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Adaptive control for a class of MIMO nonlinear time delay systems against time varying actuator failures Mahnaz Hashemi n, Jafar Ghaisari, Javad Askari Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 14 September 2014 Received in revised form 19 February 2015 Accepted 20 February 2015 This paper was recommended for publication by Prof. A.B. Rad
This paper investigates an adaptive controller for a class of Multi Input Multi Output (MIMO) nonlinear systems with unknown parameters, bounded time delays and in the presence of unknown time varying actuator failures. The type of considered actuator failure is one in which some inputs may be stuck at some time varying values where the values, times and patterns of the failures are unknown. The proposed approach is constructed based on a backstepping design method. The boundedness of all the closed-loop signals is guaranteed and the tracking errors are proved to converge to a small neighborhood of the origin. The proposed approach is employed for a double inverted pendulums benchmark and a chemical reactor system. The simulation results show the effectiveness of the proposed method. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Nonlinear time delay systems Time varying actuator failure Multi Input Multi Output Adaptive control Backstepping
1. Introduction Time delays are frequently encountered in many practical control systems. Because of the effect of time delays, these systems may have instability and poor performance [1,2]. So far, considerable attention has been devoted to the stability analysis and control design for time delay systems [3–7]. One of the other serious problems that often occurs in many practical systems and brings uncertainties to the systems is actuator failures. Actuator failures often cause undesired system behavior and sometimes lead to instability or even catastrophic accidents. The desired control design is expected to compensate the actuator failures by effectively compensating the uncertainties. In recent years, there have been remarkable efforts in reliable control of systems with actuator failures [8–11]. Adaptive mechanisms show suitable performance in the presence of uncertainties in failed actuators. Some important results of adaptive control of systems with actuator failures exist in [12–15] for linear systems and in [16–22] for nonlinear systems. For example, for linear systems, direct adaptive control schemes were developed in [12,13] for multi input systems with unknown parameters and stuck type actuator failures. In [14,15], model reference adaptive controllers were developed for Multi Input Single Output (MISO) linear systems with unknown actuator failures. For nonlinear systems, in [16–18] the adaptive controllers were presented for strict feedback MISO nonlinear systems with unknown parameters and stuck type actuator failures. In [19–21], the adaptive actuator failure compensation schemes were presented for a class of Multi Input Multi Output (MIMO) nonlinear systems in strict feedback form. In [22], a fuzzy adaptive actuator failure compensation approach was proposed for a class of MISO uncertain stochastic nonlinear systems in strict feedback form. The proposed fuzzy adaptive actuator failure compensator in [22] promised the boundedness of all the signals in the closed loop system; however, the tracking problem was not considered. The existence of time delays renders the actuator failure compensation problem much more complex and difficult. Many valuable research and practical results have been achieved in actuator failure compensation for linear time delay systems [23–29]. For example, in [23], an adaptive control scheme was investigated for multi input linear systems with actuator failures. In [24], a reliable controller was developed within the framework of Linear Matrix Inequalities for a class of multi input linear systems with time delay in control inputs and constant stuck type actuator failures. A direct adaptive control scheme was introduced in [25] for a class of multi input linear state n
Corresponding author. E-mail addresses: [email protected], [email protected] (M. Hashemi).
http://dx.doi.org/10.1016/j.isatra.2015.02.012 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Hashemi M, et al. Adaptive control for a class of MIMO nonlinear time delay systems against time varying actuator failures. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.02.012i
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delay systems with constant stuck failures in actuators. The same problem was solved for decentralized systems in [26]. Based on a linear matrix inequality technique and an adaptive method, [27,28] suggested adaptive reliable controllers for multi input linear time delay systems with known parameters against loss of effectiveness actuator failures. In [29,30], the adaptive controllers were designed for MISO state delay systems with unknown parameters and constant stuck failures in actuators. To the best of the authors' knowledge, relatively few papers were published regarding actuator failure compensation for nonlinear time delay systems [31–35]. In [31], an adaptive actuator failure compensation scheme was developed for a class of multi input nonlinear time delay systems with known parameters, constant delay and known constant control gains. In [32], a robust compensation approach was presented for multi input nonlinear time delay systems with known parameters, known constant control gains and unknown constant delays. The proposed compensation approach in [32] promised the boundedness of all the signals in the closed loop system; however, the tracking problem was not considered. In [33], an adaptive actuator failure compensation scheme was presented for a class of parametric strict feedback MIMO nonlinear time delay systems with unknown parameters and time delays. The considered actuator faults in [31–33] were types of loss of effectiveness in which the system inputs may just lose some fraction of their effectiveness. In [34], an adaptive actuator failure compensation scheme was presented for a class of parametric strict feedback MISO nonlinear time delay systems with unknown parameters and stuck failures in actuators. In [35], an adaptive fuzzy actuator failure compensator was proposed for a class of MIMO nonlinear systems in strict feedback form with known constant control gains and known actuator failures. The values, patterns, and time occurrences of the considered actuator failure in [35] were known. This paper investigates the adaptive control problem for a class of MIMO nonlinear time delay systems with variable control gains, unknown parameters, unknown time varying state delays and in the presence of unknown time varying actuator failures. The type of considered actuator failure is one in which some unknown inputs may be stuck at some unknown time varying values where the values, times and patterns of the failures are unknown that is, during system operation it is unknown when, by how much and which actuators fail. The proposed adaptive controller is constructed based on the backstepping design method. Compared with the existing results, the main contributions of this paper are as follows: (i) The control problem is investigated for a class of MIMO nonlinear time delay systems with unknown parameters, variable control gains, unknown time varying delays and in the presence of unknown time varying actuator failures. (ii) The considered actuator failures are unknown stuck types that not only cause the system gain changes but also lead to system uncertainties. (iii) A differentiable function is used to solve the differentiability problem and to avoid the singularity problem of the intermediate controllers. (iv) Appropriate Lyapunov–Krasovskii functionals are introduced to design new adaptive laws to compensate the unknown time varying actuator failures as well as uncertainties from unknown parameters and unknown state delays. (v) The proposed systematic backstepping design method proves that, with no need for explicit failure detection not only are all the signals in the closed loop system bounded, but also the tracking errors converge to a small neighborhood of the origin. The paper is organized as follows. In Section 2, the system description is given along with the necessary assumptions. In Section 3, the design procedure of the proposed adaptive controller is explained. In Section 4, the stability analysis of the proposed controller is investigated. In Section 5, simulation results are presented to illustrate the effectiveness of the proposed control method. Finally, the paper is concluded in Section 6.
2. Problem statement Consider a class of strict-feedback nonlinear time delay systems in the following form: T T x_ j;1 ðt Þ ¼ g j;1 xj;2 ðt Þ þ θf j;1 F j;1 ðxj;1 ðtÞÞ þ θhj;1 H j;1 xj;1 ðt τj;1 ðtÞÞ T T x_ j;2 ðt Þ ¼ g j;2 xj;3 ðt Þ þ θf j;2 F j;2 xj;2 ðtÞ þ θhj;2 H j;2 xj;2 t τj;2 ðt Þ : :
: x_ j;γ j 1 ðt Þ ¼ g j;γ
x ðt Þ þ θf j;γ j 1 j;γ j T
F j;γ j 1 ðxj;γ j 1 ðtÞÞ þ θhj;γ 1 T
j
H j;γ j 1 xj;γ j 1 ðt τj;γ j 1 ðt ÞÞ 1 j
ð1Þ
T T T x_ j;γ j ðt Þ ¼ θf j;γ F j;γ j ðxðtÞÞ þ θhj;γ H j;γ j ðxðt τj;γ j ÞÞ þ β j ðxðtÞÞuðtÞ j
j
yj ðt Þ ¼ xj;1 ðt Þ;
j ¼ 1; …; q:
where x A Rγ , γ ¼ γ 1 þ γ 2 þ … þ γ q , xj;i ¼ ½x1;1 ; …; x1;γ 1 γ j þ i ; …; xl;1 ; …; xl;γ l γ j þ i ; …; xq;1 ; …:; xq;γ q γ j þ i T are the state vectors, y ¼ ½y1 ; y2 ; …; yq T A Rq is the output vector and u ¼ ½u1 ; u2 ; …; um T A Rm ; m 4 q is the control input vector whose actuators may fail during operation. βj ð:Þ; F j;i ð:Þ and H j;i ð:Þ are known smooth nonlinear function vectors, θf j;i and θhj;i are unknown constant parameter vectors, g j;i is an unknown constant parameter and τj;i ðtÞ is an unknown time varying delay of the states. For systems in which actuators may fail during the operation of the system, one common approach is to use actuator redundancy. In this way, when one actuator fails, some others could compensate the effect [16]. To deal with the actuator redundancy, the m actuators are divided into q groups corresponding to q outputs as following: q X
mi ¼ m;
mi 4 1
for
i ¼ 1; …; q
i¼1
where mi is the number of the actuators in the ith group. The grouping scheme is as follows: n n o n oo Δ : j1;1 ; j1;2 ; …; j1;m1 ; …; jk;1 ; jk;2 ; …; jk;mk ; …; jq;1 ; jq;2 ; …; jq;mq where
n
jk;1 ; jk;2 ; …; jk;mk
o
ð2Þ
f1; 2; …; mg. Thus, the actuators with same or similar physical characteristics are in the same group while the
actuators of different physical characteristics usually should not be in the same group [19]. Therefore, the following proportional actuation Please cite this article as: Hashemi M, et al. Adaptive control for a class of MIMO nonlinear time delay systems against time varying actuator failures. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.02.012i
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scheme is used: vk ¼
q X
Sk;l wl ;
k ¼ 1; …; m
ð3Þ
l¼1
where wl is the nominal control for group l to be designed in the next section and Sk;l ; k ¼ 1; …; m; l ¼ 1; …; q; are the elements of the matrix S which is a design function matrix with m rows and q columns; Each column of the matrix S has some nonzero elements with certain physical meaning that indicate the actuators of the same group. The actuator failures to be considered are modeled as uk ðt Þ ¼ uk ðt Þ;
t Z tk ;
k ¼ k1 ; k2 ; …; kp ;
1 r p r mi 1;
i ¼ 1; …; q
ð4Þ
where uk ðt Þ is given by u k ð t Þ ¼ Θk Ψ k ; T
k ¼ k1 ; k2 ; …; kp ;
1 r p r mi 1;
i ¼ 1; …; q
ð5Þ
The failure time instant t k , the failure index k and the constant vector Θk ¼ ½dk;1 ; dk;2 ; …; dk;h are unknown and the bounded signal vector Ψ k ¼ ½Ψ k;1 ; Ψ k;2 ; …; Ψ k;h T ; h Z 1 is known. The stuck type actuator failures can occur due to hydraulic failures that can produce unintended movements in the control surfaces of an aircraft [16]. For plant (1) with the actuator failure (4) and (5), the input vector can be expressed as T
uðt Þ ¼ vðt Þ þ δðuðtÞ vðt ÞÞ ; uðt Þ ¼ ½u1 ðtÞ; …; um ðtÞT ;
uðt Þ ¼ ½u1 ðtÞ; …; um ðtÞT ;
δ ¼ diag δ1 ; …; δm ;
δk ¼
vðt Þ ¼ ½v1 ðtÞ; …; vm ðtÞT ( 1; uk ðt Þ ¼ uk ðtÞ 0;
ð6Þ
uk ðt Þ a uk ðtÞ
where vk ðtÞ is the applied control input to be designed later. The control objective is to design a state feedback controller for plant (1) in order to assure that all the closed loop signals are bounded and the plant outputs yj ðt Þ; j ¼ 1; …; q; track the desired signals ydj ðt Þ; j ¼ 1; …; q; despite the presence of unknown plant parameters, unknown actuator failures (4) and (5) and unknown state delays. For this purpose, the following assumptions are considered. Assumption 1. ([12–22,24–26,29,30,35]). For plant (1) with known plant parameters and failure parameters, if any up to mi 1 actuators fail in each group, the remaining actuators can still achieve a desired control objective. j ¼ 1; …; q and its first γ j th order derivatives yðkÞ dj
Assumption 2. The desired signal ydj ðt Þ; piecewise continuous.
ðk ¼ 1; …; γ j Þ are known, bounded, and
Assumption 3. The unknown time varying delay τj;i ðt Þ is a differentiable function satisfying 0 r τj;i ðt Þ r τj;i ; τ_ j;i ðtÞ r ϑj;i o 1
ð7Þ
where τj;i and ϑj;i are known positive constants.
Assumption 4. The signs of g j;i are known and there exist unknown constants g max Z g min 4 0 such that g min r g j;i r g max . For the controller design, without loss of generality it is assumed that g j;i 4 0;
j ¼ 1; …; q;
i ¼ 1; …; γ j .
3. Adaptive controller design In this section, an adaptive actuator failure compensation scheme is developed for the MIMO nonlinear time delay system (1) by employing the backstepping design method [36]. At first, the following state transformation is considered: zj;1 ¼ xj;1 ydj ;
j ¼ 1; …; q;
zj;i ¼ xj;i αj;i 1 ;
i ¼ 2; …; γ j
ð8Þ
where αj;i 1 are the intermediate control functions to be designed later. The transformed system in the new coordination becomes T T z_ j;1 ðt Þ ¼ g j;1 zj;2 ðt Þ þ g j;1 αj;1 þ θf j;1 F j;1 xj;1 þ θhj;1 H j;1 xj;1 ðt τj;1 ðtÞÞ y_ dj T T _ j;1 z_ j;2 ðt Þ ¼ g j;2 zj;3 ðt Þ þ g j;2 αj;2 þ θf F j;2 xj;2 þ θh H j;2 xj;2 t τj;2 ðt Þ α j;2
:
j;2
ð9Þ
: : T T T _ j;γ j 1 z_ j;γ j ðt Þ ¼ θf j;γ F j;γ j ðxÞ þ θhj;γ H j;γ j xðt τj;γ j Þ þ β j ðxÞuðtÞ α j j The adaptive compensation approach is designed in the following procedures. 3.1. Adaptive controller design for zj;1 ; j ¼ 1; …; q; subsystems
In this subsection, the zj;1 ; j ¼ 1; …; q; subsystems are considered and the controller is designed for these subsystems. For the zj;1 subsystems, the following Lyapunov functions are considered: V zj;1 ¼
1 2 z ðtÞ 2g j;1 j;1
ð10Þ
Please cite this article as: Hashemi M, et al. Adaptive control for a class of MIMO nonlinear time delay systems against time varying actuator failures. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.02.012i
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V U j;1 ¼
Z
1 2ð1 ϑj;1 Þg j;1
t t τj;1 ðt Þ
eμj;1 ðξ tÞ H Tj;1 xj;1 ξ H j;1 ðxj;1 ðξÞÞdξ
ð11Þ
1 T 1 V θj;1 ¼ θ~ j;1 Γ j;1 θ~ j;1 2
ð12Þ
V j;1 ¼ V zj;1 þV U j;1 þ V θj;1
ð13Þ
where μj;1 40; j ¼ 1; …; q, Г j;1 ¼ Г j;1 4 0, g j;1 is an unknown constant parameter defined in (1), ϑj;1 is a positive constant defined in (7), H j;1 ð:Þ is a known smooth nonlinear function vector defined in (1) and θ~ j;1 ¼ θ^ j;1 θj;1 in which θ^ j;1 is the estimate of θj;1 to be defined in (15). Along system (9), the time derivative of V zj;1 satisfies ! y_ dj 1 T 1 T _ θ F x þ θhj;1 Hj;1 xj;1 ðt τj;1 ðtÞÞ V zj;1 ¼ zj;1 ðt Þ zj;2 ðt Þ þ αj;1 þ g j;1 f j;1 j;1 j;1 g j;1 g j;1 T
By using the Young's inequality [35], the time derivative of V zj;1 becomes y_ d zj;1 ðtÞ T θf j;1 F j;1 xj;1 ðt Þ zj;1 ðt Þ j V_ zj;1 r zj;1 ðt Þzj;2 ðt Þ þ zj;1 ðt Þαj;1 ðt Þ þ g j;1 g j;1 þ
eμj;1 τj;1 2 e μj;1 τj;1 T T z ðtÞθhj;1 θhj;1 þ H j;1 xj;1 t τj;1 ðt Þ H j;1 xj;1 t τj;1 ðt Þ 2g j;1 j;1 2g j;1
By applying Assumption 3, the time derivative of V U j;1 becomes H Tj;1 xj;1 ðt Þ H j;1 xj;1 ðt Þ e μj;1 τj;1 T _ H j;1 xj;1 t τj;1 ðtÞ H j;1 xj;1 t τj;1 ðtÞ V U j;1 r μj;1 V U j;1 þ 2g j;1 2 1 ϑj;1 g j;1 By considering (13), the time derivative of V j;1 becomes T T 1 _ V_ j;1 ¼ V_ zj;1 þ V_ U j;1 þ V_ θj;1 rzj;1 ðt Þzj;2 ðt Þ þzj;1 ðt Þαj;1 ðt Þ þzj;1 ðt Þθj;1 φj;1 μj;1 V U j;1 þ θ~ j;1 Γ j;1 θ^ j;1
where 2
3T
θTf θTh θh 1 θj;1 ¼ 4 j;1 ; j;1 j;1 ; 5 g j;1
"
ð14Þ
g j;1
eμj;1 τj;1 zj;1 ; 2
φj;1 ¼ F Tj;1 xj;1 ðt Þ ;
ð15Þ
g j;1
( T )#T H j;1 xj;1 ðt Þ H j;1 xj;1 ðt Þ y_ dj ðt Þ 2zj;1 ðt Þ 1 ϑj;1
ð16Þ
where F j;1 ð:Þ and H j;1 ð:Þ are known smooth nonlinear function vectors defined in (1), θf j;1 and θhj;1 are unknown constant parameter vectors, g j;1 is an unknown constant parameter and τj;1 and ϑj;1 are defined in (7). According to (14), the intermediate control function is selected as
T
1 2
αj;1 xj;1 ðt Þ ¼ θ^ j;1 φj;1 zj;1 ϖ j;1 zj;1
ð17Þ
where ϖ j;1 ; j ¼ 1; …; q; are positive constants. However, it is clear that at zj;1 ¼ 0, controller singularity occursbecause φj;1 defined in (16) is not well-defined at zj;1 ¼ 0. Hence, modification should be done to guarantee the boundedness of αj;1 xj;1 ðt Þ . Besides, the computation of the intermediate control functions, αj;i ðt Þ; j ¼ 1; …; q; i ¼ 1; …; γ j , in each step of the backstepping design procedure requires that of α_ j;i 1 ðt Þ; α€ j;i 2 ðt Þ; …; αðj;1i 1Þ : As a result, αj;i ðt Þ needs to be at least ðγ j iÞth differentiable. Therefore, in this step, αj;1 ðt Þ should be at least ðγ j 1Þth differentiable. By using function qj;1 ðzj;1 Þ defined in Lemma 1, the singularity problem of the controller can be solved and the controller will still be ðγ j 1Þth differentiable. Lemma 1. [3]: Let the function qj;i ðxÞ : R-R be defined as qj;i ðxÞ 8 1; > > > > iðγ j iÞ > R x h > > > λaj;i þ λbj;i σ σ λaj;i dσ ; < cqj;i λa j;i ¼ h i γ R λ ð j iÞ > > > cqj;i x aj;i λaj;i þ λbj;i þ σ σ þ λaj;i dσ ; > > > > > : 0;
jxj Z λaj;i þ λbj;i
λaj;i o x o λaj;i þ λbj;i λaj;i þ λbj;i o x o λaj;i jxj r λaj;i 2ðγ iÞ þ 1
where λaj;i ; λbj;i 4 0 are constant parameters and cqj;i ¼ ðð½2ðγ j iÞ þ1!Þ=ðλb j j;i limited between 0 and 1. Therefore, the intermediate controller (17) is rewritten as
T 1 αj;1 xj;1 ðt Þ ¼ qj;1 zj;1 θ^ j;1 φj;1 zj;1 ϖ j;1 zj;1 2
½ðγ j iÞ!2 ÞÞ, thus qj;i ðxÞ is ðγ j iÞth differentiable and is
ð18Þ
Please cite this article as: Hashemi M, et al. Adaptive control for a class of MIMO nonlinear time delay systems against time varying actuator failures. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.02.012i
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where ϖ j;1 is a positive constant. The updating law for parameter θ^ j;1 is chosen as _
θ^ j;1 ¼ qj;1 ðzj;1 ÞΓ j;1 ½zj;1 φj;1 σ j;1 θ^ j;1
ð19Þ
where Γ j;1 ¼ Γ j;1 4 0 and small positive constant σ j;1 is used to introduce the σ -modification for the closed loop system. For clarification the following sets are considered: T
Region 1 : fzj;i ϵR j j zj;i j Z λaj;i þ λbj;i g Region 2 : fzj;i ϵR j λaj;i o zj;i o λaj;i þ λbj;i g Region 3 : fzj;i ϵR j j zj;i j r λaj;i g
ð20Þ
Case (1): For region 1, zj;1 Z λaj;1 þ λbj;1 . In this region qj;1 zj;1 ¼ 1, hence V_ j;1 ðtÞ becomes T T 1 _ 1 1 V_ j;1 r z2j;2 ϖ j;1 z2j;1 μj;1 V U j;1 zj;1 ðt Þθ~ j;1 φj;1 þ θ~ j;1 Γ j;1 θ^ j;1 2 2
and the updating law (19) becomes _
θ^ j;1 ¼ Γ j;1 ½zj;1 φj;1 σ j;1 θ^ j;1
ð21Þ
T By using the inequality σ j;1 θ~ j;1 θ^ j;1 r 12σ j;1 j j θ~ j;1 j j 2 þ 12σ j;1 j j θj;1 j j 2 , the time derivative of V j;1 becomes
V_ j;1 r C j;1 V j;1 þ 12z2j;2 þ X j;1
!
σ j;1 ; C j;1 ¼ min g min ϖ j;1 ; μj;1 ; 1 λmax ðΓ j;1 Þ
1 X j;1 ¼ σ j;1 j j θj;1 j j 2 2
In this region, the time derivative of V j;1 is dependent on zj;2 ; therefore the boundedness of V j;1 will be proved in the next step when the boundedness of zj;2 is verified. Case (2): For region 2, λaj;1 o j zj;1 j o λaj;1 þ λbj;1 . In this region, zj;1 is bounded; according to Assumption 2, ydj is also bounded. Since xj;1 ¼ zj;1 þydj and zj;1 ; ydj are bounded, xj;1 is also bounded; hence V zj;1 and V U j;1 are bounded. By using (12) and (19) in this region, the time derivative of V θj;1 becomes T T 1 _^ V_ θj;1 ¼ θ~ j;1 Γ j;1 θj;1 ¼ qj;1 ðzj;1 Þθ~ j;1 ½zj;1 φj;1 σ j;1 θ^ j;1
ð22Þ
By applying the inequality c0j;1 T 1 qj;1 zj;1 θ~ j;1 zj;1 φj;1 r 0 qj;1 zj;1 j j θ~ j;1 j j 2 þ qj;1 zj;1 φTj;1 φj;1 z2j;1 ; c0j;1 40 2cj;1 2 and by using σ j;1 θ~ j;1 θ^ j;1 r 12σ j;1 j j θ~ j;1 j j 2 þ 12σ j;1 j j θj;1 j j 2 , Eq. (22) is rewritten as T
2 1 1 1 q ðz Þðσ Þθ~ þ qj;1 ðzj;1 Þðσ j;1 j j θj;1 j j 2 þc0j;1 φTj;1 φj;1 z2j;1 Þ V_ θj;1 r 2 j;1 j;1 j;1 c0j;1 j;1 2 In this region, φj;1 is a smooth and bounded function and qj;1 zj;1 A ð0; 1Þ. By choosing c0j;1 such that σ nj;1 ¼ σ j;1 ð1=c0j;1 Þ 4 0,
V_ θj;1 r C nθj;1 V θj;1 ðt Þ þ λθj;1 C nθj;1 ¼ qj;1 zj;1
σ nj;1
λmax Γ
1 j;1
1 ; λθj;1 ¼ qj;1 zj;1 σ j;1 j j θj;1 j j 2 þ c0j;1 φTj;1 φj;1 z2j;1 2
Since λaj;1 o j zj;1 j o λaj;1 þ λbj;1 , then h i Cn t λθ 0 r V θj;1 ðt Þ r V θj;1 ð0Þ ρθj;1 e θj;1 þ ρθj;1 ; ρθj;1 ¼ nj;1 C θj;1 Thus, V θj;1 is bounded. Eventually V j;1 ðt Þ is bounded for λaj;1 o j zj;1 j o λaj;1 þ λbj;1 . Case (3): For region 3, zj;1 r λaj;1 . In this region, zj;1 is bounded; according to assumption 2, ydj is also bounded. Since xj;1 ¼ zj;1 þ ydj and _ zj;1 ; ydj are bounded, xj;1 is also bounded. In this region, qj;1 zj;1 ¼ 0, thus based on (19), θ^ j;1 ¼ 0 and θ^ j;1 is kept unchanged in a bounded value and V θj;1 is bounded. In addition, for bounded xj;1 and zj;1 , it can be concluded that V zj;1 and V U j;1 are bounded, hence V j;1 ðtÞ is bounded. As can be seen, the singularity problem of the intermediate controller at zj;1 ¼ 0 is solved by using qj;1 zj;1 , though it is still ðγ j 1Þ-th differentiable, making it possible to carry out the backstepping design method in the next steps. Therefore, all the closed loop signals in regions 2 and 3 are bounded despite the presence of unknown parameters, time varying delays and time varying actuator failures. The boundedness of the closed loop signals in region 1 is dependent on the boundedness of the signal zj;2 to be regulated in the following. 3.2. Adaptive controller design for zj;i ; j ¼ 1; …; q; i ¼ 2; …; γ j 1 subsystems In this subsection, the zj;i ; j ¼ 1; …; q; i ¼ 2; …; γ j 1; subsystems are considered. T T _ j;i 1 ðtÞ z_ j;i ðt Þ ¼ g j;i zj;i þ 1 ðt Þ þ g j;i αj;i ðt Þ þ θf j;i F j;i xj;i þ θhj;i H j;i xj;i t τj;i ðt Þ α
ð23Þ
Please cite this article as: Hashemi M, et al. Adaptive control for a class of MIMO nonlinear time delay systems against time varying actuator failures. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.02.012i
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For these subsystems, the following Lyapunov functions are considered: V zj;i ¼
1 2 z ðtÞ 2g j;i j;i
V U j;i ¼
ð24Þ
γj þ i 1 q γ k X X l¼1
k¼1
1 þ 2ð1 ϑj;i Þg j;i
Z
eμj;i ðξ t Þ H Tk;l xk;l ξ H k;l ðxk;l ξ Þdξ 2 1 ϑk;l g j;i t τk;l ðt Þ Z t eμj;i ðξ tÞ H Tj;i xj;i ðξÞ H j;i ðxj;i ðξÞÞdξ
1
t
ð25Þ
t τ j;i ðt Þ
1 T 1 V θj;i ¼ θ~ j;i Γ j;i θ~ j;i 2
ð26Þ
V j;i ¼ V zj;i þ V U j;i þ V θj;i
ð27Þ
g j;i is an unknown constant parameter, H j;i ð:Þ is a known smooth nonlinear function vector defined in (1), ϑj;i is where μj;i 4 0, Г j;i ¼ Г defined in (7) and θ~ j;i ¼ θ^ j;i θj;i in which θ^ j;i is the estimate of θj;i to be defined in (29). The intermediate controllers αj;i 1 ðt Þ; j ¼ 1; …; q; i ¼ 2; …; γ j ; are functions of xj;i 1 ðt Þ; θ^ 1;1 ; …; θ^ 1;γ 1 γ j þ i 1 ; …; θ^ q;1 ; …:; θ^ q; γ q γ j þ _ j;i 1 ðt Þ becomes i 1; ydj ; yð1Þ …; yðidj 1Þ , hence α dj T j;i 4 0,
γj þ i 1 q γ k X X ∂αj;i 1 n
α_ j;i 1 ðt Þ ¼
k¼1
þ
l¼1
∂xk;l
o T T g k;l xk;l þ 1 ðt Þ þ θf k;l F k;l þ θhk;l H k;l xk;l t τk;l ðt Þ
γj þ i 1 q γ k X X ∂αj;i 1 _^ k¼1
l¼1
∂θ^
θk;l þ
k;l
i X ∂αj;i 1
yðkÞ ðk 1Þ dj k ¼ 1 ∂yj
ð28Þ
By applying the Young's inequality, the time derivative of V zj;i becomes zj;i ðt Þ T θ F x V_ zj;i rzj;i ðt Þzj;i þ 1 ðt Þ þ zj;i ðt Þαj;i ðt Þ þ g j;i f j;i j;i j;i þ
eμj;i τj;i 2 e μj;i τj;i T zj;i ðt ÞθThj;i θhj;i þ H x t τj;i ðt Þ 2g j;i 2g j;i j;i j;i
γj þ i 1 q γ k X i o z ðt Þ X zj;i ðt Þ X ∂αj;i 1 n ∂αj;i 1 ðkÞ j;i H j;i xj;i t τj;i ðt Þ g k;l xk;l þ 1 ðt Þ þ θTfk;l F k;l y ∂xk;l g j;i k ¼ 1 l ¼ 1 g j;i k ¼ 1 ∂yðk 1Þ dj j
γj þ i 1 γj þ i 1 q γ k X q γ k X z2j;i X zj;i ðtÞ X ∂αj;i 1 _^ ∂αj;i 1 2 T eμj;i τk;l θhk;l θhk;l θk;l þ g j;i k ¼ 1 l ¼ 1 2g j;i k ¼ 1 l ¼ 1 ∂xk;l ∂θ^ k;l þ
q γ γj þ i 1 1 X k X e μj;i τk;l H Tk;l xk;l t τk;l ðt Þ H k;l xk;l t τk;l ðt Þ 2g j;i k ¼ 1 l ¼ 1
Accordingly, the time derivative of V j;i becomes T T 1 _ V_ j;i ¼ V_ zj;i þ V_ U j;i þ V_ θj;i r zj;i ðt Þzj;i þ 1 ðt Þ þ zj;i ðt Þαj;i ðt Þ þ zj;i ðt Þθj;i φj;i μj;i V U j;i þ θ~ j;i Γ j;i θ^ j;i
where θj;i and φj;i ; j ¼ 1; …; q; i ¼ 2; …; γ j 1; are defined as 2 T θTf1;γ γ þ i 1 θTh1;γ γ þ i 1 θh1;γ1 γj þ i 1 g 1;γ γ þ i 1 θf θTh θh θTf θT θh g 1 1 j j 1 j ; θj;i ¼ 4 j;i ; j;i j;i ; 1;1 ; h1;1 1;1 ; 1;1 ; …; ; ; g j;i g j;i g j;i g j;i g j;i g j;i g j;i g j;i 3T
θTfq;γq γ þ i 1 θThq;γq γ þ i 1 θhq;γq γj þ i 1 gq;γ γ þ i 1 1 j j q j …; ; ; ; 5 g j;i
g j;i
g j;i
ð29Þ
g j;i
"
∂αj;i 1 T eμj;i τ1;1 ∂αj;i 1 2 ∂αj;i 1 eμj;i τj;i zj;i ; zj;i F 1;1 ; ; x1;2 ðt Þ; …; 2 ∂x1;1 2 ∂x1;1 ∂x1;1 !2 μ τ ∂αj;i 1 ∂αj;i 1 ∂αj;i 1 e j;i 1;γ1 γj þ i 1 zj;i F T1;γ 1 γ j þ i 1 ; ; x ðt Þ; …; ∂x1;γ 1 γ j þ i 1 2 ∂x1;γ 1 γ j þ i 1 ∂x1;γ 1 γ j þ i 1 1;γ 1 γ j þ i !2 μ τ ∂αj;i 1 ∂αj;i 1 ∂αj;i 1 e j;i q;γq γj þ i 1 T zj;i F ; ; x ðt Þ; ∂xq;γ q γ j þ i 1 q;γ q γ j þ i 1 2 ∂xq;γ q γ j þ i 1 ∂xq;γ q γ j þ i 1 q;γ q γ j þ i
φj;i ¼ F Tj;i ;
8 q
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Adaptive control for a class of MIMO nonlinear time delay systems against time varying actuator failures Mahnaz Hashemi n, Jafar Ghaisari, Javad Askari Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 14 September 2014 Received in revised form 19 February 2015 Accepted 20 February 2015 This paper was recommended for publication by Prof. A.B. Rad
This paper investigates an adaptive controller for a class of Multi Input Multi Output (MIMO) nonlinear systems with unknown parameters, bounded time delays and in the presence of unknown time varying actuator failures. The type of considered actuator failure is one in which some inputs may be stuck at some time varying values where the values, times and patterns of the failures are unknown. The proposed approach is constructed based on a backstepping design method. The boundedness of all the closed-loop signals is guaranteed and the tracking errors are proved to converge to a small neighborhood of the origin. The proposed approach is employed for a double inverted pendulums benchmark and a chemical reactor system. The simulation results show the effectiveness of the proposed method. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Nonlinear time delay systems Time varying actuator failure Multi Input Multi Output Adaptive control Backstepping
1. Introduction Time delays are frequently encountered in many practical control systems. Because of the effect of time delays, these systems may have instability and poor performance [1,2]. So far, considerable attention has been devoted to the stability analysis and control design for time delay systems [3–7]. One of the other serious problems that often occurs in many practical systems and brings uncertainties to the systems is actuator failures. Actuator failures often cause undesired system behavior and sometimes lead to instability or even catastrophic accidents. The desired control design is expected to compensate the actuator failures by effectively compensating the uncertainties. In recent years, there have been remarkable efforts in reliable control of systems with actuator failures [8–11]. Adaptive mechanisms show suitable performance in the presence of uncertainties in failed actuators. Some important results of adaptive control of systems with actuator failures exist in [12–15] for linear systems and in [16–22] for nonlinear systems. For example, for linear systems, direct adaptive control schemes were developed in [12,13] for multi input systems with unknown parameters and stuck type actuator failures. In [14,15], model reference adaptive controllers were developed for Multi Input Single Output (MISO) linear systems with unknown actuator failures. For nonlinear systems, in [16–18] the adaptive controllers were presented for strict feedback MISO nonlinear systems with unknown parameters and stuck type actuator failures. In [19–21], the adaptive actuator failure compensation schemes were presented for a class of Multi Input Multi Output (MIMO) nonlinear systems in strict feedback form. In [22], a fuzzy adaptive actuator failure compensation approach was proposed for a class of MISO uncertain stochastic nonlinear systems in strict feedback form. The proposed fuzzy adaptive actuator failure compensator in [22] promised the boundedness of all the signals in the closed loop system; however, the tracking problem was not considered. The existence of time delays renders the actuator failure compensation problem much more complex and difficult. Many valuable research and practical results have been achieved in actuator failure compensation for linear time delay systems [23–29]. For example, in [23], an adaptive control scheme was investigated for multi input linear systems with actuator failures. In [24], a reliable controller was developed within the framework of Linear Matrix Inequalities for a class of multi input linear systems with time delay in control inputs and constant stuck type actuator failures. A direct adaptive control scheme was introduced in [25] for a class of multi input linear state n
Corresponding author. E-mail addresses: [email protected], [email protected] (M. Hashemi).
http://dx.doi.org/10.1016/j.isatra.2015.02.012 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Hashemi M, et al. Adaptive control for a class of MIMO nonlinear time delay systems against time varying actuator failures. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.02.012i
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delay systems with constant stuck failures in actuators. The same problem was solved for decentralized systems in [26]. Based on a linear matrix inequality technique and an adaptive method, [27,28] suggested adaptive reliable controllers for multi input linear time delay systems with known parameters against loss of effectiveness actuator failures. In [29,30], the adaptive controllers were designed for MISO state delay systems with unknown parameters and constant stuck failures in actuators. To the best of the authors' knowledge, relatively few papers were published regarding actuator failure compensation for nonlinear time delay systems [31–35]. In [31], an adaptive actuator failure compensation scheme was developed for a class of multi input nonlinear time delay systems with known parameters, constant delay and known constant control gains. In [32], a robust compensation approach was presented for multi input nonlinear time delay systems with known parameters, known constant control gains and unknown constant delays. The proposed compensation approach in [32] promised the boundedness of all the signals in the closed loop system; however, the tracking problem was not considered. In [33], an adaptive actuator failure compensation scheme was presented for a class of parametric strict feedback MIMO nonlinear time delay systems with unknown parameters and time delays. The considered actuator faults in [31–33] were types of loss of effectiveness in which the system inputs may just lose some fraction of their effectiveness. In [34], an adaptive actuator failure compensation scheme was presented for a class of parametric strict feedback MISO nonlinear time delay systems with unknown parameters and stuck failures in actuators. In [35], an adaptive fuzzy actuator failure compensator was proposed for a class of MIMO nonlinear systems in strict feedback form with known constant control gains and known actuator failures. The values, patterns, and time occurrences of the considered actuator failure in [35] were known. This paper investigates the adaptive control problem for a class of MIMO nonlinear time delay systems with variable control gains, unknown parameters, unknown time varying state delays and in the presence of unknown time varying actuator failures. The type of considered actuator failure is one in which some unknown inputs may be stuck at some unknown time varying values where the values, times and patterns of the failures are unknown that is, during system operation it is unknown when, by how much and which actuators fail. The proposed adaptive controller is constructed based on the backstepping design method. Compared with the existing results, the main contributions of this paper are as follows: (i) The control problem is investigated for a class of MIMO nonlinear time delay systems with unknown parameters, variable control gains, unknown time varying delays and in the presence of unknown time varying actuator failures. (ii) The considered actuator failures are unknown stuck types that not only cause the system gain changes but also lead to system uncertainties. (iii) A differentiable function is used to solve the differentiability problem and to avoid the singularity problem of the intermediate controllers. (iv) Appropriate Lyapunov–Krasovskii functionals are introduced to design new adaptive laws to compensate the unknown time varying actuator failures as well as uncertainties from unknown parameters and unknown state delays. (v) The proposed systematic backstepping design method proves that, with no need for explicit failure detection not only are all the signals in the closed loop system bounded, but also the tracking errors converge to a small neighborhood of the origin. The paper is organized as follows. In Section 2, the system description is given along with the necessary assumptions. In Section 3, the design procedure of the proposed adaptive controller is explained. In Section 4, the stability analysis of the proposed controller is investigated. In Section 5, simulation results are presented to illustrate the effectiveness of the proposed control method. Finally, the paper is concluded in Section 6.
2. Problem statement Consider a class of strict-feedback nonlinear time delay systems in the following form: T T x_ j;1 ðt Þ ¼ g j;1 xj;2 ðt Þ þ θf j;1 F j;1 ðxj;1 ðtÞÞ þ θhj;1 H j;1 xj;1 ðt τj;1 ðtÞÞ T T x_ j;2 ðt Þ ¼ g j;2 xj;3 ðt Þ þ θf j;2 F j;2 xj;2 ðtÞ þ θhj;2 H j;2 xj;2 t τj;2 ðt Þ : :
: x_ j;γ j 1 ðt Þ ¼ g j;γ
x ðt Þ þ θf j;γ j 1 j;γ j T
F j;γ j 1 ðxj;γ j 1 ðtÞÞ þ θhj;γ 1 T
j
H j;γ j 1 xj;γ j 1 ðt τj;γ j 1 ðt ÞÞ 1 j
ð1Þ
T T T x_ j;γ j ðt Þ ¼ θf j;γ F j;γ j ðxðtÞÞ þ θhj;γ H j;γ j ðxðt τj;γ j ÞÞ þ β j ðxðtÞÞuðtÞ j
j
yj ðt Þ ¼ xj;1 ðt Þ;
j ¼ 1; …; q:
where x A Rγ , γ ¼ γ 1 þ γ 2 þ … þ γ q , xj;i ¼ ½x1;1 ; …; x1;γ 1 γ j þ i ; …; xl;1 ; …; xl;γ l γ j þ i ; …; xq;1 ; …:; xq;γ q γ j þ i T are the state vectors, y ¼ ½y1 ; y2 ; …; yq T A Rq is the output vector and u ¼ ½u1 ; u2 ; …; um T A Rm ; m 4 q is the control input vector whose actuators may fail during operation. βj ð:Þ; F j;i ð:Þ and H j;i ð:Þ are known smooth nonlinear function vectors, θf j;i and θhj;i are unknown constant parameter vectors, g j;i is an unknown constant parameter and τj;i ðtÞ is an unknown time varying delay of the states. For systems in which actuators may fail during the operation of the system, one common approach is to use actuator redundancy. In this way, when one actuator fails, some others could compensate the effect [16]. To deal with the actuator redundancy, the m actuators are divided into q groups corresponding to q outputs as following: q X
mi ¼ m;
mi 4 1
for
i ¼ 1; …; q
i¼1
where mi is the number of the actuators in the ith group. The grouping scheme is as follows: n n o n oo Δ : j1;1 ; j1;2 ; …; j1;m1 ; …; jk;1 ; jk;2 ; …; jk;mk ; …; jq;1 ; jq;2 ; …; jq;mq where
n
jk;1 ; jk;2 ; …; jk;mk
o
ð2Þ
f1; 2; …; mg. Thus, the actuators with same or similar physical characteristics are in the same group while the
actuators of different physical characteristics usually should not be in the same group [19]. Therefore, the following proportional actuation Please cite this article as: Hashemi M, et al. Adaptive control for a class of MIMO nonlinear time delay systems against time varying actuator failures. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.02.012i
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scheme is used: vk ¼
q X
Sk;l wl ;
k ¼ 1; …; m
ð3Þ
l¼1
where wl is the nominal control for group l to be designed in the next section and Sk;l ; k ¼ 1; …; m; l ¼ 1; …; q; are the elements of the matrix S which is a design function matrix with m rows and q columns; Each column of the matrix S has some nonzero elements with certain physical meaning that indicate the actuators of the same group. The actuator failures to be considered are modeled as uk ðt Þ ¼ uk ðt Þ;
t Z tk ;
k ¼ k1 ; k2 ; …; kp ;
1 r p r mi 1;
i ¼ 1; …; q
ð4Þ
where uk ðt Þ is given by u k ð t Þ ¼ Θk Ψ k ; T
k ¼ k1 ; k2 ; …; kp ;
1 r p r mi 1;
i ¼ 1; …; q
ð5Þ
The failure time instant t k , the failure index k and the constant vector Θk ¼ ½dk;1 ; dk;2 ; …; dk;h are unknown and the bounded signal vector Ψ k ¼ ½Ψ k;1 ; Ψ k;2 ; …; Ψ k;h T ; h Z 1 is known. The stuck type actuator failures can occur due to hydraulic failures that can produce unintended movements in the control surfaces of an aircraft [16]. For plant (1) with the actuator failure (4) and (5), the input vector can be expressed as T
uðt Þ ¼ vðt Þ þ δðuðtÞ vðt ÞÞ ; uðt Þ ¼ ½u1 ðtÞ; …; um ðtÞT ;
uðt Þ ¼ ½u1 ðtÞ; …; um ðtÞT ;
δ ¼ diag δ1 ; …; δm ;
δk ¼
vðt Þ ¼ ½v1 ðtÞ; …; vm ðtÞT ( 1; uk ðt Þ ¼ uk ðtÞ 0;
ð6Þ
uk ðt Þ a uk ðtÞ
where vk ðtÞ is the applied control input to be designed later. The control objective is to design a state feedback controller for plant (1) in order to assure that all the closed loop signals are bounded and the plant outputs yj ðt Þ; j ¼ 1; …; q; track the desired signals ydj ðt Þ; j ¼ 1; …; q; despite the presence of unknown plant parameters, unknown actuator failures (4) and (5) and unknown state delays. For this purpose, the following assumptions are considered. Assumption 1. ([12–22,24–26,29,30,35]). For plant (1) with known plant parameters and failure parameters, if any up to mi 1 actuators fail in each group, the remaining actuators can still achieve a desired control objective. j ¼ 1; …; q and its first γ j th order derivatives yðkÞ dj
Assumption 2. The desired signal ydj ðt Þ; piecewise continuous.
ðk ¼ 1; …; γ j Þ are known, bounded, and
Assumption 3. The unknown time varying delay τj;i ðt Þ is a differentiable function satisfying 0 r τj;i ðt Þ r τj;i ; τ_ j;i ðtÞ r ϑj;i o 1
ð7Þ
where τj;i and ϑj;i are known positive constants.
Assumption 4. The signs of g j;i are known and there exist unknown constants g max Z g min 4 0 such that g min r g j;i r g max . For the controller design, without loss of generality it is assumed that g j;i 4 0;
j ¼ 1; …; q;
i ¼ 1; …; γ j .
3. Adaptive controller design In this section, an adaptive actuator failure compensation scheme is developed for the MIMO nonlinear time delay system (1) by employing the backstepping design method [36]. At first, the following state transformation is considered: zj;1 ¼ xj;1 ydj ;
j ¼ 1; …; q;
zj;i ¼ xj;i αj;i 1 ;
i ¼ 2; …; γ j
ð8Þ
where αj;i 1 are the intermediate control functions to be designed later. The transformed system in the new coordination becomes T T z_ j;1 ðt Þ ¼ g j;1 zj;2 ðt Þ þ g j;1 αj;1 þ θf j;1 F j;1 xj;1 þ θhj;1 H j;1 xj;1 ðt τj;1 ðtÞÞ y_ dj T T _ j;1 z_ j;2 ðt Þ ¼ g j;2 zj;3 ðt Þ þ g j;2 αj;2 þ θf F j;2 xj;2 þ θh H j;2 xj;2 t τj;2 ðt Þ α j;2
:
j;2
ð9Þ
: : T T T _ j;γ j 1 z_ j;γ j ðt Þ ¼ θf j;γ F j;γ j ðxÞ þ θhj;γ H j;γ j xðt τj;γ j Þ þ β j ðxÞuðtÞ α j j The adaptive compensation approach is designed in the following procedures. 3.1. Adaptive controller design for zj;1 ; j ¼ 1; …; q; subsystems
In this subsection, the zj;1 ; j ¼ 1; …; q; subsystems are considered and the controller is designed for these subsystems. For the zj;1 subsystems, the following Lyapunov functions are considered: V zj;1 ¼
1 2 z ðtÞ 2g j;1 j;1
ð10Þ
Please cite this article as: Hashemi M, et al. Adaptive control for a class of MIMO nonlinear time delay systems against time varying actuator failures. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.02.012i
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V U j;1 ¼
Z
1 2ð1 ϑj;1 Þg j;1
t t τj;1 ðt Þ
eμj;1 ðξ tÞ H Tj;1 xj;1 ξ H j;1 ðxj;1 ðξÞÞdξ
ð11Þ
1 T 1 V θj;1 ¼ θ~ j;1 Γ j;1 θ~ j;1 2
ð12Þ
V j;1 ¼ V zj;1 þV U j;1 þ V θj;1
ð13Þ
where μj;1 40; j ¼ 1; …; q, Г j;1 ¼ Г j;1 4 0, g j;1 is an unknown constant parameter defined in (1), ϑj;1 is a positive constant defined in (7), H j;1 ð:Þ is a known smooth nonlinear function vector defined in (1) and θ~ j;1 ¼ θ^ j;1 θj;1 in which θ^ j;1 is the estimate of θj;1 to be defined in (15). Along system (9), the time derivative of V zj;1 satisfies ! y_ dj 1 T 1 T _ θ F x þ θhj;1 Hj;1 xj;1 ðt τj;1 ðtÞÞ V zj;1 ¼ zj;1 ðt Þ zj;2 ðt Þ þ αj;1 þ g j;1 f j;1 j;1 j;1 g j;1 g j;1 T
By using the Young's inequality [35], the time derivative of V zj;1 becomes y_ d zj;1 ðtÞ T θf j;1 F j;1 xj;1 ðt Þ zj;1 ðt Þ j V_ zj;1 r zj;1 ðt Þzj;2 ðt Þ þ zj;1 ðt Þαj;1 ðt Þ þ g j;1 g j;1 þ
eμj;1 τj;1 2 e μj;1 τj;1 T T z ðtÞθhj;1 θhj;1 þ H j;1 xj;1 t τj;1 ðt Þ H j;1 xj;1 t τj;1 ðt Þ 2g j;1 j;1 2g j;1
By applying Assumption 3, the time derivative of V U j;1 becomes H Tj;1 xj;1 ðt Þ H j;1 xj;1 ðt Þ e μj;1 τj;1 T _ H j;1 xj;1 t τj;1 ðtÞ H j;1 xj;1 t τj;1 ðtÞ V U j;1 r μj;1 V U j;1 þ 2g j;1 2 1 ϑj;1 g j;1 By considering (13), the time derivative of V j;1 becomes T T 1 _ V_ j;1 ¼ V_ zj;1 þ V_ U j;1 þ V_ θj;1 rzj;1 ðt Þzj;2 ðt Þ þzj;1 ðt Þαj;1 ðt Þ þzj;1 ðt Þθj;1 φj;1 μj;1 V U j;1 þ θ~ j;1 Γ j;1 θ^ j;1
where 2
3T
θTf θTh θh 1 θj;1 ¼ 4 j;1 ; j;1 j;1 ; 5 g j;1
"
ð14Þ
g j;1
eμj;1 τj;1 zj;1 ; 2
φj;1 ¼ F Tj;1 xj;1 ðt Þ ;
ð15Þ
g j;1
( T )#T H j;1 xj;1 ðt Þ H j;1 xj;1 ðt Þ y_ dj ðt Þ 2zj;1 ðt Þ 1 ϑj;1
ð16Þ
where F j;1 ð:Þ and H j;1 ð:Þ are known smooth nonlinear function vectors defined in (1), θf j;1 and θhj;1 are unknown constant parameter vectors, g j;1 is an unknown constant parameter and τj;1 and ϑj;1 are defined in (7). According to (14), the intermediate control function is selected as
T
1 2
αj;1 xj;1 ðt Þ ¼ θ^ j;1 φj;1 zj;1 ϖ j;1 zj;1
ð17Þ
where ϖ j;1 ; j ¼ 1; …; q; are positive constants. However, it is clear that at zj;1 ¼ 0, controller singularity occursbecause φj;1 defined in (16) is not well-defined at zj;1 ¼ 0. Hence, modification should be done to guarantee the boundedness of αj;1 xj;1 ðt Þ . Besides, the computation of the intermediate control functions, αj;i ðt Þ; j ¼ 1; …; q; i ¼ 1; …; γ j , in each step of the backstepping design procedure requires that of α_ j;i 1 ðt Þ; α€ j;i 2 ðt Þ; …; αðj;1i 1Þ : As a result, αj;i ðt Þ needs to be at least ðγ j iÞth differentiable. Therefore, in this step, αj;1 ðt Þ should be at least ðγ j 1Þth differentiable. By using function qj;1 ðzj;1 Þ defined in Lemma 1, the singularity problem of the controller can be solved and the controller will still be ðγ j 1Þth differentiable. Lemma 1. [3]: Let the function qj;i ðxÞ : R-R be defined as qj;i ðxÞ 8 1; > > > > iðγ j iÞ > R x h > > > λaj;i þ λbj;i σ σ λaj;i dσ ; < cqj;i λa j;i ¼ h i γ R λ ð j iÞ > > > cqj;i x aj;i λaj;i þ λbj;i þ σ σ þ λaj;i dσ ; > > > > > : 0;
jxj Z λaj;i þ λbj;i
λaj;i o x o λaj;i þ λbj;i λaj;i þ λbj;i o x o λaj;i jxj r λaj;i 2ðγ iÞ þ 1
where λaj;i ; λbj;i 4 0 are constant parameters and cqj;i ¼ ðð½2ðγ j iÞ þ1!Þ=ðλb j j;i limited between 0 and 1. Therefore, the intermediate controller (17) is rewritten as
T 1 αj;1 xj;1 ðt Þ ¼ qj;1 zj;1 θ^ j;1 φj;1 zj;1 ϖ j;1 zj;1 2
½ðγ j iÞ!2 ÞÞ, thus qj;i ðxÞ is ðγ j iÞth differentiable and is
ð18Þ
Please cite this article as: Hashemi M, et al. Adaptive control for a class of MIMO nonlinear time delay systems against time varying actuator failures. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.02.012i
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where ϖ j;1 is a positive constant. The updating law for parameter θ^ j;1 is chosen as _
θ^ j;1 ¼ qj;1 ðzj;1 ÞΓ j;1 ½zj;1 φj;1 σ j;1 θ^ j;1
ð19Þ
where Γ j;1 ¼ Γ j;1 4 0 and small positive constant σ j;1 is used to introduce the σ -modification for the closed loop system. For clarification the following sets are considered: T
Region 1 : fzj;i ϵR j j zj;i j Z λaj;i þ λbj;i g Region 2 : fzj;i ϵR j λaj;i o zj;i o λaj;i þ λbj;i g Region 3 : fzj;i ϵR j j zj;i j r λaj;i g
ð20Þ
Case (1): For region 1, zj;1 Z λaj;1 þ λbj;1 . In this region qj;1 zj;1 ¼ 1, hence V_ j;1 ðtÞ becomes T T 1 _ 1 1 V_ j;1 r z2j;2 ϖ j;1 z2j;1 μj;1 V U j;1 zj;1 ðt Þθ~ j;1 φj;1 þ θ~ j;1 Γ j;1 θ^ j;1 2 2
and the updating law (19) becomes _
θ^ j;1 ¼ Γ j;1 ½zj;1 φj;1 σ j;1 θ^ j;1
ð21Þ
T By using the inequality σ j;1 θ~ j;1 θ^ j;1 r 12σ j;1 j j θ~ j;1 j j 2 þ 12σ j;1 j j θj;1 j j 2 , the time derivative of V j;1 becomes
V_ j;1 r C j;1 V j;1 þ 12z2j;2 þ X j;1
!
σ j;1 ; C j;1 ¼ min g min ϖ j;1 ; μj;1 ; 1 λmax ðΓ j;1 Þ
1 X j;1 ¼ σ j;1 j j θj;1 j j 2 2
In this region, the time derivative of V j;1 is dependent on zj;2 ; therefore the boundedness of V j;1 will be proved in the next step when the boundedness of zj;2 is verified. Case (2): For region 2, λaj;1 o j zj;1 j o λaj;1 þ λbj;1 . In this region, zj;1 is bounded; according to Assumption 2, ydj is also bounded. Since xj;1 ¼ zj;1 þydj and zj;1 ; ydj are bounded, xj;1 is also bounded; hence V zj;1 and V U j;1 are bounded. By using (12) and (19) in this region, the time derivative of V θj;1 becomes T T 1 _^ V_ θj;1 ¼ θ~ j;1 Γ j;1 θj;1 ¼ qj;1 ðzj;1 Þθ~ j;1 ½zj;1 φj;1 σ j;1 θ^ j;1
ð22Þ
By applying the inequality c0j;1 T 1 qj;1 zj;1 θ~ j;1 zj;1 φj;1 r 0 qj;1 zj;1 j j θ~ j;1 j j 2 þ qj;1 zj;1 φTj;1 φj;1 z2j;1 ; c0j;1 40 2cj;1 2 and by using σ j;1 θ~ j;1 θ^ j;1 r 12σ j;1 j j θ~ j;1 j j 2 þ 12σ j;1 j j θj;1 j j 2 , Eq. (22) is rewritten as T
2 1 1 1 q ðz Þðσ Þθ~ þ qj;1 ðzj;1 Þðσ j;1 j j θj;1 j j 2 þc0j;1 φTj;1 φj;1 z2j;1 Þ V_ θj;1 r 2 j;1 j;1 j;1 c0j;1 j;1 2 In this region, φj;1 is a smooth and bounded function and qj;1 zj;1 A ð0; 1Þ. By choosing c0j;1 such that σ nj;1 ¼ σ j;1 ð1=c0j;1 Þ 4 0,
V_ θj;1 r C nθj;1 V θj;1 ðt Þ þ λθj;1 C nθj;1 ¼ qj;1 zj;1
σ nj;1
λmax Γ
1 j;1
1 ; λθj;1 ¼ qj;1 zj;1 σ j;1 j j θj;1 j j 2 þ c0j;1 φTj;1 φj;1 z2j;1 2
Since λaj;1 o j zj;1 j o λaj;1 þ λbj;1 , then h i Cn t λθ 0 r V θj;1 ðt Þ r V θj;1 ð0Þ ρθj;1 e θj;1 þ ρθj;1 ; ρθj;1 ¼ nj;1 C θj;1 Thus, V θj;1 is bounded. Eventually V j;1 ðt Þ is bounded for λaj;1 o j zj;1 j o λaj;1 þ λbj;1 . Case (3): For region 3, zj;1 r λaj;1 . In this region, zj;1 is bounded; according to assumption 2, ydj is also bounded. Since xj;1 ¼ zj;1 þ ydj and _ zj;1 ; ydj are bounded, xj;1 is also bounded. In this region, qj;1 zj;1 ¼ 0, thus based on (19), θ^ j;1 ¼ 0 and θ^ j;1 is kept unchanged in a bounded value and V θj;1 is bounded. In addition, for bounded xj;1 and zj;1 , it can be concluded that V zj;1 and V U j;1 are bounded, hence V j;1 ðtÞ is bounded. As can be seen, the singularity problem of the intermediate controller at zj;1 ¼ 0 is solved by using qj;1 zj;1 , though it is still ðγ j 1Þ-th differentiable, making it possible to carry out the backstepping design method in the next steps. Therefore, all the closed loop signals in regions 2 and 3 are bounded despite the presence of unknown parameters, time varying delays and time varying actuator failures. The boundedness of the closed loop signals in region 1 is dependent on the boundedness of the signal zj;2 to be regulated in the following. 3.2. Adaptive controller design for zj;i ; j ¼ 1; …; q; i ¼ 2; …; γ j 1 subsystems In this subsection, the zj;i ; j ¼ 1; …; q; i ¼ 2; …; γ j 1; subsystems are considered. T T _ j;i 1 ðtÞ z_ j;i ðt Þ ¼ g j;i zj;i þ 1 ðt Þ þ g j;i αj;i ðt Þ þ θf j;i F j;i xj;i þ θhj;i H j;i xj;i t τj;i ðt Þ α
ð23Þ
Please cite this article as: Hashemi M, et al. Adaptive control for a class of MIMO nonlinear time delay systems against time varying actuator failures. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.02.012i
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For these subsystems, the following Lyapunov functions are considered: V zj;i ¼
1 2 z ðtÞ 2g j;i j;i
V U j;i ¼
ð24Þ
γj þ i 1 q γ k X X l¼1
k¼1
1 þ 2ð1 ϑj;i Þg j;i
Z
eμj;i ðξ t Þ H Tk;l xk;l ξ H k;l ðxk;l ξ Þdξ 2 1 ϑk;l g j;i t τk;l ðt Þ Z t eμj;i ðξ tÞ H Tj;i xj;i ðξÞ H j;i ðxj;i ðξÞÞdξ
1
t
ð25Þ
t τ j;i ðt Þ
1 T 1 V θj;i ¼ θ~ j;i Γ j;i θ~ j;i 2
ð26Þ
V j;i ¼ V zj;i þ V U j;i þ V θj;i
ð27Þ
g j;i is an unknown constant parameter, H j;i ð:Þ is a known smooth nonlinear function vector defined in (1), ϑj;i is where μj;i 4 0, Г j;i ¼ Г defined in (7) and θ~ j;i ¼ θ^ j;i θj;i in which θ^ j;i is the estimate of θj;i to be defined in (29). The intermediate controllers αj;i 1 ðt Þ; j ¼ 1; …; q; i ¼ 2; …; γ j ; are functions of xj;i 1 ðt Þ; θ^ 1;1 ; …; θ^ 1;γ 1 γ j þ i 1 ; …; θ^ q;1 ; …:; θ^ q; γ q γ j þ _ j;i 1 ðt Þ becomes i 1; ydj ; yð1Þ …; yðidj 1Þ , hence α dj T j;i 4 0,
γj þ i 1 q γ k X X ∂αj;i 1 n
α_ j;i 1 ðt Þ ¼
k¼1
þ
l¼1
∂xk;l
o T T g k;l xk;l þ 1 ðt Þ þ θf k;l F k;l þ θhk;l H k;l xk;l t τk;l ðt Þ
γj þ i 1 q γ k X X ∂αj;i 1 _^ k¼1
l¼1
∂θ^
θk;l þ
k;l
i X ∂αj;i 1
yðkÞ ðk 1Þ dj k ¼ 1 ∂yj
ð28Þ
By applying the Young's inequality, the time derivative of V zj;i becomes zj;i ðt Þ T θ F x V_ zj;i rzj;i ðt Þzj;i þ 1 ðt Þ þ zj;i ðt Þαj;i ðt Þ þ g j;i f j;i j;i j;i þ
eμj;i τj;i 2 e μj;i τj;i T zj;i ðt ÞθThj;i θhj;i þ H x t τj;i ðt Þ 2g j;i 2g j;i j;i j;i
γj þ i 1 q γ k X i o z ðt Þ X zj;i ðt Þ X ∂αj;i 1 n ∂αj;i 1 ðkÞ j;i H j;i xj;i t τj;i ðt Þ g k;l xk;l þ 1 ðt Þ þ θTfk;l F k;l y ∂xk;l g j;i k ¼ 1 l ¼ 1 g j;i k ¼ 1 ∂yðk 1Þ dj j
γj þ i 1 γj þ i 1 q γ k X q γ k X z2j;i X zj;i ðtÞ X ∂αj;i 1 _^ ∂αj;i 1 2 T eμj;i τk;l θhk;l θhk;l θk;l þ g j;i k ¼ 1 l ¼ 1 2g j;i k ¼ 1 l ¼ 1 ∂xk;l ∂θ^ k;l þ
q γ γj þ i 1 1 X k X e μj;i τk;l H Tk;l xk;l t τk;l ðt Þ H k;l xk;l t τk;l ðt Þ 2g j;i k ¼ 1 l ¼ 1
Accordingly, the time derivative of V j;i becomes T T 1 _ V_ j;i ¼ V_ zj;i þ V_ U j;i þ V_ θj;i r zj;i ðt Þzj;i þ 1 ðt Þ þ zj;i ðt Þαj;i ðt Þ þ zj;i ðt Þθj;i φj;i μj;i V U j;i þ θ~ j;i Γ j;i θ^ j;i
where θj;i and φj;i ; j ¼ 1; …; q; i ¼ 2; …; γ j 1; are defined as 2 T θTf1;γ γ þ i 1 θTh1;γ γ þ i 1 θh1;γ1 γj þ i 1 g 1;γ γ þ i 1 θf θTh θh θTf θT θh g 1 1 j j 1 j ; θj;i ¼ 4 j;i ; j;i j;i ; 1;1 ; h1;1 1;1 ; 1;1 ; …; ; ; g j;i g j;i g j;i g j;i g j;i g j;i g j;i g j;i 3T
θTfq;γq γ þ i 1 θThq;γq γ þ i 1 θhq;γq γj þ i 1 gq;γ γ þ i 1 1 j j q j …; ; ; ; 5 g j;i
g j;i
g j;i
ð29Þ
g j;i
"
∂αj;i 1 T eμj;i τ1;1 ∂αj;i 1 2 ∂αj;i 1 eμj;i τj;i zj;i ; zj;i F 1;1 ; ; x1;2 ðt Þ; …; 2 ∂x1;1 2 ∂x1;1 ∂x1;1 !2 μ τ ∂αj;i 1 ∂αj;i 1 ∂αj;i 1 e j;i 1;γ1 γj þ i 1 zj;i F T1;γ 1 γ j þ i 1 ; ; x ðt Þ; …; ∂x1;γ 1 γ j þ i 1 2 ∂x1;γ 1 γ j þ i 1 ∂x1;γ 1 γ j þ i 1 1;γ 1 γ j þ i !2 μ τ ∂αj;i 1 ∂αj;i 1 ∂αj;i 1 e j;i q;γq γj þ i 1 T zj;i F ; ; x ðt Þ; ∂xq;γ q γ j þ i 1 q;γ q γ j þ i 1 2 ∂xq;γ q γ j þ i 1 ∂xq;γ q γ j þ i 1 q;γ q γ j þ i
φj;i ¼ F Tj;i ;
8 q
