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

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.