Reliable H∞ control on saturated linear Markov jump system with uncertain transition rates and asynchronous jumped actuator failure

Accepted Manuscript

Reliable H∞ control on saturated linear Markov jump system with uncertain transition rates and asynchronous jumped actuator failure Zhaohui Chen , Zhong Cao , Qi Huang , Stephen L. Campbell PII: DOI: Reference:

S0016-0032(18)30172-8 10.1016/j.jfranklin.2018.02.029 FI 3375

To appear in:

Journal of the Franklin Institute

Received date: Revised date: Accepted date:

27 July 2017 9 January 2018 23 February 2018

Please cite this article as: Zhaohui Chen , Zhong Cao , Qi Huang , Stephen L. Campbell , Reliable H∞ control on saturated linear Markov jump system with uncertain transition rates and asynchronous jumped actuator failure, Journal of the Franklin Institute (2018), doi: 10.1016/j.jfranklin.2018.02.029

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ACCEPTED MANUSCRIPT Reliable 𝐻∞ control on saturated linear Markov jump system with uncertain transition rates and asynchronous jumped actuator failure Zhaohui Chena,d, Zhong Caob, Qi Huangc, and Stephen L. Campbelld,* a Department

of Mathematics and Physics, Chongqing University of Science and Technology, Chongqing 401331, China of Physics and Electronic Engineering, Guangzhou University, Guangzhou 510006, China c Sichuan Provincial key Lab of Power System Wide-area Measurement and Control, University of Electronic Science and Technology of China, Chengdu 611731, China d Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, USA

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b School

Abstract: This paper is concerned with reliable 𝐻∞ control for saturated linear Markov jump systems with uncertain transition rates and asynchronous jumped actuator failure. The actuator failures are assumed to occur randomly under the Markov process with a different jumping mode from the system jumping mode. In considering the mixed-mode-dependent state feedback controller, both 𝐻∞ stochastic stability analysis for closed-loop system with completely accessible transition rates and uncertain transition rates are investigated. Moreover, based on the

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obtained stability conditions, the 𝐻∞ control problems are investigated, and the controller gains can be obtained by solving a convex optimization problem with minimizing 𝐻∞ performance as objective and linear matrix inequalities (LMIs) as constraints. The problem of designing state feedback controllers such that the estimate of the domain of attraction is enlarged is also formulated and solved as an optimization problem with LMI constraints. Simulation results are presented to illustrate the effectiveness of the proposed results.

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Keywords: Linear Markov jump system, asynchronous jumped actuator failure, actuator

1.

Introduction

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saturation, reliable 𝐻∞ control, domain of attraction.

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Markov jump systems have been received considerable research attention over the past decades. The reason is mainly that they can model many practical systems whose structures may be

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subjected to abrupt changes such as component failures, repairs, and sudden environmental changes, etc [1-5]. Taking power system as an example, load profile switching processes can be described using a Markov chain, and the power system dynamics have different equilibrium

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under different load conditions [6]. Consequently there has been a lot of effort on analyzing the stability, stabilization, filtering, fault detection and isolation for Markov jump systems [7-11]. For the dynamics of Markov jump systems, the transition rates are critical parameters, which

usually cannot be precisely obtained since they are determined by numerical simulations or experimental tests [12]. Thus in the literature, the uncertainty of transition rates is a significant issue. The uncertain transition rates are often classified into three different types: polytopic description where the transition rate matrix is assumed to be in a convex hull with known vertices [13-20], element-wise description where the elements of the transition rate matrix are

*

Corresponding author.

E-mail Address: [email protected] (Stephen L. Campbell), [email protected] (Zhaohui Chen), [email protected] (Zhong Cao), [email protected] (Qi Huang).

1

ACCEPTED MANUSCRIPT assumed in practice and error bounds are given [21-27], and general uncertain where each transition rate may be unknown or only its estimate is known [28-32]. In this paper, we consider the element-wise uncertainty in the transition rate for the saturated linear Markov jump system. In practice the transition rates are measured, the estimate values and the estimate error bounds of the transition rates are given. On the robust control of Markov jump system with uncertain transition rates, a key problem is to bind the uncertain terms [27]. For the Markov jump linear system with uncertain jumping probabilities, [22] and [24] considered the relationship among the transition rates, the controller design methods were provided in terms of a class of nonlinear Matrix inequalities (NLMIs), and then the sequential linear programming method [33] was

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employed to solve the NLMIs. To reduce the conservatism of the results on robust stabilization, [26] and [27] developed a convex method and improved synthesis method for the design of a robust state-feedback controller, respectively.

On the other hand, actuator saturation and actuator failure are important sources of system instability and can severely degrade the performance of a closed-loop system. The analysis and synthesis of control systems with actuator saturation [34-37] or actuator failure [38-42] have received considerable attention. The reliable controller design methods are based on the

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assumption that control component failures are specified, and usually are modeled as outages or part outages. For the stabilization of Markov jump systems, very few works have been done considering both actuator saturation and actuator failures [43-44]. [45] focuses on the problem of reliable control for interval time-varying delay systems subjected to actuator saturation and stochastic failure, in which a new actuator fault model is proposed by assuming that the fault factor obeys a certain probabilistic distribution in an interval, and the expectation and variance of

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stochastic failure factors are utilized to analysis the stability of the closed-loop system. For a Markov jump system subject to actuator faults, it is reasonable to assume that the failures occur under a Markov process, which may be different from the Markov process of the system. Then,

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therefore, the closed-loop system becomes a mixed-mode Markov jump system with asynchronous jumped failure. [46] and [47] study the estimation of discrete-time Markov jump linear system and the stability of Markov jump neural networks, respectively. In [46-47] the jump

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systems are with piecewise homogeneous Markov chain, for the considered transition probabilities are time-varying but invariant for the same conditional mode. [48] constructs a

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novel filter which is asynchronous with system modes, such that the filtering error system is a discrete-time stochastic Markov jump system with mixed-mode. The existence criterion of the desired asynchronous filter with piecewise homogeneous Markov chain is proposed. Similarly, for

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a Markov jump linear system subject to asynchronous actuator faults, the closed-loop system becomes a jump system with piecewise homogeneous Markov chain. However, to the best of our knowledge, there are no results on continuous Markov jump systems subject to actuator saturation and asynchronous jumped failure in the existing literature, which is the motivation for this work and is its major contribution. In this paper, we assume that the failure is caused by some specific random event. The failure occurs with a constant value in a short time interval, and then recovers or is changed to another constant value. As an example, the computer controlling an actuator on a robotic mission could be hit by a cosmic ray. The response, depending on current activities could be to switch to a sleep or reduced power mode, followed a short time later to reset. Failure of multiple thrusters is another example [49]. 2

ACCEPTED MANUSCRIPT The main contribution of this paper is to introduce a new actuator fault model in which the failure scale factors of each actuator are governed by a Markov chain that is asynchronous to the system jumping modes, and then investigate the 𝐻∞ control problem for the considered system subject to both actuator saturation and asynchronous jumped actuator failure. By virtue of Lyapunov theory, the stability conditions of the closed-loop system with mixed modes are obtained for the situations of completely known and uncertain transition rates, respectively. Based on the results, the problem of designing mixed-mode-dependent state feedback controller that not only ensures the stochastic stability of the closed-loop system but also ensures an optimal 𝐻∞ performance in terms of LMIs constraints is focused upon. Furthermore, the

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obtained results are used with the aim of enlarging the estimate of the domain of attraction for the closed-loop system.

Notations: The notations used throughout this paper are fairly standard. 𝑋 < 0 (𝑋 > 0) means the matrix X is negative definite (positive definite). The superscript “ T” stands for matrix transposition. Symmetric terms in the matrix are denoted by “*”. 𝑆𝑦𝑚(𝑋) stands for 𝑋 + 𝑋 𝑇 . And 𝑑𝑖𝑎𝑔(⋯ ) represents a block-diagonal matrix. 𝐑𝑛 denotes the 𝑛-dimensional Euclidean Space, 𝐑𝑚×𝑛 is the set of all 𝑚 × 𝑛 real matrices. 𝑳𝟐 [0, ∞) is the space of square-integratable

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vector functions over [0, ∞). (Ω, ℱ, 𝒫) is a complete probability space with Ω the sample space, ℱ the σ-algebra of subsets of Ω, 𝒫 the probability measure on ℱ. 𝐄{·} denotes the expectation operator with respect to some probability measure 𝒫.

2.

Problem formulation and preliminaries

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Consider the following linear Markov jump system (1) defined in a probability space (Ω, ℱ, 𝒫). 𝑥̇ (𝑡) = 𝐴(𝑟𝑡 )𝑥(𝑡) + 𝐵(𝑟𝑡 )𝑠𝑎𝑡(𝑢(𝑡)) + 𝐵𝜔 (𝑟𝑡 )𝜔(𝑡), 𝑛

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𝑧(𝑡) = 𝐶(𝑟𝑡 )𝑥(𝑡) + 𝐷𝜔 (𝑟𝑡 )𝜔(𝑡),

where 𝑥(𝑡) ∈ 𝐑 is the state; 𝑧(𝑡) ∈ 𝐑

𝑚

(1a) (1b) 𝑝

is the control output; 𝑢(𝑡) ∈ 𝐑 is the input; 𝜔(𝑡) ∈ 𝐑𝑞

is the disturbance input which belongs to 𝑳𝟐 [0, ∞); and {𝑟𝑡 , 𝑡 ≥ 0} is a right-continuous Markov

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chain on the probability space (Ω, ℱ, 𝒫) which takes values in the finite set 𝚼 = {1,2, … , 𝑟} with transition rate matrix [π𝑖𝑗 ], 𝑖, 𝑗 ∈ 𝚼. The probability is given by

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𝜋𝑖𝑗 ∆ + 𝑜(∆), 𝑗≠𝑖 𝐏𝐫 {𝑟𝑡+∆ = 𝑗|𝑟𝑡 = 𝑖} = { , 1 + 𝜋𝑖𝑖 ∆ + 𝑜(∆), 𝑗 = 𝑖

(2)

where Δ>0, o(Δ)/Δ→0(Δ→0), 𝜋𝑖𝑗 ≥ 0 is the transition rate from mode 𝑖 at time 𝑡 to mode 𝑗

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at time 𝑡 + ∆ if 𝑖 ≠ 𝑗 and 𝜋𝑖𝑖 = − ∑𝑗≠𝑖 𝜋𝑖𝑗 ; 𝐴(𝑟𝑡 ), 𝐵(𝑟𝑡 ), 𝐵𝜔 (𝑟𝑡 ), 𝐷(𝑟𝑡 ), and 𝐷𝜔 (𝑟𝑡 ) are known system matrices with appropriate dimensions. For each possible mode 𝑟𝑡 = 𝑖 ∈ 𝚼, they will be denoted by 𝐴𝑖 , 𝐵𝑖 , 𝐵𝜔𝑖 , 𝐷𝑖 , and 𝐷𝜔𝑖 , respectively. The function 𝑠𝑎𝑡(∙): 𝐑𝑝 → 𝐑𝑝 is a vector valued standard saturation function defined as

follows: 𝑠𝑎𝑡(𝑢(𝑡)) = [𝜎(𝑢1 (𝑡)) where, 𝑢(𝑡) = [𝑢1 (𝑡)

𝜎(𝑢2 (𝑡)) ⋯

𝑢2 (𝑡) ⋯

𝜎(𝑢𝑝 (𝑡))]𝑇 ,

(3)

𝑢𝑝 (𝑡)]𝑇 is the control input vector; 𝜎: 𝐑1 → 𝐑1 denotes the

scalar valued saturation function, and 𝑠𝑔𝑛(𝑢ϑ (𝑡))𝑢̅ϑ , 𝜎(𝑢ϑ (𝑡)) = { 𝑢ϑ (𝑡),

𝑖𝑓 |𝑢ϑ (𝑡)| ≥ 𝑢̅ϑ , ϑ = 1,2, ⋯ , 𝑝 𝑖𝑓 |𝑢ϑ (𝑡)| < 𝑢̅ϑ

(4) 3

ACCEPTED MANUSCRIPT where 𝑢̅ϑ denotes the saturation level of 𝜎(𝑢ϑ (𝑡)). Suppose the unexpected actuator failures occur independently and randomly, under a Markov jump mode that is different from system jumping, and the mode-dependent actuator gains are known. The asynchronous jumped actuator failure model is: 𝑢𝐹 (t) = Ξ(𝑠𝑡 )𝑠𝑎𝑡(𝑢(t)),

(5)

where Ξ(𝑠𝑡 ) = diag{𝜉1 (𝑠𝑡 ), 𝜉2 (𝑠𝑡 ), ⋯ , 𝜉𝑝 (𝑠𝑡 )} , and 𝜉1 (𝑠𝑡 ), 𝜉2 (𝑠𝑡 ), ⋯ , 𝜉𝑝 (𝑠𝑡 ) denote the failure scale factor of each channel. {𝑠𝑡 , 𝑡 ≥ 0} is a right-continuous Markov chain on the probability space (Ω, ℱ, 𝒫) and takes values in the finite set 𝐒 = {1,2, … , 𝑠} with transition rate matrix

𝑟

𝐏𝐫 {𝑠𝑡+∆ = 𝑙|𝑠𝑡 = 𝑘} = {

𝜆𝑘𝑙𝑡 ∆ + 𝑜(∆), 1+

𝑟𝑡 𝜆𝑘𝑘 ∆

𝑙≠𝑘

+ 𝑜(∆), 𝑙 = 𝑘

,

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𝑟

[𝜆𝑘𝑙𝑡 ], 𝑘, 𝑙 ∈ 𝐒. The probability is given by

(6)

𝑟

where Δ>0, o(Δ)/Δ→0(Δ→0), 𝜆𝑘𝑙𝑡 ≥ 0 is the transition rate from mode 𝑘 at time 𝑡 to mode 𝑙 𝑟

𝑟

𝑡 at time 𝑡 + ∆ if 𝑘 ≠ 𝑙, and 𝜆𝑘𝑘 = − ∑𝑘≠𝑙 𝜆𝑘𝑙𝑡 . The Probability 𝐏𝐫 {𝑠𝑡+∆ = 𝑙|𝑠𝑡 = 𝑘} depends on

the current system mode 𝑟𝑡 . In stochastic jumping, the Markov chain {𝑟𝑡 , 𝑡 ≥ 0} is assumed to

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be independent on σ(𝑠𝜏 , 0 ≤ 𝜏 < 𝑡), which is a σ-algebra.

In this paper, we suppose the transition rates information is not accurately accessible. That is, 𝑟

the value of elements in matrices [π𝑖𝑗 ] and [𝜆𝑘𝑙𝑡 ] are uncertain, π𝑖𝑗 = π ̃𝑖𝑗 + ∆π𝑖𝑗 (𝑖, 𝑗 ∈ Υ), and 𝑟𝑡 𝑟𝑡 𝑟𝑡 ̃ 𝜆𝑘𝑙 = 𝜆𝑘𝑙 + ∆𝜆𝑘𝑙 ( 𝑘, 𝑙 ∈ S). Here π ̃𝑖𝑗 and ∆𝜋𝑖𝑗 ( |∆𝜋𝑖𝑗 | ≤ 𝜀𝑖𝑗 < |π ̃𝑖𝑗 |) represent the estimate value and the estimate error value of the transition rate π𝑖𝑗 , respectively. They satisfy 𝜋̃𝑖𝑖 = 𝑟 𝑟 𝑟 𝑟 𝑟 − ∑𝑗≠𝑖 𝜋̃𝑖𝑗 , ∆𝜋𝑖𝑖 = − ∑𝑗≠𝑖 ∆𝜋𝑖𝑗 . In addition 𝜆̃ 𝑡 and ∆𝜆 𝑡 ( |∆𝜆 𝑡 | ≤ 𝛿 𝑡 < |𝜆̃ 𝑡 | ) represent the 𝑘𝑙

𝑘𝑙

𝑘𝑘

𝑘𝑙

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estimate value and the estimate error value of the 𝑟 𝑟 𝑟 𝑟 𝜆̃ 𝑡 = − ∑𝑘≠𝑙 𝜆̃ 𝑡 , ∆𝜆 𝑡 = − ∑𝑘≠𝑙 ∆𝜆 𝑡 . The values of 𝑘𝑘

𝑘𝑙

𝑘𝑙

𝑘𝑙 𝑘𝑙 𝑟𝑡 transition rate 𝜆𝑘𝑙 , respectively. They satisfy 𝑟 𝑟 π ̃𝑖𝑗 , 𝜀𝑖𝑗 , 𝜆̃𝑘𝑙𝑡 , and 𝛿𝑘𝑙𝑡 are known a priori.

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Remark 1. In Equation (5), stochastic variables 𝜉ϑ (𝑠𝑡 ) for ϑ = 1,2, ⋯ , 𝑝 denote the failure scale factor of the ϑth channel, and indicate the actuator faults occur randomly under the Markov process {𝑠𝑡 , 𝑡 ≥ 0}. In particular, for each possible mode 𝑠𝑡 = 𝑘 ∈ 𝐒, if 𝜉ϑ𝑘 ≡ 0, it corresponds to

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ϑth channel.

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the ϑth channel with complete failure, and if

𝜉ϑ𝑘 ≡ 1, it corresponds to the normal case for the

Remark 2. From (6), it can be shown that the Markov chain {𝑠𝑡 , 𝑡 ≥ 0} is finite piecewise 𝑟

homogeneous, because the transition rate matrix [𝜆𝑘𝑙𝑡 ] is time-varying but invariant with the

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same system mode 𝑟𝑡 = 𝑖. Remark 3. It can be shown that when 𝐒 = {1}, the asynchronous jumped actuator fault (5) reduces to the type studied in [1]. That is to say, the fault (5) is more general. For the system (1) with asynchronous jumped actuator failure (5), we are interested in designing the mixed mode-dependent state feedback controller 𝑢(𝑡) = 𝐾(𝑟𝑡 , 𝑠𝑡 )𝑥(t),

(7)

where 𝐾(𝑟𝑡 , 𝑠𝑡 ) ∈ 𝐑𝑝×𝑛 are the controller gains to be determined. To facilitate the estimation of the domain of attraction and saturation control design, we will 4

ACCEPTED MANUSCRIPT introduce the convex subset 𝜓(𝐹𝑖𝑘 ) ≜ {𝑥(𝑡) ∈ 𝐑𝑛 : |𝐹𝑖𝑘 𝑥(𝑡)| ≤ 𝑢̅},

(8)

where 𝐹𝑖𝑘 ∈ 𝐑𝑝×𝑛 (𝑖 ∈ 𝚼, 𝑘 ∈ 𝐒) and 𝑢̅ = [𝑢1

𝑢2 ⋯

𝑢𝑝 ]𝑇 .

𝜗 Let 𝐹𝑖𝑘 be the 𝜗th row of the

matrix 𝐹𝑖𝑘 . For any positive matrix 𝑃𝑖𝑘 ∈ 𝐑𝑛×𝑛 , we define a contractively invariant ellipsoid ℰ(𝑃𝑖𝑘 , 1) ≜ {𝑥(𝑡) ∈ 𝐑𝑛 : |𝑥 𝑇 (𝑡)𝑃𝑖𝑘 𝑥(𝑡)| ≤ 1}.

(9)

Let 𝒰 be the set of 𝑝 × 𝑝 diagonal matrices with diagonal entries of 1 or 0. For any 𝛤𝜃 ∈ 𝒰 [50], and it will be playing an important role in our main results.

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(𝜃 = 1,2, ⋯ , 2𝑝 ), if we define 𝛤𝜃− ≜ 𝐼 − 𝛤𝜃 , then 𝛤𝜃− ∈ 𝒰 . The following Lemma 1 was given in Lemma 1 ([50]). For given matrices 𝐾𝑖𝑘 ∈ 𝐑𝑝×𝑛 and 𝐹𝑖𝑘 ∈ 𝐑𝑝×𝑛 , if 𝑥(𝑡) ∈ 𝜓(𝐹𝑖𝑘 ), then 𝑝

𝜎(𝐾𝑖𝑘 𝑥(𝑡)) = ∑2𝜃=1 𝜂𝜃 (𝛤𝜃 𝐾𝑖𝑘 + 𝛤𝜃− 𝐹𝑖𝑘 )𝑥(𝑡) , 𝑝

where ∑2𝜃=1 𝜂𝜃 = 1, 0 ≤ 𝜂𝜃 ≤ 1, 𝑖 ∈ 𝚼, and 𝑘 ∈ 𝐒.

(10)

For any possible mode 𝑠𝑡 = 𝑘 ∈ 𝐒, the actuator failure gains Ξ(𝑠𝑡 ) and scales 𝜉ϑ (𝑠𝑡 ) will be

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denoted by Ξ𝑘 and 𝜉ϑ𝑘 (ϑ = 1,2, ⋯ , 𝑝) , respectively. For convenience, let us define Λϑ = diag{0, ⏟⋯ ,0 , 1, 0, ⏟⋯ ,0}. Then (5) can be rewritten as ϑ−1

𝑝−ϑ 𝑝

𝑢𝐹 (t) = ∑ϑ=1 Λϑ 𝜉ϑ𝑘 𝑠𝑎𝑡(𝑢(t)).

(11)

Then, under the controller (9), for any possible modes 𝑟𝑡 = 𝑖 ∈ 𝚼, and 𝑠𝑡 = 𝑘 ∈ 𝐒, if 𝑥(𝑡) ∈ 𝑝

𝑝

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𝜓(𝐹𝑖𝑘 ), the closed-loop system can be described as:

(12a)

𝑧(𝑡) = 𝐶𝑖 𝑥(𝑡) + 𝐷𝜔𝑖 𝜔(𝑡).

(12b)

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𝑥̇ (𝑡) = ∑2𝜃=1 𝜇𝜃 [𝐴𝑖 + 𝐵𝑖 ∑ϑ=1 Λϑ 𝜉ϑ𝑘 (𝛤𝜃 𝐾𝑖𝑘 + 𝛤𝜃− 𝐹𝑖𝑘 )]𝑥(𝑡) + 𝐵𝜔𝑖 𝜔(𝑡),

Let us recall the following definitions and lemmas which will be used throughout the paper. Definition 1 ([51]). A set 𝔇 ⊂ 𝐑𝑛 is called a domain of attraction in mean-square sense of the satisfies

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system (1), if for any initial mode 𝑟0 ∈ Υ and initial state 𝑥0 ∈ 𝔇, the solution 𝑥(𝑡, 𝑥0 , 𝑟0 ) of (1)

𝑡

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lim𝑡𝑓→∞ 𝑬 {∫0 𝑓 𝑥 𝑇 (𝑡, 𝑥0 , 𝑟0 )𝑥(𝑡, 𝑥0 , 𝑟0 )𝑑𝑡 |𝑥0 , 𝑟0 } ≤ 𝑥0𝑇 𝑅𝑥0

(13)

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for all 𝑡𝑓 > 0 and some 𝑅 > 0. Definition 2 ([52]). The system (1) with 𝑢(𝑡) = 0 is said to be stochastically stable with γ-disturbance attenuation, if (13) holds, and under zero initial condition, for all admissible disturbance input 𝜔(𝑡), and 𝑡𝑓 > 0, the control output 𝑧(𝑡) satisfies 𝑡

𝑡

𝐄 {∫0 𝑓 𝑧 𝑇 (𝑡)𝑧(𝑡)𝑑𝑡 |𝑥0 , 𝑟0 } < 𝛾 2 𝐄 {∫0 𝑓 𝜔𝑇 (𝑡)𝜔(𝑡)𝑑𝑡 }.

(14)

T T Lemma 2 ([53]). Given matrices S11 =S11 , S22 =S22 and𝑆12 , of appropriate dimensions, the -1

T inequality S11 -S12 S22 S12 < 0 holds, if and only if

5

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[

S11 T S12

3.

T S12 S S12 ] <0 or [ 22 ] <0. S22 S12 S11

(15)

Main results

In this section, our goal is to design the mixed-mode-dependent controller in (7) for system (1) subject to asynchronous jumped fault (5), such that the closed-loop system (12) with estimated information on transition rates, is stochastically stable with optimal 𝐻∞ performance γmin . On of attraction would be the optimal objective.

3.1 Stochastic stability analysis for the closed-loop system

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the other hand, if an acceptable γ > 0 could be prescribed, maximizing the estimation of domain

First, based on the assumption that all the transition rate matrices are given, we present a sufficient condition under which the closed-loop system (12) is stochastically stable with 𝐻∞ performance. Then, under the assumption that the transition rate matrices are uncertain, the

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stability analysis is investigated.

Theorem 1. For the closed-loop Markov jump system (12) with asynchronous jumped actuator faults, given system jumping transition rate matrices [𝜋𝑖𝑗 ] and fault jumping transition rate 𝑟

matrices [𝜆𝑘𝑙𝑡 ], if there exist scalar γ > 0, a set of symmetric positive definite matrices 𝑃𝑖𝑘 ∈ 𝐑𝑛×𝑛 and appropriate matrices 𝐹𝑖𝑘 ∈ 𝐑𝑝×𝑛 , 𝐾𝑖𝑘 ∈ 𝐑𝑝×𝑛 , such that for any (𝑖, 𝑘) ∈ (𝚼, 𝐒), and

𝑃𝑖𝑘 𝐵𝜔𝑖 −𝛾 2 𝐼 ∗

𝐶𝑖𝑇 𝑇 ] < 0, 𝐷𝜔𝑖 −𝐼

ℰ(𝑃𝑖𝑘 , 1) ⊂ 𝜓(𝐹𝑖𝑘 ),

ED

𝜃 𝛨𝑖𝑘 [ ∗ ∗

M

𝜃 = 1,2, ⋯ , 2𝑝 ,

PT

𝜃 Where 𝛨𝑖𝑘 = (𝐴𝜃𝑖𝑘 )𝑇 𝑃𝑖𝑘 + 𝑃𝑖𝑘 𝐴𝜃𝑖𝑘 + ∑𝑟𝑗=1 𝜋𝑖𝑗 𝑃𝑗𝑘 + ∑𝑠𝑙=1 𝜆𝑖𝑘𝑙 𝑃𝑖𝑙 ,

(17) (18) 𝑝

𝐴𝜃𝑖𝑘 = 𝐴𝑖 + 𝐵𝑖 ∑ϑ=1 Λϑ 𝜉ϑ𝑘 (𝛤𝜃 𝐾𝑖𝑘 +

𝛤𝜃− 𝐹𝑖𝑘 ), then the closed-loop system (12) is stochastically stable with 𝐻∞ performance γ in ⋂𝑟𝑖=1 ⋂𝑠𝑘=1 ℰ(𝑃𝑖𝑘 , 1), which is contained in the domain of attraction in the mean square sense of

CE

the system (12).

Proof. Let the mode of system at time 𝑡 be 𝑟𝑡 = 𝑖, the mode of faults at time 𝑡 be 𝑠𝑡 = 𝑘. We first show the stochastic stability of the closed-loop system (12) with 𝜔(𝑡) = 0. Assume

AC

𝑥(𝑡) ∈ ⋂𝑟𝑖=1 ⋂𝑠𝑘=1 ℰ(𝑃𝑖𝑘 , 1). Then (18) implies 𝑥(𝑡) ∈ 𝜓(𝐹𝑖𝑘 ), and according to Lemma 1, the closed-loop system can be described as (12). Take the following mode-dependent Lyapunov function candidate for system (12): 𝑉(𝑥(𝑡), 𝑟𝑡 , 𝑠𝑡 ) = 𝑥 𝑇 (𝑡)𝑃(𝑟𝑡 , 𝑠𝑡 )𝑥(𝑡),

(19)

where 𝑃(𝑟𝑡 , 𝑠𝑡 ) is positive definite matrix which will be denoted as 𝑃𝑖𝑘 for 𝑟𝑡 = 𝑖 and 𝑠𝑡 = 𝑘. In terms of the weak infinitesimal operator ℒ of the stochastic process {𝑥(𝑡), 𝑟𝑡 , 𝑠𝑡 , 𝑡 ≥ 0}, we have 1

ℒ(𝑉(𝑥(𝑡), 𝑟𝑡 , 𝑠𝑡 )) = lim∆→0 𝐄{𝑉(𝑥(𝑡 + ∆), 𝑟𝑡+∆ , 𝑠𝑡+∆ ) − 𝑉(𝑥(𝑡), 𝑟𝑡 , 𝑠𝑡 )} ∆

6

ACCEPTED MANUSCRIPT

1

= lim∆→0 E{𝑥 𝑇 (𝑡 + ∆)𝑃(𝑟𝑡+∆ , 𝑠𝑡+∆ )𝑥(𝑡 + ∆) − 𝑥 𝑇 (𝑡)𝑃(𝑟𝑡 , 𝑠𝑡 )𝑥(𝑡)}. ∆

(20)

Notice the fact that 𝜋𝑖𝑗 ∆ + 𝑜(∆), 𝑗≠𝑖 Pr {𝑟𝑡+∆ = 𝑗|𝑟𝑡 = 𝑖, 𝑠𝑡 = 𝑘} = Pr {𝑟𝑡+∆ = 𝑗|𝑟𝑡 = 𝑖} = { , 1 + 𝜋𝑖𝑖 ∆ + 𝑜(∆), 𝑗 = 𝑖 and Pr {𝑠𝑡+∆ = 𝑙|𝑟𝑡+∆ = 𝑗, 𝑟𝑡 = 𝑖, 𝑠𝑡 = 𝑘} = Pr {𝑠𝑡+∆ = 𝑙|𝑟𝑡 = 𝑖, 𝑠𝑡 = 𝑘} = {

𝜆𝑖𝑘𝑙 ∆ + 𝑜(∆), 𝑙≠𝑘 . 𝑖 1 + 𝜆𝑘𝑘 ∆ + 𝑜(∆), 𝑙 = 𝑘

Pr {𝑟𝑡+∆ = 𝑗, 𝑠𝑡+∆ = 𝑙|𝑟𝑡 = 𝑖, 𝑠𝑡 = 𝑘} = Pr {𝑠𝑡+∆ = 𝑙|𝑟𝑡+∆ = 𝑗, 𝑟𝑡 = 𝑖, 𝑠𝑡 = 𝑘} ∙ Pr {𝑟𝑡+∆ = 𝑗|𝑟𝑡 = 𝑖, 𝑠𝑡 = 𝑘} = Pr {𝑠𝑡+∆ = 𝑙|𝑟𝑡 = 𝑖, 𝑠𝑡 = 𝑘} ∙ Pr {𝑟𝑡+∆ = 𝑗|𝑟𝑡 = 𝑖, 𝑠𝑡 = 𝑘} [𝜆𝑖𝑘𝑙 ∆ + 𝑜(∆)][𝜋𝑖𝑗 ∆ + 𝑜(∆)], =

[1 +

𝜆𝑖𝑘𝑘 ∆

𝑗 ≠ 𝑖, 𝑙 ≠ 𝑘

+ 𝑜(∆)][𝜋𝑖𝑗 ∆ + 𝑜(∆)],

𝑗 ≠ 𝑖, 𝑙 = 𝑘

[𝜆𝑖𝑘𝑙 ∆ + 𝑜(∆)][1 + 𝜋𝑖𝑖 ∆ + 𝑜(∆)],

𝑗 = 𝑖, 𝑙 ≠ 𝑘

.

CR IP T

Then we have

AN US

𝑖 𝑗 = 𝑖, 𝑙 = 𝑘 { [1 + 𝜆𝑘𝑘 ∆ + 𝑜(∆)][1 + 𝜋𝑖𝑖 ∆ + 𝑜(∆)], Thus, according to the definition of expectation, along the trajectory of (12) with 𝜔(𝑡) = 0, for

any possible 𝑟𝑡 = 𝑖 ∈ 𝚼 and 𝑠𝑡 = 𝑘 ∈ 𝐒, we can get that 1

ℒ(𝑉(𝑥(𝑡), 𝑖, 𝑘)) = lim∆→0 {∑𝑗≠𝑖 ∑𝑙≠𝑘[𝜋𝑖𝑗 ∆ + 𝑜(∆)][𝜆𝑖𝑘𝑙 ∆ + 𝑜(∆)]𝑥 𝑇 (𝑡 + ∆)𝑃𝑗𝑙 𝑥(𝑡 + ∆) + ∆

∑𝑗≠𝑖[𝜋𝑖𝑗 ∆ + 𝑜(∆)][1 + 𝜆𝑖𝑘𝑘 ∆ + 𝑜(∆)] 𝑥 𝑇 (𝑡 + ∆)𝑃𝑗𝑘 𝑥(𝑡 + ∆) +

M

∑𝑙≠𝑘[1 + 𝜋𝑖𝑖 ∆ + 𝑜(∆)][𝜆𝑖𝑘𝑙 ∆ + 𝑜(∆)] 𝑥 𝑇 (𝑡 + ∆)𝑃𝑖𝑙 𝑥(𝑡 + ∆) +

ED

[1 + 𝜋𝑖𝑖 ∆ + 𝑜(∆)][1 + 𝜆𝑖𝑘𝑘 ∆ + 𝑜(∆)]𝑥 𝑇 (𝑡 + ∆)𝑃𝑖𝑘 𝑥(𝑡 + ∆) − 𝑥 𝑇 (𝑡)𝑃𝑖𝑘 𝑥(𝑡)} =𝑥̇ 𝑇 (𝑡)𝑃𝑖𝑘 𝑥(𝑡) + 𝑥 𝑇 (𝑡)𝑃𝑖𝑘 𝑥̇ (𝑡) + ∑𝑟𝑗=1 𝜋𝑖𝑗 𝑥 𝑇 (𝑡)𝑃𝑗𝑘 𝑥(𝑡) + ∑𝑠𝑙=1 𝜆𝑖𝑘𝑙 𝑥 𝑇 (𝑡)𝑃𝑖𝑙 𝑥(𝑡) 𝑝

PT

=𝑥 𝑇 (𝑡) ∑2𝜃=1 𝜇𝜃 ((𝐴̃𝜃𝑖𝑘 )𝑇 𝑃𝑖𝑘 + 𝑃𝑖𝑘 𝐴̃𝜃𝑖𝑘 + ∑𝑟𝑗=1 𝜋𝑖𝑗 𝑃𝑗𝑘 + ∑𝑠𝑙=1 𝜆𝑖𝑘𝑙 𝑃𝑖𝑙 )𝑥(𝑡) 𝑝

𝜃 ≜ 𝑥 𝑇 (𝑡) ∑2𝜃=1 𝜇𝜃 𝛨𝑖𝑘 𝑥(𝑡).

(21)

CE

𝜃 For all 𝑥(𝑡) ≠ 0, according to Lemma 2, (17) implies 𝛨𝑖𝑘 < 0, and ℒ(𝑉(𝑥(𝑡), 𝑖, 𝑘)) < 0. As a

result, for all 𝑥(𝑡) ≠ 0, it follows that

AC

ℒ(𝑉(𝑥(𝑡), 𝑖, 𝑘)) =

𝑝

𝜃 𝑥 𝑇 (𝑡) ∑2 𝜃=1 𝜇𝜃 𝛨𝑖𝑘 𝑥(𝑡)

𝑥 𝑇 (𝑡)𝑃𝑖𝑘 𝑥(𝑡)

𝑉(𝑥(𝑡), 𝑖, 𝑘) ≤ −α𝑉(𝑥(𝑡), 𝑖, 𝑘)

(22)

𝑝

𝜃 where α = min𝑖∈𝛶,𝑘∈𝑆 {λmin (− ∑2𝜃=1 𝜇𝜃 𝛨𝑖𝑘 )/λmax (𝑃𝑖𝑘 )} > 0. Then, applying the Dynkin’s formula,

we have

𝑡

𝑡

𝐄{𝑉(𝑥(𝑡), 𝑟𝑡 , 𝑠𝑡 )} − 𝑉(𝑥0 , 𝑟0 , 𝑠0 ) = 𝐄 {∫0 ℒ(𝑉(𝑥(𝜏), 𝑟𝜏 , 𝑠𝜏 ))d𝜏} ≤ −α ∫0 𝐄{𝑉(𝑥(𝜏), 𝑟𝜏 , 𝑠𝜏 )}d𝜏.

(23)

Thus, for all 𝑟0 ∈ 𝚼, 𝑠0 ∈ 𝐒, 𝐄{𝑥 𝑇 (𝑡)𝑃𝑖𝑘 𝑥(𝑡)|𝑥0 , 𝑟0 , 𝑠0 } = 𝐄{𝑉(𝑥(𝑡), 𝑖, 𝑘)|𝑥0 , 𝑟0 , 𝑠0 } ≤ 𝑒 −𝛼𝑡 𝑉(𝑥0 , 𝑖, 𝑘).

(24)

Then, it can be shown that 7

ACCEPTED MANUSCRIPT

𝑡

𝑡

1

𝐄 {∫0 𝑓 𝑥 𝑇 (𝑡)𝑃𝑖𝑘 𝑥(𝑡)d𝑡 |𝑥0 , 𝑟0 , 𝑠0 } ≤ ∫0 𝑓 𝑒 −𝛼𝑡 𝑉(𝑥0 , 𝑖, 𝑘) d𝑡 = − (𝑒 −𝛼𝑡𝑓 − 1)𝑥0𝑇 𝑃𝑖𝑘 𝑥0 . 𝛼

(25)

Taking the limit as 𝑡𝑓 → ∞, we have 𝑡

1

lim𝑡𝑓→∞ 𝐄 {∫0 𝑓 𝑥 𝑇 (𝑡)𝑃𝑖𝑘 𝑥(𝑡)d𝑡 |𝑥0 , 𝑟0 , 𝑠0 } ≤ 𝑥0𝑇 𝑃𝑖𝑘 𝑥0 .

(26)

𝛼

Notice that 𝑃𝑖𝑘 > 0, so we have 𝑡

lim𝑡𝑓→∞ E {∫0 𝑓 𝑥 𝑇 (𝑡)𝑥(𝑡)d𝑡 |𝑥0 , 𝑟0 , 𝑠0 } ≤ 𝑥0𝑇 𝑅𝑥0 ,

CR IP T

(27)

where 𝑅 = max𝑖∈𝚼,𝑘∈𝐒 {𝜆max (𝑃𝑖𝑘 )/𝛼𝜆min (𝑃𝑖𝑘 )}, which implies that the closed-loop system (12) with 𝜔(𝑡) = 0 is stochastically stable, and the set ⋂𝑟𝑖=1 ⋂𝑠𝑘=1 ℰ(𝑃𝑖𝑘 , 1) is contained in the domain of attraction in mean square sense of the closed-loop system (12).

On the other hand, for system (12), in which 𝜔(𝑡) is no longer taken to be zero, considering (17) and (22), it follows that 𝑡

AN US

E{∫0 𝑓[𝑧 𝑇 (𝑡)𝑧(𝑡) − 𝛾 2 𝜔𝑇 (𝑡)𝜔(𝑡)]𝑑𝑡 } 𝑡

≤E{∫0 𝑓[𝑧 𝑇 (𝑡)𝑧(𝑡) − 𝛾 2 𝜔𝑇 (𝑡)𝜔(𝑡) + ℒ(𝑉(𝑥(𝑡), 𝑖, 𝑘))]𝑑𝑡 } 𝜔𝑇 (𝑡)]𝛭[𝑥 𝑇 (𝑡) 𝑝

where 𝛭 = ∑2𝜃=1 𝜇𝜃 [

𝜔𝑇 (𝑡)]𝑇 𝑑𝑡},

𝜃 𝛨𝑖𝑘 + 𝐶𝑖𝑇 𝐶𝑖 ∗

(28)

𝑃𝑖𝑘 𝐵𝜔𝑖 ]. By Lemma 2, (17) implies 𝑀 < 0. Therefore, 𝑇 −𝛾 𝐼 + 𝐷𝜔𝑖 𝐷𝜔𝑖 2

M

𝑡

≤E{∫0 𝑓[𝑥 𝑇 (𝑡)

for all 𝑡𝑓 > 0, 𝜔(𝑡) ∈ 𝑳𝟐 [0, ∞), we have 𝑡

E{∫0 𝑓[𝑧 𝑇 (𝑡)𝑧(𝑡) − 𝛾 2 𝜔𝑇 (𝑡)𝜔(𝑡)]𝑑𝑡 } < 0.

ED

(29)

According to Definition 1 and Definition 2, the system (12) is stochastically stable with 𝐻∞ □

PT

performance γ in ⋂𝑟𝑖=1 ⋂𝑠𝑘=1 ℰ(𝑃𝑖𝑘 , 1). This completes the proof.

Remark 4. Inspired by [48], we extend the asynchronous jump of a discrete Markov jump system

CE

to continuous Markov jump systems, such that, the closed-loop system is a kind of jump system with piecewise homogeneous Markov chain. With analysis similar to that in [48], the stochastic stability condition is obtained. The difference is the jump mode of the filter depends on the mode

AC

of the system at k+1 in [48], while, the jump mode of fault depends on the current mode of system is more reasonable in this paper. 𝑟

Theorem 2. For the given Markov jump system (12) with uncertain transition rates 𝜋𝑖𝑗 and 𝜆𝑘𝑙𝑡 , 𝑟 𝑟 given scalars 𝜋̃𝑖𝑗 , 𝜆̃ 𝑡 , 𝜀𝑖𝑗 > 0 and 𝛿 𝑡 > 0, if there exist scalar γ > 0, a set of symmetric 𝑘𝑙

𝑘𝑙

positive definite matrices 𝑃𝑖𝑘 ∈ 𝐑𝑛×𝑛 , positive definite matrices 𝑊𝑖𝑗𝑘 ∈ 𝐑𝑛×𝑛 , 𝑇𝑖𝑘𝑙 ∈ 𝐑𝑛×𝑛 and appropriate matrices 𝐹𝑖𝑘 ∈ 𝐑𝑝×𝑛 , 𝐾𝑖𝑘 ∈ 𝐑𝑝×𝑛 , 𝑄𝑖𝑘 ∈ 𝐑𝑛×𝑛 , 𝑅𝑖𝑘 ∈ 𝐑𝑛×𝑛 , such that for any (𝑖, 𝑘) ∈ (𝚼, 𝐒), and 𝜃 = 1,2, ⋯ , 2𝑝 , condition (18) and the following matrix inequalities hold, (𝐴𝜃𝑖𝑘 )𝑇 𝑃𝑖𝑘 + 𝑃𝑖𝑘 𝐴𝜃𝑖𝑘 + Ψ𝑖𝑘 + Ω𝑖𝑘 [ ∗ ∗

𝑃𝑖𝑘 𝐵𝜔𝑖 −𝛾 2 𝐼 ∗

𝐶𝑖𝑇 𝑇 ] < 0, 𝐷𝜔𝑖 −𝐼

(30) 8

ACCEPTED MANUSCRIPT 𝑃𝑗𝑘 − 𝑄𝑖𝑘 ≤ 𝑊𝑖𝑗𝑘 , ∀𝑗 ∈ Υ, 𝑗 ≠ 𝑖,

(31)

𝑃𝑖𝑘 − 𝑄𝑖𝑘 ≥ 0,

(32)

𝑃𝑖𝑙 − 𝑅𝑖𝑘 ≤ 𝑇𝑖𝑘𝑙 , ∀𝑙 ∈ S, 𝑙 ≠ 𝑘,

(33)

𝑃𝑖𝑘 − 𝑅𝑖𝑘 ≥ 0,

(34)

where, Ψ𝑖𝑘 = ∑𝑗≠𝑖 [(𝜋̃𝑖𝑗 − 𝜀𝑖𝑗 )(𝑃𝑗𝑘 − 𝑄𝑖𝑘 ) + 2𝜀𝑖𝑗 𝑊𝑖𝑗𝑘 ] + (𝜋̃𝑖𝑖 + 𝜀𝑖𝑖 )(𝑃𝑖𝑘 − 𝑄𝑖𝑘 ), and 𝑖 𝑖 𝑖 Ω𝑖𝑘 = ∑𝑙≠𝑘[(𝜆̃𝑖𝑘𝑙 − 𝛿𝑘𝑙 )(𝑃𝑖𝑙 − 𝑅𝑖𝑘 ) + 2𝛿𝑘𝑙 𝑇𝑖𝑘𝑙 ] + (𝜆̃𝑖𝑘𝑘 + 𝛿𝑘𝑘 )(𝑃𝑖𝑘 − 𝑅𝑖𝑘 ) , then the closed-loop system (12) is stochastically stable with 𝐻∞ performance γ in ⋂𝑟𝑖=1 ⋂𝑠𝑘=1 ℰ(𝑃𝑖𝑘 , 1), which is contained in domain of attraction in mean square sense of the system (12).

CR IP T

Proof. For the Markov jump system (12), given perturbed transition rates, we only need to prove that (30)-(34) guarantee that (17) holds. Noticing that 𝜋𝑖𝑗 = 𝜋̃𝑖𝑗 + ∆𝜋𝑖𝑗 , 𝜆𝑖𝑘𝑙 = 𝜆̃𝑖𝑘𝑙 + ∆𝜆𝑖𝑘𝑙 , 𝑖 |∆𝜋𝑖𝑗 | ≤ 𝜀𝑖𝑗 < |π ̃𝑖𝑗 |, |∆𝜆𝑖𝑘𝑙 | ≤ 𝛿𝑘𝑙 < |𝜆̃𝑖𝑘𝑙 |, 𝜋̃𝑖𝑖 = − ∑𝑗≠𝑖 𝜋̃𝑖𝑗 , ∆𝜋𝑖𝑖 = − ∑𝑗≠𝑖 ∆𝜋𝑖𝑗 , 𝜆̃𝑖𝑘𝑘 = − ∑𝑘≠𝑙 𝜆̃𝑖𝑘𝑙 ,

∆𝜆𝑖𝑘𝑘 = − ∑𝑘≠𝑙 ∆𝜆𝑖𝑘𝑙

,

thus,

we

have

∑𝑟𝑗=1 𝜋𝑖𝑗 𝑃𝑗𝑘 = ∑𝑟𝑗=1 𝜋𝑖𝑗 (𝑃𝑗𝑘 − 𝑄𝑖𝑘 )

∑𝑠𝑙=1 𝜆𝑖𝑘𝑙 𝑃𝑖𝑙 = ∑𝑠𝑙=1 𝜆𝑖𝑘𝑙 (𝑃𝑖𝑙 − 𝑅𝑖𝑘 ), and then

AN US

𝜃 𝛨𝑖𝑘 = (𝐴𝜃𝑖𝑘 )𝑇 𝑃𝑖𝑘 + 𝑃𝑖𝑘 𝐴𝜃𝑖𝑘 + ∑𝑟𝑗=1 𝜋𝑖𝑗 𝑃𝑗𝑘 + ∑𝑠𝑙=1 𝜆𝑖𝑘𝑙 𝑃𝑖𝑙

,

= (𝐴𝜃𝑖𝑘 )𝑇 𝑃𝑖𝑘 + 𝑃𝑖𝑘 𝐴𝜃𝑖𝑘 + ∑𝑟𝑗=1(𝜋̃𝑖𝑗 + ∆𝜋𝑖𝑗 )(𝑃𝑗𝑘 − 𝑄𝑖𝑘 ) + ∑𝑠𝑙=1(𝜆̃𝑖𝑘𝑙 + ∆𝜆𝑖𝑘𝑙 )(𝑃𝑖𝑙 − 𝑅𝑖𝑘 ) = (𝐴𝜃𝑖𝑘 )𝑇 𝑃𝑖𝑘 + 𝑃𝑖𝑘 𝐴𝜃𝑖𝑘 + ∑𝑗≠𝑖(𝜋̃𝑖𝑗 + ∆𝜋𝑖𝑗 )(𝑃𝑗𝑘 − 𝑄𝑖𝑘 ) + (𝜋̃𝑖𝑖 + ∆𝜋𝑖𝑖 )(𝑃𝑖𝑘 − 𝑄𝑖𝑘 )+ ∑𝑘≠𝑙(𝜆̃𝑖𝑘𝑙 + ∆𝜆𝑖𝑘𝑙 )(𝑃𝑖𝑙 − 𝑅𝑖𝑘 ) + (𝜆̃𝑖𝑘𝑘 + ∆𝜆𝑖𝑘𝑘 )(𝑃𝑖𝑘 − 𝑅𝑖𝑘 )

M

= (𝐴𝜃𝑖𝑘 )𝑇 𝑃𝑖𝑘 + 𝑃𝑖𝑘 𝐴𝜃𝑖𝑘 + ∑𝑗≠𝑖(𝜋̃𝑖𝑗 − 𝜀𝑖𝑗 + 𝜀𝑖𝑗 + ∆𝜋𝑖𝑗 )(𝑃𝑗𝑘 − 𝑄𝑖𝑘 ) + (𝜋̃𝑖𝑖 + ∆𝜋𝑖𝑖 )(𝑃𝑖𝑘 − 𝑄𝑖𝑘 )+ 𝑖 𝑖 ∑𝑙≠𝑘(𝜆̃𝑖𝑘𝑙 − 𝛿𝑘𝑙 + 𝛿𝑘𝑙 + ∆𝜆𝑖𝑘𝑙 )(𝑃𝑖𝑙 − 𝑅𝑖𝑘 ) + (𝜆̃𝑖𝑘𝑘 + ∆𝜆𝑖𝑘𝑘 )(𝑃𝑖𝑘 − 𝑅𝑖𝑘 )

ED

≤ (𝐴𝜃𝑖𝑘 )𝑇 𝑃𝑖𝑘 + 𝑃𝑖𝑘 𝐴𝜃𝑖𝑘 + ∑𝑗≠𝑖 [(𝜋̃𝑖𝑗 − 𝜀𝑖𝑗 )(𝑃𝑗𝑘 − 𝑄𝑖𝑘 ) + 2𝜀𝑖𝑗 𝑊𝑖𝑗𝑘 ] + (𝜋̃𝑖𝑖 + 𝜀𝑖𝑖 )(𝑃𝑖𝑘 − 𝑄𝑖𝑘 )+ 𝑖 𝑖 𝑖 ∑𝑙≠𝑘[(𝜆̃𝑖𝑘𝑙 − 𝛿𝑘𝑙 )(𝑃𝑖𝑙 − 𝑅𝑖𝑘 ) + 2𝛿𝑘𝑙 𝑇𝑖𝑘𝑙 ]+ (𝜆̃𝑖𝑘𝑘 + 𝛿𝑘𝑘 )(𝑃𝑖𝑘 − 𝑅𝑖𝑘 )

PT

≜ (𝐴𝜃𝑖𝑘 )𝑇 𝑃𝑖𝑘 + 𝑃𝑖𝑘 𝐴𝜃𝑖𝑘 + Ψ𝑖𝑘 + Ω𝑖𝑘 .

By Lemma 2, it follows that (30) implies that (17) holds. This completes the proof.

(35) □

CE

Remark 5. In the results of Theorem 2, if 𝐒 = {1}, 𝑄𝑖𝑘 = 𝑃𝑖𝑘 and 𝑅𝑖𝑘 = 𝑃𝑖𝑘 , then the items Ψ𝑖𝑘 and Ω𝑖𝑘 would be described as

Ψ𝑖𝑘 = ∑𝑗≠𝑖 [(𝜋̃𝑖𝑗 − 𝜀𝑖𝑗 )(𝑃𝑗 − 𝑃𝑖 ) + 2𝜀𝑖𝑗 𝑊𝑖𝑗 ] , and Ω𝑖𝑘 = 0 ,

respectively. In this case, the results are similar to the results of [27], which have less

AC

conservatism than [24]. As we can see, our results are more general. For the uncertain transition rates, a key problem is to bind the uncertain items. In our results, Theorem 2 not only fully considered the property and the characteristic of the perturbed transition rates, 𝜋̃𝑖𝑖 + ∆𝜋𝑖𝑖 < 0, and 𝜆̃𝑖𝑘𝑘 + ∆𝜆𝑖𝑘𝑘 < 0 to bind the uncertain items, but also took into account the purpose of synthesis, and introduced the matrices in (31)-(34), to avoid the coupling items in [24].

3.2 Mixed-mode-dependent controller Based on the stability analysis result presented in subsection 3.1, we investigate the 𝐻∞ control problem for the closed-loop system. 9

ACCEPTED MANUSCRIPT 𝑟

Theorem 3. Assume you are given transition rates 𝜋𝑖𝑗 and 𝜆𝑘𝑙𝑡 . For the system (1) subject to asynchronous jumped fault in (5), it is possible to design the mixed mode-dependent controller in (7), such that the closed-loop system (12) is stochastically stable in ⋂𝑟𝑖=1 ⋂𝑠𝑘=1 ℰ(𝑃𝑖𝑘 , 1), which is contained in domain of attraction in mean square sense of the system (12), if there exist scalar γ > 0, a set of symmetric positive definite matrices 𝑋𝑖𝑘 ∈ 𝐑𝑛×𝑛 , and appropriate matrices 𝑌𝑖𝑘 ∈ 𝐑𝑝×𝑛 , 𝑍𝑖𝑘 ∈ 𝐑𝑝×𝑛 , such that for any (𝑖, 𝑘) ∈ (𝚼, 𝐒), and 𝜃 = 1,2, ⋯ , 2𝑝 , the following linear matrix inequalities hold, 𝑢𝜗 ∗

Φ𝑖𝑘

𝜗 𝑍𝑖𝑘 ] ≥ 0, 𝑢𝜗 𝑋𝑖𝑘

Φ11 𝑖𝑘 ∗ = ∗ ∗ [ ∗

𝐵𝜔𝑖 −𝛾 2 𝐼 ∗ ∗ ∗

(36) 𝑋𝑖𝑘 𝐶𝑖𝑇 𝑇 𝐷𝜔𝑖 −𝐼 ∗ ∗

Φ14 𝑖𝑘 0 0 44 Φ𝑖𝑘 ∗

Φ15 𝑖𝑘 0 0 < 0, 0 55 Φ𝑖𝑘 ]

CR IP T

[

(37)

AN US

𝑝 𝑖 − ̃𝜃 ̃𝜃 where, Φ11 𝑖𝑘 = 𝑠𝑦𝑚(𝐴𝑖𝑘 ) + 𝜋𝑖𝑖 𝑋𝑖𝑘 + 𝜆𝑘𝑘 𝑋𝑖𝑘 , 𝐴𝑖𝑘 = 𝐴𝑖 𝑋𝑖𝑘 + 𝐵𝑖 ∑ϑ=1 Λ ϑ 𝜉ϑ𝑘 (𝛤𝜃 𝑌𝑖𝑘 + 𝛤𝜃 𝑍𝑖𝑘 ),

Φ14 𝑖𝑘 = [√𝜋𝑖1 , ⋯ , √𝜋𝑖(𝑖−1) , √𝜋𝑖(𝑖+1) , ⋯ , √𝜋𝑖𝑟 ]𝑋𝑖𝑘 ,

𝑖 𝑖 𝑖 𝑖 Φ15 𝑖𝑘 = [√𝜆𝑘1 , ⋯ , √𝜆𝑘(𝑘−1) , √𝜆𝑘(𝑘+1) , ⋯ , √𝜆𝑘𝑠 ] 𝑋𝑖𝑘 ,

44 Φ𝑖𝑘 = 𝑑𝑖𝑎𝑔(−𝑋1𝑘 , ⋯ , −𝑋(𝑖−1)𝑘 , −𝑋(𝑖+1)𝑘 , ⋯ , −𝑋𝑟𝑘 ), 55 Φ𝑖𝑘 = diag(−𝑋𝑖1 , ⋯ , −𝑋𝑖(𝑘−1) , −𝑋𝑖(𝑘+1) , ⋯ , −𝑋𝑖𝑠 ).

M

𝜗 Here 𝑍𝑖𝑘 represents the ϑth row of 𝑍𝑖𝑘 . In this case, the controller gain is given by 𝐾𝑖𝑘 = −1 −1 𝑌𝑖𝑘 𝑋𝑖𝑘 . Moreover, the matrix 𝐹𝑖𝑘 in (12a) is given by 𝐹𝑖𝑘 = 𝑍𝑖𝑘 𝑋𝑖𝑘 .

ED

−1 ) to the left of inequality (36), and Proof. Apply the congruent transformation 𝑇1 = 𝑑𝑖𝑎𝑔(𝐼, 𝑋𝑖𝑘 −1 then let 𝑋𝑖𝑘 ≜ 𝑃𝑖𝑘 , it follows that [

𝑢𝜗 ∗

𝜗 𝐹𝑖𝑘 ] ≥ 0, which is equivalent to (18). 𝑢𝜗 𝑃𝑖𝑘

PT

Assume 𝑥(𝑡) ∈ ⋂𝑟𝑖=1 ⋂𝑠𝑘=1 ℰ(𝑃𝑖𝑘 , 1). Then the closed-loop system (12) can be obtained for the system of (1) with the controller of (5). According to Lemma 2, (37) is equivalent to 𝐵𝜔𝑖 −𝛾 2 𝐼 ∗

𝑋𝑖𝑘 𝐶𝑖𝑇 𝑇 ] < 0, 𝐷𝜔𝑖 −𝐼

CE

𝛱 [∗ ∗

(38)

AC

−1 where, 𝛱 = 𝑠𝑦𝑚(𝐴̃𝜃𝑖𝑘 ) + ∑𝑟𝑗=1 𝜋𝑖𝑗 𝑋𝑖𝑘 𝑋𝑗𝑘 𝑋𝑖𝑘 + ∑𝑠𝑙=1 𝜆𝑖𝑘𝑙 𝑋𝑖𝑘 𝑋𝑖𝑙−1 𝑋𝑖𝑘 . Then applying the congruent −1 −1 transformation 𝑇2 = 𝑑𝑖𝑎𝑔(𝑋𝑖𝑘 ≜ 𝑃𝑗𝑘 , 𝐼, 𝐼) to the left of inequality (38), and letting 𝑋𝑗𝑘

(𝑗 = 1,2, ⋯ , 𝑟), it can be shown that (17) holds. This completes the proof.

□

𝑟 Theorem 4. Assume estimated transition rates 𝜋̃𝑖𝑗 , 𝜆̃𝑘𝑙𝑡 , and their estimate error values 𝜀𝑖𝑗 > 0, 𝑟

𝛿𝑘𝑙𝑡 > 0 are given. For the system (1) subject to asynchronous jumped fault in (5), it is possible to design the mixed mode-dependent controller in (7), such that the closed-loop system (12) is stochastically stable in ⋂𝑟𝑖=1 ⋂𝑠𝑘=1 ℰ(𝑃𝑖𝑘 , 1), which is contained in the domain of attraction in mean square sense of the system (12), if there exist scalar γ > 0, a set of symmetric positive ̃𝑖𝑗𝑘 ∈ 𝐑𝑛×𝑛 , 𝑇̃𝑖𝑘𝑙 ∈ 𝐑𝑛×𝑛 and definite matrices 𝑋𝑖𝑘 ∈ 𝐑𝑛×𝑛 , positive definite matrices 𝑊 appropriate matrices 𝑌𝑖𝑘 ∈ 𝐑𝑝×𝑛 , 𝑍𝑖𝑘 ∈ 𝐑𝑝×𝑛 , 𝑄̃𝑖𝑘 ∈ 𝐑𝑛×𝑛 , 𝑅̃𝑖𝑘 ∈ 𝐑𝑛×𝑛 , such that for any 10

ACCEPTED MANUSCRIPT (𝑖, 𝑘) ∈ (𝚼, 𝐒), and 𝜃 = 1,2, ⋯ , 2𝑝 , the inequality (36) and the following linear matrix inequalities hold,

[

𝐵𝜔𝑖 −𝛾 2 𝐼 ∗ ∗ ∗

̃𝑖𝑗𝑘 −𝑋𝑖𝑘 − 𝑊 𝑋𝑖𝑘

𝑋𝑖𝑘 𝐶𝑖𝑇 𝑇 𝐷𝜔𝑖 −𝐼 ∗ ∗

̃ 14 Φ 𝑖𝑘 0 0 44 Φ𝑖𝑘 ∗

̃ 15 Φ 𝑖𝑘 0 0 < 0, 0 55 Φ𝑖𝑘 ]

𝑋𝑖𝑘 ] < 0, ∀𝑗 ∈ 𝚼, 𝑗 ≠ 𝑖, −𝑋𝑗𝑘

(40)

𝑋𝑖𝑘 − 𝑄̃𝑖𝑘 > 0, [

−𝑋𝑖𝑘 − 𝑇̃𝑖𝑘𝑙 𝑋𝑖𝑘

(39)

(41)

𝑋𝑖𝑘 ] < 0, ∀𝑙 ∈ 𝐒, 𝑙 ≠ 𝑘, −𝑋𝑖𝑙

CR IP T

̃ 𝑖𝑘 Φ

̃ 11 Φ 𝑖𝑘 ∗ = ∗ ∗ [ ∗

(42)

𝑋𝑖𝑘 − 𝑅̃𝑖𝑘 > 0,

(43)

𝑖 ̃𝜃 ̃ 11 ̃𝑖𝑗𝑘 − where, Φ ̃ 𝑖𝑖 + 𝜀𝑖𝑖 )(𝑋𝑖𝑘 − 𝑄̃𝑖𝑘 ) + (𝜆̃𝑖𝑘𝑘 + 𝛿𝑘𝑘 )(𝑋𝑖𝑘 − 𝑅̃𝑖𝑘 ) + ∑𝑗≠𝑖 [2𝜀𝑖𝑗 𝑊 𝑖𝑘 = 𝑠𝑦𝑚(𝐴𝑖𝑘 ) + (𝜋

AN US

𝑖 ̃ 𝑖 (𝜋̃𝑖𝑗 − 𝜀𝑖𝑗 )𝑄̃𝑖𝑘 ] + ∑𝑙≠𝑘[2𝛿𝑘𝑙 𝑇𝑖𝑘𝑙 − (𝜆̃𝑖𝑘𝑙 − 𝛿𝑘𝑙 )𝑅̃𝑖𝑘 ],

̃ 14 Φ ̃ 𝑖1 − 𝜀𝑖1 , ⋯ , √𝜋̃𝑖(𝑖−1) − 𝜀𝑖(𝑖−1) , √𝜋̃𝑖(𝑖+1) − 𝜀𝑖(𝑖+1) , ⋯ , √𝜋̃𝑖𝑟 − 𝜀𝑖𝑟 ]𝑋𝑖𝑘 , 𝑖𝑘 = [√𝜋 𝑖 𝑖 𝑖 𝑖 ̃𝑖 ̃𝑖 ̃𝑖 ̃𝑖 ̃ 15 Φ 𝑖𝑘 = [√𝜆𝑘1 − 𝛿𝑘1 , ⋯ , √𝜆𝑘(𝑘−1) − 𝛿𝑘(𝑘−1) , √𝜆𝑘(𝑘+1) − 𝛿𝑘(𝑘+1) , ⋯ , √𝜆𝑘𝑠 − 𝛿𝑘𝑠 ] 𝑋𝑖𝑘 . −1 In this case, the controller gain is given by 𝐾𝑖𝑘 = 𝑌𝑖𝑘 𝑋𝑖𝑘 . Moreover, the matrix 𝐹𝑖𝑘 is given by

M

−1 𝐹𝑖𝑘 = 𝑍𝑖𝑘 𝑋𝑖𝑘 .

Proof. Assume 𝑥(𝑡) ∈ ⋂𝑟𝑖=1 ⋂𝑠𝑘=1 ℰ(𝑃𝑖𝑘 , 1), then the closed-loop system (12) can be obtained for

𝐵𝜔𝑖 −𝛾 2 𝐼 ∗

where,

𝑋𝑖𝑘 𝐶𝑖𝑇 𝑇 ] < 0, 𝐷𝜔𝑖 −𝐼

(44)

PT

Σ [∗ ∗

ED

the system of (1) with the controller of (5). According to Lemma 2, (39) is equivalent to

CE

−1 ̃𝑖𝑗𝑘 ]𝑋𝑖𝑘 + Σ = 𝑠𝑦𝑚(𝐴̃𝜃𝑖𝑘 ) + ∑𝑟𝑗=1 𝑋𝑖𝑘 [(𝜋̃𝑖𝑗 − 𝜀𝑖𝑗 )(𝑋𝑗𝑘 − 𝑄̃𝑖𝑘 ) + 2𝜀𝑖𝑗 𝑊 𝑖 𝑖 ̃ ∑𝑠𝑙=1 𝑋𝑖𝑘 [(𝜆̃𝑖𝑘𝑙 − 𝛿𝑘𝑙 )(𝑋𝑖𝑙−1 − 𝑅̃𝑖𝑘 ) + 2𝛿𝑘𝑙 𝑇𝑖𝑘𝑙 ] 𝑋𝑖𝑘 .

AC

−1 Then applying the congruent transformation 𝑇2 = 𝑑𝑖𝑎𝑔(𝑋𝑖𝑘 ,

𝐼, 𝐼) to the left side of

−1 ̃𝑖𝑗𝑘 𝑋𝑖𝑘 ≜ 𝑊𝑖𝑗𝑘 (𝑗 ≠ inequality (44), for all 𝑖, 𝑗 ∈ 𝚼, 𝑘, 𝑙 ∈ 𝐒, denote 𝑋𝑗𝑘 ≜ 𝑃𝑗𝑘 (𝑗 = 1,2, ⋯ , 𝑟), 𝑋𝑖𝑘 𝑊

𝑖), 𝑋𝑖𝑘 𝑇̃𝑖𝑘𝑙 𝑋𝑖𝑘 ≜ 𝑇𝑖𝑘𝑙 (𝑘 ≠ 𝑙), 𝑋𝑖𝑘 𝑄̃𝑖𝑘 𝑋𝑖𝑘 ≜ 𝑄𝑖𝑘 , 𝑋𝑖𝑘 𝑅̃𝑖𝑘 𝑋𝑖𝑘 ≜ 𝑅𝑖𝑘 . Then noticing the characteristic of the perturbed transition rates, it can be shown that (44) implies (30). That’s to say, (39) implies −1 (30). Applying the congruent transformation 𝑇3 = 𝑑𝑖𝑎𝑔(𝑋𝑖𝑘 , 𝐼) to the left of inequality (40), −1 (42), and applying the congruent transformation 𝑇4 = 𝑋𝑖𝑘 to the left of inequality (41), (43), it

can be shown that the results meet (31)-(34). This completes the proof.

□ 11

ACCEPTED MANUSCRIPT

Remark 6. Note that the conditions in Theorem 3 and Theorem 4 are all LMIs. So, it is convenient to obtain a set of 𝐻∞ performance controllers and an estimate of the domain of attraction using the set ⋂𝑟𝑖=1 ⋂𝑠𝑘=1 ℰ(𝑃𝑖𝑘 , 1). Remark 7. For the system (12) with completely accessible transition rates, and with uncertain transition rates, to obtain controllers with good 𝐻∞ performance, the 𝐻∞ control problem can be formulated as the following minimization problems, respectively. OP1: min γ2

CR IP T

(45) 𝑝

s.t. LMIs (36), and (37) with 𝑋𝑖𝑘 > 0 for all 𝑖 ∈ 𝚼, 𝑘 ∈ 𝐒, and 𝜃 = 1,2, ⋯ , 2 . OP2: min γ2

(46) ̃ ̃ s.t. LMIs (36), and (39)-(43) with 𝑋𝑖𝑘 > 0 , 𝑊𝑖𝑗𝑘 > 0 (𝑗 ≠ 𝑖) , 𝑇𝑖𝑘𝑙 > 0 (𝑙 ≠ 𝑘) for all

𝑖, 𝑗 ∈ 𝚼, 𝑘, 𝑙 ∈ 𝐒, and 𝜃 = 1,2, ⋯ , 2𝑝 .

Remark 7. Theorem 3 and Theorem 4 investigate a sufficient condition in terms of LMIs which

AN US

guarantee the mean-square asymptotically stability of the closed-loop system (12) with completely accessible transition rates and uncertain transition rates, respectively. Now, we will use the same optimization method in [51] to find the largest estimate of the domain of attraction for the closed-loop system under the two different situations. Given an acceptable 𝐻∞ −1 performance γ > 0, we will measure the size of ⋂𝑟𝑖=1 ⋂𝑠𝑘=1 ℰ(𝑋𝑖𝑘 , 1) with respect to a shape −1 reference set 𝜒𝐺 as the largest 𝜌 such that 𝜌𝜒𝐺 ⊂ ⋂𝑟𝑖=1 ⋂𝑠𝑘=1 ℰ(𝑋𝑖𝑘 , 1), where 𝜒𝐺 = ℰ(𝐺, 1) =

M

−1 {𝑥(𝑡) ∈ R𝑛 : 𝑥 𝑇 (𝑡)𝐺𝑥(𝑡) ≤ 1, 𝐺 > 0} . Obviously, if 𝜌 > 1 , then 𝜒𝐺 ⊂ ⋂𝑟𝑖=1 ⋂𝑠𝑘=1 ℰ(𝑋𝑖𝑘 , 1) . The

𝐻∞ control problem can be formulated as the following optimization problems, respectively.

s.t. 𝜌𝜒𝐺 ⊂

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OP3: max 𝜌

−1 ℰ(𝑋𝑖𝑘 , 1),

𝑝

(47)

and LMIs (36), (37) with 𝑋𝑖𝑘 > 0 for all 𝑖 ∈ 𝚼, 𝑘 ∈ 𝐒, and 𝜃 =

OP3’: min 𝜂 s.t. [

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1,2, ⋯ , 2 . By Lemma 2, OP3 is equivalent to the following minimization problem (OP3’).

𝜂𝐺 𝐼

(48)

𝐼 ] >0, and LMIs (36), (37) with 𝑋𝑖𝑘 > 0 for all 𝑖 ∈ 𝚼, 𝑘 ∈ 𝐒, and 𝜃 = 𝑋𝑖𝑘

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1,2, ⋯ , 2𝑝 , where 𝜂 = 1/𝜌2 .

(49) −1 ̃𝑖𝑗𝑘 > 0 (𝑗 ≠ 𝑖) , 𝑇̃𝑖𝑘𝑙 > s.t. 𝜌𝜒𝐺 ⊂ ℰ(𝑋𝑖𝑘 , 1) , and LMIs (36), (39)-(43) with 𝑋𝑖𝑘 > 0 , 𝑊

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OP4: max 𝜌

0 (𝑙 ≠ 𝑘) for all 𝑖, 𝑗 ∈ 𝚼, 𝑘, 𝑙 ∈ 𝐒, and 𝜃 = 1,2, ⋯ , 2𝑝 . By Lemma 2, OP4 is equivalent to the following minimization problem (OP4’). OP4’: min 𝜂 s.t. [

𝜂𝐺 𝐼

(50) 𝐼 ̃𝑖𝑗𝑘 > 0 (𝑗 ≠ 𝑖), 𝑇̃𝑖𝑘𝑙 > 0 (𝑙 ≠ ]>0, and LMIs (36), (39)-(43) with 𝑋𝑖𝑘 > 0, 𝑊 𝑋𝑖𝑘

𝑘) for all 𝑖, 𝑗 ∈ 𝚼, 𝑘, 𝑙 ∈ 𝐒, and 𝜃 = 1,2, ⋯ , 2𝑝 , where 𝜂 = 1/𝜌2 .

4.

Numerical examples

In this section, examples are given to illustrate the effectiveness of the results developed in this 12

ACCEPTED MANUSCRIPT paper. Example 1. Consider the Markov jump system (1) with two jumping modes and completely accessible transition rate, the system matrices of which are as follows: 0.22 𝐴1 = [ 1.04 C1 = [0.1

−0.76 −1.25 ], 𝐴2 = [ −0.19 0.85 −0.1], C2 = [0.12

0.6 0.12 0.08 0.83 0.35 ], 𝐵1 = [ ], 𝐵2 = [ ], 𝐵𝜔1 = [ ], 𝐵𝜔2 = [ ], 0.4 0.10 0.15 0.15 0.52

−0.1], 𝐷𝜔1 = 0.01, 𝐷𝜔2 = 0.012.

The transition rate matrix is [πij ] = [

−1.2 0.8

1.2 ], and the saturation level 𝑢̅ = 1. Suppose the −0.8

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jumped fault has three modes, the failure scales are 𝜉1 = 1.1, 𝜉2 = 0.5, 𝜉3 = 0.69, and the transition rate matrices are: −1.4 [𝜆1𝑘𝑙 ] = [ 0.5 0.6

0.8 −1.2 0.4

0.6 −1.3 0.7], [𝜆2𝑘𝑙 ] = [ 0.5 −1 0.7

0.7 −1.3 0.5

0.6 0.8 ]. −1.2

For simulation purposes, suppose the initial operating points of 𝑥0 = [1.2, −0.3]𝑇 , and the disturbance input is 𝜔 (𝑡) = 0.1sin2𝑡. By the results in Theorem 3, using the YALMIP solver in

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MATLAB to solve the optimization problem (45), it is found that γmin = 0.7434. For prescribed γ = 1, consider the problem of enlarging the invariant set. Choose the shape reference set as a unit ball, i.e. 𝐺 = ℰ(𝐼, 1). By solving the optimization problem (47), it is found that 𝜌max = 1.1445, the invariant set is the intersection in Fig.1, and controller gains are: −4.9085], 𝐾12 = 1.0𝑒 + 5[−1.2358

−0.9937],

𝐾13 = 1.0𝑒 + 4[−4.8577

−3.9340], 𝐾21 = 1.0𝑒 + 4[−2.8356

−4.4663],

−1.2850], 𝐾23 = 1.0𝑒 + 4[−5.0494

−8.0064].

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𝐾22 = 1.0𝑒 + 5[−0.8033

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𝐾11 = 1.0𝑒 + 3[−5.9742

Fig.2 describes the closed-loop system jump mode with initial mode 𝑟0 = 1 and fault jump mode with initial mode 𝑠0 = 2. The simulation results of the state response of the closed-loop system

4 3 2

2

1

x

AC

CE

PT

are given in Fig.3. It is shown that the state trajectories converge towards the origin.

0 -1 -2 -3 -4 -4

-3

-2

-1

0 x1

1

2

3

4

Fig.1 the invariant set for example 1 13

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Jump mode rt

3 2 1

0

10

20

30

40

30

40

50

Time(s)

3 2 1 0

10

20 Time(s)

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Jump mode st

4

50

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Fig. 2 the jump modes for closed-loop system in Example 1

1.4

x1

1.2

x2

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0.8 0.6 0.4

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State renponse

1

0.2

0

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-0.2

CE

-0.4

0

5

10

15

20

25

30

35

40

45

50

Time(s)

Fig.3 the state response of closed-loop system in example 1

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Remark 8. In the simulation, we get the γmin=0.7434 with respect to the very large controller gains. But if we let γ=0.8, the results are K11 = [−15.9096 − 13.5798]; K12 = [ −48.4840 − 39.0049]; K13 = [ −29.0028 − 23.9537]; K 21 = [ −10.8854 − 16.7575]; K 22 = [ −31.8083 − 48.9581]; K 23 = [ −19.2539 − 29.5983]. In practical physical systems, we can use the acceptable solution instead of the optimal solution. Example 2. Consider the Markov jump system (1) with the same parameters in Example 1. Suppose the estimated transition rate of system is [π ̃𝑖𝑗 ] = [

−1.2 0.8

1.2 ], the estimate error −0.8 14

ACCEPTED MANUSCRIPT values of the transition rates satisfy |∆𝜋𝑖𝑗 | ≤ 0.1𝜋̃𝑖𝑗 ≜ 𝜀𝑖𝑗 , and suppose the estimated transition rate of faults are −1.4 [𝜆̃1𝑘𝑙 ] = [ 0.5 0.6

0.8 −1.2 0.4

0.6 −1.3 0.7], [𝜆̃2𝑘𝑙 ] = [ 0.5 −1 0.7

0.7 −1.3 0.5

0.6 0.8 ], −1.2

𝑗 𝑗 𝑗 and the estimate error values of the transition rates satisfy |∆𝜆𝑘𝑙 | ≤ 0.1𝜆̃𝑘𝑙 ≜ 𝛿𝑘𝑙 .

By the results in Theorem 4, using the YALMIP solver in MATLAB to solve the optimization problem (46), it is found that γmin = 0.8708. For prescribed γ = √2, consider the problem of enlarging the invariant set. Choose the shape reference set as a unit ball, i.e. 𝐺 = ℰ(𝐼, 1). By

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solving the optimization problem (49), it is found that 𝜌max = 1.004, the invariant set is the intersection in Fig.4, and controller gains are: 𝐾11 = 1.0𝑒 + 3[−7.2290

−6.1253], 𝐾12 = 1.0𝑒 + 4[−2.2153

−1.8331],

𝐾13 = 1.0𝑒 + 5[−1.9598

−1.6622], 𝐾21 = 1.0𝑒 + 3[−4.9451

−7.7331],

𝐾22 = 1.0𝑒 + 4[−1.4216

−2.2725], 𝐾23 = 1.0𝑒 + 4[−0.9375

−1.4950].

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The simulation results of the state response of the closed-loop system in 100 random samplings are given in Fig.5. It is shown that the state trajectories converge towards the origin.

5.

Conclusions

This paper has studied the stability and 𝐻∞ control problem for a class of linear Markov jump

M

systems with uncertain transition rates, actuator saturation and jumped actuator failure. Sufficient conditions have been provided for the stochastic stability of the closed-loop system,

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which has mixed asynchronous jumping modes with transition rates that are either completely accessible or have uncertainty. By solving a minimized convex optimization problem with LMIs constraints, the controller gains can be found. The 𝐻∞ performance optimization problem and

PT

the estimate of the domain of the attraction are both considered.

3 2

2

1

x

AC

CE

4

0 -1 -2 -3 -4 -4

-3

-2

-1

0 x1

1

2

3

4

Fig.4 the invariant set for Example 2

15

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ACCEPTED MANUSCRIPT

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Fig.5 the state response of the closed-loop system with 100 random samplings for Example 2

Acknowledgements

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This work was supported by the National Natural Science Foundation of China under grant No. 51277022, Chongqing Education Council Foundation under grant No. KJ1501312 and KJ1601310, Chongqing University of Science and Technology doctoral Foundation under grant No.

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CK2016B17, and China Scholarship Council under grant No.201608505169.

[1]

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