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Composite anti-disturbance control for uncertain Markovian jump systems with actuator saturation based disturbance observer and adaptive neural network
Available online at www.sciencedirect.com
Journal of the Franklin Institute 356 (2019) 6926–6945 www.elsevier.com/locate/jfranklin
Composite anti-disturbance control for uncertain Markovian jump systems with actuator saturation based disturbance observer and adaptive neural network Yunliang Wei a,b,∗, Guo-Ping Liu b,c, Guangdeng Zong d, Hao Shen e a School
of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, PR China b School of Engineering, University of South Wales, Cardiff 1DL, CF37, U.K. c CTGT Center, Harbin Institute of Technology, Harbin 150001, China d School of Engineering, Qufu Normal University, Rizhao, Shandong 276826, PR China e School of Electrical and Information Engineering, Anhui University of Technology, Maanshan 243002, China Received 30 November 2018; received in revised form 2 June 2019; accepted 8 June 2019 Available online 20 June 2019
Abstract This paper studies the problem of composite control for a class of uncertain Markovian jump systems (MJSs) with partial known transition rates, multiple disturbances and actuator saturation. Compared with the existing results, a novel robust composite control scheme is put forward by virtue of adaptive neural network technique. For MJSs, the partial unknown information on transition rates and the actuator saturation influence the design of disturbance observer and the robust H∞ controller. Firstly, without taking account of external disturbances, the network reconstruction error and saturation, a novel robust adaptive control strategy is established to ensure that all the signals of the closed-loop system are asymptotically bounded in mean square. Secondly, the solvability condition for ensuring the robust H∞ performance is given by using a modified adaptive law, where the saturation is treated as a disturbancelike signal. Finally, the simulations for a numerical example and an application example are performed to validate the effectiveness of the proposed results. © 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.
https://doi.org/10.1016/j.jfranklin.2019.06.006 0016-0032/© 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.
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1. Introduction In the past few years, lots of research activities have been focused on the study of Markovian jump systems (MJSs), since many industrial plants have been modeled successfully as MJSs, such as communication systems, aerospace systems, manufacturing systems, robotic systems and chemical process [1–8]. As a kind of typical stochastic control system, the transition probabilities of the Markov process can largely determine the behaviors of MJSs. But in most cases it should be very hard or costly to get the exact value of each element in the transition rate matrix [9,10]. In this case, the MJSs with partially unknown transition rates have been studied and attracted many considerable attentions in the control fields [11–14]. It is noteworthy that the MJSs mentioned above do not suffer from the uncertainties. As we know, the uncertainties widely exist in many actual processes and practical systems due to load changing, external surroundings, modelling error and so on [15]. Therefore, the anti-disturbance control is an everlasting research subject in the field of control theory and engineering, and so many outstanding control schemes are developed to attenuate and reject the disturbances, such as H∞ control [3,13], sliding mode control [16–19], disturbance compensation control [20]. In the modern engineering practice, the control precision is becoming higher and higher. To meet with the high-precision control requirement, composite hierarchical anti-disturbance control (CHADC) has recently been proposed, which is being targeted to achieve different types of disturbances attenuation and rejection respectively [15,20,21]. In [20], the problem of anti-disturbance control for a class of nonlinear uncertain systems with an exosystem generated disturbance is investigated, the techniques of disturbance-observer-based control (DOBC) and robust observer design are combined effectively, which partly provides the foundation for later development of CHADC. Recently, a CHADC technique combining the DOBC with H∞ control is studied for the system with matched exosystem generated disturbances and mismatched H2 -norm bounded disturbances [22]. It is noteworthy that the neural network model can well approximate unknown uncertainty [23–28]. Due to the physical constraint of the control channel in practice, actuator saturation is one of the ineluctable phenomena, and can lead to negative impact on the performance and stability of the controlled system. Therefore, it is of great importance to consider the existence of actuator saturation in the controller design. For MJSs, some fundamental methods have been proposed for tackling saturation problems, such as optimal control, anti-windup control [29–31]. By the optimal control theory, [32] and [33] present the criteria of the stochastic local stabilization for continuous-time and discrete-time MJSs with saturating actuator, respectively. In [13,34], the problem of anti-windup compensation for MJSs with partially known transition probabilities is studied, and the anti-windup compensator can be synthesised by an optimal strategy. It is worth pointing out that the above mentioned results are local because the saturation nonlinearity is treated as sector nonlinearity model or polytopic differential inclusion, and the optimal strategy needs to be designed to obtain the estimation of the domain of attraction. However, the above treatment of the saturation could not be applied effectively, when the saturation is added on the actuator of the system with a composite controller based on the disturbance observer and adaptive neural network. If the error between the control input and ∗ Corresponding author at: School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, PR China. E-mail addresses: [email protected] (Y. Wei), [email protected] (G.-P. Liu), lovelyletian@ gmail.com (G. Zong), [email protected] (H. Shen).
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saturated input is defined as a disturbance-like signal, we can try to utilize anti-disturbance control scheme to minimize the effects of this signal on controlled output. In [35] and [36], the problems of anti-disturbance control for MJSs with partial known transition probabilities are studied, the proposed composite control methods are mainly based on disturbance observer and H∞ control technique to reject matched exosystem generated disturbances and attenuate unmatched disturbances. When the matched unknown nonlinear uncertainty and actuator saturation appear in the above-mentioned system, the control methods in [35,36] might be ineffective. In [37], a neural network-based DOBC for a class of nonlinear systems in strict-feedback form is studied by combining a neural network scheme and backstepping method. Inspired by [37], a novel method of the composite control for a class of uncertain MJSs subject to actuator saturation and multiple disturbances is put forward, in which the unmatched model-free disturbances, matched exosystem generated disturbances and unknown nonlinear uncertainties are considered. The factors of disturbances and uncertainties are more comprehensive than the existing results in [35,36,38]. Based on the radial basis function neural network and disturbance observer, a novel composite controller is designed for the system under consideration. The error between the control input and saturated input is considered as a disturbance-like signal. Firstly, when model-free disturbances, the network reconstruction error and disturbance-like signal are not considered, a criterion of semi-globally stochastically uniformly ultimate boundedness for the closed-loop system is established and an adaptive law is proposed. Secondly, by combining robust H∞ control with the above composite control scheme, some conditions with a modified adaptive law are given to guarantee that all of the disturbances and uncertainties in the closed-loop system can be attenuated and rejected. Further, some solvable design conditions of composite controller are put forward, respectively. The technique employed in this paper is in combination with matrix inequality technique and adaptive neural network control method, which is different from the case in [37] and can be used for more general systems. It is worth pointing out that the proposed results are applicable to the cases of the traditional MJSs and switched systems under arbitrary switching law. Notation: The standard notation is used throughout the paper. The superscript ‘T’ stands for matrix transposition; L2 [0, ∞] is the space of square-integrable vector functions over [0, ∞); Rn denotes the n-dimensional Euclidean space, and Rm × Rn is the set of all real m × n matrices; E{·} stands for mathematical expectation; In symmetric block matrices, ∗ is used as an ellipsis for the terms that are induced by symmetry; The notation P > 0 (P ≥ 0) means that P is real symmetric and positive definite (semi-positive definite); Given any two symmetric real matrices A and B, A > B refers to the fact that A − B is positive definite; The identity matrix of order m is denoted by Im (or simply I if no confusion arises); sign{·} is the signum function. 2. Preliminaries and problem statement Define the probability space (, F, P ), where is the sample space, F denotes the subσ -algebra of , and P represents the probability measure on F. Consider the uncertain MJSs under (, F, P ) as: x˙(t ) = A(t , r(t ))x(t ) + B(r(t ))[sat (u(t )) + d0 (t ) + f (x)] + E (r(t ))d1 (t ),
(1)
where x(t ) ∈ Rn is the system state; u(t ) ∈ R is the control input; d1 (t ) ∈ R is the external disturbance in L2 [0, ∞]; sat (·) = sign(·) min (·, ρ) with ρ being saturation level; d0 (t ) ∈ R
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is a kind of disturbances generated by an exogenous system as follows: ω˙ (t ) = W (r(t ))ω(t ) + G(r(t ))d2 (t ), d0 (t ) = V (r(t ))ω(t ),
(2)
where ω(t ) ∈ Rr is the disturbance system state; d2 (t) is the additional noise in L2 [0, ∞], which reflects modeling uncertainties and perturbations. {r(t)} is a continuous-time, discrete-state homogeneous Markov process, which takes values in S = {1, 2, . . . , s} and has the mode transition probabilities as if j = i, λ h + o(h), Pr{rt+h = j|rt = i} = i j (3) 1 + λii h + o(h), if j = i, with h > 0, limh→0 (o(h)/h) = 0, and λij ≥ 0, ∀i, j ∈ S, j = i are the transition rates from mode i at time t to mode j at time t + h satisfying s
λii = −
λi j .
(4)
j =1, j =i
Here, the assumption of the transition rates of the Markov process be partially known is made, and the transition rate matrix (TRM) with the notation? of unknown element can be represented by ⎡ ⎤ λ11 ? · · · λ1 s ⎢λ21 λ22 · · · ?⎥ ⎢ ⎥ =⎢ . . .. ⎥. .. .. ⎣ .. . . ⎦ ?
λs2
···
λss
For any i ∈ S, we denote i Ski { j : λi j is known}, Suk { j : λi j is unknown}.
(5)
A(t, r(t)), B(r(t)), D(r(t)), E(r(t)), W(r(t)), V(r(t)) and G(r(t)) are matrix functions and for r(t ) = i, i ∈ S, we rewrite A(t, r(t )) = Ai + Ai (t ), B(r(t )) = Bi , D(r(t )) = Di , E (r(t )) = Ei , W (r(t )) = Wi , V (r(t )) = Vi , G(r(t )) = Gi , where Ai , Bi , Di , Ei , Wi , Vi , Gi are real constant matrices with appropriate dimensions and Ai (t) is norm-bounded uncertainty satisfying Ai (t ) = Mi F (t )Ni ,
(6)
with real constant matrices Mi , Ni and an unknown matrix function F(t) satisfying FT (t)F(t) ≤ 1. f (·) : → R is an unknown smooth nonlinear function. It can be approximated by radial basis function (RBF) neural networks in the form of θ T ξ (x), where θ ∈ RN and ξ (x) is a vector valued function defined in Rn . Lemma 1 ([23],[24]). For a compact set x ∈ R p , x ∈ x ∈ R p, ξ (x) = [ρ1 (x ), ρ2 (x ), . . . , ρl (x)]T is the Gaussian basis function vector in the form
x − ζ j 2 , σ ≥ 0, j = 1, . . . , N, ρ j (x) = ex p − (7) 2σ 2
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where ζ j ∈ Rn and σ ∈ R are called the center and width of the Gaussian function, respectively. If the integer N is chosen as sufficiently large, there exists θ ∗ ∈ RN such that f (x) = θ ∗ ξ (x) + δ ∗ , x ∈ x , ∗
(8) ∗
with |δ | ≤ δ m , where δ m ∈ R and δ is the network reconstruction error. Assumption 1 ([37,40]). For positive constants θ and λ, the assumptions |θ ∗ | ≤ θ and ξ (x)T ξ (x) < λ are made throughout this paper. In what follows, the notation θˆ is used to denote the estimation of θ ∗ , and an adaptive ˆ By introducing error θ˜ = θ ∗ − θˆ, system law will be developed to update the parameter θ. (1) can be rewritten as x˙(t ) = (Ai + Ai (t ))x(t ) + Bi [sat (u(t )) + d0 (t ) + θˆT ξ (x) + (θ ∗ T ξ (x) − θˆT ξ (x)) +( f (x) − θ ∗ T ξ (x))] + Ei d1 (t ) = (Ai + Ai (t ))x(t ) + Bi [sat (u(t )) + d0 (t ) + θˆT ξ (x)] + Bi θ˜T ξ (x) + Bi δ ∗ + Ei d1 (t ). (9) Here, we construct a disturbance observer for d0 (t) as follows: dˆ0 (t ) = Vi ωˆ (t ), ωˆ (t ) = v(t ) − L x (t ),
v˙(t ) = (Wi + LBiVi )ωˆ (t ) + L (Ai x (t ) + Bi u(t ) + Bi θˆT ξ (x)).
(10)
Further, a controller for system (9) shall be designed in the form of u(t ) = −dˆ0 (t ) − θˆT ξ (x) + Ki x(t ).
(11)
Substituting Eq. (11) into Eq. (9) yields x˙(t ) = (Ai + Ai (t ))x(t ) + Bi (Vi eω (t ) + Ki x(t )) + Bi θ˜T ξ (x) + Bi δ ∗ + Ei d1 (t ) + Bi d3 (t ), (12) where eω = ω(t ) − ωˆ (t ) and d3 (t ) = sat (u(t )) − u(t ). Remark 1. The DOBC problem for a class of nonlinear systems subject to input saturation has been investigated, in which the system is subjected to system input saturation [39]. In this paper, the saturation constraint is only added onto the control inputs of the system. Due to the complexity of the controller structure, the classical treatments of the saturation nonlinearity could not be applied effectively like in [39]. Here, the error between the saturated control input and the unconstrained control input is regarded as disturbance-like signal of the system, which can be attenuated by the anti-disturbance control scheme, though it is conservative to some extent. In the future, we would make efforts to obtain less conservative method. Let x˜T (t ) = [x T (t ) eTω (t )], from Eqs. (9) and (10), we have ˙˜ ) = (A˜ i + M˜ i F (t )N˜ i )x˜(t ) + B˜ i θ˜T ξ (x) + E˜i d˜(t ), x(t
(13)
where
BiVi A + Bi Ki −Bi Mi Bi Ei 0 Bi , B˜ i = , M˜ i = , E˜i = , A˜ i = i 0 Wi + LBiVi −LBi LMi LBi LEi Gi LBi N˜ i = Ni 0 , K˜i = Ki Vi , d˜T (t ) = δ ∗ d1 (t ) d2 (t ) d3 (t ) T .
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Definition 1. Let p > 0. System (13) with d˜ = 0 is said to be asymptotically bounded in pth moment if there is a positive constant H such that lim sup E |x˜(t; t0 , x0 , r0 )| p ≤ H,
(14)
t→∞
for all (t0 , x˜0 ) ∈ R+ × Rn+r , when p = 2 we say Eq. (13) with d˜ = 0 to be asymptotically bounded in mean square. The control objective is to obtain some conditions such that: (1) The closed-loop system (13) with (2) For give γ > 0, the following H∞
T T E 0 [z˜T (t )z˜(t )]d t ≤ γ 2 E 0
d˜ = 0 is asymptotically bounded in mean square. performance for system (13) can be achieved
[d¯T (t )d¯(t )]dt . (15)
where d¯T (t ) = [d˜T (t ), θ˜T (t )]T . Remark 2. In [41], the performance inequality is proposed to solve the problem of H∞ control for a class of nonlinear systems in strict-feedback form. To investigate the composite control for system (1), we extend it as case of Eq. (15). For further development, the following lemmas will be given: Lemma 2. Assume that there exist functions V ∈ C 2,1 (Rn+r × R+ × S; R+ ), k ∈ Kv ⊂ K∞ and positive numbers p, β, α, such that k(|x˜| p ) ≤ V (x˜(t ), t , i) and A V (x˜, t, i) ≤ −αV (x˜, t, i) + β, for all (x˜, t, i) ∈ Rn+r × R+ × S . Then
β lim sup E |x˜(t; t0 , x0 , r0 )| p ≤ k −1 t→∞ α for all (x˜0 , t0 , r0 ) ∈ Rn+r × R+ × S . That is, system (13) with d˜ = 0 is asymptotically bounded in pth moment. Lemma 3 ([42]). Let D, S and F be real matrices of appropriate dimensions with FT F ≤ I. Then, for any scalar λ > 0, we have DF S + (DF S)T ≤ λ−1 DDT + λS T S. Lemma 4 ([43]). Let the function y(t) be absolutely continuous for t ≥ t0 and let its derivative satisfies the inequality y˙(t ) ≤ k(t )y(t ) + h(t ), for almost all t ≥ t0 , where k(t) and h(t) are almost everywhere continuous functions integrable over every finite interval. Then for t ≥ t0 , t t t t0 k(s)ds y(t ) ≥ y(t0 )e + e s k(u)du h(s)ds. t0
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3. Main results 3.1. stability analysis In the following theorem, we give some sufficient conditions and an adaptive law to guarantee system (13) without disturbance d˜(t ) to be asymptotically bounded in mean square. Theorem 1. For given scalar λ > 0, if there exist scalars q > 0, μ > 0, μi > 0, τ i > 0, matrices P1i > 0, P2 > 0, Q > 0 and symmetric matrices Ti , i ∈ S satisfying the following conditions: i P1 j + Ti ≤ 0, j ∈ Suk , j = i,
(16)
i P1 j + Ti ≥ 0, j ∈ Suk , j = i,
(17)
⎡
⎤ ˜ 1i + qP˜i 0 N˜ iT ⎣ ∗ (τi λ − μ)I + qQ 0 ⎦ < 0, i ∈ S, ∗ ∗ −μi I ˜ 1i = P˜i A˜ i + A˜ Ti P˜i + j∈Si λi j (P¯ j + T¯i ) + μi P˜i M˜ i M˜ iT P˜i + τ −1 i , where i k
0 0 ¯ 0 P P T 0 0 , P¯i = 1i , Ti = i , i = , P˜i = 1i ∗ P2 ∗ 0 ∗ 0 ∗ P2 L Bi BiT L P2
(18)
then for bounded initial conditions, system (13) without disturbance d˜(t ) is asymptotically bounded in mean square under the following adaptive law: ˙ θˆ = Q−1 (−ξ (x )x T (t )P1i Bi − μθˆ ). Proof. Consider the following Lyapunov function candidate as: V (x˜(t ), θ˜ (t ), i) = x˜T (t )P˜i x˜(t ) + θ˜T (t )Qθ˜ (t ),
(19)
where Q > 0 and
P1i 0 ˜ , Pi = ∗ P2 with P1i > 0, P2 > 0. Under the assumption of disturbance d˜(t ) ≡ 0, the weak infinitesimal operator A of the stochastic process {(x˜, i)} along with system (13) can be shown as follows: ⎛ ⎞ s ˙ ˙ ˙˜ ) + x˜T (t )⎝ A V (x˜(t ), θ˜ (t ), i) = 2x˜T (t )P˜i x(t λi j P˜ j ⎠x˜(t ) + 2θˆT (t )Qθˆ (t ) − 2θ ∗ T Qθˆ ⎛ ˙˜ ) + x˜T (t )⎝ = 2x˜T (t )P˜i x(t
j=1 s
⎞ ˙ λi j P˜ j ⎠x˜(t ) − 2θ˜T (t )Qθˆ.
(20)
j=1
Noting s j=1
s j=1
λi j P˜ j =
λi j = 0, for any i ∈ S, we have
s j=1
λi j P¯ j .
(21)
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By Lemma (3), there exists μi > 0 such that A V (x˜(t ), θ˜ (t ), i) = 2x˜T (t )P˜i [(A˜ i + M˜ i F (t )N˜ i )x˜(t ) + B˜ i θ˜T ξ (x)] ⎛ ⎞ s ˙ +x˜T (t )⎝ λi j P¯ j ⎠x˜(t ) − 2θ˜T (t )Qθˆ ⎛
j=1
≤ x˜T (t )⎝P˜i A˜ i + A˜ Ti P˜i +
s
⎞ ˜ ˜ ˜T ˜ ˜T ˜ ⎠ λi j P¯ j + μ−1 i Pi Mi Mi Pi + μi Ni Ni x˜ (t )
j=1
˙ +2x˜T (t )P˜i B˜ i θ˜T (t )ξ (x) − 2θ˜T (t )Qθˆ.
(22)
Based on Assumption 1 and Lemma 3, one can get the inequality 2x˜T (t )P˜i B˜ i θ˜T (t )ξ (x) ≤ −2x(t )T P1i Bi θ˜T (t )ξ (x) + τi−1 eTω P2 LBi BiT LP2 eω +τi ξ (x)T θ˜ (t )θ˜T (t )ξ (x) ≤ −2x(t )T P1i Bi θ˜T (t )ξ (x) + τi−1 eTω P2 LBi BiT LP2 eω + τi λθ˜T (t )θ˜ (t ). (23) where τ i > 0. Design an adaptive law as ˙ θˆ = Q−1 (−ξ (x )x T (t )P1i Bi − μθˆ ).
(24)
From Eq. (24), it is easy to get that ˙ −2θ˜T (t )Qθˆ = −2θ˜T (t )QQ−1 (−ξ (x )x T (t )P1i Bi − μθˆ ) = 2θ˜T (t )ξ (x )x T (t )P1i Bi + 2μθ˜T (t )θˆ.
(25)
By completing the following square 2μθ˜T (t )θˆ ≤ −μθ˜T θ˜ + μθ ∗ T θ ∗ ,
(26)
and substituting Eqs. (23), (25) and (26) into Eq. (22), we have ⎛ ⎞ s −1 ⎠ ˜ ˜ ˜T ˜ ˜T ˜ A V (x˜(t ), θ˜ (t ), i) ≤ x˜T (t )⎝P˜i A˜ i + A˜ Ti P˜i + λi j P¯ j + μ−1 i Pi Mi Mi Pi + μi Ni Ni + τi i x˜ (t ) j=1
˜T
+(τi λ − μ)θ θ˜ + μθ where 0 i = ∗
∗T
θ ∗,
(27)
0 . P2 L Bi BiT L P2
s some elements of TRM can not be accessible, one has j=1 λi j Ti = Note that i λi j Ti = 0 for any symmetric matrix Ti . j∈Ski λi j Ti + j∈Suk From Eqs. (22) and (27), it can verify that ⎛ ˜T ˜ A V (x˜(t ), θ˜ (t ), i) ≤ x˜T (t )⎝P˜i A˜ i + A˜ Ti P˜i + λi j P¯ j + λi j P¯ j + μi P˜i M˜ i M˜ iT P˜i + μ−1 i Ni Ni j∈Ski
i j∈Suk
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⎞ + τi−1 i ⎠x˜(t ) + (τi λ − μ)θ˜T θ˜ + μθ ∗ T θ ∗ ⎛ = x˜T (t )⎝P˜i A˜ i + A˜ Ti P˜i +
λi j (P¯ j + T¯i ) +
j∈Ski
⎞
λi j (P¯ j + T¯i ) + μi P˜i M˜ i M˜ iT P˜i
i j∈Suk
−1 T ∗T ∗ ˜T ˜ ⎠ ˜T ˜ ˜ ˜ ˜ + μ−1 θ . i Ni Ni +τi i x˜ (t ) + 2x˜ (t )Pi Ei d (t ) + (τi λ − μ)θ θ + μθ
(28) From Eqs. (16) and (17) and under the assumption of d˜(t ) = 0, Eq. (28) can be further rewritten as ¯ 1i x¯(t ) + μθ ∗ T θ ∗ , A V (x˜(t ), θ˜ (t ), i) ≤ x¯T (t )
(29)
where x¯(t ) = x˜T (t ) θ˜T T ,
0 1i ¯ , 1i = ∗ (τi λ − μ)I with 1i = P˜i A˜ i + A˜ Ti P˜i +
−1 ˜T ˜ ˜ ˜ ˜T ˜ λi j (P¯ j + T¯i ) + μ−1 i Ni Ni + μi Pi Mi Mi Pi + τi i .
j∈Ski
Under Assumption 1, using Schur complement formula to Eq. (18) gives rise to A V (x˜(t ), θ˜ (t ), i) ≤ −qV (x˜(t ), θ˜ (t ), i) + p,
(30)
where p = μθ 2 . By Lemma 2, it is easy to get that system (13) with d˜ = 0 asymptotically bounded in mean square. Applying Lemma 4 to Eq. (30) yields
p p −qt e + , ∀t > 0. 0 ≤ E[V (x˜(t ), θ˜ (t ), i)] ≤ V (x˜(t0 ), θ˜ (t0 ), rt0 ) − (31) q q Finially, we have p 0 ≤ E[V (x˜(t ), θ˜ (t ), i)] ≤ V ((x˜(t0 ), θ˜ (t0 ), rt0 ) + . q
(32)
From Eq. (19), when the initial conditions are bounded, we know that Eq. (32) implies that the signals x(t), ew (t ) and θ˜ (t ) are bounded in probability, which implies θˆ is also bounded in probability. Furthermore, the states x˜ and neural network estimates θˆ converge to a neighborhood of the origin. The conditions (18) in Theorem 1 are in the framework of bilinear matrix inequalities which can not be solved directly. In the following theorem, the design conditions of disturbances observer, adaptive law and controller will be further presented via solving a set of LMIs.
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Theorem 2. For prescribed positive scalars λ, q, μi , if there exist positive scalars τ i , μ, positive definite matrices X1i , P2 , Q, symmetric matrices Si , and matrices R2 , R1i , i ∈ S satisfying the following conditions:
X1i Si i ≤ 0, j ∈ Suk , j = i, (33) ∗ −X1 j i X1i + Si ≥ 0, j ∈ Suk , j = i,
⎡
i11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗
BiVi + μi Mi MiT R2T i22 ∗ ∗ ∗ ∗ ∗
(34) 0 0 i33 ∗ ∗ ∗ ∗
X1i NiT 0 0 −μi I ∗ ∗ ∗
0 R2 Mi 0 0 −μ−1 i I ∗ ∗
0 R2 Bi 0 0 0 −τi I ∗
⎤ i17 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ < 0, i ∈ S, 0 ⎥ ⎥ 0 ⎦ −i77
(35)
where Ski {k1i k2i · · · kgi }, g ∈ S and ⎧ T i i ⎨H e{Ai X1i + Bi R1i } + (λii + q)X1i + j∈Ski λi j Si + μi Mi Mi , i ∈ Sk , kq = i, i q ∈ {1, . . . , g}, 11 = ⎩ i H e{Ai X1i + Bi R1i } + j∈Si λi j Si + μi Mi MiT + qX1i , i ∈ Suk , k
i17 = λik1i X1i · · · λikgi X1i , i22 = H e{P2Wi + R2 BiVi } + qP2 , i33 = (λτi − μ)I + qQ, i77 = diag{X1k1i · · · X1kgi }, −1 then system (13) with d˜(t ) = 0 and gains Ki = R1i X1−1 i , L = P2 R2 is uniformly ultimately ˙ bounded under an adaptive law θˆ = Q−1 (−ξ (x )x T (t )X −1 Bi − μθˆ ). 1i
Proof. For any i ∈ S, set X1i = P1−1 i , S1i = X1i Ti X1i , R1i = Ki X1i and R2 = P2 L. First, using the Schur complement formula and performing a congruence transformation by P1i , we can deduce that Eq. (33) is equivalent to Eq. (16). On the other hand, pre-multiply and post-multiply Eq. (34) by P1i , we can derive Eq. (17). Finally, using the Schur complement for Eqs. (33), (34) and performing a congruence transformation by diag{P1i , I} yield Eq. (18). Therefore, the proof can be completed by Theorem 1. Remark 3. In the past decades, the adaptive control has been investigated extensively [44,45]. So far the existing results of adaptive neural network control for the nonlinear system in strictfeedback form are generally combined with the Back-stepping technique [25,37,40]. However, as another effective method, linear matrix inequations (LMIs) have been applied widely to handle control system problems. In Theorem 1, the condition of asymptotically bounded in mean square for the composite system (13) with d˜0 = 0 is proposed by matrix inequations. Further, Theorem 2 shows some solvable sufficient conditions in terms of LIMs to obtain the gains of composite controller (11) and the adaptive law.
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3.2. H∞ performance analysis In this section, a reference output z˜(t ) = C˜i x˜(t ) with C˜i = [C1i C2i ] is first introduced. In the following theorems, the results of H∞ performance analysis for system (13) will be presented with full consideration of the effect of disturbances. Meanwhile, a new adaptive law will be proposed. Theorem 3. For a given scalar γ > 0, if there exist scalars μi > 0, matrices P˜i > 0, Q > 0 and symmetric matrices Ti , i ∈ S satisfying the following conditions: i P1 j + Ti ≤ 0, j ∈ Suk , j = i,
(36)
i P1 j + Ti ≥ 0, j ∈ Suk , j = i,
(37)
P˜i E˜i < 0, i ∈ S −γ
ˆ 1i + C˜i T C˜i ∗
(38)
ˆ 1i = P˜i A˜ i + A˜ Ti P˜i + j∈Si λi j (P¯ j + T¯i ) + μ−1 N˜ iT N˜ i + μi P˜i M˜ i M˜ iT P˜i + i , then under where i k ˙ an adaptive law θˆ = −Q−1 ξ (x )x T (t )P1i Bi , the H∞ performance inequality (15) of the closedloop system (13) is achieved with a prescribed attenuation level γ . Proof. Here, we choose the Lyapunov function candidate (19). It is easy to compute the weak infinitesimal operator A of the stochastic process {(x˜, i)} along with system (13) as A V (t, i) = 2x˜T (t )P˜i [(A˜ i + M˜ i F (t )N˜ i )x˜(t ) + B˜ i θ˜T ξ (x) + E˜i d˜(t )] ⎛ ⎞ s ˙ +x˜T (t )⎝ λi j P˜ j ⎠x˜(t ) − 2θ˜T (t )Qθˆ j=1
≤ x˜ (t )[P˜i A˜ i + A˜ Ti P˜i + T
˜T ˜ ˜ ˜ ˜T ˜ λi j (P¯ j + T¯i ) + μ−1 i Ni Ni + μi Pi Mi Mi Pi ]x˜ (t )
j∈Ski
˙ +2x˜T (t )P˜i E˜i d˜(t ) + 2x˜T (t )P˜i B˜ i θ˜T ξ (x) − 2θ˜T (t )Qθˆ.
(39)
˙ Under conditions (36),(37) and the adaptive law θˆ = −Q−1 ξ (x )x T (t )P1i Bi , we can obtain ˆ 1i x˜(t ) + 2x˜T (t )P˜i E˜i d˜(t ) + λθ˜T θ˜. A V (t, i) ≤ x˜T (t )
(40)
For the given positive scalar γ , one has ˆ 1i + C˜i T C˜i )x˜(t ) + 2x˜T (t )P˜i E˜i d˜(t ) z˜T (t )z˜(t ) − γ d˜T (t )d˜(t ) + A V (t, i) ≤ x˜T (t )( +λθ˜T θ˜ − γ d˜T (t )d˜(t ) ≤ ηT (t )i η(t ) + λθ˜T θ˜, where
x˜(t ) , η(t ) = ˜ d (t )
ˆ ˜T ˜ i = 1i + Ci Ci ∗
P˜i E˜i . −γ
(41)
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Under the condition (38), inequality (41) becomes z˜T (t )z˜(t ) − γ 2 d˜T (t )d˜(t ) + A V (t, i) ≤ λθ˜T θ˜.
(42)
Under zero initial condition, defining d¯T (t ) = [d˜T (t ), θ˜T (t )]T , setting γ 2 = λ and using Dynkin’s formula for Eq. (42) yields T
T
T 2 ¯T ¯ E [z˜ (t )z˜(t )]dt ≤ E [γ d (t )d (t ) − A V (t, r(t ))]dt 0 0 T
[d¯T (t )d¯(t )]dt − E V (T , r(T )) ≤ E γ2 0 T
[d¯T (t )d¯(t )]dt . (43) ≤ E γ2 0
Therefore, the H∞ performance inequality (15) can be achieved. The controller design technique will be summarized in the following theorem.
Theorem 4. For prescribed positive scalars γ and μi , i ∈ S, if there exist positive definite matrices X1i , P2 , Q, symmetric matrices Si , and matrices R1i , R2 satisfying the following conditions:
X1i Si i ≤ 0, j ∈ Suk , j = i, (44) ∗ −X1 j i X1i + Si ≥ 0, j ∈ Suk , j = i,
⎡
i ϒ11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗
BiVi + μi Mi MiT R2T i ϒ22 ∗ ∗ ∗ ∗ ∗ ∗
(45)
i ϒ13 i ϒ23 −γ 2 I ∗ ∗ ∗ ∗ ∗
X1i NiT 0 0 −μi I ∗ ∗ ∗ ∗
0 R2 Mi 0 0 −μ−1 i I ∗ ∗ ∗
0 R2 Bi 0 0 0 −I ∗ ∗
i17 0 0 0 0 0 −i77 ∗
⎤ X1iC1Ti C2iT ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ < 0, i ∈ S, 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ −I (46)
where Ski {k1i k2i · · · kgi }, g ∈ S and H e{Ai X1i + Bi R1i } + λii X1i + j∈Si λi j Si + μi Mi MiT , i ∈ Ski , kqi = i, q ∈ {1, . . . , g}, i k ϒ11 = i H e{Ai X1i + Bi R1i } + j∈Si λi j Si + μi Mi MiT , i ∈ Suk , k
i ϒ13 = Bi Ei 0 Bi , i ϒ22 = H e{P2Wi + R2 BiVi },
i ϒ23 = R2 Bi R2 Ei P2 Gi R2 Bi
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˙ then under an adaptive law θˆ = −Q−1 ξ (x )x T (t )X1−1 i Bi , the H∞ performance of the closed−1 loop system (13) under composite controller (11) with gains Ki = R1i X1−1 i , L = P2 R2i is achieved with a prescribed attenuation level γ . Proof. By some mathematical operations and the Schur complement lemma, Theorem 4 can be obtained from Theorem 3. Hence, we omit the proof. Remark 4. In this paper, several control methods are coupled and applied to reject and attenuate the disturbances and nonlinear uncertainty, such as disturbance observer design, adaptive neural network control, disturbance compensation control and robust H∞ control. Therefore, the analysis of the stability and H∞ control performance is complicated and difficult. Based on the conditions obtained in this paper, the gains of the disturbance observer and the controller (11) can be solved, the composite anti-disturbance control design procedure can be achieved with the adaptive laws in Theorems 1 and 2. Remark 5. As described in [13], the results in this paper can be applied to the cases of the traditional Markovian jump systems with completely know transition rates and the switching systems under arbitrary switching law, which are general. 4. Numerical example In this section, two examples will be presented to illustrate the effectiveness of the proposed method. 4.1. Example 1 We consider the MJS (1) and disturbance system (2) with three operation modes, the parameters of which are set as
−2.2 1.5 −1.5 1.2 0.1 0.1 T , B1 = , E1 = , M1 = , N1 = Mode1 : A1 = , 0 1.2 2.0 1.0 0 0
0.5 0.5 T 0.5 T 0 2.0 T 0.1 , V1 = , C11 = , C21 = , W1 = , G1 = 0.1 0.1 −0.5 0 0 0.1
0.3 1.3 1 0.9 0 0 T , B2 = , E2 = , M2 = , N2 = Mode2 : A2 = , 0.4 2 1.7 1.2 0.1 0.1
0.5 0.7 T 0.5 T 0 1.2 T 0.1 , V2 = , C12 = , C22 = , W2 = , G2 = 0.2 0.1 −0.5 0 0 0.1
0.5 1.5 1.2 0.7 0.1 0.1 T , B3 = , E3 = , M3 = , N3 = Mode3 : A3 = , 0.1 1 1.7 1.0 0.1 0.1
0.5 0.2 T 0.3 T 0 1.0 T 0.1 , V3 = . (47) C13 = , C23 = , W3 = , G3 = 0.5 0.2 −0.5 0 0 0.1 Here, the partially known TRM is given as follows: ⎡ ⎤ ? 0.5 ? 0.1 ⎦. 1 = ⎣0.1 −0.2 ? ? −0.1
(48)
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Switching signal
3
2
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Fig. 1. Evolution of the mode r(t).
Fig. 2. The state trajectories of the closed-loop system (13) when d˜(t ) = 0.
Setting λ = μi = 1, i = 1, 2, 3 and solving Theorem 3, the controller gains (11) with disturbance observer (10) and the corresponding adaptive law are obtained as follows: K1 = −1.0586 −2.4541 , K2 = −0.8070 −1.5322 , K3 = −1.3134 −1.0938 ,
−0.8977 0.5844 , μ = 82.7604, Q = 75.6268. L= (49) −0.2218 0.3407 On the other hand, by setting the H∞ attenuation level as γ = 1 and μi = 1, the following parameters can be obtained by solving Theorem 4: K1 = −2.6714 −11.8414 , K2 = −8.3945 −3.7902 ,
−0.4371 0.2255 . K3 = −3.3006 −4.4669 , L = (50) −0.0573 0.0916 Under the system mode evolution described in Fig. 1, the states of system (13) are simulated with parameters (47) and (49) to illustrate the actual performance of the results. First, for the case of the system without the disturbances d˜(t ), the curves of the states of the closed-loop system are shown in Fig. 2 with the initial value x˜0 = [−0.1 0.1 0.1 0.1]T . Fig. 3
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0.02
0.01
0
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Fig. 3. The approximation error trajectory of nonlinearity f(x) under adaptive law in Theorem 1.
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x
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x
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e
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1
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Fig. 4. The state trajectories of the closed-loop system (13) when d¯(t ) = 0.
shows the approximation error of nonlinearity f(x) between real value and the estimated value based on the neural network under the adaptive law in Theorem 1. Figs. 2 and 3 show that the controller based on the neural network and disturbance observer is effective to guarantee above-mentioned system to be stochastically uniformly ultimately bounded when the effect from disturbance d˜(t ) is not considered. Next, H∞ control performance is illustrated under the attenuation levels set as γ = 1. The initial condition is chosen the same as those in above simulation. In Fig. 4, all state variables of system (13) are globally uniformly asymptotically stable when d¯(t ) = 0. when the saturation level and disturbances are set to ρ = 0.1, d1 = d2 = 0.1sin(et ), the simulation results of the state trajectories of the closed-loop system (13) are presented in Fig. 5, while the curve of the approximation error of nonlinearity is depicted in Fig. 6. One the other hand, Fig. 7 shows the corresponding trajectories of the system output. The above simulation results of Figs. 4–7 indicates that the H∞ performance of the closed-loop system (13) can be achieved.
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Fig. 5. The state trajectories of the closed-loop system (13) subject to disturbances d¯(t ).
0.02
f (x) − θˆT ξ(x)
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Fig. 6. The approximation error trajectory of nonlinearity f(x) under adaptive law in Theorem 2.
0.06
Reference output: z(t)
0.04
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0
−0.02
−0.04
−0.06
0
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Fig. 7. The curve of the system output.
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4.2. Example 2 The single link robot arm systems can be described as the following Markovian jump systems [35]:
0 0 0 1 x˙(t ) = mi gL sin(x1 (t )), D + Mi F (t )Ni x(t ) + 1 [sat (u(t )) + d0 (t ) + f (x)] + 0 − Ji − Ji Ji z(t ) = x(t ), i = 1, 2, 3.
(51)
where x(t ) = [x1T (t ), x2T (t )]T with x1 (t ) = ς (t ), x2 (t ) = ς˙ (t ) and ς (t) is the angle position of the arm; u(t) is the control input; d1 (t) and f(x) are exosystem generated disturbances and nonlinear disturbances, respectively; D is the viscous friction; g is the acceleration of gravity; mi is the mass of the payload; Ji is the moment of inertia. The corresponding parameters are shown as follows: L = 0.2, D = 1, g = 9.8, and
0.5 0 0 T 0 , N1 = , Mode1 : J1 = 15, m1 = 0.5, M1 = , W1 = 0.1 0.1 −0.5 0 1 T 0.1 , V1 = , G1 = 0 0
0.5 0.1 0.1 T 0 , N2 = , Mode2 : J2 = 5, m2 = 0.2, M2 = , W2 = 0 0 −0.5 0 1.2 T 0 , V2 = , G2 = 0 0.1
0.5 0.1 0.1 T 0 , N3 = , Mode3 : J3 = 10, m3 = 0.3, M3 = , W3 = 0.1 0.1 −0.5 0 1 T 0.1 . V3 = , G3 = (52) 0 0.1 Remark 6. In this example, the single link robot arm system is described as the uncertain Markovian jump systems with actuator saturation. Here, the harmonic disturbance and nonlinear disturbance are considered. The control scheme proposed in this paper will be used to achieve the disturbance attenuation and rejection. Under the transition rates matrix in Eq. (48) and based on the results in Theorems 2 and 4, we can get the following gains of disturbance observer (10) and controller (11): K1 = −41.4571 −39.0761 , K2 = −17.9910 −15.3560 ,
−1.7909 1.2258 . K3 = −32.8456 −28.4849 , L = (53) −0.0765 0.1504 In Fig. 8, the simulation results for Example 2 are given with f (x) = x1 ∗ sin(x2 ) and d2 (t ) = sin(ex2 ). The switching signal is shown in Fig. 8(a), which can depict the system mode evolution. From Fig. 8 (b), we can see the nonlinearity f(x) can be approximated effectively by neural network under the adaptive law in Theorems 2 and 4. One the other hand, the disturbance observer (10) with gains Eq. (53) can estimate the exosystem generated disturbances effectively, which can be demonstrated by the curves of the error d0 (t ) − dˆ0 in Fig. 8(c). Fig. 8(d) shows that, under the disturbance observed and adaptive neural networked
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a Switching signal
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c
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Fig. 8. Simulation results for Example 2. (a) System mode evolution; (b) The approximation error trajectory of nonlinearity f(x) under adaptive law in Theorem 2; (c) Estimation error of the exosystem generated disturbances under the disturbances observer (10) with gains (53); (d) States of the system (51) under controller (11) with gains (53).
based controller (11) with gains Eq. (53), the system (51) with parameters (52) has a good control performance. Remark 7. Although the system is subject to norm bounded disturbance, exosystem generated disturbance, nonlinear uncertainty and actuator saturation, the simulations of the examples show that the proposed anti-disturbance control strategy can reject and attenuate the multiple disturbances effectively, in which the error between control input and saturated input is treated as a disturbance-like signal. 5. Conclusion This paper is concerned with the problem of the composite control for a class of uncertain Markovian jump systems with partial information on transition rates, saturating actuator and multiple disturbances, which is in combination with disturbance observer based control, adaptive neural network control and robust H∞ control. Based on the different design approaches to the adaptive law, two cases of anti-disturbance control schemes have been presented, which can guarantee the closed-loop system to be stochastically uniformly ultimately bounded and achieve H∞ performance, respectively. Simulation results have been shown to demonstrate the effectiveness of the proposed method. In the future, we will investigate the Markovian jump systems with input saturation and multiple disturbances including not only deterministic disturbances but also the stochastic disturbance, and the other advanced control methods will be considered like event-triggered mechanism [17,46]. On the other hand, The Markovian jump systems with more general transition rate will be further considered. Acknowledgments This work was supported in part by a research grant from the National Natural Science Foundation of China :61703233, 61773144, 61690212, the Postdoctoral Science Foundation of
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China :2016M602112, the Natural Science Foundation of Shandong Province: ZR2016FQ09, the Open Fund of Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education under Grant: MCCSE2016A04, Science and Technology Planning Project of Qufu Normal University under Grant:xkj201513. References [1] G.J. Olsder, R. Suri, Time-optimal control of parts-routing in a manufacturing system with failure-prone machines, in: Proceedings of the 19th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes, 1980, volume 19, IEEE, 1980, pp. 722–727. [2] P. Shi, E.K. Boukas, R.K. Agarwal, Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delay, IEEE Trans. Autom. Control 44 (11) (1999) 2139–2144. [3] H.R. Karimi, A sliding mode approach to h∞ synchronization of master slave time-delay systems with Markovian jumping parameters and nonlinear uncertainties, J. Frankl. Inst. 349 (4) (2012) 1480–1496. [4] L. Wu, X. Su, P. Shi, Output feedback control of Markovian jump repeated scalar nonlinear systems, IEEE Trans. Autom. Control 59 (1) (2013) 199–204. [5] L. Zhang, Y. Zhu, W.X. Zheng, Energy-to-peak state estimation for Markov jump RNNs with time-varying delays via nonsynchronous filter with nonstationary mode transitions, IEEE Trans. Neural Netw. Learn. Syst. 26 (10) (2015) 2346–2356. [6] J. Cui, T. Liu, Y. Wang, New stability criteria for a class of Markovian jumping genetic regulatory networks with time-varying delays, Int. J. Innov. Comput. Inf. Control 13 (3) (2017) 809–822. [7] H. Shen, F. Li, S. Xu, V. Sreeram, Slow state variables feedback stabilization for semi-Markov jump systems with singular perturbations, IEEE Trans. Autom. Control 63 (8) (2018) 2709–2714. [8] S. Jiao, H. Shen, Y. Wei, X. Huang, Z. Wang, Further results on dissipativity and stability analysis of Markov jump generalized neural networks with time-varying interval delays, Appl. Math. Comput. 336 (2018) 338–350. [9] D.P. De Farias, J.C. Geromel, J.B.R. Do Val, et al., Output feedback control of Markov jump linear systems in continuous-time, IEEE Trans. Autom. Control 45 (5) (2000) 944–949. [10] L. Zhang, E.K. Boukas, J. Lam, Analysis and synthesis of Markov jump linear systems with time-varying delays and partially known transition probabilities, IEEE Trans. Autom. Control 53 (10) (2008) 2458–2464. [11] L. Zhang, J. Lam, Necessary and sufficient conditions for analysis and synthesis of Markov jump linear systems with incomplete transition descriptions, IEEE Trans. Autom. Control 55 (7) (2010) 1695–1701. [12] Q. Ma, S. Xu, Y. Zou, Stability and synchronization for Markovian jump neural networks with partly unknown transition probabilities, Neurocomputing 74 (17) (2011) 3404–3411. [13] G. Zong, D. Yang, L. Hou, et al., Robust finite-time h∞ control for Markovian jump systems with partially known transition probabilities, J. Frankl. Inst. 350 (6) (2013) 1562–1578. [14] X. Li, H.R. Karimi, Y. Wang, et al., Robust fault estimation and fault-tolerant control for Markovian jump systems with general uncertain transition rates, J. Frankl. Inst. 355 (8) (2018) 3508–3540. [15] W.H. Chen, J. Yang, L. Guo, et al., Disturbance-observer-based control and related methods an overview, IEEE Trans. Ind. Electron. 63 (2) (2016) 1083–1095. [16] L. Wu, Y. Gao, J. Liu, et al., Event-triggered sliding mode control of stochastic systems via output feedback, Automatica 82 (2017) 79–92. [17] X. Su, X. Liu, P. Shi, Y. Song, Sliding mode control of hybrid switched systems via an event-triggered mechanism, Automatica 90 (2018) 294–303. [18] G. Sun, L. Wu, Z. Kuang, et al., Practical tracking control of linear motor via fractional-order sliding mode, Automatica 94 (2018) 221–235. [19] R. Yang, W. Zheng, Model transformation based sliding mode control of discrete-time two-dimensional Fornasini-Marchesini systems, J. Frankl. Inst. 356 (5) (2019) 2463–2473. [20] L. Guo, W.H. Chen, Disturbance attenuation and rejection for systems with nonlinearity via DOBC approach, Int. J. Robust Nonlinear Control 15 (3) (2005) 109–125. [21] L. Guo, S. Cao, Anti-Disturbance Control for Systems with Multiple Disturbances, CRC Press, 2013. [22] X. Wei, L. Guo, Composite disturbance-observer-based control and h∞ control for complex continuous models, Int. J. Robust Nonlinear Control 20 (1) (2010) 106–118. [23] J. Park, I.W. Sandberg, Universal approximation using radial-basis-function networks, Neural Comput. 3 (2) (1991) 246–257.
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Journal of the Franklin Institute 356 (2019) 6926–6945 www.elsevier.com/locate/jfranklin
Composite anti-disturbance control for uncertain Markovian jump systems with actuator saturation based disturbance observer and adaptive neural network Yunliang Wei a,b,∗, Guo-Ping Liu b,c, Guangdeng Zong d, Hao Shen e a School
of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, PR China b School of Engineering, University of South Wales, Cardiff 1DL, CF37, U.K. c CTGT Center, Harbin Institute of Technology, Harbin 150001, China d School of Engineering, Qufu Normal University, Rizhao, Shandong 276826, PR China e School of Electrical and Information Engineering, Anhui University of Technology, Maanshan 243002, China Received 30 November 2018; received in revised form 2 June 2019; accepted 8 June 2019 Available online 20 June 2019
Abstract This paper studies the problem of composite control for a class of uncertain Markovian jump systems (MJSs) with partial known transition rates, multiple disturbances and actuator saturation. Compared with the existing results, a novel robust composite control scheme is put forward by virtue of adaptive neural network technique. For MJSs, the partial unknown information on transition rates and the actuator saturation influence the design of disturbance observer and the robust H∞ controller. Firstly, without taking account of external disturbances, the network reconstruction error and saturation, a novel robust adaptive control strategy is established to ensure that all the signals of the closed-loop system are asymptotically bounded in mean square. Secondly, the solvability condition for ensuring the robust H∞ performance is given by using a modified adaptive law, where the saturation is treated as a disturbancelike signal. Finally, the simulations for a numerical example and an application example are performed to validate the effectiveness of the proposed results. © 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.
https://doi.org/10.1016/j.jfranklin.2019.06.006 0016-0032/© 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.
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1. Introduction In the past few years, lots of research activities have been focused on the study of Markovian jump systems (MJSs), since many industrial plants have been modeled successfully as MJSs, such as communication systems, aerospace systems, manufacturing systems, robotic systems and chemical process [1–8]. As a kind of typical stochastic control system, the transition probabilities of the Markov process can largely determine the behaviors of MJSs. But in most cases it should be very hard or costly to get the exact value of each element in the transition rate matrix [9,10]. In this case, the MJSs with partially unknown transition rates have been studied and attracted many considerable attentions in the control fields [11–14]. It is noteworthy that the MJSs mentioned above do not suffer from the uncertainties. As we know, the uncertainties widely exist in many actual processes and practical systems due to load changing, external surroundings, modelling error and so on [15]. Therefore, the anti-disturbance control is an everlasting research subject in the field of control theory and engineering, and so many outstanding control schemes are developed to attenuate and reject the disturbances, such as H∞ control [3,13], sliding mode control [16–19], disturbance compensation control [20]. In the modern engineering practice, the control precision is becoming higher and higher. To meet with the high-precision control requirement, composite hierarchical anti-disturbance control (CHADC) has recently been proposed, which is being targeted to achieve different types of disturbances attenuation and rejection respectively [15,20,21]. In [20], the problem of anti-disturbance control for a class of nonlinear uncertain systems with an exosystem generated disturbance is investigated, the techniques of disturbance-observer-based control (DOBC) and robust observer design are combined effectively, which partly provides the foundation for later development of CHADC. Recently, a CHADC technique combining the DOBC with H∞ control is studied for the system with matched exosystem generated disturbances and mismatched H2 -norm bounded disturbances [22]. It is noteworthy that the neural network model can well approximate unknown uncertainty [23–28]. Due to the physical constraint of the control channel in practice, actuator saturation is one of the ineluctable phenomena, and can lead to negative impact on the performance and stability of the controlled system. Therefore, it is of great importance to consider the existence of actuator saturation in the controller design. For MJSs, some fundamental methods have been proposed for tackling saturation problems, such as optimal control, anti-windup control [29–31]. By the optimal control theory, [32] and [33] present the criteria of the stochastic local stabilization for continuous-time and discrete-time MJSs with saturating actuator, respectively. In [13,34], the problem of anti-windup compensation for MJSs with partially known transition probabilities is studied, and the anti-windup compensator can be synthesised by an optimal strategy. It is worth pointing out that the above mentioned results are local because the saturation nonlinearity is treated as sector nonlinearity model or polytopic differential inclusion, and the optimal strategy needs to be designed to obtain the estimation of the domain of attraction. However, the above treatment of the saturation could not be applied effectively, when the saturation is added on the actuator of the system with a composite controller based on the disturbance observer and adaptive neural network. If the error between the control input and ∗ Corresponding author at: School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, PR China. E-mail addresses: [email protected] (Y. Wei), [email protected] (G.-P. Liu), lovelyletian@ gmail.com (G. Zong), [email protected] (H. Shen).
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saturated input is defined as a disturbance-like signal, we can try to utilize anti-disturbance control scheme to minimize the effects of this signal on controlled output. In [35] and [36], the problems of anti-disturbance control for MJSs with partial known transition probabilities are studied, the proposed composite control methods are mainly based on disturbance observer and H∞ control technique to reject matched exosystem generated disturbances and attenuate unmatched disturbances. When the matched unknown nonlinear uncertainty and actuator saturation appear in the above-mentioned system, the control methods in [35,36] might be ineffective. In [37], a neural network-based DOBC for a class of nonlinear systems in strict-feedback form is studied by combining a neural network scheme and backstepping method. Inspired by [37], a novel method of the composite control for a class of uncertain MJSs subject to actuator saturation and multiple disturbances is put forward, in which the unmatched model-free disturbances, matched exosystem generated disturbances and unknown nonlinear uncertainties are considered. The factors of disturbances and uncertainties are more comprehensive than the existing results in [35,36,38]. Based on the radial basis function neural network and disturbance observer, a novel composite controller is designed for the system under consideration. The error between the control input and saturated input is considered as a disturbance-like signal. Firstly, when model-free disturbances, the network reconstruction error and disturbance-like signal are not considered, a criterion of semi-globally stochastically uniformly ultimate boundedness for the closed-loop system is established and an adaptive law is proposed. Secondly, by combining robust H∞ control with the above composite control scheme, some conditions with a modified adaptive law are given to guarantee that all of the disturbances and uncertainties in the closed-loop system can be attenuated and rejected. Further, some solvable design conditions of composite controller are put forward, respectively. The technique employed in this paper is in combination with matrix inequality technique and adaptive neural network control method, which is different from the case in [37] and can be used for more general systems. It is worth pointing out that the proposed results are applicable to the cases of the traditional MJSs and switched systems under arbitrary switching law. Notation: The standard notation is used throughout the paper. The superscript ‘T’ stands for matrix transposition; L2 [0, ∞] is the space of square-integrable vector functions over [0, ∞); Rn denotes the n-dimensional Euclidean space, and Rm × Rn is the set of all real m × n matrices; E{·} stands for mathematical expectation; In symmetric block matrices, ∗ is used as an ellipsis for the terms that are induced by symmetry; The notation P > 0 (P ≥ 0) means that P is real symmetric and positive definite (semi-positive definite); Given any two symmetric real matrices A and B, A > B refers to the fact that A − B is positive definite; The identity matrix of order m is denoted by Im (or simply I if no confusion arises); sign{·} is the signum function. 2. Preliminaries and problem statement Define the probability space (, F, P ), where is the sample space, F denotes the subσ -algebra of , and P represents the probability measure on F. Consider the uncertain MJSs under (, F, P ) as: x˙(t ) = A(t , r(t ))x(t ) + B(r(t ))[sat (u(t )) + d0 (t ) + f (x)] + E (r(t ))d1 (t ),
(1)
where x(t ) ∈ Rn is the system state; u(t ) ∈ R is the control input; d1 (t ) ∈ R is the external disturbance in L2 [0, ∞]; sat (·) = sign(·) min (·, ρ) with ρ being saturation level; d0 (t ) ∈ R
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is a kind of disturbances generated by an exogenous system as follows: ω˙ (t ) = W (r(t ))ω(t ) + G(r(t ))d2 (t ), d0 (t ) = V (r(t ))ω(t ),
(2)
where ω(t ) ∈ Rr is the disturbance system state; d2 (t) is the additional noise in L2 [0, ∞], which reflects modeling uncertainties and perturbations. {r(t)} is a continuous-time, discrete-state homogeneous Markov process, which takes values in S = {1, 2, . . . , s} and has the mode transition probabilities as if j = i, λ h + o(h), Pr{rt+h = j|rt = i} = i j (3) 1 + λii h + o(h), if j = i, with h > 0, limh→0 (o(h)/h) = 0, and λij ≥ 0, ∀i, j ∈ S, j = i are the transition rates from mode i at time t to mode j at time t + h satisfying s
λii = −
λi j .
(4)
j =1, j =i
Here, the assumption of the transition rates of the Markov process be partially known is made, and the transition rate matrix (TRM) with the notation? of unknown element can be represented by ⎡ ⎤ λ11 ? · · · λ1 s ⎢λ21 λ22 · · · ?⎥ ⎢ ⎥ =⎢ . . .. ⎥. .. .. ⎣ .. . . ⎦ ?
λs2
···
λss
For any i ∈ S, we denote i Ski { j : λi j is known}, Suk { j : λi j is unknown}.
(5)
A(t, r(t)), B(r(t)), D(r(t)), E(r(t)), W(r(t)), V(r(t)) and G(r(t)) are matrix functions and for r(t ) = i, i ∈ S, we rewrite A(t, r(t )) = Ai + Ai (t ), B(r(t )) = Bi , D(r(t )) = Di , E (r(t )) = Ei , W (r(t )) = Wi , V (r(t )) = Vi , G(r(t )) = Gi , where Ai , Bi , Di , Ei , Wi , Vi , Gi are real constant matrices with appropriate dimensions and Ai (t) is norm-bounded uncertainty satisfying Ai (t ) = Mi F (t )Ni ,
(6)
with real constant matrices Mi , Ni and an unknown matrix function F(t) satisfying FT (t)F(t) ≤ 1. f (·) : → R is an unknown smooth nonlinear function. It can be approximated by radial basis function (RBF) neural networks in the form of θ T ξ (x), where θ ∈ RN and ξ (x) is a vector valued function defined in Rn . Lemma 1 ([23],[24]). For a compact set x ∈ R p , x ∈ x ∈ R p, ξ (x) = [ρ1 (x ), ρ2 (x ), . . . , ρl (x)]T is the Gaussian basis function vector in the form
x − ζ j 2 , σ ≥ 0, j = 1, . . . , N, ρ j (x) = ex p − (7) 2σ 2
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where ζ j ∈ Rn and σ ∈ R are called the center and width of the Gaussian function, respectively. If the integer N is chosen as sufficiently large, there exists θ ∗ ∈ RN such that f (x) = θ ∗ ξ (x) + δ ∗ , x ∈ x , ∗
(8) ∗
with |δ | ≤ δ m , where δ m ∈ R and δ is the network reconstruction error. Assumption 1 ([37,40]). For positive constants θ and λ, the assumptions |θ ∗ | ≤ θ and ξ (x)T ξ (x) < λ are made throughout this paper. In what follows, the notation θˆ is used to denote the estimation of θ ∗ , and an adaptive ˆ By introducing error θ˜ = θ ∗ − θˆ, system law will be developed to update the parameter θ. (1) can be rewritten as x˙(t ) = (Ai + Ai (t ))x(t ) + Bi [sat (u(t )) + d0 (t ) + θˆT ξ (x) + (θ ∗ T ξ (x) − θˆT ξ (x)) +( f (x) − θ ∗ T ξ (x))] + Ei d1 (t ) = (Ai + Ai (t ))x(t ) + Bi [sat (u(t )) + d0 (t ) + θˆT ξ (x)] + Bi θ˜T ξ (x) + Bi δ ∗ + Ei d1 (t ). (9) Here, we construct a disturbance observer for d0 (t) as follows: dˆ0 (t ) = Vi ωˆ (t ), ωˆ (t ) = v(t ) − L x (t ),
v˙(t ) = (Wi + LBiVi )ωˆ (t ) + L (Ai x (t ) + Bi u(t ) + Bi θˆT ξ (x)).
(10)
Further, a controller for system (9) shall be designed in the form of u(t ) = −dˆ0 (t ) − θˆT ξ (x) + Ki x(t ).
(11)
Substituting Eq. (11) into Eq. (9) yields x˙(t ) = (Ai + Ai (t ))x(t ) + Bi (Vi eω (t ) + Ki x(t )) + Bi θ˜T ξ (x) + Bi δ ∗ + Ei d1 (t ) + Bi d3 (t ), (12) where eω = ω(t ) − ωˆ (t ) and d3 (t ) = sat (u(t )) − u(t ). Remark 1. The DOBC problem for a class of nonlinear systems subject to input saturation has been investigated, in which the system is subjected to system input saturation [39]. In this paper, the saturation constraint is only added onto the control inputs of the system. Due to the complexity of the controller structure, the classical treatments of the saturation nonlinearity could not be applied effectively like in [39]. Here, the error between the saturated control input and the unconstrained control input is regarded as disturbance-like signal of the system, which can be attenuated by the anti-disturbance control scheme, though it is conservative to some extent. In the future, we would make efforts to obtain less conservative method. Let x˜T (t ) = [x T (t ) eTω (t )], from Eqs. (9) and (10), we have ˙˜ ) = (A˜ i + M˜ i F (t )N˜ i )x˜(t ) + B˜ i θ˜T ξ (x) + E˜i d˜(t ), x(t
(13)
where
BiVi A + Bi Ki −Bi Mi Bi Ei 0 Bi , B˜ i = , M˜ i = , E˜i = , A˜ i = i 0 Wi + LBiVi −LBi LMi LBi LEi Gi LBi N˜ i = Ni 0 , K˜i = Ki Vi , d˜T (t ) = δ ∗ d1 (t ) d2 (t ) d3 (t ) T .
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Definition 1. Let p > 0. System (13) with d˜ = 0 is said to be asymptotically bounded in pth moment if there is a positive constant H such that lim sup E |x˜(t; t0 , x0 , r0 )| p ≤ H,
(14)
t→∞
for all (t0 , x˜0 ) ∈ R+ × Rn+r , when p = 2 we say Eq. (13) with d˜ = 0 to be asymptotically bounded in mean square. The control objective is to obtain some conditions such that: (1) The closed-loop system (13) with (2) For give γ > 0, the following H∞
T T E 0 [z˜T (t )z˜(t )]d t ≤ γ 2 E 0
d˜ = 0 is asymptotically bounded in mean square. performance for system (13) can be achieved
[d¯T (t )d¯(t )]dt . (15)
where d¯T (t ) = [d˜T (t ), θ˜T (t )]T . Remark 2. In [41], the performance inequality is proposed to solve the problem of H∞ control for a class of nonlinear systems in strict-feedback form. To investigate the composite control for system (1), we extend it as case of Eq. (15). For further development, the following lemmas will be given: Lemma 2. Assume that there exist functions V ∈ C 2,1 (Rn+r × R+ × S; R+ ), k ∈ Kv ⊂ K∞ and positive numbers p, β, α, such that k(|x˜| p ) ≤ V (x˜(t ), t , i) and A V (x˜, t, i) ≤ −αV (x˜, t, i) + β, for all (x˜, t, i) ∈ Rn+r × R+ × S . Then
β lim sup E |x˜(t; t0 , x0 , r0 )| p ≤ k −1 t→∞ α for all (x˜0 , t0 , r0 ) ∈ Rn+r × R+ × S . That is, system (13) with d˜ = 0 is asymptotically bounded in pth moment. Lemma 3 ([42]). Let D, S and F be real matrices of appropriate dimensions with FT F ≤ I. Then, for any scalar λ > 0, we have DF S + (DF S)T ≤ λ−1 DDT + λS T S. Lemma 4 ([43]). Let the function y(t) be absolutely continuous for t ≥ t0 and let its derivative satisfies the inequality y˙(t ) ≤ k(t )y(t ) + h(t ), for almost all t ≥ t0 , where k(t) and h(t) are almost everywhere continuous functions integrable over every finite interval. Then for t ≥ t0 , t t t t0 k(s)ds y(t ) ≥ y(t0 )e + e s k(u)du h(s)ds. t0
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3. Main results 3.1. stability analysis In the following theorem, we give some sufficient conditions and an adaptive law to guarantee system (13) without disturbance d˜(t ) to be asymptotically bounded in mean square. Theorem 1. For given scalar λ > 0, if there exist scalars q > 0, μ > 0, μi > 0, τ i > 0, matrices P1i > 0, P2 > 0, Q > 0 and symmetric matrices Ti , i ∈ S satisfying the following conditions: i P1 j + Ti ≤ 0, j ∈ Suk , j = i,
(16)
i P1 j + Ti ≥ 0, j ∈ Suk , j = i,
(17)
⎡
⎤ ˜ 1i + qP˜i 0 N˜ iT ⎣ ∗ (τi λ − μ)I + qQ 0 ⎦ < 0, i ∈ S, ∗ ∗ −μi I ˜ 1i = P˜i A˜ i + A˜ Ti P˜i + j∈Si λi j (P¯ j + T¯i ) + μi P˜i M˜ i M˜ iT P˜i + τ −1 i , where i k
0 0 ¯ 0 P P T 0 0 , P¯i = 1i , Ti = i , i = , P˜i = 1i ∗ P2 ∗ 0 ∗ 0 ∗ P2 L Bi BiT L P2
(18)
then for bounded initial conditions, system (13) without disturbance d˜(t ) is asymptotically bounded in mean square under the following adaptive law: ˙ θˆ = Q−1 (−ξ (x )x T (t )P1i Bi − μθˆ ). Proof. Consider the following Lyapunov function candidate as: V (x˜(t ), θ˜ (t ), i) = x˜T (t )P˜i x˜(t ) + θ˜T (t )Qθ˜ (t ),
(19)
where Q > 0 and
P1i 0 ˜ , Pi = ∗ P2 with P1i > 0, P2 > 0. Under the assumption of disturbance d˜(t ) ≡ 0, the weak infinitesimal operator A of the stochastic process {(x˜, i)} along with system (13) can be shown as follows: ⎛ ⎞ s ˙ ˙ ˙˜ ) + x˜T (t )⎝ A V (x˜(t ), θ˜ (t ), i) = 2x˜T (t )P˜i x(t λi j P˜ j ⎠x˜(t ) + 2θˆT (t )Qθˆ (t ) − 2θ ∗ T Qθˆ ⎛ ˙˜ ) + x˜T (t )⎝ = 2x˜T (t )P˜i x(t
j=1 s
⎞ ˙ λi j P˜ j ⎠x˜(t ) − 2θ˜T (t )Qθˆ.
(20)
j=1
Noting s j=1
s j=1
λi j P˜ j =
λi j = 0, for any i ∈ S, we have
s j=1
λi j P¯ j .
(21)
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By Lemma (3), there exists μi > 0 such that A V (x˜(t ), θ˜ (t ), i) = 2x˜T (t )P˜i [(A˜ i + M˜ i F (t )N˜ i )x˜(t ) + B˜ i θ˜T ξ (x)] ⎛ ⎞ s ˙ +x˜T (t )⎝ λi j P¯ j ⎠x˜(t ) − 2θ˜T (t )Qθˆ ⎛
j=1
≤ x˜T (t )⎝P˜i A˜ i + A˜ Ti P˜i +
s
⎞ ˜ ˜ ˜T ˜ ˜T ˜ ⎠ λi j P¯ j + μ−1 i Pi Mi Mi Pi + μi Ni Ni x˜ (t )
j=1
˙ +2x˜T (t )P˜i B˜ i θ˜T (t )ξ (x) − 2θ˜T (t )Qθˆ.
(22)
Based on Assumption 1 and Lemma 3, one can get the inequality 2x˜T (t )P˜i B˜ i θ˜T (t )ξ (x) ≤ −2x(t )T P1i Bi θ˜T (t )ξ (x) + τi−1 eTω P2 LBi BiT LP2 eω +τi ξ (x)T θ˜ (t )θ˜T (t )ξ (x) ≤ −2x(t )T P1i Bi θ˜T (t )ξ (x) + τi−1 eTω P2 LBi BiT LP2 eω + τi λθ˜T (t )θ˜ (t ). (23) where τ i > 0. Design an adaptive law as ˙ θˆ = Q−1 (−ξ (x )x T (t )P1i Bi − μθˆ ).
(24)
From Eq. (24), it is easy to get that ˙ −2θ˜T (t )Qθˆ = −2θ˜T (t )QQ−1 (−ξ (x )x T (t )P1i Bi − μθˆ ) = 2θ˜T (t )ξ (x )x T (t )P1i Bi + 2μθ˜T (t )θˆ.
(25)
By completing the following square 2μθ˜T (t )θˆ ≤ −μθ˜T θ˜ + μθ ∗ T θ ∗ ,
(26)
and substituting Eqs. (23), (25) and (26) into Eq. (22), we have ⎛ ⎞ s −1 ⎠ ˜ ˜ ˜T ˜ ˜T ˜ A V (x˜(t ), θ˜ (t ), i) ≤ x˜T (t )⎝P˜i A˜ i + A˜ Ti P˜i + λi j P¯ j + μ−1 i Pi Mi Mi Pi + μi Ni Ni + τi i x˜ (t ) j=1
˜T
+(τi λ − μ)θ θ˜ + μθ where 0 i = ∗
∗T
θ ∗,
(27)
0 . P2 L Bi BiT L P2
s some elements of TRM can not be accessible, one has j=1 λi j Ti = Note that i λi j Ti = 0 for any symmetric matrix Ti . j∈Ski λi j Ti + j∈Suk From Eqs. (22) and (27), it can verify that ⎛ ˜T ˜ A V (x˜(t ), θ˜ (t ), i) ≤ x˜T (t )⎝P˜i A˜ i + A˜ Ti P˜i + λi j P¯ j + λi j P¯ j + μi P˜i M˜ i M˜ iT P˜i + μ−1 i Ni Ni j∈Ski
i j∈Suk
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⎞ + τi−1 i ⎠x˜(t ) + (τi λ − μ)θ˜T θ˜ + μθ ∗ T θ ∗ ⎛ = x˜T (t )⎝P˜i A˜ i + A˜ Ti P˜i +
λi j (P¯ j + T¯i ) +
j∈Ski
⎞
λi j (P¯ j + T¯i ) + μi P˜i M˜ i M˜ iT P˜i
i j∈Suk
−1 T ∗T ∗ ˜T ˜ ⎠ ˜T ˜ ˜ ˜ ˜ + μ−1 θ . i Ni Ni +τi i x˜ (t ) + 2x˜ (t )Pi Ei d (t ) + (τi λ − μ)θ θ + μθ
(28) From Eqs. (16) and (17) and under the assumption of d˜(t ) = 0, Eq. (28) can be further rewritten as ¯ 1i x¯(t ) + μθ ∗ T θ ∗ , A V (x˜(t ), θ˜ (t ), i) ≤ x¯T (t )
(29)
where x¯(t ) = x˜T (t ) θ˜T T ,
0 1i ¯ , 1i = ∗ (τi λ − μ)I with 1i = P˜i A˜ i + A˜ Ti P˜i +
−1 ˜T ˜ ˜ ˜ ˜T ˜ λi j (P¯ j + T¯i ) + μ−1 i Ni Ni + μi Pi Mi Mi Pi + τi i .
j∈Ski
Under Assumption 1, using Schur complement formula to Eq. (18) gives rise to A V (x˜(t ), θ˜ (t ), i) ≤ −qV (x˜(t ), θ˜ (t ), i) + p,
(30)
where p = μθ 2 . By Lemma 2, it is easy to get that system (13) with d˜ = 0 asymptotically bounded in mean square. Applying Lemma 4 to Eq. (30) yields
p p −qt e + , ∀t > 0. 0 ≤ E[V (x˜(t ), θ˜ (t ), i)] ≤ V (x˜(t0 ), θ˜ (t0 ), rt0 ) − (31) q q Finially, we have p 0 ≤ E[V (x˜(t ), θ˜ (t ), i)] ≤ V ((x˜(t0 ), θ˜ (t0 ), rt0 ) + . q
(32)
From Eq. (19), when the initial conditions are bounded, we know that Eq. (32) implies that the signals x(t), ew (t ) and θ˜ (t ) are bounded in probability, which implies θˆ is also bounded in probability. Furthermore, the states x˜ and neural network estimates θˆ converge to a neighborhood of the origin. The conditions (18) in Theorem 1 are in the framework of bilinear matrix inequalities which can not be solved directly. In the following theorem, the design conditions of disturbances observer, adaptive law and controller will be further presented via solving a set of LMIs.
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Theorem 2. For prescribed positive scalars λ, q, μi , if there exist positive scalars τ i , μ, positive definite matrices X1i , P2 , Q, symmetric matrices Si , and matrices R2 , R1i , i ∈ S satisfying the following conditions:
X1i Si i ≤ 0, j ∈ Suk , j = i, (33) ∗ −X1 j i X1i + Si ≥ 0, j ∈ Suk , j = i,
⎡
i11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗
BiVi + μi Mi MiT R2T i22 ∗ ∗ ∗ ∗ ∗
(34) 0 0 i33 ∗ ∗ ∗ ∗
X1i NiT 0 0 −μi I ∗ ∗ ∗
0 R2 Mi 0 0 −μ−1 i I ∗ ∗
0 R2 Bi 0 0 0 −τi I ∗
⎤ i17 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ < 0, i ∈ S, 0 ⎥ ⎥ 0 ⎦ −i77
(35)
where Ski {k1i k2i · · · kgi }, g ∈ S and ⎧ T i i ⎨H e{Ai X1i + Bi R1i } + (λii + q)X1i + j∈Ski λi j Si + μi Mi Mi , i ∈ Sk , kq = i, i q ∈ {1, . . . , g}, 11 = ⎩ i H e{Ai X1i + Bi R1i } + j∈Si λi j Si + μi Mi MiT + qX1i , i ∈ Suk , k
i17 = λik1i X1i · · · λikgi X1i , i22 = H e{P2Wi + R2 BiVi } + qP2 , i33 = (λτi − μ)I + qQ, i77 = diag{X1k1i · · · X1kgi }, −1 then system (13) with d˜(t ) = 0 and gains Ki = R1i X1−1 i , L = P2 R2 is uniformly ultimately ˙ bounded under an adaptive law θˆ = Q−1 (−ξ (x )x T (t )X −1 Bi − μθˆ ). 1i
Proof. For any i ∈ S, set X1i = P1−1 i , S1i = X1i Ti X1i , R1i = Ki X1i and R2 = P2 L. First, using the Schur complement formula and performing a congruence transformation by P1i , we can deduce that Eq. (33) is equivalent to Eq. (16). On the other hand, pre-multiply and post-multiply Eq. (34) by P1i , we can derive Eq. (17). Finally, using the Schur complement for Eqs. (33), (34) and performing a congruence transformation by diag{P1i , I} yield Eq. (18). Therefore, the proof can be completed by Theorem 1. Remark 3. In the past decades, the adaptive control has been investigated extensively [44,45]. So far the existing results of adaptive neural network control for the nonlinear system in strictfeedback form are generally combined with the Back-stepping technique [25,37,40]. However, as another effective method, linear matrix inequations (LMIs) have been applied widely to handle control system problems. In Theorem 1, the condition of asymptotically bounded in mean square for the composite system (13) with d˜0 = 0 is proposed by matrix inequations. Further, Theorem 2 shows some solvable sufficient conditions in terms of LIMs to obtain the gains of composite controller (11) and the adaptive law.
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3.2. H∞ performance analysis In this section, a reference output z˜(t ) = C˜i x˜(t ) with C˜i = [C1i C2i ] is first introduced. In the following theorems, the results of H∞ performance analysis for system (13) will be presented with full consideration of the effect of disturbances. Meanwhile, a new adaptive law will be proposed. Theorem 3. For a given scalar γ > 0, if there exist scalars μi > 0, matrices P˜i > 0, Q > 0 and symmetric matrices Ti , i ∈ S satisfying the following conditions: i P1 j + Ti ≤ 0, j ∈ Suk , j = i,
(36)
i P1 j + Ti ≥ 0, j ∈ Suk , j = i,
(37)
P˜i E˜i < 0, i ∈ S −γ
ˆ 1i + C˜i T C˜i ∗
(38)
ˆ 1i = P˜i A˜ i + A˜ Ti P˜i + j∈Si λi j (P¯ j + T¯i ) + μ−1 N˜ iT N˜ i + μi P˜i M˜ i M˜ iT P˜i + i , then under where i k ˙ an adaptive law θˆ = −Q−1 ξ (x )x T (t )P1i Bi , the H∞ performance inequality (15) of the closedloop system (13) is achieved with a prescribed attenuation level γ . Proof. Here, we choose the Lyapunov function candidate (19). It is easy to compute the weak infinitesimal operator A of the stochastic process {(x˜, i)} along with system (13) as A V (t, i) = 2x˜T (t )P˜i [(A˜ i + M˜ i F (t )N˜ i )x˜(t ) + B˜ i θ˜T ξ (x) + E˜i d˜(t )] ⎛ ⎞ s ˙ +x˜T (t )⎝ λi j P˜ j ⎠x˜(t ) − 2θ˜T (t )Qθˆ j=1
≤ x˜ (t )[P˜i A˜ i + A˜ Ti P˜i + T
˜T ˜ ˜ ˜ ˜T ˜ λi j (P¯ j + T¯i ) + μ−1 i Ni Ni + μi Pi Mi Mi Pi ]x˜ (t )
j∈Ski
˙ +2x˜T (t )P˜i E˜i d˜(t ) + 2x˜T (t )P˜i B˜ i θ˜T ξ (x) − 2θ˜T (t )Qθˆ.
(39)
˙ Under conditions (36),(37) and the adaptive law θˆ = −Q−1 ξ (x )x T (t )P1i Bi , we can obtain ˆ 1i x˜(t ) + 2x˜T (t )P˜i E˜i d˜(t ) + λθ˜T θ˜. A V (t, i) ≤ x˜T (t )
(40)
For the given positive scalar γ , one has ˆ 1i + C˜i T C˜i )x˜(t ) + 2x˜T (t )P˜i E˜i d˜(t ) z˜T (t )z˜(t ) − γ d˜T (t )d˜(t ) + A V (t, i) ≤ x˜T (t )( +λθ˜T θ˜ − γ d˜T (t )d˜(t ) ≤ ηT (t )i η(t ) + λθ˜T θ˜, where
x˜(t ) , η(t ) = ˜ d (t )
ˆ ˜T ˜ i = 1i + Ci Ci ∗
P˜i E˜i . −γ
(41)
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Under the condition (38), inequality (41) becomes z˜T (t )z˜(t ) − γ 2 d˜T (t )d˜(t ) + A V (t, i) ≤ λθ˜T θ˜.
(42)
Under zero initial condition, defining d¯T (t ) = [d˜T (t ), θ˜T (t )]T , setting γ 2 = λ and using Dynkin’s formula for Eq. (42) yields T
T
T 2 ¯T ¯ E [z˜ (t )z˜(t )]dt ≤ E [γ d (t )d (t ) − A V (t, r(t ))]dt 0 0 T
[d¯T (t )d¯(t )]dt − E V (T , r(T )) ≤ E γ2 0 T
[d¯T (t )d¯(t )]dt . (43) ≤ E γ2 0
Therefore, the H∞ performance inequality (15) can be achieved. The controller design technique will be summarized in the following theorem.
Theorem 4. For prescribed positive scalars γ and μi , i ∈ S, if there exist positive definite matrices X1i , P2 , Q, symmetric matrices Si , and matrices R1i , R2 satisfying the following conditions:
X1i Si i ≤ 0, j ∈ Suk , j = i, (44) ∗ −X1 j i X1i + Si ≥ 0, j ∈ Suk , j = i,
⎡
i ϒ11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗
BiVi + μi Mi MiT R2T i ϒ22 ∗ ∗ ∗ ∗ ∗ ∗
(45)
i ϒ13 i ϒ23 −γ 2 I ∗ ∗ ∗ ∗ ∗
X1i NiT 0 0 −μi I ∗ ∗ ∗ ∗
0 R2 Mi 0 0 −μ−1 i I ∗ ∗ ∗
0 R2 Bi 0 0 0 −I ∗ ∗
i17 0 0 0 0 0 −i77 ∗
⎤ X1iC1Ti C2iT ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ < 0, i ∈ S, 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ −I (46)
where Ski {k1i k2i · · · kgi }, g ∈ S and H e{Ai X1i + Bi R1i } + λii X1i + j∈Si λi j Si + μi Mi MiT , i ∈ Ski , kqi = i, q ∈ {1, . . . , g}, i k ϒ11 = i H e{Ai X1i + Bi R1i } + j∈Si λi j Si + μi Mi MiT , i ∈ Suk , k
i ϒ13 = Bi Ei 0 Bi , i ϒ22 = H e{P2Wi + R2 BiVi },
i ϒ23 = R2 Bi R2 Ei P2 Gi R2 Bi
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˙ then under an adaptive law θˆ = −Q−1 ξ (x )x T (t )X1−1 i Bi , the H∞ performance of the closed−1 loop system (13) under composite controller (11) with gains Ki = R1i X1−1 i , L = P2 R2i is achieved with a prescribed attenuation level γ . Proof. By some mathematical operations and the Schur complement lemma, Theorem 4 can be obtained from Theorem 3. Hence, we omit the proof. Remark 4. In this paper, several control methods are coupled and applied to reject and attenuate the disturbances and nonlinear uncertainty, such as disturbance observer design, adaptive neural network control, disturbance compensation control and robust H∞ control. Therefore, the analysis of the stability and H∞ control performance is complicated and difficult. Based on the conditions obtained in this paper, the gains of the disturbance observer and the controller (11) can be solved, the composite anti-disturbance control design procedure can be achieved with the adaptive laws in Theorems 1 and 2. Remark 5. As described in [13], the results in this paper can be applied to the cases of the traditional Markovian jump systems with completely know transition rates and the switching systems under arbitrary switching law, which are general. 4. Numerical example In this section, two examples will be presented to illustrate the effectiveness of the proposed method. 4.1. Example 1 We consider the MJS (1) and disturbance system (2) with three operation modes, the parameters of which are set as
−2.2 1.5 −1.5 1.2 0.1 0.1 T , B1 = , E1 = , M1 = , N1 = Mode1 : A1 = , 0 1.2 2.0 1.0 0 0
0.5 0.5 T 0.5 T 0 2.0 T 0.1 , V1 = , C11 = , C21 = , W1 = , G1 = 0.1 0.1 −0.5 0 0 0.1
0.3 1.3 1 0.9 0 0 T , B2 = , E2 = , M2 = , N2 = Mode2 : A2 = , 0.4 2 1.7 1.2 0.1 0.1
0.5 0.7 T 0.5 T 0 1.2 T 0.1 , V2 = , C12 = , C22 = , W2 = , G2 = 0.2 0.1 −0.5 0 0 0.1
0.5 1.5 1.2 0.7 0.1 0.1 T , B3 = , E3 = , M3 = , N3 = Mode3 : A3 = , 0.1 1 1.7 1.0 0.1 0.1
0.5 0.2 T 0.3 T 0 1.0 T 0.1 , V3 = . (47) C13 = , C23 = , W3 = , G3 = 0.5 0.2 −0.5 0 0 0.1 Here, the partially known TRM is given as follows: ⎡ ⎤ ? 0.5 ? 0.1 ⎦. 1 = ⎣0.1 −0.2 ? ? −0.1
(48)
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Switching signal
3
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Fig. 1. Evolution of the mode r(t).
Fig. 2. The state trajectories of the closed-loop system (13) when d˜(t ) = 0.
Setting λ = μi = 1, i = 1, 2, 3 and solving Theorem 3, the controller gains (11) with disturbance observer (10) and the corresponding adaptive law are obtained as follows: K1 = −1.0586 −2.4541 , K2 = −0.8070 −1.5322 , K3 = −1.3134 −1.0938 ,
−0.8977 0.5844 , μ = 82.7604, Q = 75.6268. L= (49) −0.2218 0.3407 On the other hand, by setting the H∞ attenuation level as γ = 1 and μi = 1, the following parameters can be obtained by solving Theorem 4: K1 = −2.6714 −11.8414 , K2 = −8.3945 −3.7902 ,
−0.4371 0.2255 . K3 = −3.3006 −4.4669 , L = (50) −0.0573 0.0916 Under the system mode evolution described in Fig. 1, the states of system (13) are simulated with parameters (47) and (49) to illustrate the actual performance of the results. First, for the case of the system without the disturbances d˜(t ), the curves of the states of the closed-loop system are shown in Fig. 2 with the initial value x˜0 = [−0.1 0.1 0.1 0.1]T . Fig. 3
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Fig. 3. The approximation error trajectory of nonlinearity f(x) under adaptive law in Theorem 1.
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x
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Fig. 4. The state trajectories of the closed-loop system (13) when d¯(t ) = 0.
shows the approximation error of nonlinearity f(x) between real value and the estimated value based on the neural network under the adaptive law in Theorem 1. Figs. 2 and 3 show that the controller based on the neural network and disturbance observer is effective to guarantee above-mentioned system to be stochastically uniformly ultimately bounded when the effect from disturbance d˜(t ) is not considered. Next, H∞ control performance is illustrated under the attenuation levels set as γ = 1. The initial condition is chosen the same as those in above simulation. In Fig. 4, all state variables of system (13) are globally uniformly asymptotically stable when d¯(t ) = 0. when the saturation level and disturbances are set to ρ = 0.1, d1 = d2 = 0.1sin(et ), the simulation results of the state trajectories of the closed-loop system (13) are presented in Fig. 5, while the curve of the approximation error of nonlinearity is depicted in Fig. 6. One the other hand, Fig. 7 shows the corresponding trajectories of the system output. The above simulation results of Figs. 4–7 indicates that the H∞ performance of the closed-loop system (13) can be achieved.
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Fig. 5. The state trajectories of the closed-loop system (13) subject to disturbances d¯(t ).
0.02
f (x) − θˆT ξ(x)
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Fig. 6. The approximation error trajectory of nonlinearity f(x) under adaptive law in Theorem 2.
0.06
Reference output: z(t)
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Fig. 7. The curve of the system output.
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4.2. Example 2 The single link robot arm systems can be described as the following Markovian jump systems [35]:
0 0 0 1 x˙(t ) = mi gL sin(x1 (t )), D + Mi F (t )Ni x(t ) + 1 [sat (u(t )) + d0 (t ) + f (x)] + 0 − Ji − Ji Ji z(t ) = x(t ), i = 1, 2, 3.
(51)
where x(t ) = [x1T (t ), x2T (t )]T with x1 (t ) = ς (t ), x2 (t ) = ς˙ (t ) and ς (t) is the angle position of the arm; u(t) is the control input; d1 (t) and f(x) are exosystem generated disturbances and nonlinear disturbances, respectively; D is the viscous friction; g is the acceleration of gravity; mi is the mass of the payload; Ji is the moment of inertia. The corresponding parameters are shown as follows: L = 0.2, D = 1, g = 9.8, and
0.5 0 0 T 0 , N1 = , Mode1 : J1 = 15, m1 = 0.5, M1 = , W1 = 0.1 0.1 −0.5 0 1 T 0.1 , V1 = , G1 = 0 0
0.5 0.1 0.1 T 0 , N2 = , Mode2 : J2 = 5, m2 = 0.2, M2 = , W2 = 0 0 −0.5 0 1.2 T 0 , V2 = , G2 = 0 0.1
0.5 0.1 0.1 T 0 , N3 = , Mode3 : J3 = 10, m3 = 0.3, M3 = , W3 = 0.1 0.1 −0.5 0 1 T 0.1 . V3 = , G3 = (52) 0 0.1 Remark 6. In this example, the single link robot arm system is described as the uncertain Markovian jump systems with actuator saturation. Here, the harmonic disturbance and nonlinear disturbance are considered. The control scheme proposed in this paper will be used to achieve the disturbance attenuation and rejection. Under the transition rates matrix in Eq. (48) and based on the results in Theorems 2 and 4, we can get the following gains of disturbance observer (10) and controller (11): K1 = −41.4571 −39.0761 , K2 = −17.9910 −15.3560 ,
−1.7909 1.2258 . K3 = −32.8456 −28.4849 , L = (53) −0.0765 0.1504 In Fig. 8, the simulation results for Example 2 are given with f (x) = x1 ∗ sin(x2 ) and d2 (t ) = sin(ex2 ). The switching signal is shown in Fig. 8(a), which can depict the system mode evolution. From Fig. 8 (b), we can see the nonlinearity f(x) can be approximated effectively by neural network under the adaptive law in Theorems 2 and 4. One the other hand, the disturbance observer (10) with gains Eq. (53) can estimate the exosystem generated disturbances effectively, which can be demonstrated by the curves of the error d0 (t ) − dˆ0 in Fig. 8(c). Fig. 8(d) shows that, under the disturbance observed and adaptive neural networked
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b
a Switching signal
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Fig. 8. Simulation results for Example 2. (a) System mode evolution; (b) The approximation error trajectory of nonlinearity f(x) under adaptive law in Theorem 2; (c) Estimation error of the exosystem generated disturbances under the disturbances observer (10) with gains (53); (d) States of the system (51) under controller (11) with gains (53).
based controller (11) with gains Eq. (53), the system (51) with parameters (52) has a good control performance. Remark 7. Although the system is subject to norm bounded disturbance, exosystem generated disturbance, nonlinear uncertainty and actuator saturation, the simulations of the examples show that the proposed anti-disturbance control strategy can reject and attenuate the multiple disturbances effectively, in which the error between control input and saturated input is treated as a disturbance-like signal. 5. Conclusion This paper is concerned with the problem of the composite control for a class of uncertain Markovian jump systems with partial information on transition rates, saturating actuator and multiple disturbances, which is in combination with disturbance observer based control, adaptive neural network control and robust H∞ control. Based on the different design approaches to the adaptive law, two cases of anti-disturbance control schemes have been presented, which can guarantee the closed-loop system to be stochastically uniformly ultimately bounded and achieve H∞ performance, respectively. Simulation results have been shown to demonstrate the effectiveness of the proposed method. In the future, we will investigate the Markovian jump systems with input saturation and multiple disturbances including not only deterministic disturbances but also the stochastic disturbance, and the other advanced control methods will be considered like event-triggered mechanism [17,46]. On the other hand, The Markovian jump systems with more general transition rate will be further considered. Acknowledgments This work was supported in part by a research grant from the National Natural Science Foundation of China :61703233, 61773144, 61690212, the Postdoctoral Science Foundation of
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