Accepted Manuscript
A new design of sliding mode control for Markovian jump systems based on stochastic sliding surface Jianyu Zhang, Qingling Zhang, Yingying Wang PII: DOI: Reference:
S0020-0255(17)30414-0 10.1016/j.ins.2017.02.005 INS 12728
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Information Sciences
Received date: Revised date: Accepted date:
28 December 2015 19 August 2016 2 February 2017
Please cite this article as: Jianyu Zhang, Qingling Zhang, Yingying Wang, A new design of sliding mode control for Markovian jump systems based on stochastic sliding surface, Information Sciences (2017), doi: 10.1016/j.ins.2017.02.005
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ACCEPTED MANUSCRIPT
A new design of sliding mode control for Markovian jump systems based on stochastic sliding surface
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Jianyu Zhanga,d , Qingling Zhanga,b,∗, Yingying Wangc a
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Institute of Systems Science, Northeastern University, Shenyang 110819, PR China b State Key Laboratory of Synthetical Automation for Process Industries, Shenyang c College of Information Science and Engineering, Northeastern University, Shenyang d Guidaojiaotong Polytechnic Institute, Shenyang 110023, PR China
Abstract
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With the jumping transfer matrix unknown, partly unknown or known, the stabilization problems for Markovian jump systems are considered by use of sliding mode control method in this paper. Firstly, a new integral sliding mode surface named stochastic sliding surface (SSS) is introduced. The SSS is characterized that the state trajectories are on the sliding surface all the time even at the time of system switching. So the problem of the state trajectories moving among several sliding surfaces by use of traditional sliding mode control method is settled perfectly. Based on the new sliding mode surface, sufficient conditions for the stability of the sliding mode dynamics are derived. Secondly, sliding mode control (SMC) law is synthesized to avoid the state trajectories escaping from the surface. Finally, some simulations are provided to illustrate the validity of the proposed method.
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Keywords: Stochastic sliding surface; Markovian jump systems; Time delay; Integral sliding mode control; Uncertainty. 1. Introduction
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In the last years, there has been significant improvements in control field, e.g., stabilization [21], [36], sliding mode control [1], [7], [11], [14], [16], fuzzy ∗
Corresponding author at: Institute of Systems Science, Northeastern University, Shenyang 110819, PR China. Email addresses:
[email protected] (Jianyu Zhang),
[email protected] (Qingling Zhang),
[email protected] (Yingying Wang) Preprint submitted to Information Sciences
February 2, 2017
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control [4], [22], network control [17], [31], and so on. Among all the publications, Markovian jump systems (MJSs) have received increasing attention, and a lot of results have been proposed for MJSs [3], [8], [9], [13], [24]-[30], [35]. Recently, the application on integral sliding mode control (SMC) is also extended to MJSs [17], [2], [19], since integral SMC has been recognized as an effective robust control approach. By use of traditional method, the integral sliding mode surface is dependent on system mode. Theoretically, SMC system is twofold. Firstly, the SMC law can drive the dynamics of the controlled system into a designated sliding mode surface in finite time. Secondly, the SMC system converges to zero on the surface. In [26], the authors used system output information to stabilize a class of nonlinear interconnected systems with mismatched uncertainty by means of SMC. In [27], based on sliding mode techniques, a static output feedback control law was synthesized to stabilize a class of nonlinear systems with delay disturbances. But, a problem may arise by Markovian jump process. After the system state trajectories reach the sliding mode surface s(t, i) = 0, it probably jumps to another sliding mode surface s(t, j) at time t + ∆t. When the system state trajectories reach s(t + ∆t, j) = 0, the condition s(t + ∆t, i) = 0 is destroyed. Thus, the jumping effect is generated. How to deal with this problem? By use of the jumping transfer matrix, the jumping effect is settled. Let π = (αij )ν×ν (i, j=1,2,. . .,ν) represent the jumping transfer matrix. By means of jumping transfer matrix, a lot of results about MJSs have been published. Shi et al. [20] considered the design of SMC for stochastic jump system. Kao et al. [10] considered H∞ SMC for uncertain neutral-type stochastic systems with Markovian jumping parameters. In [12], the paper was focused on designing an H∞ SMC for a class of neutral-type stochastic systems with Markovian switching parameters. In [19], the authors dealt with the problem of SMC for a class of nonlinear uncertainty stochastic systems with Markovian switching. In [15], the author studied the stabilization problem for a class of Markovian stochastic jump systems against sensor fault, actuator fault and input disturbances simultaneously. Some results about MJSs with time-delay have also been published. In paper [6], the authors concerned with the state estimation and the SMC for a class of MJSs with mixed mode-dependent time delays and input nonlinearity. In [5], the paper is concerned with the problem of finite-time H∞ control for a class of MJSs with mode-dependent time-varying delay. However, it is noted that in all of the related results of MJSs, the jumping transfer matrix is assumed 2
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to be known or partly known [33]. Hence, there are still some unfavorable circumstances. In fact, it isn’t an easy thing to obtain a transfer matrix, since the complete techniques on the transfer matrix maybe time-consuming or expensive got, or even not acquired at all [32], [34]. And a transfer matrix can be changed when the background of the system is changed. It will make the transfer matrix unknown for us. For example, the transfer matrix of market share can be changed because every manager improves the quality of products or boosts advertising effectiveness. So the managers can’t know the transfer matrix in time. Paper [33] proposed a method to make the Markovian linear system stabilization with the transfer matrix unknown. But how to deal with the system with external disturbance? This question is worth studying. In this paper, a new method is proposed to deal with this system with transfer matrix unknown. And other two theorems are proposed to deal with the system with transfer matrix partly unknown. This paper will consider the problem of SMC for MJSs in which the jumping transfer matrix is unknown, partly unknown or known. Instead of traditional several sliding mode surfaces related to every mode, a new sliding mode surface, e.g., stochastic sliding surface (SSS), is presented for MJSs. This new sliding mode surface is characterized that the trajectories of the system are on the surface all the time even at the time of system switching. In another word, there is not a process of reaching the sliding mode surface. Now, let us represent the sliding mode surface’s forming process. At the beginning, the trajectories move on the surface which passes through the equilibrium point and the starting point. At one point, the system switches. Then the trajectories move on another surface which passes through the equilibrium point and the point which is the state at the time the system just switching. The process continues to form the SSS. This is the forming process of SSS. Next, sufficient conditions for the stability of the sliding mode dynamics based on the new sliding mode surface are derived. And then SMC law is synthesized to avoid the state trajectories leaving away from the sliding mode surface. The rest of the paper is organized as follows. Section 2 presents related preliminaries and lemmas. In section 3, a new sliding mode surface named SSS is introduced. Based on the SSS, the SMC stabilization problem is addressed. And a SMC controller is synthesized to avoid the trajectories running away from the surface. In section 4, some simulations are provided to illustrate the effect of the proposed method. Finally, the paper is concluded in section 5. Notations: k · k denotes the Euclidean norm of a vector or its induced 3
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2. PRELIMINARIES
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matrix norm. For a real symmetric matrix, R > 0 means that the matrix R is positive definite and R = RT . Matrices are assumed to be compatible for algebraic operations if their dimensions are not explicitly presented.
Consider the following uncertain time-delay system with matched external disturbance: x(t) ˙ = (A(r(t)) + ∆A(r(t)))x(t) + (Ad (r(t)) + ∆Ad (r(t)))x(t − d)
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+B(r(t))(u(t) + f (x(t), t, (r(t)))) x(t) = ϕ(t), t ∈ [−d 0]
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where x(t) ∈ Rn is the state, u(t) ∈ Rl is the control input. The matrices A(r(t)) ∈ Rn×n , Ad (r(t)) ∈ Rn×n , and B(r(t)) ∈ Rn×l are known real constant matrices, ∆A(t)(r(t)) and ∆Ad (r(t)) are unknown time-varying matrices representing system parameter uncertainties. Time delay d is a known real constant scalar satisfying 0 < t < d < ∞. f (x(t), t, (r(t))) ∈ Rl is an unknown nonlinear function satisfying kf (x(t), t, (r(t)))k ≤ µ, where µ is a known positive scalar. {r(t), t ≥ 0} is a right-continuous Markovian process on the probability space which takes values in a finite state space S = {1, 2, ..., ν}, generator αij with transition probability from mode i at time t to mode j at time t + δ, i, j ∈ S αij δ + o(δ) i 6= j P r{r(t + δ) = j|r(t) = i} = 1 + αii δ + o(δ) i=j
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αii = −
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j=1,j6=i
αij , αij ≥ 0, ∀i, j ∈ S, i 6= j
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where δ > 0 and limδ→0 o(δ)/δ = 0. In this paper, the transition matrix of the jumping process are assumed to be unknown, partly unknown or known. In other words, some elements in matrix π are unknown, partly unknown or known. For example, if there are three operation modes in system (1)-(2), the corresponding jumping transfer matrix π may be described as:
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α11 α12 α13 π = α21 α22 α23 α31 α32 α33
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and
? ? ? π= ? ? ? ? ? ? ? α12 ? ? π = α21 ? ? α32 α33
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where ”?” denotes the unknown elements. In addition, it is assumed that the input matrix B(r(t)) has full column rank. And ∆A(r(t)), ∆Ad (r(t)) are assumed to be norm bounded:
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∆A(r(t)) = E(r(t))F1 (t, (r(t)))H(r(t))
∆Ad (r(t)) = E( r(t))F2 (t, (r(t)))Hd (r(t))
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where E(r(t)), Ed (r(t)), H(r(t)), Hd (r(t)) are known real constant matrices, and F1 (t, (r(t))), F2 (t, (r(t))) are unknown time-varying matrices satisfying
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FiT (t, (r(t)))Fi (t, (r(t))) ≤ I
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for all t, where i = 1, 2. Before proceeding further, we provide the following lemmas which will be needed in our proof. Lemma 1[23]: Let E, F , and H be real matrices with appropriate dimensions, and let F satisfy F T F ≤ I. Then, for any scalar ε > 0, the following matrix inequality holds: EF H + H T F T E T ≤ ε−1 EE T + εH T H.
Lemma 2: For any B ∈ Rn×l and P > 0, the following matrix inequality holds: −B T P B ≤ ±BlT ± Bl + Pl−1 5
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where l ≤ n, Bl = I1 B, Pl−1 = I1 P −1 I1T , with I1 = (Il 0l×(n−l) ). ¯ = (B 0) ∈ Rn×n , since P > 0, Proof. Define B ¯T P B ¯ ±B ¯T ± B ¯ + P −1 ≥ 0 B ¯T P B ¯ ≤ ±B ¯T ± B ¯ + P −1 −B T B PB 0 ¯T ± B ¯ + P −1 − ≤ ±B 0 0
Hence, one can conclude
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−B T P B ≤ ±BlT ± Bl + Pl−1 .
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¯ ± P −1 )T P (B ¯ ± P −1 ) ≥ 0 (B
3. The design of SMC
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The aim of this paper is to design a new sliding mode surface named SSS s(t) and a SMC law u(t) for a Markovian jump system (1)-(2). Under this SMC, the trajectories of the Markovian jump system are on the designed SSS s(t) = 0 for all the time. Firstly, the sliding mode surface for the Markovian jump system will be firstly constructed and the sufficient conditions for the asymptotic stability of sliding mode dynamics will be derived. Secondly, the sliding mode controller will be designed such that the trajectories of the system state will be on the sliding surface for all the time despite Markovian switching. Here, it is worth noting that there are several modes in a Markovian jump system. If so, we have to answer the following questions during the design of SMC system. Q1: How to design only one integral sliding mode surface for a finite-state Markovian jump system (1)-(2)? Q2: How is the sliding mode controller updated so as to ensure the attraction of the sliding mode surface under Markovian switching?
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3.1. Stochastic sliding surface design In this part, we propose a new design method of integral sliding mode surface for MJSs. The integral sliding mode surface is designed as follows: Z t s(t) = −G(r(t)) (A(r(t)) + B(r(t))K(r(t)))x(z)dz T (t)
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Step function T(t)
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Fig 1. The trajectory of T(t).
+G(r(t))x(t) − G(r(t))X0 (t)
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where T (t) ∈ R+ is a step function, T (t) = t when r(t) ˙ 6= 0. X0 (t) ∈ Rn is a step function too, X0 (t) = x(t) when r(t) ˙ 6= 0. For example, if the time t which the MJSs switch at is 0, 5, 20, 25, 35, the simulation results of T (t) is in Fig. 1. Definition 1. The integral sliding mode function s(t) like equation (6) is said to be stochastic sliding surface (SSS) if s(t) = 0 at the moment of the MJSs switching (r(t) ˙ 6= 0). Remark 1: From equation (6), it can be seen that the sliding surface function contains two step functions T (t) and X0 (t). The values of step functions T (t) and X0 (t) can be changed when the Markovian system switches. Because the switch of the system is stochastic, the sliding surface s(t) is modified stochastically. That’s why the sliding surface is called stochastic sliding surface. This is also one difference from the traditional integral sliding surface. Now, the problem is whether the equation (6) is zero at the moment of the MJSs switching (r(t) ˙ 6= 0). When r(t) ˙ 6= 0, we have X0 (t) = x(t), T (t) = t. And Z t s(t) = −G(r(t)) (A(r(t)) + B(r(t))K(r(t)))x(z)dz t
+G(r(t))x(t) − G(r(t))x(t) =0 7
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Remark 2: Using traditional sliding mode surface, it is well known that the state trajectories can be driven onto the traditional sliding mode surface in finite time. By use of the SSS, the state x(t) is on the surface all the time. It needs not to take time to reach the sliding mode surface. This is the other difference from the traditional integral sliding surface. Q1 is answered.
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3.2. Stabilization results with the jumping transfer matrix unknown For each r(t) = i ∈ S, A(r(t)) = A(i), Ad (r(t)) = Ad (i), ∆A(r(t)) = ∆A(i), ∆Ad (r(t)) = ∆Ad (i), B(r(t)) = B(i), f (x(t), t, r(t)) = f (i). Then system (1) becomes x(t) ˙ = (A(i) + ∆A(i))x(t) + (Ad (i) + ∆Ad (i))x(t − d) +B(i)(u(t) + f (i)), i ∈ S
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T (t)
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By equation (7), one can have the solution of x(t) as follows: Z t x(t) = x(T (t)) + ((A(i) + ∆A(i))x(z) + (Ad (i) + ∆Ad (i))x(z − d)
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+B(i)(u + f (i)))dz Z t = X0 (t) + (((A(i) + ∆A(i))x(z) + B(i)(u + f (i)) T (t)
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+(Ad (i) + ∆Ad (i))x(z − d))dz
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It follows from Equation (6) and (8) that Z t s(t) = G(i) (∆A(i) − B(i)K(i))x(z) + (Ad (i) + ∆Ad (i))x(z − d) T (t)
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According to the SMC theory, the equivalent control law can be obtained from s(t) ˙ = 0, thus we have ueq = −(G(i)B(i))−1 G(i)(∆A(i) − G(i)B(i)K(i))x(t) − f (i) −(G(i)B(i))−1 G(i)(Ad (i) + ∆Ad (i))x(t − d) 8
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By substituting equation (10) into equation (7), the sliding mode dynamics yields as
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x(t) ˙ = (A(i) + B(i)K(i) + ∆A(i) + (Ad (i) + ∆Ad (i) −B(i)(G(i)B(i))−1 G(i)(Ad (i) + ∆Ad (i)))x(t − d) −B(i)(G(i)B(i))−1 G(i)∆A(i))x(t)
3.2.1. Performance of the sliding mode dynamics
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The following theorem gives a sufficient condition to guarantee the asymptotic stability of the sliding mode dynamics (11). Theorem 1: Consider the Markovian jumping system (1)-(2) and the stochastic sliding surface (6). The sliding mode dynamics (11) is asymp¯ a totically stable, if there exist symmetrical positive-definite matrices P¯ , Z, matrix M (i), scalars ε1 (i) > 0, ε2 (i) > 0, ε3 (i) > 0, ε4 (i) > 0, satisfying Γ1 Γ2 χ(i) = <0 (12) ∗ Γ3 where
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√ X + X T + Y Ad (i)P¯ 2B(i) Γ1 = ∗ Z¯ − 2P¯ 0 T ¯ ∗ ∗ −B(i)l − B(i)l + Pl 0 P¯ 0 P¯ H T (i) P¯ H T (i) 0 0 0 0 P¯ HdT (i) P¯ HdT (i) Γ2 = 0 0 P¯ ATd (i) 0 0 0 0 0 0 0
¯ −P¯ + ε4 (i)Ed (i)E T (i), Γ3 = diag{−P¯ + ε3 (i)E(i)E T (i), −Z, d −ε1 (i), −ε3 (i), −ε2 (i), −ε4 (i)}
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with Y = ε1 (i)E(i)E T (i) + ε2 (i)Ed (i)EdT (i), X = A(i)P¯ + B(i)M (i), Bl = I1 B, P¯l = I1 P¯ I1T , I1 = (Il 0l×(n−l) ), l ≤ n, i ∈ S. Moreover, the feedback gain can be obtained: K(i) = M (i)P¯ −1 . Proof: Choose the following Lyapunov function candidate: V (t) = V1 (t) + V2 (t) 9
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V1 (t) = xT (t)P x(t) Z t V2 (t) = xT (ω)Zx(ω)dω t−d
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where
where P , Z is symmetrical positive-definite matrices. Then calculating the time derivative of V (t) along the solution of system (11) obtains: V˙ 1 = 2xT P (A(i) + B(i)K(i) + ∆A(i) +2xT P (Ad (i) + ∆Ad (i)
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−B(i)(G(i)B(i))−1 G(i)∆A(i))x(t)
−B(i)(G(i)B(i))−1 G(i)(Ad (i) + ∆Ad (i)))x(t − d)
V˙ 2 = x Zx(t) − x (t − d)Zx(t − d) T
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Based on (13) and (14), one can obtain the following differential equation T x(t) Ω + ΩT + Z Φ x(t) ˙ V = x(t − d) ∗ −Z x(t − d) where
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Ω = P A(i) + P B(i)K(i) + P ∆A(i) − P B(i)(G(i)B(i))−1 G(i)∆A(i) Φ = P Ad (i) + P ∆Ad (i) − P B(i)(G(i)B(i))−1 G(i)(Ad (i) + ∆Ad (i))
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If the following inequality holds: Ω + ΩT + Z Φ <0 ∗ −Z
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(15)
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the sliding mode dynamics (11) is asymptotically stable. Defining −1 P 0 Υ= 0 P −1 and pre-multiplying and post-multiplying (15) by Υ, one can conclude that (15) is equivalent to Ω1 + ΩT1 + P −1 ZP −1 Φ1 <0 (16) ∗ −P −1 ZP −1 10
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where
Ω2 = A(i)P −1 + B(i)K(i)P −1 + ∆A(i)P −1 Φ1 = Ad (i)P −1 + ∆Ad (i)P −1
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Ω1 = Ω2 − B(i)(G(i)B(i))−1 G(i)∆A(i)P −1
−B(i)(G(i)B(i))−1 G(i)(Ad (i) + ∆Ad (i))P −1
Note that G(i) = B T (i)P ,
−2xT B(i)(G(i)B(i))−1 G(i)∆A(i)P −1 x
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−2xT B(i)(G(i)B(i))−1 G(i)(Ad (i) + ∆Ad (i))P −1 x(t − d)
≤ xT B(i)(G(i)B(i))−1 B T (i)x + xT (t − d)P −1 (ATd (i) +∆ATd (i))P (Ad (i) + ∆Ad (i))P −1 x(t − d)
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Substituting (17)-(18) into (16) , one can derive that (16) is no larger than Ω3 Ad (i)P −1 + ∆Ad (i)P −1 (19) ∗ Ψ where
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Ω3 = Ω2 + ΩT2 + 2B(i)(G(i)B(i))−1 B T (i) + P −1 ∆AT (i)P −1 + P −1 ZP −1
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Ψ = −P −1 ZP −1 + P −1 Z(ATd (i) + ∆ATd (i))P (Ad (i) + ∆Ad (i))P −1 Let P¯ = P −1 . By Schur complement lemma, one can concluded that (19) is equivalent to Ω4 Γ2 (20) ∗ Γ3
where
Ω5 + ΩT5 + ε1 E(i)E T (i) + ε2 Ed (i)EdT (i) ∗ Ω4 = ∗ 11
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√ Ad (i)P −1 2B −P¯ Z P¯ 0 ∗ −G(i)B(i)
Since Z > 0, the following inequality holds: (P¯ − Z −1 )Z(P¯ − Z −1 ) < 0
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Ω5 = A(i)P¯ + B(i)K(i)P¯
Let Z¯ = Z −1 . Then the following inequality is derived −P¯ Z P¯ ≤ Z¯ − 2P¯
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Based on Lemma 2 and inequality (21), one can conclude Γ1 ≥ Ω4 So Γ1 Γ2 ∗ Γ3
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the sliding mode dynamics (11) is asymptotically stable. The proof is completed.
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3.2.2. Sliding mode controller design
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it can be shown that the trajectories of the considered system can approach the surface (SSS) s(t) = 0 by an SMC law:
−sgn(s(t))(ρ(t) + µ + λ)
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u(t) = K(i)x(t) − (G(i)B(i))−1 G(i)Ad (i)x(t − d) (22)
1 V = sT (t)(G(i)B(i))−1 s(t) 2 Choose
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where λ is a small positive scalar, ρ(t) = k(G(i)B(i))−1 G(i)E(i)k kH(i)x(t)k+ k(G(i)B(i))−1 G(i)Ed (i)kkHd (i)x(t − d)k and kf (i)k < µ. Proof: First, consider s(t) 6= 0. Consider the following Lyapunov function candidate:
u1 (t) = K(i)x(t) − (G(i)B(i))−1 G(i)Ad (i)x(t − d) s(t) (ρ(t) + (µ + λ)) ks(t)k
From (9), one can have
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+G(i)∆A(i)x(t) + G(i)∆Ad (i)x(t − d)
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V˙ = sT (t)(G(i)B(i))−1 s(t) ˙
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Substitute (24) into (25) V˙ = sT (t)((G(i)B(i))−1 G(i)∆A(i)x(t) + (G(i)B(i))−1 G(i)∆Ad (i)x(t − d) −sgn(s(t))(ρ(t) + (µ + λ)) + f (i))
≤ ks(t)kk(G(i)B(i))−1 G(i)E(i)kkH(i)x(t)k 13
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+ks(t)kk(G(i)B(i))−1 G(i)Ed (i)kkHd (i)x(t − d)k −ks(t)k(ρ(t) + (µ + λ)) + ks(t)kkf (i)k = −ks(t)k(µ + λ) + ks(t)kkf (i)k
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≤ −ks(t)k(µ + λ) + ks(t)kµ = −λks(t)k <0
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This implies that the trajectories of the time-delay system with uncertainty will approach s(t) = 0. At the time of system switching, the state trajectories are on the SSS s(t) = 0. Next, consider s(t) = 0. For convenience, let s(t) ∈ R. From (24), one can have lim s(t) ˙ = G(i)∆A(i)x(t) + G(i)∆Ad (i)x(t − d)
s(t)→0+
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lim s(t) ˙ = G(i)∆A(i)x(t) + G(i)∆Ad (i)x(t − d)
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+G(i)B(i)(ρ(t) + µ + λ)
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(27)
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In order to avoid going away fast from s(t) = 0 for the trajectories of the considered system, the control u2 (t) is chosen as (28)
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u2 (t) = K(i)x(t) − (G(i)B(i))−1 G(i)Ad (i)x(t − d)
Substitute (28) into (23)
AC
s(t) ˙ = G(i)∆A(i)x(t) + G(i)∆Ad (i)x(t − d) + G(i)B(i)f (i)
It can be concluded that, when s(t) = 0, s(t) ˙ is between lims(t)→0+ s(t) ˙ and lims(t)→0− s(t). ˙ Thus, the trajectories will not go away from s(t) = 0 fast. From u1 (t) and u2 (t), the control u(t) is chosen as u(t) = K(i)x(t) − (G(i)B(i))−1 G(i)Ad (i)x(t − d) 14
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−sgn(s(t))(ρ(t) + (µ + λ))
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The proof is completed. Q2 is answered. Remark 3: Without disturbance f , the method proposed in paper [33] can make the MJSs stable with the jumping transfer matrix unknown. But, when the external disturbance is concluded in the considered MJSs, the result in [33] can not be used to stabilize the system. Using Theorem 1 and Theorem 2 in this paper, we can make the MJSs with matched external disturbance f stable. By use of the SSS, the problem that the state trajectories of MJSs moving around from one sliding surface to other surface is settled. And there isn’t the jumping transfer rate αij in the SMC law. So the SMC law isn’t constrained by the jumping transfer rate. When the time-delay term x(t−d) is not included in the considered system, system (1)-(2) is reduced to the following one: x(t) ˙ = (A(i) + ∆A(i))x(t) + B(i)(u(t) + f (i))
(29)
M
where i ∈ S. Accordingly, following the similar procedure as from (6) to (11), the sliding mode dynamics is obtained: x(t) ˙ = (A(i) + B(i)K(i) + ∆A(i) − B(i)(G(i)B(i))−1 G(i)∆A(i))x(t) (30)
CE
PT
ED
Corollary 1: Consider the Markovian jumping system (29) and the stochastic sliding surface (6). The sliding mode dynamics (30) is asymptotically stable, if there exists a symmetrical positive-definite matrix P¯ , a matrix M (i), scalars ε1 (i) > 0, ε2 (i) > 0, satisfying Γ1 Γ2 <0 (31) χ(i) = ∗ Γ3 where
X + XT + Y ∗
AC
B(i) Γ1 = T −B (i)l − B(i)l + P¯l 0 P¯ H T (i) P¯ H T (i) Γ2 = 0 0 0
Γ3 = diag{−P¯ + ε2 (i)E(i)E T (i), −ε1 (i), −ε2 (i)} 15
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u(t) = K(i)x(t) − sgn(s(t))(ρ(t) + (µ + λ))
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with Y = ε1 (i)E(i)E T (i), X = A(i)P¯ + B(i)M (i), Bl = I1 B, P¯l = I1 P¯ I1T , I1 = (Il 0l×(n−l) ), l ≤ n, i ∈ S. Moreover, the feedback gain can be obtained: K(i) = M (i)P¯ −1 . Corollary 2: Consider the uncertain system (29). Suppose that the LMI (31) is feasible and that the stochastic sliding surface is given by (6) with G(i) = B T (i)P , where P = P¯ −1 is the solution of LMI (31). Then it can be shown that the trajectories of the uncertain system (29) can approach the surface (SSS) s(t) = 0 by an SMC law: (32)
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where λ is a small positive scalar, ρ(t) = k(G(i)B(i))−1 G(i)E(i)kkH(i)x(t)k and kf (i)k < µ.
AC
CE
PT
ED
M
3.3. Stabilization results with the jumping transfer matrix partly unknown Maybe in practice application, we are luck enough to know the transfer matrix partly beforehand. In this case, how to stabilize the MJSs based on the partly unknown transfer matrix by using the SSS? For this purpose, we extended corollaries 1-2 to the following results. Accordingly, the following question is answered in this subsection. Q3: what is the benefit using the SSS in MJSs with partly unknown jumping transfer matrix? (i) (i) For convenience, we denote `K = {j : αij is known}, `U K = {j : αij is unknown}, and {k1 , k2 , · · · , kq } = {j : αij is known, j 6= i}. Theorem 3: Consider the Markovian jumping system (29) and the stochastic sliding surface (6). The sliding mode dynamics (30) is asymptotically stable, if there exists a symmetrical positive-definite matrix P¯ (i), a matrix M (i), scalars ε1 (i) > 0, ε2 (i) > 0, satisfying Γ1 Γ2 <0 (33) ∗ Γ3 X + X T P¯ (i) (i) ≤ 0, ∀j ∈ `U K , j 6= i (34) ¯ ∗ −P (j) X + X T + P¯ (i) ≥ 0,
where
Γ1 =
(1 +
P
(i)
j∈`K
(i)
∀i ∈ `U K
αij )(X + X T ) + Y ∗ ∗ 16
(35) P¯ (i)H T (i) −ε1 (i) ∗
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B(i) , ∀i ∈ `U(i)K 0 −B(i)Tl − B(i)l + P¯ (i)l P (1 + j∈`(i) αij )(X + X T ) + Y + αii P¯ (i) P¯ (i)H T (i) K Γ1 = ∗ −ε1 (i) ∗ ∗ B(i) , ∀i ∈ `(i) 0 K −B(i)Tl − B(i)l + P¯ (i)l 0 P¯ (i)H T (i) Ξ 0 0 Γ2 = 0 0 0 0
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V (i) = xT (t)P (i)x(t)
= I1 B(i), P¯ (i)l = √ ( αik1 P¯ (i), · · · , the feedback gain
M
Γ3 = diag{−P¯ (i) + ε2 (i)E(i)E T (i), −ε2 (i), Λ} with Y = ε1 (i)E(i)E T (i). X = A(i)P¯ (i)+B(i)M (i), B(i)l I1 P¯ (i)I1T , with I1 = (Il 0l×(n−l) ), l ≤ n, i ∈ S. Ξ = √ αikq P¯ (i)). Λ = diag{−P¯ (k1 ), · · · , −P¯ (kq )}. Moreover, can be obtained: K(i) = M (i)P¯ −1 (i). Proof: The stochastic Lyapunov function is chosen as
PT
where P (i) is a symmetrical positive-definite matrix. From Mao [18], let L is the weak infinitesimal generator along the solution (30). Then for each i ∈ S LV (i) = 2xT (t)P (i)x(t) ˙ +
ν X
αij xT (t)P (j)x(t)
j=1
CE
= xT (t)(P (i)A(i) + P (i)B(i)K(i) + P (i)∆A(i)
AC
+(P (i)A(i) + P (i)B(i)K(i) + P (i)∆A(i))T −2P (i)B(i)(G(i)B(i))−1 G(i)∆A(i))x(t) ν X + αij xT (t)P (j)x(t) j=1
Using congruence transformation to (36) by P −1 (i) A(i)P −1 (i) + B(i)K(i)P −1 (i) + ∆A(i)P −1 (i) 17
(36)
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+(A(i)P −1 (i) + B(i)K(i)P −1 (i) + ∆A(i)P −1 (i))T −2B(i)(G(i)B(i))−1 G(i)∆A(i)P −1 (i) αij P −1 (i)P (j)P −1 (i)
(37)
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+
ν X j=1
Let P¯ (i) = P −1 (i). By Schur complement lemma, Lemma 1 and (17), it can be concluded from (37) Γ4 Γ5 χ(i) = ∗ Γ6
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where
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(1 + Σνj=1 αij )(X + X T ) + Y + Σνj=1 αij P¯ (i)P (j)P¯ (i) P¯ (i)H T (i) ∗ −ε1 (i) Γ4 = ∗ ∗ B(i) 0 T ¯ −B(i)l − B(i)l + P (i)l 0 P¯ (i)H T (i) 0 Γ5 = 0 0 0 Γ6 = diag{−P¯ (i) + ε2 (i)E(i)E T (i), −ε2 (i)}
AC
CE
PT
If the LMIs (33)-(35) hold, χ(i) < 0. And one can have LV < 0. The sliding mode dynamics (30) is asymptotically stable. The proof is completed. Theorem 4: Consider the uncertain system (29). Suppose that the LMIs (33)-(35) are feasible and that the stochastic sliding surface is given by (6) with G(i) = B T (i)P (i), where P (i) = P¯ −1 (i) is the solution of LMIs (33)(35). Then it can be shown that the trajectories of the uncertain system (29) can approach the surface (SSS) s(t) = 0 by an SMC law (32), where λ is a small positive scalar, ρ(t) = k(G(i)B(i))−1 G(i)E(i)kkH(i)x(t)k and kf (i)k < µ. Proof: The proof is similar to Theorem 2. So it is omitted here. Remark 4: Using the SSS, there is no process to reach the surface throughout the whole stabilization procedure. It can be seen clearly from the simulations, e.g. Fig 4, Fig 8 and Fig 13. The method provided in this paper 18
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can make the system stable fast because there needs no time to reach the surface. Q3 is answered. Next, according to Theorem 3 and Theorem 4, corollary 3 and corollary 4 are given about stabilization results with the jumping transfer matrix completely known in the following. Corollary 3: Consider the Markovian jumping system (29) and the stochastic sliding surface (6). The sliding mode dynamics (30) is asymptotically stable, if there exists a symmetrical positive-definite matrix P¯ (i), a matrix M (i), scalars ε1 (i) > 0, ε2 (i) > 0, satisfying for each i ∈ S Γ1 Γ2 <0 (38) ∗ Γ3 where
X + X T + Y + αii P¯ (i) P¯ (i)H T (i) B(i) ∗ −ε1 (i) 0 Γ1 = T ∗ ∗ −B(i)l − B(i)l + P¯ (i)l 0 P¯ (i)H T (i) Ξ 0 0 Γ2 = 0 0 0 0
M
AC
CE
PT
ED
Γ3 = diag{−P¯ (i) + ε2 (i)E(i)E T (i), −ε2 (i), Λ} with Y = ε1 (i)E(i)E T (i). X = A(i)P¯ (i)+B(i)M (i), B(i)l = I1 B(i), P¯ (i)l = √ √ I1 P¯ (i)I1T , with I1 = (Il 0l×(n−l) ), l ≤ n. Ξ = ( αiq1 P¯ (i), · · · , αiqν−1 P¯ (i)). Λ = diag{−P¯ (q1 ), · · · , −P¯ (qν−1 )}, where {q1 , q2 , · · · , qν−1 } = {1, 2, · · · , i− 1, i + 1, · · · , ν}. Moreover, the feedback gain can be obtained: K(i) = M (i)P¯ −1 (i). Proof: The proof is similar to Theorem 3. So it is omitted here. Corollary 4: Consider the uncertain system (29). Suppose that the LMI (38) is feasible and that the stochastic sliding surface is given by (6) with G(i) = B T (i)P (i), where P (i) = P¯ −1 (i) is the solution of LMI (38). Then it can be shown that the trajectories of the uncertain system (29) can approach the surface (SSS) s(t) = 0 by an SMC law (32), where λ is a small positive scalar, ρ(t) = k(G(i)B(i))−1 G(i)E(i)kkH(i)x(t)k and kf (i)k < µ. Proof: The proof is similar to Theorem 2. So it is omitted here.
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4. SIMULATIONS
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In this section, some numerical examples are provided to illustrate the effectiveness of the above results. In Example 1, a Markovian jump system with time-delay is considered. With the jumping transfer matrix unknown, the considered system could be SMC controlled using Theorems 1-2. And several simulation figures are given to present the controlled results clearly. In Example 2, a Markovian jump system with partly unknown jumping transfer matrix is considered. By use of Theorems 3-4, the system could be SMC controlled. Three according figures are given to complete it. In Example 3, we consider the system from [35] with a minor modification which makes the system more complex, e.g., adding a matched disturbance f to the system. unlike [35] which controlled the system based on a known jumping transfer matrix, we can stabilize the system based on SSS by use of corollaries 1-2 with minor modification. The given figures make the results more clearly. Example 1: Consider the MJSs (1)-(2) with jumping transfer matrix unknown. In this simulation, time delay d = 1, and for any i ∈ S, S = {1, 2}, the jumping transfer matrix can be arbitrary. The system parameters are as follows: −1 0.2 0.5 0.1 A(1) = , Ad (1) = −0.3 −0.9 0.1 0.8 −0.4 0.2 −0.03 −0.02 E(1) = , H(1) = −0.6 0.1 −0.05 0.01 0.02 −0.05 0.02 −0.03 Ed (1) = , Hd (1) = 0.03 −0.01 −0.01 0.09 sin(t) 0 cos(t) 0 F1 (1) = , F2 (1) = 0 sin(t) 0 cos(t) −2 0.6 0.2 −0.1 A(2) = , Ad (2) = 0.6 −0.5 0.3 0.4 −0.6 0.8 −0.4 −0.17 E(2) = , H(2) = −0.3 0.5 −0.4 0.5 0.2 −0.4 0.62 −0.53 Ed (2) = , Hd (2) = 0.13 −0.21 −0.21 0.04 20
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4000 X1
3500
X
2
2500 2000 1500 1000 500 0
0
5
10
15
20 25 Time t(sec)
30
35
40
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−500
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Open−loop state responses
3000
Fig 2. State trajectories of the open-loop system. 1.5
X1 X
2
M
0.5
0
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Closed−loop state responses
1
−0.5
0
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−1
2
4
6
8
10
Time t(sec)
Fig 3. State trajectories of the closed-loop system.
sin(t) 0 cos(t) 0 F1 (2) = , F2 (2) = 0 sin(t) 0 cos(t) T T B(1) = 1 5 , B(2) = 2 1 , f (1) = sin(t), f (2) = cos(t)
AC
CE
Solving the LMI (12), the feedback gains can be obtained: 0.69405 −0.030102 1.5662 −0.12481 ¯ ¯ P = , Z= −0.030102 0.46544 −0.12481 2.1559 21
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−3
4
x 10
2 1 0 −1 −2 −3 −4
0
2
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6
8
10
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Time t(sec)
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Stochastic sliding surface s(t)
3
Fig 4. Stochastic sliding surface. 8
5
6
0
−5 1.999 2
2
2.001
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Control input u(t)
4
0
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−2
−4
0
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−6
2
4
6
8
10
Time t(sec)
Fig 5. Control input u(t).
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−2.3923 −10.756 , M (2) = −4.3369 −4.267 K(1) = −1.3366 −4.9342 , K(2) = −2.8816 −1.8555
AC
M (1) =
The simulation results are given in Figs. 2-5, which show the effectiveness of Theorem 1 and Theorem 2. Remark 5: From Fig. 4, it is obviously that the state trajectories are on the stochastic sliding surface for all the time including the moment of system switching. And there is not the process of reaching the SSS throughout the whole stabilization procedure. The problem of the state trajectories moving 22
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or
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−1.3 0.2 0.2 0.9 4.7 −5.3 0.3 0.3 π= 0.6 0.4 −1.5 0.5 0.4 0.6 0.7 −1.7
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among several sliding surfaces is settled perfectly. And without the jumping transfer matrix, some Markovian systems can be stabilized. Example 2: With partly unknown or completely known transition probabilities, we consider the system (29). For any i ∈ S, S = {1, 2, 3, 4}, in this simulation, the jumping transfer matrix with the interval time δ = 0.1 is −1.3 0.2 0.2 0.9 ? ? 0.3 0.3 π= (39) 0.6 ? −1.5 ? 0.4 ? ? −1.7
(40)
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PT
ED
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and the other system parameters are as follows: −7.25 −17.75 −0.15 −0.49 A(1) = , A(2) = −3.1 −1.5 3 −1.1 −3 −0.15 −0.9 −0.34 A(3) = , A(4) = 1.5 0.2 1.5 −1.65 T E(1) = 0.2 0 , H(1) = 0 0.1 T E(2) = 0 0.1 , H(2) = 0.005 0 T E(3) = 0 0.1 , H(3) = 0.1 0 T E4 = 0.1 0 , H(4) = 0.05 0.01
AC
F1 (1) = F1 (2) = F1 (3) = F1 (4) = sin(t) T T B(1) = 6 3 , B(2) = 3 0 T T B(3) = −2 2 , B(4) = −2 0
f (1) = sin(t), f (2) = cos(t), f (3) = sin(t)cos(t), f (4) = cos(t) With partly unknown transition probabilities (39), solving the LMIs (33)23
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3000 X1 X2
1000 0 −1000 −2000 −3000 −4000 −5000
0
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6
8
10
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Time t(sec)
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Open−loop state responses
2000
Fig 6. State trajectories of the open-loop system. 2.5
X1
2
X
2
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1 0.5 0 −0.5
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Closed−loop state responses
1.5
−1
−1.5
0
PT
−2
2
4
6
8
10
Time t(sec)
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Fig 7. State trajectories of the closed-loop system.
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(35), the feedback gains can be obtained: 11.5663 2.6777 2.5029 0.1003 ¯ ¯ P (1) = , P (2) = 2.6777 1.3197 0.1003 0.4659 M (1) = K(1) =
−19.8605 −15.5778 , M (2) = −0.2688 −2.4155
1.9154 −15.6903 , K(2) = 0.1011 −5.2062 24
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−3
2
x 10
1 0.5 0 −0.5 −1 −1.5 −2 −2.5
0
2
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6
8
10
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Time t(sec)
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Stochastic sliding surface s(t)
1.5
Fig 8. Stochastic sliding surface. 3
X1 X
2
2
M
1.5 1 0.5
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Closed−loop state responses
2.5
0
−0.5
0
2
PT
−1
4
6
8
10
Time t(sec)
Fig 9. State trajectories of the closed-loop system.
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2.4468 0.0991 0.0991 0.8065 M (3) = 104.7117 −111.1022 , M (4) = −0.3571 1.9164 K(3) = 20.9254 −54.9295 , K(4) = −0.2434 2.4061 3.7923 −0.4615 −0.4615 1.8467
, P¯ (4) =
AC
P¯ (3) =
The simulation results are given in Figs. 6-8, which show the effectiveness of Theorem 3 and Theorem 4. With completely known transition probabilities (40), solving the LMI (38),
25
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−3
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x 10
1 0.5 0 −0.5 −1 −1.5 −2 −2.5
0
2
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Time t(sec)
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Stochastic sliding surface s(t)
1.5
Fig 10. Stochastic sliding surface.
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the feedback gains can be obtained: 5.0165 −0.46601 11.505 2.7865 ¯ ¯ P (1) = , P (2) = 2.7865 1.4705 −0.46601 1.1404 M (1) = 10.247 −1.4418 , M (2) = −16.079 −1.1017 K(1) = 2.0851 −4.9316 , K(2) = −3.4251 −2.3657 3.5307 −0.60882 3.5351 −0.45856 ¯ ¯ P (3) = , P (4) = −0.60882 3.7708 −0.45856 3.574 M (3) = 7.2636 −16.148 , M (4) = 21.577 1.3772 K(3) = 1.3566 −4.0634 , K(4) = 6.2579 1.1883
AC
CE
With the jumping transfer matrix completely known the simulation results are given in Figs. 9-10, which show the effectiveness of Corollary 3 and Corollary 4. Example 3: The important virtue by use of stochastic sliding surface is that Morkovian jump system with disturbance may be ensured asymptotically stable when the jumping transfer matrix completely unknown. As far as the authors knowledge, this problem has not been settled by use of sliding mode control in published literature. The system data coming from paper [35] are used to show the effectiveness of the stochastic sliding surface in this example. In paper [35], the jumping transfer matrix is known, e.g. 26
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−8 4 2 2 1 −1 0 0 in the simulation, and there is no external (αij ) = 2 1 −4 1 0 2 2 −4 disturbance. But if the jumping transfer matrix is unknown and there is a matched disturbance, how to ensure the Markovian jump system asymptotically stable? Consider system (29) without the uncertainty
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x(t) ˙ = A(i)x(t) + B(i)(u + f (i))
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the according system parameters are as 0 2
0 1 0 2
, f (1) = sin(t) , f (2) = cos(t)
, f (3) = (sin(t) + cos(t))/2
ED
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For any i ∈ S, S = {1, 2, 3, 4}, follows: 0 1 A(1) = , B(1) = 3 5 0 1 A(2) = , B(2) = 2 4 0 1 A(3) = , B(3) = 5 3 0 1 A(4) = , B(4) = 4 2
(41)
0 1
, f (4) = sin(t)cos(t)
PT
With minor modification to Corollary 1, condition (31) could be changed into Υ + ΥT < 0,
(42)
AC
CE
where Υ = A(i)P¯ + B(i)M (i). The feedback gain can be obtained: K(i) = M (i)P¯ −1 , i ∈ S, S = {1, 2, 3, 4}. Solving the LMI (42), the feedback gains can be obtained: K(1) = −2.2615 −3.0885 , K(2) = −3.5231 −5.1769 K(3) = −3.2615 −2.0885 , K(4) = −5.5231 −3.1769
Then the according SMC law can be obtained: u = K(i)x(t)−sgn(s(t)(λ+ µ)) where λ = 1, µ = 1. From this simulation, we can stabilize the system (41) with the jumping transfer matrix unknown. The responses are shown in Figs. 11-13, which shows the effectiveness of the proposed methods. 27
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6
x 10
X1 X2
4
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2
1
0
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10 Time t(sec)
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Open−loop state responses
5
Fig 11. State trajectories of the open-loop system. 1
X1
0.8
X
2
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0.4 0.2 0 −0.2
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Closed−loop state responses
0.6
−0.4 −0.6
0
PT
−0.8
5
10 Time t(sec)
15
20
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Fig 12. State trajectories of the closed-loop system.
5. conclusion
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In this paper, a new sliding mode surface has been presented. By use of this surface, the state trajectories of MJSs are on it all the time. So the process to reach the sliding mode surface is not needed here. The new sliding mode surface is called stochastic sliding surface (SSS). Compared with the traditional sliding mode surface, the method provided in this paper can avoid the problem that the state trajectories of MJSs moving from one sliding mode surface towards the other sliding mode surface. Some Markovian systems 28
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−5
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1 0.5 0 −0.5 −1 −1.5
0
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10 Time t(sec)
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−2
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Stochastic sliding surface s(t)
1.5
Fig 13. Stochastic sliding surface.
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Acknowledgment
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can be stabilized with jumping transfer matrix unknown, partly unknown or known. And a controller has been synthesized to ensure that the state trajectories do not leave away from the SSS. Finally, some examples have been provided to illustrate the validity of the proposed method and the obtained results.
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This work was supported by the National Natural Science Foundation of China under grant numbers 61673099, 61273003, 61273008, 60974004, and 61104003, respectively. References
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[1] B. Chen, Y.G. Niu, and Y.Y. Zou, Sliding mode control for stochastic Markovian jumping systems subject to successive packet losses, J. Franklin Inst. 351(4)(2014) 2169-2184.
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[2] B. Chen, Y.G. Niu, and Y.Y. Zou, Adaptive sliding mode control for stochastic Markovian jumping systems with actuator degradation, Automatica 49(6)(2013) 1748-1754. [3] J. Cheng, H.L. Xiang, H.L. Wang, Z.J. Liu and L.Y. Hou, Finite-time stochastic contractive boundedness of markovian jump systems subject to input constraints, ISA Trans. 60 (2016) 74-81. 29
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[4] J. Cheng, J.H. Park, Y.J. Liu, Z.J. Liu and L.M. Tang, Finite-time H∞ fuzzy control of nonlinear markovian jump delayed systems with partly uncertain transition descriptions, Fuzzy sets syst. (2016) June 15, doi:10.1016/j.fss.2016.06.007. [5] J. Cheng, H. Zhu, S.M. Zhong, Y. Zeng and X.C. Dong, Finitetime H∞ control for a class of markovian jump systems with modedependent time-varying delays via new lyapunov functionals, ISA Trans. 52(6)(2013) 768-774.
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[6] L.J. Gao, D.D. Wang, and Y.Q. Wu, Non-fragile observer-based sliding mode control for Markovian jump systems with mixed mode-dependent time delays and input nonlinearity, Appl. math. comput. 229(2014) 374395.
[7] J. Hu, Z.D. Wang, Y.G. Niu, and H.J. Gao, Sliding mode control for uncertain discrete-time systems with Markovian jumping parameters and mixed delays, J. Franklin Inst. 351(4)(2014) 2185-2202.
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[8] H.L. Hu, D. Zhao, and Q.L. Zhang, Singular value decomposition-based method for sliding mode control and optimization of nonlinear neutral systems, J. Appl. Math. 2013(2013) 1-11.
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[9] R. Huang, Y. Lin, and Z.W. Lin, Sliding mode H∞ control design for uncertain nonlinear stochastic state-delayed Markovian jump systems with actuator failures, Nonlinear Anal.: Hybrid Syst. 5(4)(2011) 692703.
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