Robust model reference adaptive output feedback ...

ISA Transactions 57 (2015) 51–56

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Robust model reference adaptive output feedback tracking for uncertain linear systems with actuator fault based on reinforced dead-zone modification H.M. Bagherpoor n, Farzad R. Salmasi 1 School of Electrical and Computer Engineering, Faculty of Engineering, University of Tehran, Tehran 14395, Iran

art ic l e i nf o

a b s t r a c t

Article history: Received 27 May 2014 Received in revised form 2 October 2014 Accepted 7 February 2015 Available online 3 March 2015 This paper was recommended for publication by Dr. Q.-G. Wang

In this paper, robust model reference adaptive tracking controllers are considered for Single-Input Single-Output (SISO) and Multi-Input Multi-Output (MIMO) linear systems containing modeling uncertainties, unknown additive disturbances and actuator fault. Two new lemmas are proposed for both SISO and MIMO, under which dead-zone modification rule is improved such that the tracking error for any reference signal tends to zero in such systems. In the conventional approach, adaption of the controller parameters is ceased inside the dead-zone region which results tracking error, while preserving the system stability. In the proposed scheme, control signal is reinforced with an additive term based on tracking error inside the dead-zone which results in full reference tracking. In addition, no Fault Detection and Diagnosis (FDD) unit is needed in the proposed approach. Closed loop system stability and zero tracking error are proved by considering a suitable Lyapunov functions candidate. It is shown that the proposed control approach can assure that all the signals of the close loop system are bounded in faulty conditions. Finally, validity and performance of the new schemes have been illustrated through numerical simulations of SISO and MIMO systems in the presence of actuator faults, modeling uncertainty and output disturbance. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Robust adaptive Model Reference Control Actuator fault Output feedback Dead-zone Uncertain system

1. Introduction Conventional robust adaptive controllers can guarantee stability in uncertain control systems; however, these controllers generally lead to tracking error. Robust adaptive tracking controllers are eligible substitutes and have gained considerable attention. Marino and Tomei [1] proposed such controllers for linear time varying systems with additive disturbance, but they did not consider modeling uncertainties. An adaptive state feedback flight tracking controller is proposed in [2], in which actuator's loss of effectiveness is considered. In [3], an adaptive output feedback controller is addressed for a class of nonlinear time-delay systems. Although the controller approach is based on output feedback but robustness to modeling uncertainties and actuator faults have not been taken into account. In [4], an adaptive model reference output feedback tracking controller is proposed, which can compensate actuator failure. Moreover, robust adaptive state feedback tracking for uncertain dynamical systems is considered in the literature [5,6]. Availability of all the states is a major setback for n

Corresponding author. Tel.: þ 98 9365243213. E-mail addresses: [email protected] (H.M. Bagherpoor), [email protected], [email protected] (F.R. Salmasi). 1 Tel.: þ 98 21 82089735; fax: þ 98 21 88776830. http://dx.doi.org/10.1016/j.isatra.2015.02.007 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.

development of such methods. Adaptive output feedback tracking controllers for linear MIMO and a class of nonlinear SISO systems with unknown and time-varying state delay are introduced in [7], while there are no other modeling mismatches, rather than the unknown state delay. In [8–15] adaptive fault tolerant controllers (FTC) were introduced to compensate actuator efficiency losses; however in these works, external disturbances have not been taken into account. The proposed fault tolerant controller in [16] is designed based on an adaptive sliding mode method for uncertain systems to guarantee the closed-loop stability. However, reference tracking is not achievable by this controller. Zhang and Yu proposed a fault tolerant controller for discrete-time switched linear systems, which guarantees closed-loop stability of the system with actuator saturation, while modeling uncertainty is not taken into account [17]. A novel output feedback tracking controller based on sliding mode and tracking error observer is presented in [18]; however, the designed controller fails to accommodate actuator failures. In [19,20] robust adaptive controllers are used to handle loss of actuator effectiveness for systems with mismatched parameters uncertainty and external disturbance, but they do not guarantee reference tracking. Robust tracking problem for stochastic uncertain systems is reported in [21], while modeling uncertainties are not considered. A robust adaptive attitude tracking control is introduced in [22] for an orbiting flexible space craft. In [23], a robust output-feedback

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model predictive control scheme is proposed for systems with unstructured uncertainty, which considers multi-model uncertainty and can achieve piecewise constant reference tracking. Paulo et al. introduced an output feedback model reference sliding mode control for uncertain multivariable systems with input disturbance [24]. Robust adaptive controllers for nonlinear systems are also reported in the literature. A robust adaptive fault tolerant controller based on output feedback for a class of nonlinear systems with actuator failure is introduced in [25]. Nevertheless, the proposed approach cannot attain zero tracking error and the output tracking error converges to a specified bound, when the system is subjected to dynamic uncertainties. A robust fault-tolerant state feedback tracking controller for spacecraft under control input saturation is developed in [26], which can compensate loss of actuator effectiveness and bounded disturbance. Tong et al. introduced an adaptive fuzzy back-stepping controller for strict feedback nonlinear systems with unknown dead-zones. In this approach system states are estimated based on fuzzy filters, and close loop stability of the system is guaranteed using an adaptive back-stepping recursive technique [27]. However, the proposed controller is designed based on strict feedback and cannot support many of practical nonlinear systems. Furthermore, since actuator failure and modeling uncertainty have not been considered, the proposed method will not guarantee closed-loop stability in the presence of faults. A robust fault tolerant controller based on sliding mode and feedback linearization is proposed in [28] for nonlinear affine systems with modeling uncertainty and actuator faults, while the unstructured uncertainty and external disturbance are not considered. In [29], a new robust adaptive output feedback control scheme is reported for uncertain discrete-time nonlinear systems without consideration of additive disturbance. Peixoto et al. proposed a global tracking sliding mode controller for uncertain time varying nonlinear systems [30]. However, stability and convergence of the overall control system are not guaranteed in the presence of external disturbance. There are two renowned robust modification schemes for adaptive systems: dead-zone and projection. The former shows better performance “if the knowledge of the uncertainty level is sufficiently conservative” as stated in [31]. Nevertheless, these conventional schemes do not ensure asymptotic tracking of reference signals. In [32], the robust model reference adaptive control (MRAC) with conventional dead-zone is modified to improve the closed loop performance, in which the controller signal is chosen based on the tracking error inside the dead-zone. It has been shown that the output regulation and tracking of step-like commanded signals can be achieved with the proposed improved dead-zone modification rule for linear time invariant (LTI) SISO and two input-two output (TITO) systems. A new fault tolerant controller for nonlinear systems with partial and total actuator faults based on MRAC is considered in [33]. The proposed approach has the capability to guarantee the closed-loop stability of the system in the presence of partial and total actuator faults. But the control structure is designed based on state feedback. In this paper, a new reinforced dead-zone modification MRAC algorithm is presented such that the controller forces the system output to track any reference signal generated by the reference model for SISO and MIMO systems, with modeling uncertainties, imposed unknown additive disturbances and actuator faults. To the best of our knowledge, no control scheme with such capabilities has been reported so far. The rest of the paper is organized as follows. First, conventional approach of robust MRAC for the SISO system is described in Section 2. Section 3 presents a new controller based on measuring tracking error inside the dead-zone, which results in zero tracking error of the SISO system even in faulty conditions. The proposed scheme relies on stopping adaptation of controller parameters inside the dead-zone and using new

reinforced algorithm to remove the tracking error for any reference signal. A new extended lemma for MIMO systems is proposed in Section 4. Closed loop system stability and output tracking are ensured, for both SISO and MIMO systems, by analyzing suitable Lyapunov candidates based on the proposed controllers. Numerical simulation results of new proposed lemmas are presented in Section 5. Finally, Section 6 concludes the paper.

2. Problem statement In this paper we assume that the plant model is given by

 yp ¼ GðsÞ up þ d ; GðsÞ ¼ G0 ðsÞ I þ Δm ðsÞ ;

ð1Þ

where G0 ðsÞ is a nominal plant, Δm ðsÞ A RH 1 is unknown uncertainty with actuator failures such that j j Δm j j 1 r δ, and d A l1 is the disturbance vector. This model can be simplified as follows: yðt Þ ¼ G0 ðsÞup ðt Þ þ ζ ðt Þ; in which




ð2Þ 

ζ ðt Þ ¼ G0 ðsÞΔm ðsÞ up þ d þ G0 ðsÞd: The control objective is to design an output feedback control signal up ðt Þ A ℝN , such that the plant output yp A ℝN asymptotically tracks the generated output ym ðtÞ A ℝN from the stable rational reference model W m ðsÞ and the bounded reference input signal r ðt Þ A ℝN , i.e. ym ðt Þ ¼ W m ðsÞr ðt Þ: The standard control signal for SISO MRAC systems is given by up ¼ θ ω ¼ θe yp ðt Þ þ θu xu ðt Þ þ θy xy ðt Þ þ θr r ðt Þ; T

T

T

ð3Þ

where xu ðt Þ ¼ H ðsÞup ; xy ðt Þ ¼ H ðsÞyp ; H ðsÞ ¼

 n2 T s …s 1 V H ðsÞ ¼ ; RH ðsÞ RH ðsÞ

RH ðsÞ ¼ sn  1 þ … þ h1 s þ h0 ; 

ω ¼ yp ;

V H ðsÞ V H ðsÞ y ; up ; r RH ðsÞ p RH ðsÞ

T :

whereas θu ; θy ; θe ; θr A ℝ1 are controller parameters. Besides, RH ðsÞ is an arbitrary Hurwitz polynomial of degree n  1. If there is no modeling uncertainty and output disturbance with actuator fault, nT the ideal control signal unp ðt Þ ¼ θ ω can be determined from the matching equations developed in classical MRAC theory, in which h i θn ¼ θne ; θnuT ; θnyT ; θnr is the ideal controller parameter vector. By using the bilinear model, the output tracking error eðt Þ ¼ yp ðt Þ  ym ðtÞ, and the normalized output tracking error es ðt Þ for the perturbed system (2) are given as follows [34]:  T e ¼ W m ðsÞ ρn θ~ ω þ ζ ;  T es ¼ e W m ðsÞ es n2s ¼ W m ðsÞ ρn θ~ ω þ ρn ζ  es n2s ; ð4Þ in which ρn ¼ K p =K r where K p and K r are respectively high frequency gain of the system and model reference,  T T T T θ~ ¼ θ~ e ; θ~ u ; θ~ y ; θ~ r is the vector of controller parameter deviation from the ideal vector and ns is the normalizing signal. If all the controller parameters are updated based on the deadzone modification rule [34] we have
 ð5Þ θ_ ¼ Γωðes þ gÞsgn ρn ;

H.M. Bagherpoor, F.R. Salmasi / ISA Transactions 57 (2015) 51–56

in which symmetric positive definite Γ is the adaptation gain, then the closed loop stability is guaranteed. In this case, the steady state tracking error is significant enough to demand for better solution. In the above adaptation rules using continuous dead-zone, we have 8 if es m o  g 0 > < g 0 =m g ¼  g 0 =m if es m 4 g 0 ð6Þ > : e if jes j r g 0 =m s where m2 ¼ 1 þ n2s is the dynamic normalizing signal and g 0 is a constant parameter, which represents the dead-zone band. The stability properties of conventional dead-zone modification are discussed in [34].

3. Output tracking controller based on reinforced dead-zone modification: SISO case If a Strictly Positive Real (SPR) reference model is chosen such



 as W m ¼ ðsI  Am Þ  1 , and Am ¼ diag  γ i ; Re γ i 40, then we can write following state space representation for tracking error:

nT ð7Þ e_ s ¼ Am es þ ρn up  θ ω þ ζ  ρn  1 es n2s ; The following lemma is about asymptotic stability of the above system. Lemma 1. For the system given by (2) and the control signal outside the dead-zone given by (5), if the control signal is updated inside the dead-zone using the following rule
 es ðt Þ T  udz ðt Þ ¼ θdz ωðt Þ  μ sgn ρn  ; es ðt Þ þ γ ðt Þ

ð8Þ

in which θdz stands for the last update of controller parameters just before entering the dead-zone region, and γ ðt Þ : ℝ þ - ℝ þ is a _ continuous function such in   that   the dead-zone region γ ðt Þ o 0; limt-1 γ ðt Þ ¼ 0 and γ ðt Þ⪢es ðt Þ, then the output tracking error tends to zero if   
 
 ð9Þ μ 4 γ m n2s  g 0 =mρn  þ sgn ρn θTdz ω þ ζ ; where



γ m ¼ maxi γ i

Proof. Since closed-loop stability analysis outside the dead-zone is similar to the proposed approach in [34]. Stability analysis and zero steady state tracking error are the only concerns which will be resolved in this note due to using continuous dead-zone region. If we use the following Lyapunov function 1 V ðes Þ ¼ pc e2s ; pc Z0 A ℝ; ð10Þ 2

  e2s γ 2ðt Þ ¼ es ðtÞ  γ ðt Þ þ  ; jes jþ γ ðt Þ es ðt Þ þ γ ðt Þ

 and W m ¼ diag 1=ðs þ γ i Þ ; then   V_ r  μqes ðt Þ þ μqγ ðt Þ  μq

2
 γ 2ðt Þ þ pc γ m es ðt Þ es ðt Þ þ γ ðt Þ

   T  2  þ ρn pc es ðt Þθ~ dz ω þ ζ  pc es ðt Þ n2s

  T    

   ¼  es ðt Þ μq  pc γ m  n2s es ðt Þ  ρn pc θ~ dz ω þ ζ  þ μqφðtÞ ð12Þ where

! γ 2 ðt Þ  φðt Þ ¼ γ ðt Þ   ; es ðt Þ þ γ ðt Þ   Consequently, since inside the dead-zone es ðt Þ og 0 =m,    

  T  V_ r  es ðt Þ μq  g 0 =m pc γ m  n2s   ρn pc θ~ dz ωþ ζ  þ μqφðtÞ ð13Þ Since function such that in the dead-zone  γ ðtÞ is  a continuous  region γ ðt Þ⪢es ðt Þ and γ 2 ðt Þ converges to zero fast enough, then limt-1 φðt Þ ¼ 0; so we conclude that if  

  T  μq 4 g 0 =m pc γ m  n2  þ ρn pc θ~ ω þζ ; s

dz

which is equivalent to (9), thus there is t F and μ4 0 such that 8 t Z t F we have V_ o0. This concludes that limt-1 es ðt Þ ¼ 0: Remark 1. Assuming that at t ¼ t F , the tracking
þ  error
  enters the dead-zone region, γðtÞ is chosen such that u t dz  u t dz o ε; where  ε denotes a very small constant. This is achieved if γ ðt dz Þ⪢ es ðt dz Þ:

4. Output tracking controller based on reinforced dead-zone modification: MIMO case Model reference adaptive controller for MIMO systems is given as   up ¼ ΘT1 Ω1 ΘT2 Ω2 … ΘTN ΩN ; ð14Þ where  T 1N ; Θ1 ðt Þ ¼ θT ðt Þθ12 u ðt Þ … θ u ðt Þ  T  T 23 2N Θ2 ðt Þ ¼ θ ðt Þθu ðt Þ… θu ðt Þ ; ⋮  T ΘN ðt Þ ¼ θT ðt Þ ; and   ΩT1 ¼ ωT u2 u3 ⋯uN ;   ΩT2 ¼ ωT u3 ⋯uN ; ⋮ ΩTN

  ¼ ωT ;

where

then, based on (7) and (8) we have
 T e2s V_ ¼  μq þ pc Am e2s þ ρn pc es ðθ~ dz ω þ ζ Þ  pc e2s n2s ; jes j þ γ ðt Þ   n such that q ¼ ρn pc 40 and θ~ dz ¼ θdz  θ . Since

53

ð11Þ

  ω ¼ ωT1 ; ωT2 ; yT ; r T ;

    ω1 ; ω2 A ℝNðν  1Þ ; ω1 ¼ AðsÞ=ΛðsÞ u; ω2 ¼ AðsÞ=ΛðsÞ y;  T AðsÞ ¼ s Isν  2 Isν  3 ⋯Is I ; I A ℝNN ; θu A ℝNðν  1Þ : ΛðsÞ is a monic Hurwitz polynomial of order ν 1 and ν is the observability index of G0 ðsÞ: In [34], it is shown that the high frequency gain can be factored as K p ¼ lims-1 sW m ðsÞ ¼ SDU where S is a symmetric positive definite matrix, D ¼ diagðdi Þ; i ¼ 1; 2; … ; N and U is a unity upper

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triangular matrix. Then the tracking error is given by,
 e ¼ W m ðsÞSD up  Θn T Ω þ ζ ;

ð15Þ

So, the normalized tracking error is generated as follows: es ¼ e  W m ðsÞ es n2s ;

ð16Þ



n T es ¼ W m ðsÞ SD up  Θ Ω þ ζ  ðSDÞ  1 es n2s : Therefore, the state space representation for tracking error is given by

n T ð17Þ e_ s ¼ Am es þ ðSDÞ up  Θ Ω þ ζ  ðSDÞ  1 es n2s ; Lemma 2. For the MIMO system given by (17), if inside the dead-zone the control signal is updated by udz ðt Þ ¼ Θdz Ω  ΞH; T

ð18Þ

where H¼β

es ðtÞ ; j j es ðtÞj j þ j j ϒ ðtÞj j

Fig. 1. System output and tracking error for the first scenario in the SISO system.

in which ϒ ðt Þ : ℝ þ -ℝnþ is continuous decreasing function such that in the dead-zone region limt-1 j j ϒ ðtÞj j ¼ 0; j j ϒ ðtÞj j ⪢ j j es ðtÞ j j and Ξ is such that SDΞ is positive definite (PD), then limt-1 j j ess ðtÞj j ¼ 0; if
 γ  n2 ‖g‖ λmax ðSDÞ ~ T
 ‖Θ dz Ω þ ζ ‖: þ β Z m
s ð19Þ λmin Ψ m λmin Ψ Remark 2. For MIMO systems, we define the dead-zone region  where j j es ðtÞj j r j j gj j =m; in which g ¼ g 1 … g m and g i is the designated band for each channel. Proof. With the proposed controller, we have  e s ðt Þ ~ T Ω þ ζ  ðSDÞ  1 e n2 þΘ e_ s ¼ Am es þðSDÞ  β s s dz ‖es ðt Þ‖þ ‖ϒ ðt Þ‖ ð20Þ For stability analysis, the following Lyapunov function is used: 1 V ðes Þ ¼ ‖e2s ‖ 2

ð21Þ


T Since SDΞ is PD, we define Ψ  SDΞ þ SDΞ 4 0. The time derivative of this Lyapunov function is given by V_ ¼  β

T

eTs Ψ es ~ Ω þ ζ  ‖e2 ‖n2 þ eTs Am es þ eTs ðSDÞ Θ s s dz ‖es ðt Þ‖ þ‖ϒ ðt Þ‖ ð22Þ

On the other hand, we have

Remark 3. The controller gains μ and β are not related to the tracking error which results in less computational complexity in real time implementation of this controller. 5. Illustrative example To illustrate performance of the proposed scheme, while comparing with the conventional dead-zone approach, we consider two case studies: a SISO system and a MIMO system. First, a SISO system is selected which is given by following transfer function   0:3 2s þ 5 Y ðsÞ ¼ U ðsÞ; sþ1 s þ 0:5 The reference model is chosen to be, W m ðsÞ ¼

1 ; sþ3

In this test, two scenarios are considered: a. Actuator efficiency loss and a delay with τ ¼ 0:2 s, which imposed at t ¼ 125 s, such that the system model is changed to   0:03 2s þ 5  0:2s e Y ðsÞ ¼ U ðsÞ; sþ1 s þ 0:5 b. In addition to the above fault, a step additive disturbance signal with amplitude du ¼ 10 is imposed at t ¼ 50 s. The main controller parameters are chosen to beRH ðsÞ ¼ ðs þ 10Þ; g ¼ 0:03;



T 2 2 e Ψe  β j j es j js þ jsj ϒ j j r  βλmin ðΨ Þ j j esjj jjeþs j jj j ϒ j j r  βλmin ðΨ Þ j j es j j  j j ϒ j j þ j j esjj jj ϒþ jj jj ϒ j j

2 ~ T Ω þζj j j j es j j n2 j j es j j 2 V_ r  βλmin ðΨ Þ j j es j j  j j ϒ j j þ j j esjj jj ϒþ jj jj ϒ j j þ γ m j j es j j 2 þ λmax ðSDÞj j Θ s dz

T γ m  n2s λmax ðSDÞ ~ Ω þ ζj j þ βλmin ðΨ ÞΦðtÞ ¼  λmin ðΨ Þj j es j j β  λmin ðΨ Þ j j es j j  λmin ðΨ Þ j j Θ dz

where

 Φðt Þ ¼ j j ϒ ðtÞj j  ‖ϒ ðt Þ‖2 = ‖es ðt Þ‖ þ ‖ϒ ðt Þ‖ and inside the dead-zone we have j j ϒ ðtÞj j ⪢j j es ðtÞj j . By choosing β as (19) and considering the fact that j j ϒ ðt Þj j 2 tends to zero fast enough, then limt-1 Φðt Þ ¼ 0 and based on (23) there is a t s such that 8 t Z t S we have V_ r 0; or equivalently limt-1 j j es ðtÞj j ¼ 0:

ð23Þ

and μ ¼ 2. Simulation results are given in Figs. 1 and 3 where the robustness and good performance of the suggested control law for MRAC in comparison with conventional dead-zone approach are demonstrated. It can be observed from Fig. 1 that before 125 s, the system operates in normal case. After 125 s actuator loses its efficiency and delay is imposed on the system. It can be inferred that using the proposed approach the control signal is updated

H.M. Bagherpoor, F.R. Salmasi / ISA Transactions 57 (2015) 51–56

55

based on tracking error to compensate actuator efficiency loss and imposed delay. However, using conventional dead-zone approach, the control signal is updated based on defined controller parameters, which resulted in tracking error in faulty conditions. The results in second scenario are similar to the first scenario. When disturbance signal applies to first scenario at 50 s, the control signal utilizes tracking error to retrieve zero tracking error but the older dead-zone scheme cannot support zero tracking. Furthermore, adaptations of controller parameters for two scenarios are shown in Figs. 2 and 4, which are switched off inside the dead-zone according to new proposed scheme. From Fig. 2 it can be seen that using proposed approach for first scenario, after 125 s the controller parameters adaptation is ceased and the control signal is updated using tracking error. However, in conventional dead-zone approach the control signal is determined based on updating controller parameters which cannot guarantee the zero tracking error.

Fig. 4. Controllers parameters for conventional dead-zone and improved deadzone modification for the second scenario.

Fig. 2. Controllers parameters for dead-zone and improved dead-zone modification for the first scenario.

Fig. 5. System outputs for the first scenario in the MIMO system.

Fig. 3. System output and tracking error for the second scenario in the SISO system.

Fig. 6. System outputs for the second scenario in the MIMO system.

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H.M. Bagherpoor, F.R. Salmasi / ISA Transactions 57 (2015) 51–56

Next, a MIMO system is chosen with the following transfer matrix: 3" " # 2 # 20  200 U 1 ðsÞ Y 1 ðsÞ sþ1 ðs þ 0:5Þðs þ 5Þ 5 ¼ 4 1:6ðs þ 0:2Þ ; 18 Y 2 ðsÞ U 2 ðsÞ ðs þ 0:6Þ ðs þ 1Þðs þ 4Þ In order to design the new adaptive controller we chose the reference model as follows: 2 3 1 0 sþ5 5; W m ðsÞ ¼ 4 0 ðs þ1 5Þ Similar to the numerical tests for SISO systems, two cases are tested for the above MIMO system. In the first scenario, actuator efficiency loss is assumed and a delay with τ ¼ 0:5 s is imposed at t ¼ 35 s, such that the system model is changed to 3 " # 2 # 1  0:5s  70  0:5s " Y 1 ðsÞ U 1 ðsÞ s þ 1e ðs þ 0:5Þðs þ 5Þe 5 ¼ 4 0:1ðs þ 0:2Þ  0:5s ; 100 Y 2 ðsÞ U 2 ðsÞ ðs þ 1Þðs þ 4Þe ðs þ 0:6Þ In the second scenario, in addition to the above fault and delay, an additive disturbance du ¼ ½20 20 is imposed at t ¼ 15 s. The main controller parameters are chosen to beΛðsÞ ¼ ðs þ 10Þ2 , g ¼ ½0:05 0:05; and β ¼ ½5 5. Simulation results are given in Figs. 5 and 6, where we show output signals of conventional deadzone, improved dead-zone modification method and desired output. In the simulation results, it is clear that by using new adaptation scheme, the robustness and good performance are guaranteed despite the unknown plant uncertainty and actuator fault. 6. Conclusion In this paper, an adaptive fault-tolerant control scheme for SISO and MIMO systems described by uncertain linear input–output models is considered. The proposed robust adaptive controllers utilize the improved dead-zone modification approach to eliminate tracking error in the presence of faulty conditions. The main concept of proposed controllers is to switch of the controllers parameters adaptation inside the dead-zone and updating control signal based on output tracking error. It is shown that the closedloop system remains stable and tracking error tends to zero for any input generated by the reference model. The proposed closed loop system based on robust adaptive control approach is capable of tracking any reference signal for uncertain linear SISO and MIMO systems, with output disturbance and actuator failures. Simulation results validate performance of the proposed schemes. References [1] Marino R, Tomei P. Robust adaptive regulation of linear time-varying systems. IEEE Trans Autom Control 2000;45(7):1301–11. [2] Ye D, Yang G. Adaptive fault tolerant tracking control against actuator faults with application to flight control. IEEE Trans Control Syst Technol 2006; 14(6):1088–96. [3] Zhai J, Zha W. Global adaptive output feedback control for a class of nonlinear time-delay systems. ISA Trans 2014;53:2–9. [4] Tao G, Chen S, Joshi S. An adaptive actuator failure compensation controller using output feedback. IEEE Trans Autom Control 2002;47(3):506–11. [5] Wu H. Adaptive robust tracking and model following of uncertain dynamical systems with multiple time delays. IEEE Trans Autom Control 2004; 49(4):611–6.

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