Observer-Based Controller Design for a Class of Nonlinear Networked ...

Hindawi Journal of Control Science and Engineering Volume 2017, Article ID 1523825, 13 pages https://doi.org/10.1155/2017/1523825

Research Article Observer-Based Controller Design for a Class of Nonlinear Networked Control Systems with Random Time-Delays Modeled by Markov Chains Yanfeng Wang, Peiliang Wang, Zuxin Li, and Huiying Chen School of Engineering, Huzhou University, Huzhou, Zhejiang 313000, China Correspondence should be addressed to Yanfeng Wang; [email protected] Received 3 January 2017; Revised 14 February 2017; Accepted 1 March 2017; Published 30 March 2017 Academic Editor: Lei Zou Copyright © 2017 Yanfeng Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper investigates the observer-based controller design problem for a class of nonlinear networked control systems with random time-delays. The nonlinearity is assumed to satisfy a global Lipschitz condition and two dependent Markov chains are employed to describe the time-delay from sensor to controller (S-C delay) and the time-delay from controller to actuator (C-A delay), respectively. The transition probabilities of S-C delay and C-A delay are both assumed to be partly inaccessible. Sufficient conditions on the stochastic stability for the closed-loop systems are obtained by constructing proper Lyapunov functional. The methods of calculating the controller and the observer gain matrix are also given. Two numerical examples are used to illustrate the effectiveness of the proposed method.

1. Introduction Networked control systems (NCSs) are spatially distributed systems where the communication between sensor, controller, and actuator is carried out by a shared band limited digital network [1, 2]. NCSs are used in a wide range of areas such as robots, industrial manufacturing plants, and remote surgery due to their advantages in practical applications, for example, flexible architectures, the reduced weight, simple installation, and maintenance as well as high flexibility and reliability [3, 4]. However, the communication networks also present some constraints such as time-delays and packet dropouts result from the limited bandwidth. It is generally known that the time-delay maybe degrades the performance or even causes instability [5, 6]. The Markov chain which is a discrete-time stochastic process with the Markov property can be effectively used to model the time-delay in NCSs. The random time-delays in NCSs modeled as Markov chains have been researched in the past several years, and many results have been reported [7– 17]. In [7], the time-delay of NCSs was modeled as a Markov

chain, and further a LQG optimal controller design method was proposed. In [8], the NCSs were molded as Markov jump linear systems (MJLSs) where the S-C delay was molded as a finite state Markov chain, and a V-K iteration method was proposed to get a stabilizing controller. In [9], a buffer was added ahead of the actuator, and the time-delays from sensor to actuator were lumped together which was molded as a Markov chain, and then the mean-square stability of the closed-loop system was derived. In [10–12], for the NCSs with S-C delay, the problem of 𝐻∞ control was investigated using the Bounded Real Lemma and the Markov jump theory. In [13, 14], the S-C delay and C-A delay were modeled as two independent Markov chains. The resulting closed-loop systems were transformed to control systems which contain two Markov chains. The sufficient and necessary conditions for the stochastic stability of the resulting closed-loop systems were established, and the mode-dependent state feedback and output feedback controller were designed, respectively. The transition probabilities of time-delays in [7–14] were assumed to be completely accessible. However, in practical applications, this assumption is too ideal and hence will limit

2

Journal of Control Science and Engineering

Actuator

Controlled plant

Sensor yk

ũk


휏k

dk uk

Controller

x̂k

Observer

yk−
휏
푘

uk

Figure 1: Diagram of the NCSs.

the application of the derived results due to the difficulty in obtaining all the transition probabilities of time-delay precisely. Some results have been obtained when the transition probabilities of the time-delays (data packet dropout) are partly inaccessible. In [15], the 𝐻∞ control problem was investigated for the NCSs with random data packet dropouts. The closed-loop systems were modeled as MJLSs with four modes and partly inaccessible transition probabilities. In [16, 17], the closed-loop systems were modeled as MJLSs with partly inaccessible transition probabilities of the S-C delay, and the stabilization controller was designed though the linear matrix inequality (LMI) method. In [18], the transition probabilities of the time-delay were assumed to be partly inaccessible, and the fault-tolerant controller for the discretetime NCSs was designed. Unfortunately, in [16–18], only the S-C delay was considered and an improved controller should take both S-C delay and C-A delay into consideration. It is well known that nonlinearities usually exist in practical systems. Hence, research about nonlinear NCSs is important in both application and theory. To the best of the authors’ knowledge, up to now, involving both S-C delay and C-A delay to design the controller for nonlinear NCSs when transition probabilities of S-C delay and C-A delay are both partly inaccessible has not been investigated, which motivates our investigation. In this paper, we propose two controller design methods for a kind of nonlinear NCSs with both S-C delay and C-A delay based on observer. Compared to the previous relevant works, the main contribution of this paper is that the proposed methods can deal with the situations of both complete accessible transition probabilities and partly inaccessible transition probabilities. The rest of this paper is organized as follows. The closed-loop system model with Markov delays is obtained in Section 2. The main results and proofs are given in Section 3. Section 4 presents the simulation results, and the conclusions are provided in Section 5.

2. Problem Formulation The configuration of the NCSs considering time-delays is depicted in Figure 1 where 𝜏𝑘 and 𝑑𝑘 denote the S-C delay and C-A delay, respectively. In this paper, 𝜏𝑘 and 𝑑𝑘 are modeled as two homogeneous Markov chains which take value in the set Ω = {𝜏𝑚 , . . . , 𝜏𝑀} and Γ = {𝑑𝑚 , . . . , 𝑑𝑀}, and their transition probability

matrices are Ξ = [𝜆 𝑖𝑗 ] and Π = [𝜋𝑟𝑠 ], respectively. That is, 𝜏𝑘 and 𝑑𝑘 jump from mode 𝑖 to 𝑗 and from 𝑟 to 𝑠, respectively, with probabilities 𝜆 𝑖𝑗 and 𝜋𝑟𝑠 , which are defined by 𝜆 𝑖𝑗 = Pr(𝜏𝑘+1 = 𝑗 | 𝜏𝑘 = 𝑖), 𝜋𝑟𝑠 = Pr(𝑑𝑘+1 = 𝑠 | 𝑑𝑘 = 𝑟), where 𝜏𝑀 𝑑𝑀 𝜆 𝑖𝑗 ≥ 0, 𝜋𝑟𝑠 ≥ 0, and ∑𝑗=𝜏 𝜆 𝑖𝑗 = 1, ∑𝑠=𝑑 𝜋𝑟𝑠 = 1, for all 𝑚 𝑚 𝑖, 𝑗 ∈ Ω and 𝑟, 𝑠 ∈ Γ. In this paper, the transition probabilities of 𝜏𝑘 and 𝑑𝑘 are both considered to be partly accessible; that is, some elements in matrix Ξ and Π are unknown. For notational clarity, ∀𝑖 ∈ Ω, we denote Ω = Ω𝑖𝑘 + Ω𝑖𝑢𝑘 with Ω𝑖𝑘 = {𝑗 : 𝜆 𝑖𝑗 is known}, Ω𝑖𝑢𝑘 = {𝑗 : 𝜆 𝑖𝑗 is unknown}. Moreover, if Ω𝑖𝑘 ≠ ⌀, it is further described as Ω𝑖𝑘 = {Ω𝑘1𝑖 , Ω𝑘2𝑖 , . . . , Ω𝑘𝜇𝑖 } (1 ≤ 𝜇 ≤ 𝑒),

and Ω𝑖𝑢𝑘 is described as Ω𝑖𝑢𝑘 = {Ω𝑘𝑖 , Ω𝑘𝑖 , . . . , Ω𝑘𝑖 }, where 𝑒 1

2

𝑒−𝜇

is the number of elements in the set Ω. 𝑟 Similarly, ∀𝑟 ∈ Γ, we denote Γ = Γ𝑘𝑟 + Γ𝑢𝑘 with Γ𝑘𝑟 = {𝑠 : 𝑟 𝜋𝑟𝑠 is known}, Γ𝑢𝑘 = {𝑠 : 𝜋𝑟𝑠 is unknown}. If Γ𝑘𝑟 ≠ ⌀, it is further described as Γ𝑘𝑟 = {Γ𝑘1𝑟 , Γ𝑘2𝑟 , . . . , Γ𝑘𝜃𝑟 } (1 ≤ 𝜃 ≤ 𝜙) and 𝑟 𝑟 Γ𝑢𝑘 is described as Γ𝑢𝑘 = {Γ𝑘𝑟 , Γ𝑘𝑟 , . . . , Γ𝑘𝑟 }, where 𝜙 is the 1

2

𝜙−𝜃

number of elements in the set Γ. Considering the following controlled plant after sampling, 𝑥𝑘+1 = 𝐴𝑥𝑘 + 𝑓 (𝑘, 𝑥𝑘 ) + 𝐵̃ 𝑢𝑘 , 𝑦𝑘 = 𝐶𝑥𝑘 ,

(1)

where 𝑥𝑘 ∈ 𝑅𝑛 is the state vector, 𝑢̃𝑘 ∈ 𝑅𝑚 is the control input, and 𝑦𝑘 ∈ 𝑅𝑝 is the measured output. 𝐴, 𝐵, and 𝐶 are known real constant matrices with appropriate dimensions. 𝑓(𝑘, 𝑥𝑘 ) is a nonlinear vector function which satisfies the following global Lipschitz conditions: 󵄩󵄩󵄩𝑓 (𝑘, 𝑥𝑘 )󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑔𝑥𝑘 󵄩󵄩󵄩 , 󵄩 󵄩 󵄩 󵄩 (2) 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩𝑓 (𝑘, 𝑥𝑘 ) − 𝑓 (𝑘, 𝑧𝑘 )󵄩󵄩 ≤ 󵄩󵄩𝑔 (𝑥𝑘 − 𝑧𝑘 )󵄩󵄩󵄩 , where 𝑔 is a known real scalar. The dynamic observer-based control scheme is given by {𝑥̂𝑘+1 = 𝐴𝑥̂𝑘 + 𝑓 (𝑘, 𝑥̂𝑘 ) + 𝐵𝑢𝑘 + 𝐿 (𝑦𝑘−𝜏𝑘 − 𝑦̂𝑘−𝜏𝑘 ) , Observer: { 𝑦̂ = 𝐶𝑥̂𝑘 , { 𝑘

{𝑢𝑘 = 𝐾𝑥̂𝑘 , Controller: { 𝑢̃ = 𝑢𝑘−𝑑𝑘 , { 𝑘

(3)

(4)

where 𝑥̂𝑘 ∈ 𝑅𝑛 , 𝑢𝑘 ∈ 𝑅𝑚 , and 𝑦̂𝑘 ∈ 𝑅𝑝 are the state vector, control input, and output vector of the observer, respectively. 𝐾 ∈ 𝑅𝑚×𝑛 is the controller gain and 𝐿 ∈ 𝑅𝑛×𝑝 is the observer gain. Remark 1. It should be pointed out that the control input 𝑢̃𝑘 of the controlled plant (1) is different from the control input 𝑢𝑘 of the observer (3) due to the existence of the C-A delay 𝑑𝑘 , while, in the most of the observer-based controller design problem, 𝑢̃𝑘 and 𝑢𝑘 were assumed to be identical. Define the state estimate errors as 𝑒𝑘 = 𝑥𝑘 − 𝑥̂𝑘 .

(5)

Journal of Control Science and Engineering

3

Substituting (3) and (4) into (1) and (5), the closed-loop system can be obtained as follows: 𝑥𝑘+1 = 𝐴𝑥𝑘 + 𝑓 (𝑘, 𝑥𝑘 ) + 𝐵𝐾 (𝑥𝑘−𝜏𝑘 − 𝑒𝑘−𝜏𝑘 ) , 𝑒𝑘+1 = (𝐴 + 𝐵𝐾) 𝑒𝑘 + 𝐹𝑘 − 𝐵𝐾𝑥𝑘 + 𝐵𝐾 (𝑥𝑘−𝑑𝑘 − 𝑒𝑘−𝑑𝑘 ) (6) − 𝐿𝐶𝑒𝑘−𝜏𝑘 ,

Lemma 4 (see [20], S-procedure). Let 𝑇𝑙 ∈ 𝑅𝑛×𝑛 (𝑙 = 0, 1, . . . , 𝑧) be symmetric matrices. The conditions on 𝜉𝑇 𝑇0 𝜉 < 0, ∀𝜉 ≠ 0 s.t. 𝜉𝑇 𝑇𝑠 𝜉 ≤ 0, (𝑙 = 0, 1, . . . , 𝑧) hold if there exist scalars 𝜀𝑙 ≥ 0 (𝑙 = 0, 1, . . . , 𝑧) such that 𝑇0 − ∑𝑧𝑙=1 𝜀𝑙 𝑇𝑙 < 0. Lemma 5 (see [21]). For given scalars 𝜆 𝑖 ≥ 0 (𝑖 = 1, . . . , 𝑁1 ), 𝜋𝑟 ≥ 0 (𝑟 = 1, . . . , 𝑁2 ) and matrix 𝑃𝑖,𝑟 ≥ 0, the following inequality always holds:

where 𝐹𝑘 = 𝑓(𝑘, 𝑥𝑘 ) − 𝑓(𝑘, 𝑥̂𝑘 ). 𝜂𝑘𝑇

[𝑥𝑘𝑇

= the closed-loop system (6) By defining can be written in a compact form as follows: 𝜂𝑘+1 = (𝐴 − 𝐵1 𝐾𝐼1 ) 𝜂𝑘 + 𝐵2 𝐾𝐼1 𝜂𝑘−𝑑𝑘 − 𝐼2 𝐿𝐶𝜂𝑘−𝜏𝑘 + 𝐹𝑘 ,

(7)

where 𝐴=[

𝐴 0 0 𝐴

],

0

𝐵1 = [ ] , 𝐵 𝐵 𝐵2 = [ ] , 𝐵 𝐼1 = [𝐼 −𝐼] ,

(8)

0 𝐼2 = [ ] , 𝐼

𝑓 (𝑘, 𝑥𝑘 ) ]. 𝐹𝑘

𝑁1 𝑁2

𝑖=1 𝑟=1

𝑖=1 𝑟=1

𝑖=1 𝑟=1

(10)

In the following, a sufficient condition such that the closed-loop system (7) is stochastically stable will be derived. Theorem 6. Taking the controller gain matrix 𝐾 and observer gain matrix 𝐿, if there exist positive-definite matrices 𝑃𝑖,𝑟 > 0, 𝑃𝑗,𝑠 > 0, 𝑄1 > 0, 𝑄2 > 0, 𝑄3 > 0, 𝑄4 > 0, 𝑄5 > 0, 𝑄6 > 0, 𝑍1 > 0, 𝑍2 > 0, 𝑍3 > 0, 𝑍4 > 0 and scalar 𝜀1 ≥ 0, 𝜀2 ≥ 0 such that the following inequality Φ11 [ [Φ21 [ [ [Φ31 [ [ [Φ41 Φ≜[ [𝑍 [ 2 [ [ 0 [ [ [ 𝑍4 [ [ 0

∗

∗

∗

∗

∗

∗

Φ22

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

Φ32 Φ33

Φ42 Φ43 Φ44 𝑍1

0

0

Φ55

∗

∗

𝑍1

0

0

0

Φ66

∗

0

𝑍3

0

0

0

Φ77

0

𝑍3

0

0

0

0

∗

] ∗ ] ] ] ∗ ] ] ] ∗ ] ] < 0, (11) ∗ ] ] ] ∗ ] ] ] ∗ ] ] Φ88 ]

𝑇

2

+ 𝑄4 + 𝑄5 + 𝑄6 + (𝑑𝑀 − 𝑑𝑚 ) (𝐴 − 𝐵1 𝐾𝐼1 − 𝐼)

Definition 2 (see [13]). System (7) is stochastically stable if, for every finite 𝜂0 and 𝜏0 ∈ Ω, 𝑑0 ∈ Γ, there exists a finite matrix 𝑅 > 0 such that

2 ⋅ 𝑍1 (𝐴 − 𝐵1 𝐾𝐼1 − 𝐼) + 𝑑𝑚 (𝐴 − 𝐵1 𝐾𝐼1 − 𝐼)

𝑇

2

⋅ 𝑍2 (𝐴 − 𝐵1 𝐾𝐼1 − 𝐼) + (𝜏𝑀 − 𝜏𝑚 ) 𝑇

∞

󵄩 󵄩2 𝐸 { ∑ 󵄩󵄩󵄩𝜂𝑘 󵄩󵄩󵄩 | 𝜂0 , 𝜏0 , 𝑑0 } < 𝜂0𝑇 𝑅𝜂0 .

𝑁1 𝑁2

where Φ11 = (𝑑𝑀 − 𝑑𝑚 + 1) 𝑄1 + (𝜏𝑀 − 𝜏𝑚 + 1) 𝑄2 + 𝑄3

𝐶 = [0 𝐶] , 𝐹=[

𝑁1 𝑁2

∑ ∑𝜆 𝑖 𝜋𝑟 𝑃𝑖,𝑟 ≤ ∑ ∑𝜆 𝑖 𝜋𝑟 ∑ ∑𝑃𝑖,𝑟 .

𝑇 𝑒𝑘𝑇 ] ,

(9)

𝑘=0

In this paper, our objective is to design the dynamic observerbased control scheme (3) and (4), such that the closedloop system (7) is stochastically stable on condition that the transition probabilities of 𝜏𝑘 and 𝑑𝑘 are both partly inaccessible.

⋅ (𝐴 − 𝐵1 𝐾𝐼1 − 𝐼) 𝑍3 (𝐴 − 𝐵1 𝐾𝐼1 − 𝐼) 𝑇

+ 𝜏𝑚2 (𝐴 − 𝐵1 𝐾𝐼1 − 𝐼) 𝑍4 (𝐴 − 𝐵1 𝐾𝐼1 − 𝐼) 𝑇

+ (𝐴 − 𝐵1 𝐾𝐼1 ) 𝑃𝑗,𝑠 (𝐴 − 𝐵1 𝐾𝐼1 ) − 𝑃𝑖,𝑟 + 𝜀1 𝑔2 𝐼3𝑇 𝐼3 + 𝜀2 𝑔2 𝐼4𝑇 𝐼4 , 2

𝑇

Φ21 = (𝑑𝑀 − 𝑑𝑚 ) (𝐵2 𝐾𝐼1 ) 𝑍1 (𝐴 − 𝐵1 𝐾𝐼1 − 𝐼)

3. Main Results and Proofs In this section, we will present the main results. To proceed, we will need the following three lemmas. Lemma 3 (see [19]). For any positive-definite matrix 𝑊, scalars 𝛿, 𝛿0 satisfying 𝛿 ≥ 𝛿0 ≥ 1 and vector function 𝜐𝑙 , one has (∑𝛿𝑙=𝛿0 𝜐𝑙 )𝑇 𝑊 ∑𝛿𝑙=𝛿0 𝜐𝑙 ≤ (𝛿 − 𝛿0 + 1) ∑𝛿𝑙=𝛿0 𝜐𝑙𝑇 𝑊𝜐𝑙 .

𝑇

2

2 + 𝑑𝑚 (𝐵2 𝐾𝐼1 ) 𝑍2 (𝐴 − 𝐵1 𝐾𝐼1 − 𝐼) + (𝜏𝑀 − 𝜏𝑚 ) 𝑇

⋅ (𝐵2 𝐾𝐼1 ) 𝑍3 (𝐴 − 𝐵1 𝐾𝐼1 − 𝐼) + 𝜏𝑚2 (𝐵2 𝐾𝐼1 ) 𝑇

⋅ 𝑍4 (𝐴 − 𝐵1 𝐾𝐼1 − 𝐼) + (𝐵2 𝐾𝐼1 ) ⋅ 𝑃𝑗,𝑠 (𝐴 − 𝐵1 𝐾𝐼1 ) ,

𝑇

4

Journal of Control Science and Engineering 2

𝑇

Φ66 = −𝑄4 − 𝑍1 ,

Φ22 = (𝑑𝑀 − 𝑑𝑚 ) (𝐵2 𝐾𝐼1 ) 𝑍1 (𝐵2 𝐾𝐼1 ) 𝑇

2

2 (𝐵2 𝐾𝐼1 ) 𝑍2 (𝐵2 𝐾𝐼1 ) + (𝜏𝑀 − 𝜏𝑚 ) (𝐵2 𝐾𝐼1 ) + 𝑑𝑚

𝑇

𝑇

⋅ 𝑍3 (𝐵2 𝐾𝐼1 ) + 𝜏𝑚2 (𝐵2 𝐾𝐼1 ) 𝑍4 (𝐵2 𝐾𝐼1 )

𝐼4 = [0 𝐼] ,

𝑇

Φ31 = − (𝑑𝑀 − 𝑑𝑚 ) (𝐼2 𝐿𝐶) 𝑍1 (𝐴 − 𝐵1 𝐾𝐼1 − 𝐼) 𝑇

𝜏𝑀

2

𝑇

𝑗=𝜏𝑚 𝑠=𝑑𝑚

(12)

𝑇

⋅ (𝐼2 𝐿𝐶) 𝑍3 (𝐴 − 𝐵1 𝐾𝐼1 − 𝐼) − 𝜏𝑚2 (𝐼2 𝐿𝐶) 𝑇

⋅ 𝑍4 (𝐴 − 𝐵1 𝐾𝐼1 − 𝐼) − (𝐼2 𝐿𝐶) 𝑃𝑗,𝑠 (𝐴 − 𝐵1 𝐾𝐼1 ) , Φ32 = − (𝑑𝑀 − 𝑑𝑚 ) (𝐼2 𝐿𝐶) 𝑍1 (𝐵2 𝐾𝐼1 )

𝑇

𝑇

⋅ 𝑍3 (𝐵2 𝐾𝐼1 ) − 𝜏𝑚2 (𝐼2 𝐿𝐶) 𝑍4 (𝐵2 𝐾𝐼1 ) − (𝐼2 𝐿𝐶) ⋅ 𝑃𝑗,𝑠 (𝐵2 𝐾𝐼) , 𝑇

2 Φ33 = (𝑑𝑀 − 𝑑𝑚 ) (𝐼2 𝐿𝐶) 𝑍1 (𝐼2 𝐿𝐶) + 𝑑𝑚 (𝐼2 𝐿𝐶)

𝑇

𝑘−1

𝑘−1

𝑙=𝑘−𝑑𝑘

𝑙=𝑘−𝜏𝑘

𝑘−1

𝑘−1

𝑙=𝑘−𝑑𝑚

𝑙=𝑘−𝑑𝑀

𝑉3 (𝜂𝑘 , 𝑑𝑘 , 𝜏𝑘 ) = ∑ 𝜂𝑙𝑇 𝑄3 𝜂𝑙 + ∑ 𝜂𝑙𝑇 𝑄4 𝜂𝑙

⋅ 𝑍2 (𝐼2 𝐿𝐶) + (𝜏𝑀 − 𝜏𝑚 ) (𝐼2 𝐿𝐶) 𝑍3 (𝐼2 𝐿𝐶) 𝑇

𝑇

𝑉4 (𝜂𝑘 , 𝑑𝑘 , 𝜏𝑘 ) = 2

2

(𝐴 − 𝐵1 𝐾𝐼1 − 𝐼)

2

2 𝑍2 (𝐵2 𝐾𝐼1 ) Φ42 = (𝑑𝑀 − 𝑑𝑚 ) 𝑍1 (𝐵2 𝐾𝐼1 ) + 𝑑𝑚

𝑉5 (𝜂𝑘 , 𝑑𝑘 , 𝜏𝑘 ) =

∑ 𝜂𝑙𝑇 𝑄1 𝜂𝑙

−𝜏𝑚

𝑘−1

∑ 𝜂𝑙𝑇 𝑄2 𝜂𝑙 ,

∑

−𝑑𝑚

∑

(13)

+ 𝑃𝑗,𝑠 (𝐵2 𝐾𝐼1 ) , 2

(𝐼2 𝐿𝐶)

∑ (𝑑𝑀 − 𝑑𝑚 ) 𝜁𝑙𝑇 𝑍1 𝜁𝑙

−1

𝑘−1

+ ∑ ∑ 𝑑𝑚 𝜁𝑙𝑇 𝑍2 𝜁𝑙 𝑛=−𝑑𝑚 𝑙=𝑘+𝑛

+

2 Φ43 = − (𝑑𝑀 − 𝑑𝑚 ) 𝑍1 (𝐼2 𝐿𝐶) − 𝑑𝑚 𝑍2 (𝐼2 𝐿𝐶)

𝑘−1

𝑛=−𝑑𝑀 +1 𝑙=𝑘+𝑛

2

− (𝜏𝑀 − 𝜏𝑚 ) 𝑍3 (𝐼2 𝐿𝐶) −

𝑘−1

𝑛=−𝑑𝑀 +1 𝑙=𝑘+𝑛

+ (𝜏𝑀 − 𝜏𝑚 ) 𝑍3 (𝐵2 𝐾𝐼1 ) + 𝜏𝑚2 𝑍4 (𝐵2 𝐾𝐼1 )

𝜏𝑚2 𝑍4

𝑙=𝑘−𝜏𝑀

𝑛=−𝜏𝑀 +1 𝑙=𝑘+𝑛

+ 𝑃𝑗,𝑠 (𝐴 − 𝐵1 𝐾𝐼1 ) ,

2

𝑙=𝑘−𝜏𝑚

∑

+

2 𝑍2 (𝐴 − 𝐵1 𝐾𝐼1 − 𝐼) + (𝜏𝑀 − 𝜏𝑚 ) + 𝑑𝑚

⋅ 𝑍3 (𝐴 − 𝐵1 𝐾𝐼1 − 𝐼) +

𝑘−1

−𝑑𝑚

Φ41 = (𝑑𝑀 − 𝑑𝑚 ) 𝑍1 (𝐴 − 𝐵1 𝐾𝐼1 − 𝐼)

𝜏𝑚2 𝑍4

𝑘−1

+ ∑ 𝜂𝑙𝑇 𝑄5 𝜂𝑙 + ∑ 𝜂𝑙𝑇 𝑄6 𝜂𝑙 ,

+ 𝜏𝑚2 (𝐼2 𝐿𝐶) 𝑍4 (𝐼2 𝐿𝐶) + (𝐼2 𝐿𝐶) 𝑃𝑗,𝑠 (𝐼2 𝐿𝐶) − 𝑄2 − 2𝑍3 ,

as

𝑉2 (𝜂𝑘 , 𝑑𝑘 , 𝜏𝑘 ) = ∑ 𝜂𝑙𝑇 𝑄1 𝜂𝑙 + ∑ 𝜂𝑙𝑇 𝑄2 𝜂𝑙 ,

𝑇

2

candidate

𝑉1 (𝜂𝑘 , 𝜏𝑘 , 𝑑𝑘 ) = 𝜂𝑘𝑇 𝑃𝜏𝑘 ,𝑑𝑘 𝜂𝑘 ,

𝑇

2

2 (𝐼2 𝐿𝐶) 𝑍2 (𝐵2 𝐾𝐼1 ) − (𝜏𝑀 − 𝜏𝑚 ) (𝐼2 𝐿𝐶) − 𝑑𝑚

2

holds for all 𝑖, 𝑗 ∈ Ω and 𝑟, 𝑠 ∈ Γ, the closed-loop system (7) is stochastically stable. Proof. Choose the Lyapunov function 𝑉(𝜂𝑘 , 𝜏𝑘 , 𝑑𝑘 ) = ∑5𝜌=1 𝑉𝜌 (𝜂𝑘 , 𝜏𝑘 , 𝑑𝑘 ), where

𝑇

𝑇

𝑑𝑀

𝑃𝑗,𝑠 = ∑ ∑ 𝜆 𝑖𝑗 𝜋𝑟𝑠 𝑃𝑗,𝑠 ,

2 (𝐼2 𝐿𝐶) 𝑍2 (𝐴 − 𝐵1 𝐾𝐼1 − 𝐼) − (𝜏𝑀 − 𝜏𝑚 ) − 𝑑𝑚

2

Φ88 = −𝑄6 − 𝑍3 , 𝐼3 = [𝐼 0] ,

𝑇

+ (𝐵2 𝐾𝐼1 ) 𝑃𝑗,𝑠 (𝐵2 𝐾𝐼1 ) − 𝑄1 − 2𝑍1 , 2

Φ77 = −𝑄5 − 𝑍4 − 𝑍3 ,

−𝜏𝑚

∑

𝑘−1

∑ (𝜏𝑀 − 𝜏𝑚 ) 𝜁𝑙𝑇 𝑍3 𝜁𝑙

𝑛=−𝜏𝑀 +1 𝑙=𝑘+𝑛 −1

𝑘−1

+ ∑ ∑ 𝜏𝑚 𝜁𝑙𝑇 𝑍4 𝜁𝑙 , 𝑛=−𝜏𝑚 𝑙=𝑘+𝑛

𝜁𝑙 = 𝜂𝑙+1 − 𝜂𝑙 .

− 𝑃𝑗,𝑠 (𝐼2 𝐿𝐶) , 2

2

2 𝑍2 + (𝜏𝑀 − 𝜏𝑚 ) 𝑍3 Φ44 = (𝑑𝑀 − 𝑑𝑚 ) 𝑍1 + 𝑑𝑚

+ 𝜏𝑚2 𝑍4 + 𝑃𝑗,𝑠 − 𝜀1 𝐼3𝑇 𝐼3 − 𝜀2 𝐼4𝑇𝐼4 , Φ55 = −𝑄3 − 𝑍2 − 𝑍1 ,

In the following, we denote 𝑃𝑑𝑘 ,𝜏𝑘 as 𝑃𝑟,𝑖 when 𝑑𝑘 = 𝑟 and 𝜏𝑘 = 𝑖. Then along the solution of system (7), we have 𝑇 𝑃𝜏𝑘+1 ,𝑑𝑘+1 𝜂𝑘+1 | 𝜏𝑘 = 𝑖, 𝑑𝑘 = 𝑟} 𝐸 {Δ𝑉1 } = 𝐸 {𝜂𝑘+1

− 𝜂𝑘𝑇 𝑃𝑖,𝑟 𝜂𝑘 = ((𝐴 − 𝐵1 𝐾𝐼1 ) 𝜂𝑘 + 𝐵2 𝐾𝐼1 𝜂𝑘−𝑑𝑘

Journal of Control Science and Engineering 𝜏𝑀

𝑇

5 2

𝐸 {Δ𝑉5 } = (𝑑𝑀 − 𝑑𝑚 ) 𝜁𝑘𝑇 𝑍1 𝜁𝑘

𝑑𝑀

− 𝐼2 𝐿𝐶𝜂𝑘−𝜏𝑘 + 𝐹𝑘 ) ∑ ∑ 𝜆 𝑖𝑗 𝜋𝑟𝑠 𝑃𝑗,𝑠

𝑘−𝑑𝑚 −1

𝑗=𝜏𝑚 𝑠=𝑑𝑚

− ∑ (𝑑𝑀 − 𝑑𝑚 ) 𝜁𝑙𝑇 𝑍1 𝜁𝑙

⋅ ((𝐴 − 𝐵1 𝐾𝐼1 ) 𝜂𝑘 + 𝐵2 𝐾𝐼1 𝜂𝑘−𝑑𝑘 − 𝐼2 𝐿𝐶𝜂𝑘−𝜏𝑘 + 𝐹𝑘 )

𝑙=𝑘−𝑑𝑀

𝑘−1

− 𝜂𝑘𝑇 𝑃𝑖,𝑟 𝜂𝑘 , 𝐸 {Δ𝑉2 } =

𝜂𝑘𝑇 𝑄1 𝜂𝑘

𝑘−1

−

∑

𝑙=𝑘+1−𝑑𝑘

−

𝑇 𝜂𝑘−𝑟 𝑄1 𝜂𝑘−𝑟

𝑘−1

+

∑

𝑙=𝑘+1−𝑑𝑘+1

𝑙=𝑘−𝑑𝑚

𝜂𝑙𝑇 𝑄1 𝜂𝑙

2

+ (𝜏𝑀 − 𝜏𝑚 ) 𝜁𝑘𝑇 𝑍3 𝜁𝑘 𝑘−𝜏𝑚 −1

𝜂𝑙𝑇 𝑄1 𝜂𝑙

𝑘−1

+

2 𝑇 + 𝑑𝑚 𝜁𝑘 𝑍2 𝜁𝑘 − 𝑑𝑚 ∑ 𝜁𝑙𝑇 𝑍2 𝜁𝑙

∑

𝑙=𝑘+1−𝜏𝑘+1

+

𝜂𝑙𝑇 𝑄2 𝜂𝑙

𝜂𝑘𝑇 𝑄2 𝜂𝑘

−

𝑘−1

−

∑

𝑙=𝑘+1−𝜏𝑘

− ∑ (𝜏𝑀 − 𝜏𝑚 ) 𝜁𝑙𝑇 𝑍3 𝜁𝑙 + 𝜏𝑚2 𝜁𝑘𝑇 𝑍4 𝜁𝑘

𝑇 𝜂𝑘−𝑖 𝑄2 𝜂𝑘−𝑖

𝑙=𝑘−𝜏𝑀

𝑘−1

− 𝜏𝑚 ∑ 𝜁𝑙𝑇 𝑍4 𝜁𝑙 .

𝜂𝑙𝑇 𝑄2 𝜂𝑙 .

𝑙=𝑘−𝜏𝑚

(14) It is noticed that 𝑘−1

∑

𝑙=𝑘+1−𝑑𝑘+1

𝜂𝑙𝑇 𝑄1 𝜂𝑙 =

≤ 𝑘−1

∑

𝑙=𝑘+1−𝜏𝑘+1

𝜂𝑙𝑇 𝑄2 𝜂𝑙 =

≤

𝑘−1

∑

𝑙=𝑘+1−𝑑𝑚

𝜂𝑙𝑇 𝑄1 𝜂𝑙 +

𝑘−𝑑𝑚

∑

𝑙=𝑘+1−𝑑𝑘+1

𝑘−1

𝑘−𝑑𝑚

𝑙=𝑘+1−𝑑𝑘

𝑙=𝑘+1−𝑑𝑀

∑ 𝜂𝑙𝑇 𝑄1 𝜂𝑙 +

∑

𝑘−𝜏𝑚

𝑙=𝑘+1−𝜏𝑚

𝑙=𝑘+1−𝜏𝑘+1

∑ 𝜂𝑙𝑇 𝑄2 𝜂𝑙 +

∑

𝑘−1

𝑘−𝜏𝑚

𝑙=𝑘+1−𝜏𝑘

𝑙=𝑘+1−𝜏𝑀

∑ 𝜂𝑙𝑇 𝑄2 𝜂𝑙 +

∑

𝜂𝑙𝑇 𝑄1 𝜂𝑙

𝑙=𝑘−𝑑𝑚

𝑙=𝑘−𝑑𝑀

𝑘−1

𝑘−𝑑𝑚 −1

𝑙=𝑘−𝑑𝑚

𝑙=𝑘−𝑟

𝑘−𝑟−1

− ∑ (𝑑𝑀 − 𝑑𝑚 ) 𝜁𝑙𝑇 𝑍1 𝜁𝑙 𝑙=𝑘−𝑑𝑀

𝑘−1

𝑘−𝑑𝑚 −1

𝑙=𝑘−𝑑𝑚

𝑙=𝑘−𝑟

(17)

𝜂𝑙𝑇 𝑄2 𝜂𝑙 .

𝑘−𝑟−1

− ∑ (𝑑𝑀 − 𝑟) 𝜁𝑙𝑇 𝑍1 𝜁𝑙 . 𝑙=𝑘−𝑑𝑀

𝑇 𝑄1 𝜂𝑘−𝑟 + 𝐸 {Δ𝑉2 } ≤ 𝜂𝑘𝑇 𝑄1 𝜂𝑘 − 𝜂𝑘−𝑟

𝑘−𝑑𝑚

∑

𝑙=𝑘+1−𝑑𝑀

𝜂𝑙𝑇 𝑄1 𝜂𝑙

𝑇 + 𝜂𝑘𝑇 𝑄2 𝜂𝑘 − 𝜂𝑘−𝑖 𝑄2 𝜂𝑘−𝑖 𝑘−𝜏𝑚

∑

𝑙=𝑘+1−𝜏𝑀

𝜂𝑙𝑇 𝑄2 𝜂𝑙 ,

𝑘−1

𝑘−𝑑𝑚 −1

𝑙=𝑘−𝑑𝑚

𝑙=𝑘−𝑑𝑀

𝑇

≤ − [𝜂𝑘 − 𝜂𝑘−𝑑𝑚 ] 𝑍2 [𝜂𝑘 − 𝜂𝑘−𝑑𝑚 ]

(18)

𝑇

− [𝜂𝑘−𝑑𝑚 − 𝜂𝑘−𝑟 ] 𝑍1 [𝜂𝑘−𝑑𝑚 − 𝜂𝑘−𝑟 ] 𝑇

− [𝜂𝑘−𝑟 − 𝜂𝑘−𝑑𝑀 ] 𝑍1 [𝜂𝑘−𝑟 − 𝜂𝑘−𝑑𝑀 ] .

𝑇 − 𝜂𝑘−𝑑 𝑄4 𝜂𝑘−𝑑𝑀 + 𝜂𝑘𝑇 𝑄5 𝜂𝑘 𝑀

Similarly, we have

𝑇 − 𝜂𝑘−𝜏 𝑄5 𝜂𝑘−𝜏𝑚 + 𝜂𝑘𝑇 𝑄6 𝜂𝑘 𝑚

𝑘−1

𝑘−𝜏𝑚 −1

− 𝜏𝑚 ∑ 𝜁𝑙𝑇 𝑍4 𝜁𝑙 − ∑ (𝜏𝑀 − 𝜏𝑚 ) 𝜁𝑙𝑇 𝑍3 𝜁𝑙

𝑇 𝜂𝑘−𝜏 𝑄6 𝜂𝑘−𝜏𝑀 , 𝑀

𝐸 {Δ𝑉4 } = (𝑑𝑀 − 𝑑𝑚 ) 𝜂𝑘𝑇 𝑄1 𝜂𝑘 −

We can obtain the following inequality by Lemma 3: − 𝑑𝑚 ∑ 𝜁𝑙𝑇 𝑍2 𝜁𝑙 − ∑ (𝑑𝑀 − 𝑑𝑚 ) 𝜁𝑙𝑇 𝑍1 𝜁𝑙

𝑇 𝐸 {Δ𝑉3 } = 𝜂𝑘𝑇 𝑄3 𝜂𝑘 − 𝜂𝑘−𝑑 𝑄3 𝜂𝑘−𝑑𝑚 + 𝜂𝑘𝑇 𝑄4 𝜂𝑘 𝑚

−

𝑘−𝑑𝑚 −1

≤ −𝑑𝑚 ∑ 𝜁𝑙𝑇 𝑍2 𝜁𝑙 − ∑ (𝑟 − 𝑑𝑚 ) 𝜁𝑙𝑇 𝑍1 𝜁𝑙

Therefore, we obtain

+

𝑘−1

= −𝑑𝑚 ∑ 𝜁𝑙𝑇 𝑍2 𝜁𝑙 − ∑ (𝑑𝑀 − 𝑑𝑚 ) 𝜁𝑙𝑇 𝑍1 𝜁𝑙 (15)

𝜂𝑙𝑇 𝑄2 𝜂𝑙

Note that − 𝑑𝑚 ∑ 𝜁𝑙𝑇 𝑍2 𝜁𝑙 − ∑ (𝑑𝑀 − 𝑑𝑚 ) 𝜁𝑙𝑇 𝑍1 𝜁𝑙

𝜂𝑙𝑇 𝑄1 𝜂𝑙 ,

𝑘−1

(16)

𝑙=𝑘−𝜏𝑚

𝑘−𝑑𝑚

∑

𝑙=𝑘+1−𝑑𝑀

+ (𝜏𝑀 − 𝜏𝑚 ) 𝜂𝑘𝑇 𝑄2 𝜂𝑘 −

𝜂𝑙𝑇 𝑄1 𝜂𝑙

𝑘−𝜏𝑚

∑

𝑙=𝑘+1−𝜏𝑀

𝑙=𝑘−𝜏𝑀

𝑇

≤ − [𝜂𝑘 − 𝜂𝑘−𝜏𝑚 ] 𝑍4 [𝜂𝑘 − 𝜂𝑘−𝜏𝑚 ] 𝑇

− [𝜂𝑘−𝜏𝑚 − 𝜂𝑘−𝑖 ] 𝑍3 [𝜂𝑘−𝜏𝑚 − 𝜂𝑘−𝑖 ] 𝜂𝑙𝑇 𝑄2 𝜂𝑙 ,

𝑇

− [𝜂𝑘−𝑖 − 𝜂𝑘−𝜏𝑀 ] 𝑍3 [𝜂𝑘−𝑖 − 𝜂𝑘−𝜏𝑀 ] .

(19)

6

Journal of Control Science and Engineering ≜ 𝜉𝑘𝑇 Λ 1 𝜉𝑘 ≤ 0,

From (14)–(19), we have 𝐸 {Δ𝑉 (𝜂𝑘 , 𝑑𝑘 , 𝜏𝑘 ) | 𝜏𝑘 = 𝑖, 𝑑𝑘 = 𝑟} ≤ 𝜉𝑘𝑇 Φ𝜉𝑘 ,

(20)

𝑇

𝐹𝑘𝑇 𝐹𝑘 − 𝑔2 𝑒𝑘𝑇 𝑒𝑘 = 𝐹𝑘 𝐼4𝑇 𝐼4 𝐹𝑘 − 𝑔2 𝜂𝑘𝑇 𝐼4𝑇 𝐼4 𝜂𝑘 −𝑔2 𝐼4𝑇 𝐼4 ∗ ∗

where Φ11

[ [Φ21 [ [ [Φ31 [ [ [Φ41 Φ=[ [ [ 𝑍2 [ [ 0 [ [ [𝑍 [ 4

∗

∗

∗

∗

∗

∗

Φ22

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

Φ32 Φ33

Φ42 Φ43 Φ44 𝑍1

0

0

Φ55

∗

∗

𝑍1

0

0

0

Φ66

∗

0

𝑍3

0

0

0

Φ77

0

𝑍3

0

0

0

0

[ 0

[ [ [ [ [ [ [ 𝑇[ = 𝜉𝑘 [ [ [ [ [ [ [ [ [

∗

] ∗ ] ] ] ∗ ] ] ] ∗ ] ], ] ∗ ] ] ∗ ] ] ] ∗ ] ] Φ88 ]

[

0 ∗

0

0 0

0

0 0

0

0 0

0

0 0

0

0 0

0

0 0

𝑇

𝑇

2 ⋅ 𝑍1 (𝐴 − 𝐵1 𝐾𝐼1 − 𝐼) + 𝑑𝑚 (𝐴 − 𝐵1 𝐾𝐼1 − 𝐼)

(21)

By the well-known S-procedure, that is, Lemma 4, we can get 𝐸{Δ𝑉(𝜂𝑘 , 𝑑𝑘 , 𝜏𝑘 )} ≤ 𝜉𝑘𝑇 Φ𝜉𝑘 < 0 with constraints (23) holding if there exist nonnegative real scalars 𝜀1 ≥ 0, 𝜀2 ≥ 0 such that Φ − 𝜀1 Λ 1 − 𝜀2 Λ 2 = Φ < 0. 𝐸 {Δ𝑉 (𝜂𝑘 , 𝑑𝑘 , 𝜏𝑘 ) | 𝜏𝑘 = 𝑖, 𝑑𝑘 = 𝑟} ≤ −𝜆 min (−Φ)

𝑇

𝑇 𝑇 ⋅ 𝜉𝑘𝑇 𝜉𝑘 = −𝜆 min (−Φ) (𝜂𝑘𝑇 𝜂𝑘 + 𝜂𝑘−𝑟 𝜂𝑘−𝑟 + 𝜂𝑘−𝑖 𝜂𝑘−𝑖

⋅ (𝐴 − 𝐵1 𝐾𝐼1 − 𝐼) 𝑍3 (𝐴 − 𝐵1 𝐾𝐼1 − 𝐼)

𝑇

𝑇 𝑇 𝑇 + 𝐹𝑘 𝐹𝑘 + 𝜂𝑘−𝑑 𝜂 + 𝜂𝑘−𝑑 𝜂 + 𝜂𝑘−𝜏 𝜂 𝑚 𝑘−𝑑𝑚 𝑀 𝑘−𝑑𝑀 𝑚 𝑘−𝜏𝑚

𝑇

+ 𝜏𝑚2 (𝐴 − 𝐵1 𝐾𝐼1 − 𝐼) 𝑍4 (𝐴 − 𝐵1 𝐾𝐼1 − 𝐼)

2

where 𝛼 = inf{−𝜆 min (−Φ)} > 0. From (25), we can see that, for any 𝑇 ≥ 1,

2

2 Φ44 = (𝑑𝑀 − 𝑑𝑚 ) 𝑍1 + 𝑑𝑚 𝑍2 + (𝜏𝑀 − 𝜏𝑚 ) 𝑍3

∞ 1 󵄩 󵄩2 𝐸 { ∑ 󵄩󵄩󵄩𝜂𝑘 󵄩󵄩󵄩 } ≤ 𝐸 {𝑉 (𝜂0 , 𝑑0 , 𝜏0 )} 𝛼 𝑘=0

+ 𝜏𝑚2 𝑍4 + 𝑃𝑗,𝑠 , 𝑇 𝑇 𝑇 𝑇 𝑇 𝑇 𝜂𝑘−𝑖 𝐹𝑘 𝜂𝑘−𝑑 𝜂𝑘−𝑑 𝜂𝑘−𝜏 𝜂𝑘−𝜏 ]. 𝜉𝑘𝑇 = [𝜂𝑘𝑇 𝜂𝑘−𝑟 𝑚 𝑀 𝑚 𝑀

−

It follows from (2) that 𝑇 𝑓 ≤ 𝑔2 𝑥𝑘𝑇 𝑥𝑘 , 𝑓𝑘,𝑥 𝑘 𝑘,𝑥𝑘

𝐹𝑘𝑇 𝐹𝑘 ≤ 𝑔2 𝑒𝑘𝑇 𝑒𝑘 , which imply that −

=

−𝑔2 𝐼3𝑇 𝐼3 ∗ ∗

[ [ [ [ [ [ [ [ = 𝜉𝑘𝑇 [ [ [ [ [ [ [ [ [ [

0

0 ∗

0

0 0

0

0 0

0

0 0

0

0 0

0

0 0

0

0 0

∗

−

𝑔2 𝜂𝑘𝑇 𝐼3𝑇 𝐼3 𝜂𝑘

∗ ∗ ∗ ∗

] ∗ ∗ ∗ ∗] ] ] ∗ ∗ ∗ ∗ ∗] ] ] 𝐼3𝑇 𝐼3 ∗ ∗ ∗ ∗ ] ] 𝜉𝑘 ] 0 0 ∗ ∗ ∗] ] 0 0 0 ∗ ∗] ] ] 0 0 0 0 ∗] ] 0 0 0 0 0] ∗

(25)

󵄩 󵄩2 𝑇 + 𝜂𝑘−𝜏 𝜂 ) ≤ −𝛼 󵄩󵄩󵄩𝜂𝑘 󵄩󵄩󵄩 , 𝑀 𝑘−𝜏𝑀

𝑇

+ (𝐴 − 𝐵1 𝐾𝐼1 ) 𝑃𝑗,𝑠 (𝐴 − 𝐵1 𝐾𝐼1 ) − 𝑃𝑖,𝑟 ,

𝑇 𝐹𝑘 𝐼3𝑇 𝐼3 𝐹𝑘

(24)

From (24), we have

2

⋅ 𝑍2 (𝐴 − 𝐵1 𝐾𝐼1 − 𝐼) + (𝜏𝑀 − 𝜏𝑚 )

𝑔2 𝑥𝑘𝑇 𝑥𝑘

] ∗ ∗ ∗ ∗] ] ] ∗ ∗ ∗ ∗ ∗] ] ] 𝐼4𝑇 𝐼4 ∗ ∗ ∗ ∗ ] ] 𝜉𝑘 ] 0 0 ∗ ∗ ∗] ] 0 0 0 ∗ ∗] ] ] 0 0 0 0 ∗] ] 0 0 0 0 0] ∗

(23)

+ 𝑄4 + 𝑄5 + 𝑄6 + (𝑑𝑀 − 𝑑𝑚 ) (𝐴 − 𝐵1 𝐾𝐼1 − 𝐼)

𝑇 𝑓 𝑓𝑘,𝑥 𝑘 𝑘,𝑥𝑘

∗ ∗ ∗ ∗

≜ 𝜉𝑘𝑇 Λ 2 𝜉𝑘 ≤ 0.

Φ11 = (𝑑𝑀 − 𝑑𝑚 + 1) 𝑄1 + (𝜏𝑀 − 𝜏𝑚 + 1) 𝑄2 + 𝑄3 2

0

∗

1 𝐸 {𝑉 (𝜂𝑇+1 , 𝑑𝑇+1 , 𝜏𝑇+1 )} 𝛼

(26)

1 𝐸 {𝑉 (𝜂0 , 𝑑0 , 𝜏0 )} . 𝛼 According to Definition 2, the closed-loop system (7) is stochastically stable, which completes the proof. ≤

(22)

Theorem 7. Taking the controller gain matrix 𝐾 and observer gain matrix 𝐿, the closed-loop system (7) is stochastically stable if there exist positive-definite matrices 𝑃𝑖,𝑟 > 0, 𝑃𝑗,𝑠 > 0, 𝐹𝑗,𝑠 > 0, 𝑄1 > 0, 𝑄2 > 0, 𝑄3 > 0, 𝑄4 > 0, 𝑄5 > 0, 𝑄6 > 0, 𝑍1 > 0, 𝑍2 > 0, 𝑍3 > 0, 𝑍4 > 0, 𝑌1 > 0, 𝑌2 > 0, 𝑌3 > 0, 𝑌4 > 0 and scalar 𝜀1 ≥ 0, 𝜀2 ≥ 0 such that ∗ Λ 11 ∗ [ [Λ 21 Λ 22 ∗ [ [Λ [ 31 0 Λ 33 [Λ 41

0

0

∗

] ∗ ] ] < 0, ∗ ] ]

(27)

Λ 44 ] 𝑃𝑗,𝑠 𝐹𝑗,𝑠 = 𝐼, 𝑌𝑙 𝑍𝑙 = 𝐼,

𝑙 ∈ {1, 2, 3, 4} ,

(28)

Journal of Control Science and Engineering

7

where

Λ 11

Λ 21

Λ 22

Λ 31

Λ 33

Λ11 ∗ ∗ ∗ [ ] [ 0 −𝑄1 − 2𝑍1 ] ∗ ∗ [ ], =[ ] ∗ 0 −𝑄2 − 2𝑍3 [ 0 ] 𝑇 𝑇 0 0 −𝜏1 𝐼3 𝐼3 − 𝜏2 𝐼4 𝐼4 ] [ 0 𝑍2 𝑍1 0 0 [ 0 𝑍 0 0] ] [ 1 =[ ], [𝑍4 0 𝑍3 0] [ 0 0 𝑍3 0] −𝑄3 − 𝑍2 − 𝑍1 ∗ ∗ ∗ [ ] − 𝑍 ∗ ∗ 0 −𝑄 [ ] 4 1 =[ ], [ ] ∗ 0 0 −𝑄5 − 𝑍4 − 𝑍3 0 0 0 −𝑄6 − 𝑍3 ] [ (𝑑𝑀 − 𝑑𝑚 ) Υ (𝑑𝑀 − 𝑑𝑚 ) 𝐵2 𝐾𝐼1 − (𝑑𝑀 − 𝑑𝑚 ) 𝐼2 𝐿𝐶 (𝑑𝑀 − 𝑑𝑚 ) 𝐼 ] [ [ 𝑑𝑚 𝐵2 𝐾𝐼1 −𝑑𝑚 𝐼2 𝐿𝐶 −𝑑𝑚 𝐼 ] 𝑑𝑚 Υ ], [ =[ ] [ (𝜏𝑀 − 𝜏𝑚 ) Υ (𝜏𝑀 − 𝜏𝑚 ) 𝐵2 𝐾𝐼1 − (𝜏𝑀 − 𝜏𝑚 ) 𝐼2 𝐿𝐶 (𝜏𝑀 − 𝜏𝑚 ) 𝐼 ] 𝜏𝑚 Υ 𝜏𝑚 𝐵2 𝐾𝐼1 −𝜏𝑚 𝐼2 𝐿𝐶 −𝜏𝑚 𝐼 ] [ ∗ ∗ −𝑌1 ∗ [ 0 −𝑌 ∗ ∗ ] [ ] 2 =[ ], [ 0 0 −𝑌3 ∗ ] 0 0 −𝑌4 ] [ 0

Λ 44 Λ11

𝑇 𝑇]

[ [√𝜆 𝑖Ω𝑘𝑖 𝜋𝑟Γ𝑘𝑟 (𝐴 − 𝐵1 𝐾𝐼1 ) ⋅ ⋅ ⋅ √(1 − ∑ 𝜆 𝑖𝑙 ) (1 − ∑ 𝜋𝑟𝑙 ) (𝐴 − 𝐵1 𝐾𝐼1 ) ] ] [ 1 1 𝑙∈Γ𝑘𝑟 𝑙∈Ω𝑖𝑘 ] [ ] [ ] [ ] [ [ 𝑇 𝑇 ] √(1 − ∑ 𝜆 𝑖𝑙 ) (1 − ∑ 𝜋𝑟𝑙 ) (𝐵2 𝐾𝐼1 ) ] ⋅⋅⋅ [ √𝜆 𝑖Ω𝑘𝑖 𝜋𝑟Γ𝑘𝑟 (𝐵2 𝐾𝐼1 ) ] [ 1 1 𝑙∈Γ𝑘𝑟 𝑙∈Ω𝑖𝑘 ] [ ] , =[ ] [ ] [ 𝑇 𝑇 ] [ ⋅ ⋅ ⋅ −√(1 − ∑ 𝜆 𝑖𝑙 ) (1 − ∑ 𝜋𝑟𝑙 ) (𝐼2 𝐿𝐶) ] [ −√𝜆 𝑖Ω𝑘𝑖 𝜋𝑟Γ𝑘𝑟 (𝐼2 𝐿𝐶) ] [ 1 1 𝑙∈Γ𝑘𝑟 𝑙∈Ω𝑖𝑘 ] [ ] [ ] [ ] [ ] [ √(1 − ∑ 𝜆 𝑖𝑙 ) (1 − ∑ 𝜋𝑟𝑙 )𝐼 ⋅⋅⋅ ] [ √𝜆 𝑖Ω𝑘𝑖 𝜋𝑟Γ𝑘1𝑟 𝐼 1 𝑙∈Γ𝑘𝑟 𝑙∈Ω𝑖𝑘 ] [ −𝐹Ω 𝑖 ,Γ𝑘𝑟 ∗ ∗ 𝑘1 1 [ ] ], 0 d ∗ =[ [ ] 0 0 −𝐹Ω 𝑖 ,Γ 𝑟 𝑘𝑒−𝜇 𝑘𝜙−𝜃 [ ] = (𝑑𝑀 − 𝑑𝑚 + 1) 𝑄1 + (𝜏𝑀 − 𝜏𝑚 + 1) 𝑄2 + 𝑄3 + 𝑄4 + 𝑄5 + 𝑄6 − 𝑍2 − 𝑍4 + 𝜀1 𝑔2 𝐼3𝑇 𝐼3 + 𝜀2 𝑔2 𝐼4𝑇 𝐼4 − 𝑃𝑖,𝑟 , 𝑇

Λ 41

(29)

Υ = 𝐴 − 𝐵1 𝐾𝐼1 − 𝐼 hold for all 𝑖, 𝑗 ∈ Ω and 𝑟, 𝑠 ∈ Γ.

∑ ∑ 𝜆 𝑖𝑗 𝜋𝑟𝑠 𝑃𝑗,𝑠 ≤ (1 − ∑ 𝜋𝑟𝑙 ) ∑ 𝜆 𝑖𝑗 ∑ ∑ 𝑃𝑗,𝑠 , (31)

𝑟 𝑗∈Ω𝑖𝑘 𝑠∈Γ𝑢𝑘

Proof. Making use of Lemma 5, we have

𝑙∈Γ𝑘𝑟

𝑗∈Ω𝑖𝑘

𝑟 𝑗∈Ω𝑖𝑘 𝑠∈Γ𝑢𝑘

∑ ∑ 𝜆 𝑖𝑗 𝜋𝑟𝑠 𝑃𝑗,𝑠

∑ ∑ 𝜆 𝑖𝑗 𝜋𝑟𝑠 𝑃𝑗,𝑠

𝑟 𝑗∈Ω𝑖𝑢𝑘 𝑠∈Γ𝑢𝑘

𝑗∈Ω𝑖𝑢𝑘 𝑠∈Γ𝑘𝑟

(30) ≤ (1 − ∑ 𝜆 𝑖𝑙 ) ∑ 𝜋𝑟𝑠 ∑ ∑ 𝑃𝑗,𝑠 , 𝑙∈Ω𝑖𝑘

𝑠∈Γ𝑘𝑟

𝑗∈Ω𝑖𝑢𝑘 𝑠∈Γ𝑘𝑟

(32) ≤ (1 − ∑ 𝜋𝑟𝑙 ) (1 − ∑ 𝜆 𝑖𝑙 ) ∑ ∑ 𝑃𝑗,𝑠 . 𝑙∈Γ𝑘𝑟

𝑙∈Ω𝑖𝑘

𝑟 𝑗∈Ω𝑖𝑢𝑘 𝑠∈Γ𝑢𝑘

8

Journal of Control Science and Engineering

−1 Letting 𝑃𝑗,𝑠 = 𝐹𝑗,𝑠 , 𝑗 ∈ Ω, 𝑠 ∈ Γ, 𝑍𝑙−1 = 𝑌𝑙 , 𝑙 ∈ {1, 2, 3, 4} and by Schur complement Lemma, (27) can be obtained from (11) and (30)–(32), which complete the proof.

∗ ∑ 𝜆 𝑖𝑗 𝜋𝑟𝑠 Λ ] < 0, ∀𝑗 ∈ Ω𝑖 , [𝑠∈Γ𝑘𝑟 𝑢𝑘 −Δ Ω𝑖𝑢𝑘 ,Γ𝑘𝑟 ] [ ΘΩ𝑖𝑢𝑘 ,Γ𝑘𝑟

(34)

∗ ∑ 𝜆 𝑖𝑗 Λ [𝑗∈Ω𝑖𝑘 ] < 0, ∀𝑠 ∈ Γ𝑟 , 𝑢𝑘 [ ΘΩ𝑖𝑘 ,Γ𝑢𝑘𝑟 −Δ Ω𝑖𝑘 ,Γ𝑢𝑘𝑟 ]

(35)

In Theorem 7, in order to get a desired controller (3)(4) for the closed-loop NCS (7) with partly inaccessible transition probabilities, Lemma 5 is used to separate 𝜆 𝑖𝑗 𝜋𝑟𝑠 from 𝜆 𝑖𝑗 𝜋𝑟𝑠 𝑃𝑗,𝑠 . However, this could lead to conservativeness to some extents as shown in the Numerical Example. In the following, a less conservative result will be given in the following theorem.

Λ 11 ∗ [Λ Λ Λ = [ 21 22 [Λ 31

Λ 11

Λ 21

[

Λ 31

Λ 33

(37)

where

∗

∗ ] ], Λ 33 ]

0 𝑍3 0] ∗

∗

∗

0

−𝑄4 − 𝑍1

∗

∗

0

0

−𝑄5 − 𝑍4 − 𝑍3

∗

0

0

0

] ] ], ] ]

−𝑄6 − 𝑍3 ]

(𝑑𝑀 − 𝑑𝑚 ) Υ (𝑑𝑀 − 𝑑𝑚 ) 𝐵2 𝐾𝐼1 − (𝑑𝑀 − 𝑑𝑚 ) 𝐼2 𝐿𝐶 (𝑑𝑀 − 𝑑𝑚 ) 𝐼 [ ] [ 𝑑𝑚 𝐵2 𝐾𝐼1 −𝑑𝑚 𝐼2 𝐿𝐶 −𝑑𝑚 𝐼 ] 𝑑𝑚 Υ [ ] =[ ], [ (𝜏 − 𝜏 ) Υ (𝜏 − 𝜏 ) 𝐵 𝐾𝐼 − (𝜏 − 𝜏 ) 𝐼 𝐿𝐶 (𝜏 − 𝜏 ) 𝐼 ] 𝑀 𝑚 2 1 𝑀 𝑚 2 𝑀 𝑚 [ 𝑀 𝑚 ] Υ 𝜏 𝐵 𝐾𝐼 −𝜏 𝐼 𝐿𝐶 −𝜏 𝐼 𝜏 𝑚 𝑚 2 1 𝑚 2 𝑚 [ ] −𝑌1 ∗ ∗ [ [ 0 −𝑌2 ∗ =[ [ 0 0 −𝑌3 [ [ 0

0

0

∗

] ∗ ] ], ∗ ] ]

−𝑌4 ]

(36)

𝑌𝑙 𝑍𝑙 = 𝐼, 𝑙 ∈ {1, 2, 3, 4} ,

−𝑄3 − 𝑍2 − 𝑍1

[

]

𝐹𝑗,𝑠 𝑃𝑗,𝑠 = 𝐼,

𝑍2 𝑍1 0 0 [ ] [ 0 𝑍1 0 0] [ ], =[ ] [𝑍4 0 𝑍3 0]

[ [ =[ [ [

] < 0, 𝑟 , ∀𝑗 ∈ Ω𝑖𝑢𝑘 , ∀𝑠 ∈ Γ𝑢𝑘

Λ11 ∗ ∗ ∗ [ ] [ 0 −𝑄1 − 2𝑍1 ] ∗ ∗ [ ] =[ ], [ 0 ] 0 −𝑄 − 2𝑍 ∗ 2 3 [ ] 𝑇 𝑇 𝐼 𝐼 − 𝜏 𝐼 𝐼 0 0 0 −𝜏 1 3 3 2 4 4] [

[0

Λ 22

0

(33)

∗

ΘΩ𝑖𝑢𝑘 ,Γ𝑢𝑘𝑟 −Δ Ω𝑖𝑢𝑘 ,Γ𝑢𝑘𝑟

Theorem 8. There exists dynamic observer-based control scheme (3)-(4) such that the resulting closed-loop system (7) is stochastically stable if, for all 𝑖, 𝑗 ∈ Ω and 𝑟, 𝑠 ∈ Γ, there exist matrices 𝑃𝑖,𝑟 > 0, 𝑃𝑗,𝑠 > 0, 𝐹𝑗,𝑠 > 0, 𝑄1 > 0, 𝑄2 > 0, 𝑄3 > 0, 𝑄4 > 0, 𝑄5 > 0, 𝑄6 > 0, 𝑍1 > 0, 𝑍2 > 0, 𝑍3 > 0, 𝑍4 > 0, 𝑌1 > 0, 𝑌2 > 0, 𝑌3 > 0, 𝑌4 > 0 and scalar 𝜀1 ≥ 0, 𝜀2 ≥ 0 such that ∗ ∑ ∑ 𝜆 𝑖𝑗 𝜋𝑟𝑠 Λ [𝑗∈Ω𝑖𝑘 𝑠∈Γ𝑘𝑟 ] < 0, −Δ Ω𝑖𝑘 ,Γ𝑘𝑟 ] ΘΩ𝑖𝑘 ,Γ𝑘𝑟 [

Λ

[

Journal of Control Science and Engineering

9

Λ11 = (𝑑𝑀 − 𝑑𝑚 + 1) 𝑄1 + (𝜏𝑀 − 𝜏𝑚 + 1) 𝑄2 + 𝑄3 + 𝑄4 + 𝑄5 + 𝑄6 − 𝑍2 − 𝑍4 + 𝜀1 𝑔2 𝐼3𝑇 𝐼3 + 𝜀2 𝑔2 𝐼4𝑇 𝐼4 − 𝑃𝑖,𝑟 , ̃ Θ𝑇Ω𝑖 ,Γ𝑟 = [√𝜆 𝑖Ω𝑘𝑖 𝜋𝑟Γ𝑘1𝑟 Ψ 𝑘

𝑇

1

𝑘

̃𝑇] , ⋅ ⋅ ⋅ √𝜆 𝑖Ω 𝑖 𝜋𝑟Γ𝑘𝑟 Ψ 𝑘𝜇 𝜃

Δ Ω𝑖𝑘 ,Γ𝑘𝑟 = diag {−𝐹Ω 𝑖 Γ𝑘𝑟 , . . . , −𝐹Ω 𝑖 Γ𝑘𝑟 } , 𝑘1

Θ𝑇Ω𝑖

𝑢𝑘

,Γ𝑘𝑟

𝑘𝜇

1

𝜃

̃𝑇 ⋅ ⋅ ⋅ 𝜋𝑟Γ 𝑟 Ψ ̃𝑇 = [√𝜋𝑟Γ𝑘1𝑟 Ψ √ 𝑘𝜃 ] ,

Δ Ω𝑖𝑢𝑘 ,Γ𝑘𝑟 = diag {−𝐹𝑗,Γ𝑘𝑟 , . . . , −𝐹𝑗,Γ𝑘𝑟 } , 1

̃ Θ𝑇Ω𝑖 ,Γ𝑟 = [√𝜆 𝑖Ω𝑘𝑖 Ψ 𝑘

𝑇

1

𝑢𝑘

𝜃

̃𝑇] , ⋅ ⋅ ⋅ √𝜆 𝑖Ω 𝑖 Ψ 𝑘𝜇

Δ Ω𝑖𝑘 ,Γ𝑢𝑘𝑟 = diag {−𝐹Ω 𝑖 ,𝑠 , . . . , −𝐹Ω 𝑖 ,𝑠 } , 𝑘1

Θ𝑇Ω𝑖

𝑢𝑘

𝑟 ,Γ𝑢𝑘

𝑘𝜇

̃𝑇] , ̃𝑇 ⋅ ⋅ ⋅ Ψ = [Ψ

Δ Ω𝑖𝑢𝑘 ,Γ𝑢𝑘𝑟 = diag {−𝐹𝑗,𝑠 , . . . , −𝐹𝑗,𝑠 } , ̃ = [Ψ 0 0] , Ψ Ψ = [𝐴 − 𝐵1 𝐾𝐼1 𝐵2 𝐾𝐼1 −𝐼2 𝐿𝐶 𝐼] , Υ = 𝐴 − 𝐵1 𝐾𝐼1 − 𝐼. (38)

Proof. Equation (11) is equivalent to 𝜏𝑀

̃𝑇 ∑ 𝜋𝑟𝑠 𝑃𝑗,𝑠 Ψ) + ∑ 𝜋𝑟𝑠 ( ∑ 𝜆 𝑖𝑗 Λ + Ψ

𝑑𝑀

̃ < 0, ̃𝑇 ∑ ∑ 𝜆 𝑖𝑗 𝜋𝑟𝑠 𝑃𝑗,𝑠 Ψ Λ+Ψ

(39)

𝑗=𝜏𝑚 𝑠=𝑑𝑚

𝑟 𝑠∈Γ𝑢𝑘

𝑗∈Ω𝑖𝑘

𝑗∈Ω𝑖𝑘

̃𝑇𝑃𝑗,𝑠 Ψ) < 0. + ∑ ∑ 𝜆 𝑖𝑗 𝜋𝑟𝑠 (Λ + Ψ 𝑟 𝑗∈Ω𝑖𝑢𝑘 𝑠∈Γ𝑢𝑘

that is

(41) ( ∑ ∑ 𝜆 𝑖𝑗 𝜋𝑟𝑠 + ∑ ∑ 𝜆 𝑖𝑗 𝜋𝑟𝑠 + ∑ ∑ 𝜆 𝑖𝑗 𝜋𝑟𝑠 𝑗∈Ω𝑖𝑘 𝑠∈Γ𝑘𝑟

𝑗∈Ω𝑖𝑢𝑘 𝑠∈Γ𝑘𝑟

By Schur complement Lemma,

𝑟 𝑗∈Ω𝑖𝑘 𝑠∈Γ𝑢𝑘

̃𝑇 𝑃𝑗,𝑠 Ψ < 0, ∑ ∑ 𝜆 𝑖𝑗 𝜋𝑟𝑠 Λ + ∑ ∑ 𝜆 𝑖𝑗 𝜋𝑟𝑠 Ψ

𝑗∈Ω𝑖𝑘 𝑠∈Γ𝑘𝑟

̃𝑇 ( ∑ ∑ 𝜆 𝑖𝑗 𝜋𝑟𝑠 𝑃𝑗,𝑠 + ∑ ∑ 𝜆 𝑖𝑗 𝜋𝑟𝑠 ) Λ + Ψ 𝑟 𝑗∈Ω𝑖𝑢𝑘 𝑠∈Γ𝑢𝑘

𝑗∈Ω𝑖𝑘 𝑠∈Γ𝑘𝑟

+ ∑ ∑ 𝜆 𝑖𝑗 𝜋𝑟𝑠 𝑃𝑗,𝑠 + ∑ ∑ 𝜆 𝑖𝑗 𝜋𝑟𝑠 𝑃𝑗,𝑠 𝑗∈Ω𝑖𝑢𝑘 𝑠∈Γ𝑘𝑟

𝑟 𝑗∈Ω𝑖𝑘 𝑠∈Γ𝑢𝑘

+ ∑ ∑ 𝜆 𝑖𝑗 𝜋𝑟𝑠 𝑃𝑗,𝑠 ) Ψ < 0.

is equivalent to the following inequality ∗ ∑ ∑𝜆 𝜋 Λ [𝑗∈Ω𝑖 𝑠∈Γ𝑟 𝑖𝑗 𝑟𝑠 ] [ 𝑘 𝑘 ] < 0, ̃ Ω𝑖 ,Γ𝑟 −Δ ΘΩ𝑖𝑘 ,Γ𝑘𝑟 𝑘 𝑘] [ 𝑘1

The above inequality can be written as ̃𝑇 ∑ ∑ 𝜆 𝑖𝑗 𝜋𝑟𝑠 𝑃𝑗,𝑠 Ψ ∑ ∑ 𝜆 𝑖𝑗 𝜋𝑟𝑠 Λ + Ψ

𝑗∈Ω𝑖𝑘 𝑠∈Γ𝑘𝑟

𝑗∈Ω𝑖𝑘 𝑠∈Γ𝑘𝑟

̃𝑇 ∑ 𝜋𝑟𝑠 𝑃𝑗,𝑠 Ψ) + ∑ 𝜆 𝑖𝑗 ( ∑ 𝜋𝑟𝑠 Λ + Ψ 𝑠∈Γ𝑘𝑟

(42)

(43)

̃ Ω𝑖 Γ𝑟 = diag{−𝑃−1 Ω Γ 𝑟 , . . . , −𝑃−1 Ω Γ 𝑟 }. Let 𝑃−1 = where Δ 𝑗,𝑠 𝑖 𝑘 𝑖 𝑘 𝑘 𝑘

𝑟 𝑗∈Ω𝑖𝑢𝑘 𝑠∈Γ𝑢𝑘

𝑗∈Ω𝑖𝑢𝑘

(40)

𝑗∈Ω𝑖𝑘 𝑠∈Γ𝑘𝑟

𝑠∈Γ𝑘𝑟

1

𝑘𝜇

𝜃

𝐹𝑗,𝑠 , 𝑗 ∈ Ω, 𝑠 ∈ Γ, and 𝑍𝑙−1 = 𝑌𝑙 , 𝑙 ∈ {1, 2, 3, 4}; we can obtain (33) and (37). Hence, (42) holds if (33) holds. Because 𝜆 𝑖𝑗 ≥ 0, 𝜋𝑟𝑠 ≥ 0, it can be concluded that if (33)–(36) holds, (39) holds, which completes the proof. The conditions in Theorem 7 are in fact a set of linear matrix inequalities (LMIs) with some inversion constraints. Though they are nonconvex which brings difficulties in

10

Journal of Control Science and Engineering

solving them using the existing convex optimization tool, we can use the cone complementarity linearization (CCL) algorithm [22] to transform this problem into the nonlinear minimization problem with LMIs constraints as follows: 𝜏𝑀

4

𝑑𝑀

min tr (∑𝑍𝑙 𝑌𝑙 + ∑ ∑ 𝑃𝑗,𝑠 𝐹𝑗,𝑠 ) 𝑗=𝜏𝑚 𝑠=𝑑𝑚

𝑙=1

s.t. 𝑍𝑙 𝐼

[ [

𝐼 𝑌𝑙

𝑃𝑗,𝑠

𝐼

𝐼

𝐹𝑗,𝑠

0.3 𝐵 = [ ], 0.2 1.5 0.7 𝐶=[ ], 0.2 0.3 𝑔 = 0.1,

(44)

𝑓 (𝑘, 𝑥𝑘 ) = 0.1 sin 𝑥𝑘 .

(27) , (45) , and (46)

(47) ] > 0, ] > 0,

𝑙 ∈ {1, 2, 3, 4} ,

(45)

𝑗 ∈ Ω, 𝑠 ∈ Γ.

(46)

Furthermore, the iterative algorithm which can be used to calculate the controller gain matrix 𝐾 and observer gain matrix 𝐿 is given as follows.

We can see that the linear part of the control plant is stable itself. The random S-C delay 𝜏𝑘 ∈ {0, 1} and C-A delay 𝑑𝑘 ∈ {0, 1}. Their transition probability matrices are as follows, respectively:

Ξ=[

0.7 0.3

Algorithm 9. Step 1. Find a feasible solution satisfying (27), (45), and (46) 0 0 , 𝐹𝑗,𝑠 , 𝐾0 , 𝐿0 ); Let and set as (𝑍10 , 𝑍20 , 𝑍30 , 𝑍40 , 𝑌10 , 𝑌20 , 𝑌30 , 𝑌40 , 𝑃𝑗,𝑠 𝑘 = 0. Step 2. Solve the following LMI optimization problem for variables (𝑍1 , 𝑍2 , 𝑍3 , 𝑍4 , 𝑌1 , 𝑌2 , 𝑌3 , 𝑌4 , 𝑃𝑗,𝑠 , 𝐹𝑗,𝑠 , 𝐾, 𝐿). 𝜏

𝑑

𝑘 𝑘 𝑀 𝑀 (𝑃𝑗,𝑠 𝐹𝑗,𝑠 + 𝑃𝑗,𝑠 𝐹𝑗,𝑠 )), min tr(∑4𝑙=1 (𝑍𝑙𝑘 𝑌𝑙 + 𝑅𝑙 𝑌𝑙𝑘 ) + ∑𝑗=𝜏 ∑𝑠=𝑑 𝑚 𝑚

subject to (27), (45), and (46); set (𝑍1𝑘 = 𝑍1 , 𝑌1𝑘 = 𝑌1 , 𝑍2𝑘 = 𝑍2 , 𝑌2𝑘 = 𝑌2 , 𝑍3𝑘 = 𝑍3 , 𝑌3𝑘 = 𝑌3 , 𝑍4𝑘 = 𝑍4 , 𝑌4𝑘 = 𝑌4 , 𝑘 𝑘 𝑃𝑗,𝑠 = 𝑃𝑗,𝑠 , 𝐹𝑗,𝑠 = 𝐹𝑗,𝑠 , 𝐾𝑘 = 𝐾, 𝐿𝑘 = 𝐿).

Step 3. If (27) and (28) are satisfied, then exit the iteration. Otherwise, let 𝑘 = 𝑘 + 1, and return to Step 2. The conditions in Theorem 8 can be solved by the similar procedures. Remark 10. Making use of Matlab LMI tool box, a feasible solution satisfying (27), (45), and (46) can be obtained by the functions feasp() and dec2mat(). Once the feasible solution 0 0 , 𝐹𝑗,𝑠 , 𝐾0 , 𝐿0 ) is obtained, (𝑍10 , 𝑍20 , 𝑍30 , 𝑍40 , 𝑌10 , 𝑌20 , 𝑌30 , 𝑌40 , 𝑃𝑗,𝑠

then the LMI optimization problem min tr(∑4𝑙=1 (𝑍𝑙𝑘 𝑌𝑙 + 𝜏𝑀 𝑑𝑀 𝑘 𝑘 (𝑃𝑗,𝑠 𝐹𝑗,𝑠 + 𝑃𝑗,𝑠 𝐹𝑗,𝑠 )), subject to (27), (45), 𝑍𝑙 𝑌𝑙𝑘 ) + ∑𝑗=𝜏 ∑𝑠=𝑑 𝑚 𝑚 and (46), can be solved by the functions defcx() and mincx().

Π=[

?

?

?

?

0.2 0.8

], (48) ].

By Theorem 7, we can obtain the controller gain matrix 𝐾 and observer gain matrix 𝐿 as follows: 𝐾 = [−0.0424 0.1725] , 𝐿=[

0.5519 −2.6383 0.1065 −0.6730

(49)

].

𝑇

𝑇

The initial value 𝑥0 = [−1 1] , 𝑥̂0 = [−0.9 −1.1] , 𝜏0 = 𝑑0 = 0. The trajectories of the closed-loop system’s states and the corresponding estimated value are shown in Figures 2 and 3 which indicate that the closed-loop system is stochastically stable. By Theorem 8, the controller gain matrix 𝐾 and observer gain matrix 𝐿 can also be obtained as follows: 𝐾 = [−0.0802 0.2425] , 𝐿=[

0.2968 −1.7036 0.1569 −1.0017

].

(50)

4. Numerical Example In this section, we present two examples to illustrate the effectiveness of the proposed method. Example 1. Consider the nonlinear controlled plant with the following parameter: 𝐴=[

0.52 −0.69 0

0.19

],

The trajectories of the closed-loop system’s states and the corresponding estimated value are omitted. Example 2. Consider the classical angular positioning system shown in Figure 4, which consists of a rotating antenna at the origin of the plane driven by a motor [23]. Assume that the angular position of the antenna 𝜑, the angular position of the moving object 𝜑𝑟 , and the angular velocity of the antenna 𝜑̇ are measurable. The state variables

Journal of Control Science and Engineering

11

1

Target object

0.5 0 Antenna −0.5 Motor

−1 0

5

10

15

20

25

30

35

40

45


휑r


휑

Figure 4: The angular positioning system.

50

k 1

x1 x̂1

0.5

Figure 2: State 𝑥1 and its estimated value 𝑥̂1 .

0 −0.5

1.2 −1

1

−1.5

0.8

−2

0.6

0

50

100 k

0.4 x1 x̂1

0.2 0 −0.2

150

Figure 5: State 𝑥1 and its estimated value 𝑥̂1 . 0

5

10

15

20

25

30

35

40

45

50

k

delay 𝑑𝑘 ∈ {0, 1}. Their transition probability matrices are as follows, respectively:

x2 x̂2

? ? Ξ=[ ], 0.6 0.4

Figure 3: State 𝑥2 and its estimated value 𝑥̂2 .

𝑇

are chosen as [𝜑 𝜑]̇ ; then state equation of the angular positioning system can be obtained with the following parameter, where 𝑓(𝑘, 𝑥𝑘 ) can be treated as the external disturbance: 𝐴=[

1 0.0995 0

0.99

],

𝐶=[ ], −0.2 0.4

?

].

For this unstable plant, we cannot get the controller gain matrix 𝐾 and observer gain matrix 𝐿 by Theorem 7 due to its conservativeness. However, by Theorem 8, we can obtain 𝐾 and 𝐿 as follows:

𝐿=[ (51)

𝑔 = 0.2, 𝑓 (𝑘, 𝑥𝑘 ) =

?

(52)

𝐾 = [−1.1957 −1.1143] ,

0.0039 𝐵=[ ], 0.0783 1.4 0.8

Π=[

0.2 0.8

0.2 sin 𝑥𝑘 . 𝑘

Obviously, it can be seen that the linear part of the control plant is unstable. The random S-C delay 𝜏𝑘 ∈ {0, 1} and C-A

0.1943 −0.2715 0.1130 0.6251 𝑇

(53)

]. 𝑇

The initial value 𝑥0 = [−1.6 1.7] , 𝑥̂0 = [−1.7 1.6] , 𝜏0 = 𝑑0 = 0. The trajectories of the closed-loop system’s states and the corresponding estimated value are shown in Figures 5 and 6 which indicate that the closed-loop system is stochastically stable. Remark 11. The results in this paper only require a part of transition probabilities while the results in [13, 14] need the full information of the transition probabilities. In view of

12

Journal of Control Science and Engineering 2 1.5 1 0.5 0 −0.5

0

50

100

150

k x2 x̂2

Figure 6: State 𝑥2 and its estimated value 𝑥̂2 .

these, the proposed controller design method is more general than that of [13, 14].

5. Conclusion The observer-based controller design problem for a class of nonlinear networked control systems with random timedelays is investigated in this paper. Two dependent Markov chains are employed to describe S-C delay and C-A delay, respectively. The transition probabilities of S-C delay and CA delay are both assumed to be partly accessible. Sufficient conditions on the stochastic stability for the closed-loop systems are obtained by the Lyapunov stability theory. The CCL algorithm is employed to calculate the controller and observer gain matrices. Finally, two examples are used to illustrate the effectiveness of the proposed method.

Conflicts of Interest The authors declare that they have no conflicts of interest.

Acknowledgments The research is supported by the National Natural Science Foundation of China (Grant nos. 61503136, 61573137, 61573136, and 61603133).

References [1] P. V. Zhivoglyadov and R. H. Middleton, “Networked control design for linear systems,” Automatica, vol. 39, no. 4, pp. 743– 750, 2003. [2] D. Yue, Q. L. Han, and C. Peng, “State feedback controller design of networked control systems,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 51, no. 11, pp. 640–644, 2004. [3] Q. Zhou, C. Wu, and P. Shi, “Observer-based adaptive fuzzy tracking control of nonlinear systems with time delay and input saturation,” Fuzzy Sets and Systems, 2016. [4] X. Tang and B. Ding, “Model predictive control of linear systems over networks with data quantizations and packet losses,” Automatica, vol. 49, no. 5, pp. 1333–1339, 2013.

[5] Y. Zhang and H. Fang, “Stabilization of nonlinear networked systems with sensor random packet dropout and time-varying delay,” Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems, vol. 35, no. 5, pp. 2253–2264, 2011. [6] E. Tian, D. Yue, and X. Zhao, “Quantised control design for networked control systems,” IET Control Theory and Applications, vol. 1, no. 6, pp. 1693–1699, 2007. [7] J. Nilsson, Real-time control systems with delays [Ph.D. thesis], Lund Institute of Technology, Lund, Sweden, 1998. [8] L. Xiao, A. Hassibi, and J. P. How, “Control with random communication delays via a discrete-time jump system approach,” in Proceedings of the American Control Conference, pp. 2199– 2204, IEEE, Chicago, Ill, USA, June 2000. [9] S. W. Gao and G. Y. Tang, “Output feedback stabilization of networked control systems with random delays,” in Proceedings of the 18th IFAC World Congress, pp. 3250–3255, Milan, Italy, 2011. [10] P. Shi and E. K. Boukas, “𝐻∞ control for markovian jumping linear systems with parametric uncertainty,” Journal of Optimization Theory and Applications, vol. 95, no. 1, pp. 75–99, 1997. [11] P. Shi, E.-K. Boukas, and R. K. Agarwal, “Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delay,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 44, no. 11, pp. 2139–2144, 1999. [12] O. L. V. Costa and R. P. Marques, “Robust H2 control for discrete-time Markovian jump linear systems,” International Journal of Control, vol. 73, no. 1, pp. 11–21, 2000. [13] L. Q. Zhang, Y. Shi, T. W. Chen, and B. Huang, “A new method for stabilization of networked control systems with random delays,” IEEE Transactions on Automatic Control, vol. 50, no. 8, pp. 1177–1181, 2005. [14] Y. Shi and B. Yu, “Output feedback stabilization of networked control systems with random delays modeled by Markov chains,” IEEE Transactions on Automatic Control, vol. 54, no. 7, pp. 1668–1674, 2009. [15] Q. Li, B. G. Xu, and S. B. Li, “H-infinity control for networked control systems with data packet dropouts and partly unknown transition probabilities,” Control Theory & Applications, vol. 28, no. 8, pp. 1105–1112, 2011. [16] S. Q. Yu and J. M. Li, “Robust stabilization of networked control systems with nonlinear perturbation,” Control and Decision, vol. 27, no. 12, pp. 1917–1920, 2012. [17] Y. F. Wang, P. L. Wang, H. Y. Chen, and M. Fang, “State feedback for networked control systems with partly unknown transition probabilities,” Control Engineering of China, vol. 22, no. 4, pp. 776–779, 2015. [18] Y. F. Wang, P. L. Wang, Z. X. Li, and H. Y. Chen, “Fault-tolerant control for networked control systems with limited information in case of actuator fault,” Mathematical Problems in Engineering, vol. 2015, Article ID 785289, 7 pages, 2015. [19] X. F. Jiang, Q.-L. Han, and X. H. Yu, “Stability criteria for linear discrete-time systems with interval-like time-varying delay,” in Proceedings of the American Control Conference (ACC ’05), pp. 2817–2822, IEEE, Portland, Ore, USA, June 2005. [20] J. G. Li, J. Q. Yuan, and J. G. Lu, “Observer-based H∞ control for networked nonlinear systems with random packet losses,” ISA Transactions, vol. 49, no. 1, pp. 39–46, 2010. [21] Y. F. Wang, P. L. Wang, and Z. D. Cai, “Robust H-infinity fault detection for networked control systems with partly

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