change in system parameters by virtue of their on-line learning capability. Notsurprisingly, many adaptive schemes have been developedto solve this problem. In [1], authors proposed a novelmodel reference adaptive control (MRAC) based framework to solve this problem for linear systems. They furtherextended their direct fault-compensation scheme to various classes of nonlinear systems in [2, 3] using backstepping based adaptive control. Another popular adaptive approach to solve this problem was multiple model adaptivecontrol (MMAC), switching and tuning [4]. An indirectfault accommodation scheme based on adaptive observerwas proposed in [5]. Such schemes require a fault detection and diagnosis (FDD) module, and will not be investigated in the present work. Note that none of these papers considered unstructured or non-parametric uncertainties, e.g., unknown nonlinearities and disturbances, whichcan be a limiting-factor of the achievable system performance. Robust adaptive control (RAC) based schemes,that can guarantee boundedness of closed loop signals inpresence of unknown modeling uncertainties and disturbances, were investigated in [6, 7]. But RAC is a variantof adaptive control and lacks two desirable properties inherent to robust control based techniques. First, there isno convenient and transparent way to attenuate the effectof non-parametric uncertainties, like external disturbances,on system response and steady-state tracking error. Second, such techniques are not well suited to suppress theundesirable transients following a sudden change in system

Shreekant Gayaka et al.: Output feedback based adaptive robust fault-tolerant control for a class of uncertain nonlinear systemsparameters due to unknown actuator faults. As poor transients in adaptive control based schemes can be attributedto the learning phase of the controller, it may appear thatincreasing the adaptation gain can improve the transientresponse as it speeds up the learning process. In fact, thisresult has been claimed in many articles (see [8, 9]). However, as projection types of robustness modifications arepresent in RAC based techniques to avoid parameter drift,the use of high adaptation gains may cause the estimatedparameters to bounce back and forth between the presentupper and lower limits. This could introduce a high frequency component in the control signal, which may ultimately excite the high-frequency ignored dynamics. Thus,even though all signals can be shown to be bounded, obtaining guaranteed transient response in a RAC frameworkstill remains a challenging problem. Robust control basedschemes [10–13], on the other hand, have guaranteed transient response in presence of various uncertainties. Furthermore, the effect of such disturbances can be attenuated to any desired extent on the steady-state tracking error. However, due to limited bandwidth of any practicalsystem and the large parametric uncertainties introduceddue to the actuator faults, such schemes could lead to unacceptably large steady-state tracking errors.A critical review of the existing literature reveals thatadaptive and robust control based fault-tolerant schemescan each address a part of the whole problem, but not allthe issues associated with actuator fault-tolerant control(FTC) when used individually. Thus, an FTC should possess the desirable properties of both the schemes — abilityto suppress undesirable transients and good final trackingaccuracy in order to satisfactorily solve the problem of output tracking in presence of actuator failures. Furthermore,when disturbances hinder the performance of the adaptation mechanism, the fault-tolerant scheme should still beable to guarantee the desired transient response, in addition to boundedness of all closed loop-signals.Given the need for stability in safety critical missions,the large parametric uncertainties introduced due to unknown actuator failures and the inherent limitations ofadaptive control, the idea of safe adaptive control is coming to forefront. Safe adaptive control ensures certain stability properties even without adaptation [14, 15]. Adaptive robust control (ARC) based schemes have already resolved this issue [16, 17] and may be classified as the socalled safe adaptive control. Switching the adaptation offat any instant converts the adaptive robust controller intoa deterministic robust controller with guaranteed transientperformance. Moreover, the design procedure allows us tocalculate explicit upper bound for tracking errors over theentire time history in terms of certain controller parameters39and achieve pre-specified final tracking accuracy. Thus,ARC based schemes are natural choices for safety sensitivesystems over conventional adaptive and robust schemes.An output feedback ARC based direct scheme was recently proposed [18] to accommodate unknown actuatorfaults in uncertain linear systems, which seamlessly integrated adaptive and robust control designs. Comparative simulation studies proved the superior performanceof the proposed design over MRAC based fault-tolerantcontroller. In this paper, we develop an output feedbackbased adaptive robust FTC (ARFTC) technique for accommodation of unknown actuator faults for a class of uncertain nonlinear systems. The proposed technique combines adaptive backstepping [19] and discontinuous projection based ARC [17]. We claim that the proposed technique can solve both the problems associated with actuatorFTC. In ARC based designs, the baseline controller is arobust controller which incorporates the bounds on parametric uncertainties and unstructured uncertainties. Thisguarantees desired transient response can be achieved evenafter the actuators fail, and an upper bound for the absolute value of transient error for the entire time-history canbe given. Furthermore, ARFTC has a learning mechanismin addition to the robust filter structure. Thus, the changein system parameters due to actuator faults can be learntover time, which leads to improved model compensationand better steady-state tracking accuracy. In fact, in presence of parametric uncertainties only, ARFTC can achieveasymptotic tracking with guaranteed transient response inthe event of an actuator failure. It is worth mentioning thatthere are fundamental differences between ARC and RACbased fault-tolerant schemes. Most importantly, ARFTCputs more emphasis on robust filter structure as a meansto attenuate the extent of modeling uncertainties. Hence,robust performance is guaranteed even when adaptation isswitched off. See [16] for detailed discussion on variousdifferences between ARC and RAC. The difference in theachievable performance will be clear from the discussionsfollowing the simulation results.Note that in this work the comparative studies are carried out with respect to a robust adaptive backsteppingbased design, whereas in our previous work [18], the comparative studies were conducted with an MRAC basedscheme. As backstepping based designs with tuning func tions take into account the estimation error transient (θ̃),it can achieve better performance than an MRAC basedscheme which relies on certainty-equivalence principle.Furthermore, as MRAC design is fundamentally differentfrom backstepping based designs, it is not possible to compare and contrast the two designs at each step. In thepresent work, however, the comparison is done between

40Journal of Systems Engineering and Electronics Vol. 22, No. 1, February 2011two backstepping based approaches, which makes it possible to emphasize the underlying subtle but important differences in the design of the two FTC strategies. Thismakes the comparative studies much more compelling.2. Problem statementIn the present work, we will consider systems which canbe represented in the following output feedback formp ẋ1 x2 ϕ0,1 (y) aj ϕ1,j (y) Δ1j 1.ẋρ 1 xρ ϕ0,ρ 1 (y) ẋρ xρ 1 ϕ0,ρ (y) p j 1p aj ϕρ,j (y) bm,j βj (y)uj (t) Δρẋn ϕ0,n (y) p (1)ηjj(1 σjj )u (t) σjj ūjβj (y)aj ϕi,j (y) Δi (y, t)for i 1, 2, . . . , ρ 1, andb0,j βj (y)uj (t) Δnẋρ xρ 1 ϕ0,ρ (y) j 1where u j represents the control command to the jth actuator, Tf is the unknown instant of failure, ūj is an unknown constant value at which the actuator gets stuck, andηjj [(ηjj )min , 1] represents actuator loss in efficiency.Without actuator redundancy and sufficient control authority, actuator faults can not be accommodated and thesame is stated formally in the following assumption.Assumption 1 System (1) is such that the desired control objective can be fulfilled with up to q 1 stuck actua-p j 1aj ϕn,j (y) where ρ n m is the relative degree, uj is the controlinput, y x1 is the measured output, ϕ0,i (y) and βj (y)are the known smooth functions of y. It will be assumedthat βj (y) 0 for any y. Δi Δi (y, t) represents uncertainties, e.g., modeling error and disturbances. ai andbi,j are unknown constants such that the sign of the highfrequency gain (sgn(bm,j )) is known.System (1) is subjected to actuator faults, which can berepresented as ūj , t Tf , if jth actuator gets stuck at Tfuj (t) (2) ηjj uj (t), t Tf , if jth actuator loses efficiency at Tf(3)for j 1, . . . , q where σjj 1 corresponds to stuck actuators, σjj 0 and (ηjj )min ηjj 1 represents actuatorloss of efficiency, σjj 0 and ηjj 1 corresponds tohealthy actuators.With this, we can rewrite the system asẋi x2 ϕ0,i (y) j 1q Control signals uj (t) are designed such that βj (y)uj (t) u (t). With a fault model (2) and the chosen actuationscheme, we can rewrite the control inputs asuj (t) j 1.3. Output feedback based ARFTCaj ϕρ 1,j (y) Δρ 1j 1q tors and any number of actuators with loss in efficiency.The problem we attempt to solve in this paper can nowbe stated as follows. For the uncertain nonlinear system(1), subjected to faults (2) the goal is to design an outputfeedback based control strategy such that the output tracking error converges exponentially to a prespecified boundand has a guaranteed transient performance.p aj ϕρ,j (y) j 1q μm,j βj (y) κm u (t) Δρ (y, t)j 1.ẋn ϕ0,n (y) p aj ϕn,j (y) j 1q μ0,j βj (y) κ0 u (t) Δn (y, t)(4)j 1whereκi q ηjj (1 σjj )bi,jj 1μi,j σjj ūj bi,ji 0, 1, . . . , m;j 1, 2, . . . , qNote that κi is the unknown measure of actuator effectiveness after faults and μi,j is the unknown measure of thefault magnitude which needs to be compensated. Thus, thesystem experiences jump in parameter values and boundeddisturbances with the occurrence of each new fault. Theproposed scheme accommodates such faults by estimatingμi,j and κi , and relies on a robust feedback scheme to deal

Shreekant Gayaka et al.: Output feedback based adaptive robust fault-tolerant control for a class of uncertain nonlinear systemswith the estimation mismatch and the jump in parametervalues.We will make the following realistic assumptions regarding the uncertainties present in the system.Assumption 2 The extent of parametric uncertaintiesand uncertain nonlinearities satisfyai Ωa {ai : (ai )min ai (ai )max }κi Ωκ {κi : (κi )min κi (κi )max }μi,j Ωμ {μi,j : (μi,j )min μi,j (μi,j )max }We need to construct a state-estimator form en i κi u ẋ A0 x k̄y ϕ0 (y) Φ(y)a where i 0, 1, . . . , m and j 1, 2, . . . , q.Due to the special structure of A0 , the order of K-filterscan be reduced by using the following two filtersλ̇ A0 λ en u (6)ϑi Ai0 λψi,j Ai0 ζj , ϕ0,1 ϕ0 (y) . , In 1 ,0. 0 ϕ1,1 .Φ(y) .ϕ0,nϕn,1(12)The estimated state can be written asx̂ ξ0 ξa m i 0κi ϑi q m μi,j ψi,j(13)j 1 i 0(14)From (5), we know there exists an unknown but boundedfunction which upperbounds Δ̃(y, t), i.e.(15)TTV̇εx εTx (A0 P P A0 )εx 2εx Δ̃(y, t) k1 k̄ . kn · · · ϕ1,p. . · · · ϕn,p εx 21 ( εx 2δ̃(t))2 2δ̃ 2 (t) 22 (8)Δ̃(y, t) [Δ1 , Δ2 , · · · , Δn ]TNote that A A0 kC T and the observer matrix A0 canbe made stable by a suitable choice of observer gain k suchthat there exists a symmetric positive definite matrix PsatisfyingP PT 0 εx 2 2 εx δ̃(t) (7)andP A0 AT0 P I,(11)where denotes the L2 norm. Then, considering a positive semi-definite (psd) function Vεx , we obtainC [1 0 0 . . . 0] k1 .A0 . kni 0, 1, . . . , m Δ̃(y, t) δ̃(t)y Cx j 1, 2, . . . , qand the following algebraic equationsj 1 i 0whereψi,j Rn 1ε̇ A0 ε Δ̃(y, t)i 0en i μi,j βj (y) Δ̃(y, t)(10)Let ε x x̂ be the estimation error. Then, the stateestimation error dynamics is given by3.1 State estimationq m ϑi Rn 1ψ̇i,j A0 ψi,j en i βj (y),ζ̇j A0 ζj en βj ,where (ai )min , (ai )max , (κi )min , (κi )max , (μi,j )min ,(μi,j )max are known and δi (t) is an unknown but boundedfunction.We will make another standard assumption which guarantees stability of the zero dynamics.Assumption 3 The polynomial κm sm κm 1 sm 1 · · · κ0 is a stable polynomial and sign(κm ) is known,irrespective of the failed actuators.ξ Rn pϑ̇i A0 ϑi en i u ,(5)Δi ΩΔ {Δi : Δi (y, t) δi (t)}ξ0 Rn 1ξ 0 A0 ξ0 ky ϕ0 (y),ξ̇ A0 ξ Φ(y),41(9)We will define the following set of filters for the purposeof state-estimation εx 2 2δ̃ 2 (t)2(16)using (15), pmin min{eig(P )}, pmax max{eig(P )}and pmin εx (t) 2 Vεx (t) pmax εx (t) 2 . Integrationboth sides of (16), and using the comparison lemma weobtain tVεx (t) exp Vεx (0) 2pmax tt τ4 exp (17)δ̃(τ )2 dτ2pmax0 pmaxt εx (0) 2 exp εx (t) 2 pmin2pmax t2t τexp δ̃(τ )2 dτpmin 02pmax

42Journal of Systems Engineering and Electronics Vol. 22, No. 1, February 2011 pmaxtexp εx (0) pmin4pmax t 2t τ2exp (18)δ̃(τ ) dτpmin 02pmax εx (t) In (18), the first term is exponentially vanishing, and thesecond term is unknown but bounded. Hence, the stateestimation error remains bounded and converges exponentially to a residual-ball whose size depends on the extent ofunknown modeling uncertainties, i.e. εi xi x̂i δεi (t),i 1, . . . , nan adaptive law like smooth projection and the full-statefeedback based ARC controller design, we propose thatthe readers refer to [17].Step 1 The derivative of the output tracking error z1 y yd is given byż1 ẏ ẏd p x2 ϕ0,1 (y) aj ϕ1,j (y) ẏd Δ1 (y, t) j 1p x̂2 ϕ0,1 (y) (19)aj ϕ1,j ẏd Δ̄1 (y, t) j 1ω0 ω T θ Δ̄1 (y, t) 3.2 Parameter projectionκm ϑm,2 ω0 ω̄ T θ Δ̄1 (y, t)Let θ̂ denote the estimate of θ and θ̃ θ̂ θ denote the estimation error. It is a well known fact that gradient based parameter estimation algorithms suffer from parameter driftin presence of disturbances, and can result in system statesgrowing unboundedly. We use discontinuous parameterprojection [20] to deal with this problem. The update lawand the projection mapping used here have the followingform (20)θ̂ Projθ̂ (Γ τ ) 0, if θ̂i θi,max and i 0Projθ̂i (21)0, if θ̂i θi,min and i 0 i , otherwisewhere Γ 0 is a diagonal matrix, and τ is any adaptation function. The projection mapping guarantees that thefollowing two properties are always satisfiedP1θ̂ Ωθ {θ̂ : θmin θ̂ θmax }(22)P2θ̃T (Γ 1 Projθ̂ (Γ τ ) τ ) 0, τ(23)3.3 Controller designThe output feedback based controller design presentedhere combines the output feedback based adaptive backstepping [19] and discontinuous projection based ARC[17], which uses state-feedback. The main idea is to synthesize a virtual control law which will drive the error to asmall residual ball. But, as in this case only a single stateis available for measurement, the synthesized virtual control law will replace the reconstructed state at each step,and the state estimation error will be dealt with via robust feedback. Also, it should be noted that the use ofdiscontinuous projection implies a tuning function basedbackstepping can not be used, and hence a stronger robustcontrol law is needed to negate the effects of parameter estimation transients. For the advantages of discontinuousprojection based technique over smooth modifications of(24)whereω0 [ξ0,2 ϕ0,1 ]ω [ξ(2) Φ(1) , ϑm,2 , ϑm 1,2 , . . . , ϑ0,2 ,ψm,1(2) , . . . , ψm,q(2) , . . . , ψ0,1(2) , . . . , ψ0,q(2) ]Tω̄ ω e p 1 ϑm,2(25)θ [a1 , a2 , . . . , ap , κm , . . . , κ0 ,μm,1 , . . . , μm,q , . . . , μ0,1 , . . . , μ0,q ]TΔ̄1 (y, t) Δ1 (y, t) ε2and e k is the kth basis vector in Rp m qm 1 .If ϑm,2 is the input, we would synthesize a virtual control law α1 to make z1 as small as possibleα1 (y, ξ0 , ξ, λ̄m 1 , ψi,j,2 , θ̂, t) α1a α1s1α1a {ω0 ω̄ T θ ẏd }κ̂m(26)In (26), α1a is the model compensation component of thecontrol law used to achieve an improved model compensation through on-line parameter adaptation. Thus, usingmodel compensation, the fault is partly accommodated as[θ̂p m 2 , . . . , θ̂p m qm 1 ]T [μm,1 , . . . , μ0,1 , . . . , μ0,q ]TNow, as the sign of κm is assumed as known, withoutloss of generality, one can assume κm 0 and it is lowerbounded by a non-zero positive constant, i.e. (κm )min (θp 1 )min 0 where (θp 1 )min is independent of the failure pattern. Also, that (θp 1 )min is known from Assumption 2. Then, the projection mapping (22) guarantees thatκm (κm )min 0, which implies that the control law(26) is well defined. Substituting (26) into (24), we getż1 κm (z1 α1s ) θ̃φ1 Δ̄1(27)The robust component is designed to compensate for the

Shreekant Gayaka et al.: Output feedback based adaptive robust fault-tolerant control for a class of uncertain nonlinear systemspotential destabilizing effect of the uncertainties on theright hand side of (27) asα1s α1s1 α1s2 α1s31α1s1 k1s z1κm,min(28)where k1s is a nonlinear gain, such thatk1s g1 Cφ1 Γ φ1 2 ,g1 0(29)in which Cφ1 is a positive definite constant diagonal matrix to be specified later. As discontinuous projection isused, tuning functions can not be used to compensate forparameter estimation-error transients.Substituting (29) in (27), we obtainż1 κm z2 κmκm,minκm (α1s2 α1s3 ) θ̃T φ1 Δ̄1(30)Define a psd function V1 z12 /2. Its derivative is given byV̇1 κm z1 z2 k1s z12 Tz1 (κm α1s2 θ̃ φ1 ) z1 (κm α1s3 Δ̄1 )(31)From Assumption 1 θ̃T φ1 θM φ1 θM θmax θmin(32)As θ̃T φ1 is bounded by a known function, there exists arobust control function satisfying the following conditions(a)z1 {κm α1s2 θ̃T φ1 } 11(b)z1 α1s2 0(33)Similarly, from Assumption 2 and (19), we have Δ̄1 ε2 Δ1 δε2 (t) δ1 (t) δ̄1 (t)Fig. 1Now, we can follow the same strategy as in (33) to design a robust control law. But, as δ̄1 (y, t) is unknown, wecan not prespecify the level of control accuracy. Hence, weseek to achieve the following relaxed conditions(a)z1 {κm α1s3 Δ̄1 (y, t)} 12 δ̄12(b)z1 α1s2 0(34)(35)Remark 1 Condition (a) of (33) shows that α1s2 issynthesized to attenuate the effect of parametric uncertainties θ̃ with the level of control accuracy being measuredby 11 . Condition (b) ensures that α1s2 is dissipative innature so that it does not interfere with the functionality ofthe adaptive control law α1a . One smooth example of α1s2satisfying (33) isα1s2 k1s z1 43h124κm,min 11z1 , h1 θM φ1 2(36)Similarly, an example of α1s3 satisfying (35), which is synthesized to attenuate the effect of unstructured uncertainties Δ̄1 (y, t), is given byα1s3 1z14κm,min 12(37)Remark 2 There are subtle but important design differences between an ARC and RAC based FTC scheme.Fig. 1 shows the underlying structure of ARC and RACbased designs. Note that the ARC based fault-tolerantcontroller, the emphasis is on the inner loop robust controller, and the adaptation mechanism in the outer loop isused to reduce the extent of modeling uncertainties. Asunknown actuator faults introduce severe estimation error (θ̃) and estimation error transients (θ̃), it is necessary tosuppress their undesirable effect on the system dynamics.The coordination mechanism ensures that the potentialdestabilizing effect of θ̃ and θ̃ are effectively suppressedby the robust controller. Furthermore, the bounded uncertainties are also attenuated to desired extent by the robustStructure of ARC and RAC based fault-tolerant controllers

44Journal of Systems Engineering and Electronics Vol. 22, No. 1, February 2011controller. Thus, desired transient response is guaranteed.On the other hand, the RAC based designs use an adaptivecontroller in conjunction with robustness modifications tothe adaptation scheme to guarantee the boundedness of allthe signals. They lack the extra design freedom present inARC due to the underlying robust controller. Thus, theycan not guarantee desired transient response.Step 2 From (11)–(13) and (24)–(26), we can obtainthe derivative of α1 asα̇1cα̇1 α̇1c α̇1u α1 α1{ω0 ω T θ̂} {A0 ξ Φ(y)} y ξ α1{A0 ξ0 ky ϕ0 (y)} ξ0q m 1m 1 α1 α1 α1(38)λ̇i ζ̇i,j λi ζi,j ti 1j 1 i 1α̇1u α1 α1 ( θ̃T ω Δ̄1 ) θ̂ y θ̂(39)where α̇1c is calculable and will be used for control function design. α̇1u , however, is not calculable and will bedealt with via certain robust terms. From (10), the derivative of z2 ϑm,2 α1 is given byż2 ϑm,3 k2 ϑm,1 α̇1c α̇1u(40)Define a psd function V2 V1 z22 /2. Then, the derivative of V2 using (31) and (40) is given byz2 (α2s3 Δ̄2 ) z2 α1 θ̂ θ̂(43)where z3 ϑm,3 α2 represents the input discrepancyandφ2 e n 1 z1 α1ω, yΔ̄2 α1Δ̄1 y(44)From (34), it follows that Δ̄2 α1 / y δ̄1 . Similar to(35) and (41), the robust control functions α2s2 and α2s3are chosen to satisfy(a)(b)(c)z2 (α2s2 θ̃T φ2 ) 21z2 (α2s3 Δ̄2 ) 22 δ̄12z2 α2s2 0, z2 α2s3 0(45)where 21 and 22 are positive design parameters. As givenin (36) and (37), α2s2 and α2s3 can be chosen as 2 α1h21α2s2 z2 , α2s3 z2 (46)4 214 21 ywhere h2 is any smooth function satisfying h2 θM 2 · φ2 2 . From (31) and h2 defined above, the derivative ofV2 satisfiesV 2 z2 z3 2 kjs zj2 z1 (κm α1s2 θ̃1 φ1 ) j 1(41)where V̇1 α1 k1s z12 z1 (κm α1s2 θ̃T φ1 ) z1 (κm α1s3 Δ̄1 ). Similar to (26), we can now define α2 for ϑm,3 asα2 (y, ξ0 , ξ, λ̄m 2 , ψi,j,3 , θ̂, t) α2a α2sα2a κ̂m z1 k2 ϑm,1 α̇1cα2s α2s1 α2s2 α2s3α2s1 k2s z2 α12 Ck2s g2 θ̂ θ2 Cφ2 Γ φ2 V̇2 V̇1 α1 z2 z3 k2s z22 z2 (α2s2 θ̃T φ2 ) z1 (κm α1s3 Δ̄1 ) z2 (α2s2 θ̃T φ2 ) V̇2 V̇1 α1 z2 {κm z1 ϑm,3 k2 ϑm,1 α̇1c α̇1u }in the proof of Theorem 1. Substituting (42) in (41), weobtain(42)where g2 0 is a constant, Cθ2 and Cφ2 are positive definite constant diagonal matrices, α2s2 and α2s3 are robustcontrol functions to be synthesized later. Due to use ofdiscontinuous projection, we can not use tuning functionswhich anticipates and compensates for the effect of parameter estimation transients. α2s1 is the robust control termwhich compensates for this loss of information. The reason for choosing this form for α2s1 will become apparentz2 (α2s3 Δ̄2 ) α1 θ̂z2 θ̂(47)Step i (3 i ρ) Mathematical induction will beused to prove the general result for all the intermediatesteps. At each step i, the ARC control function αi willbe constructed for the virtual control input ϑm,i 1 . Forany j [3, i 1], let zj ϑm,j αj 1 and recursivelydesignφj αj 1ω, yΔ̄j αj 1Δ̄1 y(48)Lemma 1 At Step i, choose the desired ARC controlfunction αi asαi (y, ξ0 , ξ, λ̄m i , ψk,j,i 1 , θ̂, t) αia αisαia zi ki ϑm,i α̇(i 1)cαis αis1 αis2 αis3αis1 kis zi αi 12 Cθi kis gi Cφi Γ φi θ̂(49)

Shreekant Gayaka et al.: Output feedback based adaptive robust fault-tolerant control for a class of uncertain nonlinear systemswhere gi 0 is a constant, and Cθi and Cφi are positivedefinite constant diagonal matrices, αis2 and αis3 are robust control functions satisfyingand(a)zi (αis2 θ̃T φi ) i1(b)zi (αis3 Δ̄i ) i2 δ̄12(c)zi αis2 0,(50)zi αis3 0 α1 α1{ω0 ω T θ̂} {A0 ξ0 ky y ξ0m 1 α1 α1ϕ0 (y)} {A0 ξ Φ} λ̇i ξ λii 1żρ ϑm,ρ 1 u kρ ϑm,1 α̇(ρ 1)c αρ 1 αρ 1( θ̃T ω Δ̄1 ) θ̂(56) y θ̂If ϑm,ρ 1 u is the virtual input, (56) would have thesame form as the intermediate Step i. Therefore, the general form, (48)–(53) and (56) is applied to Step ρ. Since u is the actual control input, it can be chosen asu αρ ϑm,ρ 1α̇(i 1)c q m 1 α1 α1ζ̇i,j ζi,j tj 1 i 1żi zi 1 zi 1 kis zi αi 1 θ̂(αis2 θ̃T φi ) (αis3 Δ̄i ) θ̂τ i (52)kjs zj2 z1 (κm α1s2 θ̃T φ1 ) j 1cφkr are the rth diagonal element of Cθj and Cφk respectively, then, the control law (55) guarantees that(i) In general the control input and all internal signalsare bounded. Furthermore, Vρ is bounded above bywhere λρj 2j 2zj (αjs3 ρ1 ρ2 δ̄1 2 [1 exp( λρ t)] (59)λρρ 2min{g1 , . . . , gρ }, ρ1 j1 , ρ2 Vρ (t) zj (κm αjs2 θ̃T φj ) z1 (κm α1s3 Δ̄1 ) i (58)φj zjIf diagonal controller gain matrices Cθj and Cφk are cho ρ 224, where cθjr andsen such that cφkr ρ1/cθjrj 1i ρ j 1and the derivative of the augmented psd function Vi Vi 1 1/2zi2 satisfiesV̇i zi zi 1 (57)where αρ is given by (53). Then, zρ 1 u ϑm,ρ 1 αρ 0.Theorem 1 Let the parameter estimates be updatedusing the adaptation law (20) in which τ is chosen as(51)Then, the ith error subsystem is45i αj 1 Δ̄j ) θ̂zj θ̂j 2j 1(53)Proof The first two steps satisfy the Lemma and canbe verified by substitution. Assume that the lemma is truefor Steps j (j i 1) and we need to prove it for Step ito complete the proof. From (52), we get αi 1 (54)δ̄1 Δ̄i yTherefore, there exist αis2 and αis3 satisfying (54) andthe control can be defined. The derivative of zi is given byρ j2 and δ̄1 2 stands for the infinity norm of δ̄1 .j 1(ii) If after a finite time t0 , Δ̄(y, t) 0 (i.e. in the presence of parametric uncertainties only) then, in addition toresults in (59), asymptotic output tracking control is alsoachieved.Proof Using (57), we known zρ 1 0. From (53),we haveαρ (y, η, λ̄n , ψ, θ̂, t) αρa αρsαρa zρ 1 kρ υm,1 α̇(ρ 1)cαρs αρs1 αρs2 αρs3żi υm,i 1 ki υm,1 α̇i 1,c αi 1 αi 1 ( θ̃T ω Δ̄1 ) θ̂ y θ̂(55)By substituting (53) and υm,i 1 zi 1 αi into (55),it can be verified that the lemma is satisfied for Step i. Step ρ In this final step, the actual control law u willbe synthesized such that ϑm,ρ tracks the desired ARC control function αρ 1 . The derivative of zρ can be obtainedaskρs(60)αρs1 kρs zρ αρ 12 Cθρ gρ Cφρ Γ φρ θ̂Substituting in (56), we obtainżρ zρ 1 kρs zρ (αρs2 θ̃T φρ ) (αρs3 Δ̄ρ ) αρ 1 θ̂ θ̂(61)

46Journal of Systems Engineering and Electronics Vol. 22, No. 1, February 2011Also, substituting i ρ 1 in (53) givesV̇ρ 1 zρ 1 zρ ρ 1 kjs zj2 z1 (bfm α1s2 θ̃T φ1 ) j 1ρ 1 j 2zj (αjs3 Δ̄j ) j 2ρ 1 αj 1 θ̂zj θ̂j 2(62)V̇ρ V̇ρ 1 αρ 1 zρ żρ kjs zj2 z1 (bfm α1s2 θ̃T φ1 ) j 1ρ zj (αjs2 θ̃T φj ) z1 (bfm α1s3 Δ̄1 ) j 2ρ zj (αjs3 Δ̄j )

of parametric and non-parametric uncertainties like dis-turbances and uncertain nonlinearities. The faults con-sidered here comprise of stuck actuators, loss in actuator Manuscript received November 18, 2010. *Corresponding author. The work was supported by the US National Science Foundation (CMMI-1052872) and the Ministry of Education of China.

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