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Journal of Control Science and Engineering
VolumeΒ 2012, Article IDΒ 734758, 7 pages
http://dx.doi.org/10.1155/2012/734758
Research Article

Robust Fault-Tolerant Control for Uncertain Networked Control Systems with State-Delay and Random Data Packet Dropout

1School of Control Science and Engineering, Shandong University, Jinan 250061, China
2Department of Electrical and Computer Engineering, Dalhousie University, Halifax, NS, Canada B3J 2X4

Received 22 November 2011; Accepted 9 February 2012

Academic Editor: YangΒ Shi

Copyright Β© 2012 Xiaomei Qi 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.

Abstract

A robust fault-tolerant controller design problem for networked control system (NCS) with random packet dropout in both sensor-to-controller link and controller-to-actuator link is investigated. A novel stochastic NCS model with state-delay, model uncertainty, disturbance, probabilistic sensor failure, and actuator failure is proposed. The random packet dropout, sensor failures, and actuator failures are characterized by a binary random variable. The sufficient condition for asymptotical mean-square stability of NCS is derived and the closed-loop NCS satisfies 𝐻∞ performance constraints caused by the random packet dropout and disturbance. The fault-tolerant controller is designed by solving a linear matrix inequality. A numerical example is presented to illustrate the effectiveness of the proposed method.

1. Introduction

Networked control systems (NCSs) are one type of distributed control systems, in which the information of control system components (sensors, controllers, actuators, etc.) is exchanged via communication networks (see Figure 1). Compared with the conventional point-to-point control systems, NCSs have many advantages, such as low cost of installation and maintenance, ease of diagnosis, and flexible architectures. However, the network in the control systems also bring many problems, such as network-induced delay, packet dropout, multiple channel transmission (as in [1–4]). Recent researches have a deep look into the controller and filter design for NCSs without faults (see [5–10]) and the references therein.

734758.fig.001
Figure 1: Schematic description of a distributed NCS.

Actually, NCSs are more vulnerable to faults than conventional control systems, due to the complexity introduced by the network. It is very significant to guarantee security and reliability of NCSs, because modern technological systems rely on sophisticated control systems to meet increased performance and safety requirement. Therefore, research on fault-tolerant control (FTC) for NCSs has attracted more and more attention from both industry and academia.

However, research on FTC for NCSs is quite different from the ones for conventional control systems in many aspects (see [11–15]). A suitable architecture for FTC of NCSs must take into consideration the dynamical behaviour of network. In most research on FTC for NCSs, the fault model is described in a static way (as in [16–20]). Actually, faults often happen in a random way, so it is suitable to be studied in a dynamic way (as in [21–25]). In [21], the probabilistic sensor reductions are modeled by using a random variable that obeys a specific distribution in a known interval. In [22], the entire sensor failures or missing measurements have been described as a Bernoulli distributed variable. The reliable control design is considered for NCSs against probabilistic actuator fault with different failure rates in [23]. But, only actuator failures or sensor failures are considered in [16–23].

As we know, many engineering control systems such as conventional oil-chemical industrial processes, nuclear reactors, long transmission lines in pneumatic, hydraulic, and rolling mill systems, NCSs contain some time-delay effects, model uncertainties (as in [26, 27]), and external disturbances. However, most of the aforementioned researches discuss FTC for NCSs without model uncertainty and disturbance(as in [16–20, 23, 24]).

Therefore, from the above description, considering random packet dropout, the robust FTC for state-delay uncertain NCSs with both probabilistic sensors failures and actuators failures is still a challenging problem.

In this paper, we study the robust FTC problem for NCSs with random packet dropout in both sensor-to-controller (S-C) link and controller-to-actuator (C-A) link. A new stochastic NCS model with fault is proposed, which includes the state-delay, model uncertainty, disturbance, random packet dropout, probabilistic sensors failures, and actuators failures. The random packet dropout, the sensor failure and the actuator failure are described as a binary random variable. The aim of this paper is to design a dynamic fault-tolerant controller for the NCS including packet dropouts, both sensor failures and actuator failures. The closed-loop NCS can be asymptotical mean-square stability and satisfies the performance constraint.

The rest of the paper is organized as follows. The problem is formulated in Section 2, a new stochastic NCS with probabilistic sensors failures and actuators failures is modelled. Section 3 presents the integral analysis of asymptotical mean-square stability for stochastic NCS with sensor and actuator faults. Section 4 designs a dynamic fault-tolerant controller. Section 5 gives a numerical example to demonstrate the effectiveness of the proposed method. Concluding remarks are made in Section 6.

2. Problem Formulation

Consider the following uncertain linear state-delay system:π‘₯π‘˜+1=(𝐴+Δ𝐴)π‘₯π‘˜+𝐴𝑑π‘₯π‘˜βˆ’π‘‘+π΅π‘’π‘˜+𝐷0πœ”π‘˜,π‘¦π‘˜=𝐢0π‘₯π‘˜,π‘§π‘˜=𝐢π‘₯π‘˜+𝐢𝑑π‘₯π‘˜βˆ’π‘‘+𝐷1πœ”π‘˜,(1) where π‘₯π‘˜βˆˆβ„π‘›,π‘¦π‘˜βˆˆβ„π‘š,π‘’π‘˜βˆˆβ„π‘,π‘§π‘˜βˆˆβ„π‘ž denote the state, the sensor measurement, the control input, and the controlled output, respectively. πœ”π‘˜βˆˆβ„π‘Ÿ is the disturbance input belonging to 𝑙2[0,∞).𝐴,𝐴𝑑,𝐡,𝐢0,𝐢,𝐢𝑑,𝐷0,𝐷1, are known real constant matrices with appropriate dimensions. 𝑑>0 is a known delay. Δ𝐴 denotes the model uncertainty, which satisfies Δ𝐴=π»πΉπ‘˜π‘,πΉπ‘‡π‘˜πΉπ‘˜<𝐼,πΉπ‘˜ represents an unknown real-valued time-varying matrix.

Figure 1 shows a typical feedback loop of NCS. Due to network congestion, traffic load balancing, or other unpredictable network behavior, the network-induced delay, data packet dropout, disorder may occur at the same time. In this paper, we focus on the data packet dropout phenomenon. Some assumptions in this paper are as follows.(1)The sensor is clock-driven, the controller and the actuator are event-driven.(2)Data packet dropouts occur in both S-C link and C-A link.(3)Data are single-packet transmission with timestamp.(4)Ignore the effects of quantization and asynchronous error in this paper.

Remark 1. A clock-driven sensor can send measurements to network periodically and is often used in real-time computing. The advantage of event-driven controller/actuator is that the controller/actuator will be updated as soon as the new data packet comes.

Remark 2. Taking the Internet as an illustration, the Transmission Control Protocol (TCP) is one of the core protocols of the Internet Protocol Suite. TCP is responsible for verifying the correct delivery of data from client to server. Data can be lost in the intermediate network. TCP adds support to detect errors or lost data and to trigger retransmission until the data is correctly and completely received. Thus, TCP is optimized for accurate delivery rather than timely delivery. Furthermore, for real-time feedback control, it is appropriate to discard the old data and transmit a new packet if it is available.

Therefore, we assume if the total network-induced delay πœπ‘˜ is larger than a sampling period, the output terminal will actively discard this packet, which means the network-induced delay problem can be considered as a packet-dropout problem. And the receiver with a buffer can rearrange the packets by reading the information of timestamp; in this way, the data disorder problem will be solved. Hence, we only focus on the data packet dropout issue in this paper.

The binary random variable πœƒπ‘˜ is an identically distributed (i.i.d.) process. πœƒπ‘˜=1 means that there is no packet dropout, and the sensors and the actuators are reliable; πœƒπ‘˜=0 means packet is lost, and the sensors and the actuators have failures. The probability distribution of πœƒπ‘˜ is 𝑃{πœƒπ‘˜=0}=𝑝,𝑃{πœƒπ‘˜=1}=1βˆ’π‘, where π‘βˆˆ(0,1) indicates the sensor/actuator failure rate and the packet dropout rate.

Considering the channel from the sensor to the controller, the sensor measurement π‘¦π‘˜ will be updated to π‘¦π‘˜ as follows:π‘¦π‘˜=πœƒπ‘˜π‘¦π‘˜+ξ€·1βˆ’πœƒπ‘˜ξ€Έπ‘¦π‘˜βˆ’1.(2)

Considering the channel from the controller to the actuator, the control output π‘’π‘˜ will be updated as follows:π‘’π‘˜=πœƒπ‘˜π‘’π‘˜+ξ€·1βˆ’πœƒπ‘˜ξ€Έπ‘’π‘˜βˆ’1.(3)

Remark 3. From expressions (2) and (3), when πœƒπ‘˜=0 at time π‘˜, the sensor measurements and control information at time π‘˜ are missing. The last available measurement π‘¦π‘˜βˆ’1 and controller output π‘’π‘˜βˆ’1 stored in a buffer are utilized to substitute the missing data, which means at least one packet can be transmitted successfully in a sampling period.

Since the random variable πœƒπ‘˜ also represents sensor failures and actuator failures, the dynamic fault-tolerant controller is designedΜ‚π‘₯π‘˜+1=𝐴𝑓̂π‘₯π‘˜+π΅π‘“π‘¦π‘˜,π‘’π‘˜=𝐢𝑓̂π‘₯π‘˜.(4)

Define πœ‚π‘˜=[π‘₯π‘‡π‘˜,Μ‚π‘₯π‘‡π‘˜]𝑇,π‘‘π‘˜=[π‘’π‘‡π‘˜βˆ’1,π‘¦π‘‡π‘˜βˆ’1]𝑇, the stochastic NCS with probabilistic sensor failures and actuator failuresπœ‚π‘˜+1=ξ‚π΄πœ‚π‘˜+ξ€·πœƒπ‘˜ξ€Έξ‚π΄βˆ’π‘2πœ‚π‘˜+ξ‚π΄π‘‘π‘πœ‚π‘˜βˆ’π‘‘+ξ€·πœƒπ‘˜ξ€Έξ‚π΅βˆ’π‘1π‘‘π‘˜+𝐡2π‘‘π‘˜+𝐷1πœ”π‘˜,π‘§π‘˜=ξ‚πΆπœ‚π‘˜+πΆπ‘‘π‘πœ‚π‘˜βˆ’π‘‘+𝐷1πœ”π‘˜,(5) where𝐴𝐴=1+ξ‚π»πΉπ‘˜ξ‚βŽ‘βŽ’βŽ’βŽ£π‘=𝐴𝑝𝐡𝐢𝑓𝑝𝐡𝑓𝐢0π΄π‘“βŽ€βŽ₯βŽ₯⎦+⎑⎒⎒⎣𝐻0⎀βŽ₯βŽ₯βŽ¦πΉπ‘˜ξ‚ƒξ‚„,𝐴𝑁02=⎑⎒⎒⎣0𝐡𝐢𝑓𝐡𝑓𝐢00⎀βŽ₯βŽ₯⎦,𝐴𝑑=βŽ‘βŽ’βŽ’βŽ£π΄π‘‘0⎀βŽ₯βŽ₯⎦,𝐡1=βŽ‘βŽ’βŽ’βŽ£βˆ’π΅00βˆ’π΅π‘“βŽ€βŽ₯βŽ₯⎦,𝐡2=⎑⎒⎒⎣(1βˆ’π‘)𝐡00(1βˆ’π‘)π΅π‘“βŽ€βŽ₯βŽ₯⎦,,𝐷𝐢=𝐢01=⎑⎒⎒⎣𝐷00⎀βŽ₯βŽ₯βŽ¦ξ‚ƒξ‚„.,𝑍=𝐼0(6)

The aim of this paper is to design a dynamic fault-tolerant controller for the NCS (5), such that for all the possible data packet dropout and failures, the system (5) satisfies the following requirements.

(Q1) The closed-loop NCS (5) is asymptotically mean-square stable.

(Q2) Under the zero-initial condition, the output π‘§π‘˜ satisfies βˆžξ“π‘˜=0π”Όξ‚†β€–β€–π‘§π‘˜β€–β€–2<𝛾2βˆžξ“π‘˜=0π”Όξ‚†β€–β€–πœ”π‘˜β€–β€–2+𝛿2βˆžξ“π‘˜=0π”Όξ‚†β€–β€–π‘‘π‘˜β€–β€–2(7) for all nonzero π‘‘π‘˜,πœ”π‘˜, where 𝛾,𝛿>0 are the scalars we will design.

3. The Stability Analysis of NCS

In this section, the stability analysis for the NCS (5) is discussed.

Lemma 4 (as in [28]). Let π‘Š=π‘Šπ‘‡<0,𝐻,𝑁 be matrices of appropriate dimensions, with 𝐹 satisfying 𝐹𝑇𝐹≀𝐼, then π‘Š+𝐻𝐹𝑁+𝑁𝑇𝐹𝑇𝐻𝑇<0 holds, if and only if there exists a πœ€>0 such that π‘Š+πœ€π»π»π‘‡+πœ€βˆ’1𝑁𝑇𝑁<0.

Definition 5. The NCS with faults given by (5) with πœ”π‘˜=0,π‘‘π‘˜=0, is asymptotically mean-square stable, if for any initial state, limπ‘˜β†’βˆžπ”Ό{β€–π‘§π‘˜β€–2}=0 holds.

Theorem 6. Given 𝛾>0,𝛿>0, the system (5) is asymptotically mean-square stable, and the output π‘§π‘˜ satisfies (7), if there exist matrices 𝑃=𝑃𝑇>0, and 𝑄=𝑄𝑇>0 satisfying βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£ξ‚ξ‚π΄βˆ’π‘ƒ0𝑃𝐴𝑃𝑑𝑃𝐷1𝐡0πœŽπ‘ƒ1𝑃𝐡20ξ‚βˆ—βˆ’πΌπΆπΆπ‘‘π·1𝐴0000βˆ—βˆ—βˆ’π‘ƒ00πœŽπ‘‡2𝑃00π‘π‘‡βˆ—βˆ—βˆ—βˆ’π‘„00000βˆ—βˆ—βˆ—βˆ—βˆ’π›Ύ2𝐼0000βˆ—βˆ—βˆ—βˆ—βˆ—βˆ’π‘ƒ000βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ’π›Ώ2𝐼00βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ’π›Ώ2𝐼0βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ’π‘„βˆ’1⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦<0.(8)

Proof. Let Ξ˜π‘˜=[πœ‚π‘‡π‘˜,πœ‚π‘‡π‘˜βˆ’1,…,πœ‚π‘‡π‘˜βˆ’π‘‘]𝑇, for all nonzero πœ‚π‘˜, consider the Lyapunov function π‘‰π‘˜(Ξ˜π‘˜)=𝑉1π‘˜+𝑉2π‘˜ with 𝑉1π‘˜=πœ‚π‘‡π‘˜π‘ƒπœ‚π‘˜,𝑉2π‘˜=π‘˜βˆ’1𝑖=π‘˜βˆ’π‘‘πœ‚π‘‡π‘–π‘π‘‡π‘„π‘πœ‚π‘–.(9) The difference of π‘‰π‘˜ is Δ𝑉1π‘˜ξ€½π‘‰=𝔼1π‘˜ξ€·Ξ˜π‘˜+1ξ€Έξ€Ύβˆ’π‘‰1π‘˜ξ€·Ξ˜π‘˜ξ€Έ=πœ‚π‘‡π‘˜ξ‚π΄π‘‡π‘ƒξ‚π΄πœ‚π‘˜+πœ‚π‘‡π‘˜ξ‚π΄π‘‡π‘ƒξ‚π΄π‘‘π‘πœ‚π‘˜βˆ’π‘‘+πœ‚π‘‡π‘˜ξ‚π΄π‘‡π‘ƒξ‚π΅2π‘‘π‘˜+πœ‚π‘‡π‘˜ξ‚π΄π‘‡π‘ƒξ‚π·1πœ”π‘˜ξ‚†ξ€·πœƒ+π”Όπ‘˜ξ€Έβˆ’π‘2ξ‚‡πœ‚π‘‡π‘˜ξ‚π΄π‘‡2𝑃𝐴2πœ‚π‘˜ξ‚†ξ€·πœƒ+π”Όπ‘˜ξ€Έβˆ’π‘2ξ‚‡πœ‚π‘‡π‘˜ξ‚π΄π‘‡2𝑃𝐡1π‘‘π‘˜+πœ‚π‘‡π‘˜βˆ’π‘‘π‘π‘‡ξ‚π΄π‘‡π‘‘π‘ƒξ‚π΄πœ‚π‘˜+πœ‚π‘‡π‘˜βˆ’π‘‘π‘π‘‡ξ‚π΄π‘‡π‘‘π‘ƒξ‚π΄π‘‘π‘πœ‚π‘˜βˆ’π‘‘+πœ‚π‘‡π‘˜βˆ’π‘‘π‘π‘‡ξ‚π΄π‘‡π‘‘π‘ƒξ‚π΅2π‘‘π‘˜+πœ‚π‘‡π‘˜βˆ’π‘‘π‘π‘‡ξ‚π΄π‘‡π‘‘π‘ƒξ‚π·1πœ”π‘˜ξ‚†ξ€·πœƒ+π”Όπ‘˜ξ€Έβˆ’π‘2ξ‚‡π‘‘π‘‡π‘˜ξ‚π΅π‘‡1𝑃𝐴2πœ‚π‘˜ξ‚†ξ€·πœƒ+π”Όπ‘˜ξ€Έβˆ’π‘2ξ‚‡π‘‘π‘‡π‘˜ξ‚π΅π‘‡1𝑃𝐡1π‘‘π‘˜+π‘‘π‘‡π‘˜ξ‚π΅π‘‡2π‘ƒξ‚π΄πœ‚π‘˜+π‘‘π‘‡π‘˜ξ‚π΅π‘‡2π‘ƒξ‚π΄π‘‘π‘πœ‚π‘˜βˆ’π‘‘+π‘‘π‘‡π‘˜ξ‚π΅π‘‡2𝑃𝐡2π‘‘π‘˜+π‘‘π‘‡π‘˜ξ‚π΅π‘‡2𝑃𝐷1πœ”π‘˜+πœ”π‘‡π‘˜ξ‚π·π‘‡1π‘ƒξ‚π΄πœ‚π‘˜+πœ”π‘‡π‘˜ξ‚π·π‘‡1π‘ƒξ‚π΄π‘‘π‘πœ‚π‘˜βˆ’π‘‘+πœ”π‘‡π‘˜ξ‚π·π‘‡1𝑃𝐡2π‘‘π‘˜+πœ”π‘‡π‘˜ξ‚π·π‘‡1𝑃𝐷1πœ”π‘˜βˆ’πœ‚π‘‡π‘˜π‘ƒπœ‚π‘˜Ξ”π‘‰2π‘˜ξ€½π‘‰=𝔼2π‘˜ξ€·Ξ˜π‘˜+1ξ€Έξ€Ύβˆ’π‘‰2π‘˜ξ€·Ξ˜π‘˜ξ€Έ=πœ‚π‘‡π‘˜ξ€·π‘π‘‡ξ€Έπœ‚π‘„π‘π‘˜βˆ’πœ‚π‘‡π‘˜βˆ’π‘‘π‘π‘‡π‘„π‘πœ‚π‘˜βˆ’π‘‘.(10)
For 𝔼{πœƒπ‘˜βˆ’π‘}=0, and denote 𝔼{(πœƒπ‘˜βˆ’π‘)2}=𝜎2, we have Ξ”π‘‰π‘˜=βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£πœ‚π‘˜π‘πœ‚π‘˜βˆ’π‘‘π‘‘π‘˜πœ”π‘˜βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦π‘‡βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£Ξ¦11Ξ¦12Ξ¦13Ξ¦14βˆ—Ξ¦22Ξ¦23Ξ¦24βˆ—βˆ—Ξ¦33Ξ¦34βˆ—βˆ—βˆ—Ξ¦44⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£πœ‚π‘˜π‘πœ‚π‘˜βˆ’π‘‘π‘‘π‘˜πœ”π‘˜βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦<πœπ‘‡π‘˜Ξ¦π‘–π‘—πœπ‘˜(𝑖,𝑗=1,…,4),(11) where πœπ‘˜=[πœ‚π‘‡π‘˜,πœ‚π‘‡π‘˜βˆ’π‘‘π‘π‘‡,π‘‘π‘‡π‘˜,πœ”π‘‡π‘˜]𝑇,Ξ¦11=𝐴𝑇𝑃𝐴+𝑍𝑇𝑄𝑍+𝜎2𝐴𝑇2𝑃𝐴2βˆ’π‘ƒ,Ξ¦12=𝐴𝑇𝑃𝐴𝑑,Ξ¦13=𝐴𝑇𝑃𝐡2+𝜎2𝐴𝑇2𝑃𝐡1,Ξ¦14=𝐴𝑇𝑃𝐷1,Ξ¦22=ξ‚π΄π‘‡π‘‘π‘ƒξ‚π΄π‘‘βˆ’π‘„,Ξ¦23=𝐴𝑇𝑑𝑃𝐡2,Ξ¦24=𝐴𝑇𝑑𝑃𝐷1,Ξ¦33=𝐡𝑇2𝑃𝐡2+𝜎2𝐡𝑇1𝑃𝐡1,Ξ¦34=𝐡𝑇2𝑃𝐷1,Ξ¦44=𝐷𝑇1𝑃𝐷1,βˆ— is implicitly defined by the fact that the matrix is symmetric.
When πœ”π‘˜=0,π‘‘π‘˜=0, (11) is rewritten as Ξ”π‘‰π‘˜,πœ”π‘˜=π‘‘π‘˜=0=βŽ‘βŽ’βŽ’βŽ£ξ‚π΄π‘‡π‘ƒξ‚π΄+𝑍𝑇𝑄𝑍+𝜎2𝐴𝑇2𝑃𝐴2ξ‚π΄βˆ’π‘ƒπ‘‡π‘ƒξ‚π΄π‘‘ξ‚π΄π‘‡π‘‘π‘ƒξ‚π΄ξ‚π΄π‘‡π‘‘π‘ƒξ‚π΄π‘‘βŽ€βŽ₯βŽ₯βŽ¦βˆ’π‘„=Ξ .(12)
From (8) and the Schur complement theorem, Ξ <0 is arrived. Therefore, for all nonzero πœ‚π‘˜, we have Ξ”π‘‰π‘˜<0, then the NCS (5) with sensor and actuator fault is asymptotically mean-square stable.
Next, For any nonzero πœ”π‘˜,π‘‘π‘˜, it follows from (5), (8), and (11) that π”Όξ€½π‘‰ξ€·Ξ˜π‘˜+1ξ€½π‘‰ξ€·Ξ˜ξ€Έξ€Ύβˆ’π”Όπ‘˜ξ€½π‘§ξ€Έξ€Ύ+π”Όπ‘‡π‘˜π‘§π‘˜ξ€Ύβˆ’π›Ύ2π”Όξ€½πœ”π‘‡π‘˜πœ”π‘˜ξ€Ύβˆ’π›Ώ2π”Όξ€½π‘‘π‘‡π‘˜π‘‘π‘˜ξ€Ύξ‚†πœβ‰€π”Όπ‘‡π‘˜Ξ¦π‘–π‘—πœπ‘˜+πœ‚π‘‡π‘˜ξ‚πΆπ‘‡ξ‚πΆπœ‚π‘˜+πœ‚π‘‡π‘˜ξ‚πΆπ‘‡πΆπ‘‘π‘πœ‚π‘˜βˆ’π‘‘+πœ‚π‘‡π‘˜ξ‚πΆπ‘‡π·1πœ”π‘˜+πœ‚π‘‡π‘˜βˆ’π‘‘π‘π‘‡πΆπ‘‡π‘‘ξ‚πΆπœ‚π‘˜+πœ‚π‘‡π‘˜βˆ’π‘‘π‘π‘‡πΆπ‘‡π‘‘πΆπ‘‘π‘πœ‚π‘˜βˆ’π‘‘+πœ‚π‘‡π‘˜βˆ’π‘‘π‘π‘‡πΆπ‘‡π‘‘π·1πœ”π‘˜+πœ”π‘‡π‘˜π·π‘‡1ξ‚πΆπœ‚π‘˜+πœ”π‘‡π‘˜π·π‘‡1πΆπ‘‘π‘πœ‚π‘˜βˆ’π‘‘+πœ”π‘‡π‘˜π·π‘‡1𝐷1πœ”π‘˜βˆ’π›Ύ2πœ”π‘‡π‘˜πœ”π‘˜βˆ’π›Ώ2π‘‘π‘‡π‘˜π‘‘π‘˜ξ€Ύ.(13)
Then, we have π”Όξ€½π‘‰ξ€·Ξ˜π‘˜+1ξ€½π‘‰ξ€·Ξ˜ξ€Έξ€Ύβˆ’π”Όπ‘˜ξ€½π‘§ξ€Έξ€Ύ+π”Όπ‘‡π‘˜π‘§π‘˜ξ€Ύβˆ’π›Ύ2π”Όξ€½πœ”π‘‡π‘˜πœ”π‘˜ξ€Ύβˆ’π›Ώ2π”Όξ€½π‘‘π‘‡π‘˜π‘‘π‘˜ξ€Ύ<0.(14) Now, summing (14) from 0 to ∞ with respect to π‘˜ yields βˆžξ“π‘˜=0π”Όξ€½π‘§π‘‡π‘˜π‘§π‘˜ξ€Ύ<𝛾2βˆžξ“π‘˜=0π”Όξ€½πœ”π‘‡π‘˜πœ”π‘˜ξ€Ύ+𝛿2βˆžξ“π‘˜=0π”Όξ€½π‘‘π‘‡π‘˜π‘‘π‘˜ξ€Ύξ€½π‘‰+𝔼0ξ€Ύξ€½π‘‰βˆ’π”Όβˆžξ€Ύ.(15) Since the system (5) is asymptotically mean-square stable, we can get that the following inequality: βˆžξ“π‘˜=0π”Όξ€½π‘§π‘‡π‘˜π‘§π‘˜ξ€Ύ<𝛾2βˆžξ“π‘˜=0π”Όξ€½πœ”π‘‡π‘˜πœ”π‘˜ξ€Ύ+𝛿2βˆžξ“π‘˜=0π”Όξ€½π‘‘π‘‡π‘˜π‘‘π‘˜ξ€Ύ(16) holds under the zero initial condition. The proof is thus, complete.

4. Robust 𝐻∞ Controller Design

In this section, a theorem will be proposed to solve the controller design problem for stochastic state-delay NCS (5).

Theorem 7. Given a scalar 𝛾>0,𝛿>0, the system (5) is asymptotically mean-square stable, and the controlled output π‘§π‘˜ satisfies the 𝐻∞ constraints (7), if there exist real scalars πœ€>0, and matrices 𝑆=𝑆𝑇>0,𝑄=𝑄𝑇>0, and 𝑅=𝑅𝑇>0, and real matrices π‘Š1,π‘Š2, and π‘Š3, such that the following inequality holds: Ω𝑖𝑗<0,𝑖,𝑗=1,…,16,(17) where Ξ©11=βˆ’π‘†βˆ’1,Ξ©12=βˆ’πΌ,Ξ©14=𝐴+π‘π΅π‘Š3,Ξ©15=𝐴,Ξ©16=𝐴𝑑,Ξ©17=𝐷0,Ξ©1,10=βˆ’πœŽπ΅,Ξ©1,12=(1βˆ’π‘)𝐡,Ξ©1,15=𝐻,Ξ©21=βˆ’πΌ,Ξ©22=βˆ’π‘…,Ξ©24=𝑅𝐴+π‘π‘Š2𝐢0+π‘Š1,Ξ©25=𝑅𝐴+π‘π‘Š2𝐢0,Ξ©26=𝑅𝐴𝑑,Ξ©27=𝑅𝐷0,Ξ©2,10=βˆ’πœŽπ‘…π΅,Ξ©2,11=βˆ’πœŽπ‘Š2,Ξ©2,12=(1βˆ’π‘)𝑅𝐡,Ξ©2,13=(1βˆ’π‘)π‘Š2,Ξ©2,15=𝑅𝐻,Ξ©33=βˆ’πΌ,Ξ©34=𝐢,Ξ©35=𝐢,Ξ©36=𝐢𝑑,Ξ©37=𝐷1,Ξ©41=Ω𝑇14,Ξ©42=Ω𝑇24,Ξ©43=𝐢𝑇,Ξ©44=βˆ’π‘†,Ξ©45=βˆ’π‘†,Ξ©48=πœŽπ‘Šπ‘‡3𝐡𝑇,Ξ©49=πœŽπ‘Šπ‘‡3𝐡𝑇𝑅+πœŽπΆπ‘‡0π‘Šπ‘‡2,Ξ©4,14=𝐼,Ξ©4,16=πœ–π‘π‘‡,Ξ©51=𝐴𝑇,Ξ©52=Ω𝑇25,Ξ©53=𝐢𝑇,Ξ©54=βˆ’π‘†,Ξ©55=βˆ’π‘…,Ξ©59=πœŽπΆπ‘‡0π‘Šπ‘‡2,Ξ©5,14=𝑄,Ξ©5,16=πœ–π‘π‘‡,Ξ©61=𝐴𝑇𝑑,Ξ©61=𝐴𝑇𝑑𝑅,Ξ©63=𝐢𝑇𝑑,Ξ©66=βˆ’π‘„,Ξ©71=𝐷𝑇0,Ξ©72=𝐷𝑇0𝑅,Ξ©73=𝐷𝑇1,Ξ©77=βˆ’π›Ύ2𝐼,Ξ©84=Ω𝑇48,Ξ©88=βˆ’π‘†,Ξ©89=βˆ’π‘†,Ξ©94=Ω𝑇49,Ξ©95=Ω𝑇59,Ξ©99=βˆ’π‘…,Ξ©10,1=βˆ’πœŽπ΅π‘‡,Ξ©10,2=βˆ’πœŽπ΅π‘‡π‘…,Ξ©10,10=Ξ©11,11=Ξ©12,12=Ξ©13,13=βˆ’π›Ώ2𝐼,Ξ©11,2=Ω𝑇2,11,Ξ©12,1=Ω𝑇12,1,Ξ©12,2=Ω𝑇2,12,Ξ©13,2=Ω𝑇2,13,Ξ©14,4=𝐼,Ξ©14,5=𝑄𝑇,Ξ©14,14=βˆ’πΌ,Ξ©15,1=𝐻𝑇,Ξ©15,2=𝐻𝑇𝑅,Ξ©15,15=Ξ©16,16=βˆ’πœ€πΌ,Ξ©16,4=Ξ©16,5=πœ–π‘, the rest of matrix entries are zero.
The fault-tolerant controller parameters are 𝐴𝑓=π‘‹βˆ’112ξ‚ƒξ€·π‘Š1βˆ’π‘π‘…π΅π‘Š3ξ€Έπ‘†βˆ’1ξ€·π‘Œπ‘‡12ξ€Έβˆ’1ξ‚„,𝐡𝑓=π‘‹βˆ’112π‘Š2,𝐢𝑓=π‘Š3π‘†βˆ’1ξ€·π‘Œπ‘‡12ξ€Έβˆ’1.(18)

Proof. The system (5) is a parameter-dependent system. By Lemma 4, (8) is rewritten asβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£ξ‚π΄βˆ’π‘ƒ0𝑃1𝑃𝐴𝑑𝑃𝐷1𝐡0πœŽπ‘ƒ1𝑃𝐡20𝑃𝐻0βˆ—βˆ’πΌπΆπΆπ‘‘π·1𝐴000000βˆ—βˆ—βˆ’π‘ƒ00πœŽπ‘‡2𝑃00𝑍𝑇𝑁0πœ–π‘‡βˆ—βˆ—βˆ—βˆ’π‘„0000000βˆ—βˆ—βˆ—βˆ—βˆ’π›Ύ2𝐼000000βˆ—βˆ—βˆ—βˆ—βˆ—βˆ’π‘ƒ00000βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ’π›Ώ2𝐼0000βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ’π›Ώ2𝐼000βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ’π‘„βˆ’1⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦00βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ’πœ–πΌ0βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ’πœ–πΌ<0.(19)Next, partition 𝑃 and π‘ƒβˆ’1 as βŽ‘βŽ’βŽ’βŽ£π‘ƒ=𝑅𝑋12𝑋𝑇12𝑋22⎀βŽ₯βŽ₯⎦,π‘ƒβˆ’1=βŽ‘βŽ’βŽ’βŽ£π‘†βˆ’1π‘Œ12π‘Œπ‘‡12π‘Œ22⎀βŽ₯βŽ₯⎦.(20) Define 𝑇1=βŽ‘βŽ’βŽ’βŽ£π‘†βˆ’1πΌπ‘Œπ‘‡120⎀βŽ₯βŽ₯⎦,𝑇2=βŽ‘βŽ’βŽ’βŽ£πΌπ‘…0𝑋𝑇12⎀βŽ₯βŽ₯⎦.(21) Obviously, we have 𝑃𝑇1=𝑇2,𝑇𝑇1𝑃𝑇1=𝑇𝑇1𝑇2. Performing the congruence transformation diag{𝑇𝑇1,𝐼,𝑇𝑇1,𝐼,𝐼,𝑇𝑇1𝐼,𝐼,𝑇𝑇1,𝐼,𝐼} to (19), we obtain the following: Λ𝑖,𝑗<0,𝑖,𝑗=1,…,16,(22) where Ξ›11=βˆ’π‘†βˆ’1,Ξ›12=βˆ’πΌ,Ξ›14=π΄π‘†βˆ’1+π‘π΅πΆπ‘“π‘Œπ‘‡12,Ξ›15=𝐴,Ξ›16=𝐴𝑑,Ξ›17=𝐷0,Ξ›1,10=βˆ’πœŽπ΅,Ξ›1,12=(1βˆ’π‘)𝐡,Ξ›1,15=𝐻,Ξ›21=βˆ’πΌ,Ξ›22=βˆ’π‘…,Ξ›24=(𝑅𝐴+𝑝𝑋12𝐡𝑓𝐢0)π‘†βˆ’1+(𝑝𝑅𝐡𝐢𝑓+𝑋12𝐴𝑓)π‘Œπ‘‡12,Ξ›25=𝑅𝐴+𝑝𝑋12𝐡𝑓𝐢0,Ξ›26=𝑅𝐴𝑑,Ξ›27=𝑅𝐷0,Ξ›2,10=βˆ’πœŽπ‘…π΅,Ξ›2,11=βˆ’πœŽπ‘‹12𝐡𝑓,Ξ›2,12=(1βˆ’π‘)𝑅𝐡,Ξ›2,13=(1βˆ’π‘)𝑋12𝐡𝑓,Ξ›2,15=𝑅𝐻,Ξ›33=βˆ’πΌ,Ξ›34=πΆπ‘†βˆ’1,Ξ›35=𝐢,Ξ›36=𝐢𝑑,Ξ›37=𝐷1,Ξ›41=Λ𝑇41,Ξ›42=Λ𝑇42,Ξ›43=π‘†βˆ’1𝐢𝑇,Ξ›44=βˆ’π‘†βˆ’1,Ξ›45=βˆ’πΌ,Ξ›48=πœŽπ‘Œ12𝐢𝑇𝑓𝐡𝑇,Ξ›49=πœŽπ‘Œ12𝐢𝑇𝑓𝐡𝑇𝑅+πœŽπ‘†βˆ’1𝐢𝑇0𝐡𝑇𝑓𝑋𝑇12,Ξ›4,14=π‘†βˆ’1,Ξ›4,16=πœ–π‘†βˆ’1𝑁𝑇,Ξ›51=𝐴𝑇,Ξ›52=Λ𝑇25,Ξ›53=𝐢𝑇,Ξ›54=βˆ’π‘†,Ξ›55=βˆ’π‘…,Ξ›59=πœŽπΆπ‘‡0𝐡𝑇𝑓𝑋𝑇12,Ξ›5,12=𝐼,Ξ›5,16=πœ–π‘π‘‡,Ξ›61=𝐴𝑇𝑑,Ξ›62=𝐴𝑇𝑑𝑅,Ξ›63=𝐢𝑇𝑑,Ξ›66=βˆ’π‘„,Ξ›71=𝐷𝑇0,Ξ›72=𝐷𝑇0𝑅,Ξ›73=𝐷𝑇1,Ξ›77=βˆ’π›Ύ2𝐼,Ξ›84=Λ𝑇48,Ξ›88=βˆ’π‘†βˆ’1,Ξ›89=βˆ’πΌ,Ξ›94=Λ𝑇49,Ξ›95=Λ𝑇59,Ξ›99=βˆ’π‘…,Ξ›10,1=βˆ’πœŽπ΅π‘‡,Ξ›10,2=βˆ’πœŽπ΅π‘‡π‘…,Ξ›11,2=βˆ’πœŽπ‘‹12𝐡𝑓,Ξ›12,1=(1βˆ’π‘)𝐡𝑇,Ξ›12,2=(1βˆ’π‘)𝐡𝑇𝑅,Ξ›13,2=(1βˆ’π‘)𝑋12𝐡𝑓,Ξ›14,4=π‘†βˆ’1,Ξ›14,5=𝐼,Ξ›10,10=Ξ›11,11=Ξ›12,12=Ξ›13,13=βˆ’π›Ώ2𝐼,Ξ›14,14=βˆ’π‘„βˆ’1,Ξ›15,15=Ξ›16,16=βˆ’πœ–πΌ,Ξ›15,1=𝐻𝑇,Ξ›15,2=𝐻𝑇𝑅,Ξ›16,4=πœ–π‘π‘†βˆ’1,Ξ›16,5=πœ–π‘, the rest of matrix entries are zero.
Applying the congruence transformation diag{𝐼,𝐼,𝐼,𝑆,𝐼,𝐼,𝐼,𝑆,𝐼,𝐼,𝐼,𝐼,𝐼,𝑄,𝐼,𝐼} to (22) again, then (17) is achieved. Therefore, by Theorem 6, the desired result follows immediately.
Next, we will design the scalars 𝛾 and 𝛿 by solving the optimization problem, minπœ–,𝑆,𝑅,𝑄>0,π‘Š1,π‘Š2,π‘Š3𝛾,𝛿,subjectto(17).(23) Using the robust control toolbox, we will get the optimal value of 𝛾 and 𝛿, and the proof is thus, complete.

Remark 8. From Theorems 6 and 7, we know if (17) is feasible, πΌβˆ’π‘…π‘†βˆ’1=𝑋12π‘Œπ‘‡12<0, hence, the square and nonsingular matrices 𝑋12 and π‘Œ12 can be always found (as in [29]). Then, the fault-tolerant controller parameters (18) are obtained.

5. Simulation Example

Consider the system (1) with parameters (as in [22]) as follows:⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦𝐴=βˆ’0.30βˆ’0.300.60.20.500.7,𝐴𝑑=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,⎑⎒⎒⎒⎒⎣012⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βˆ’0.10000.10000.2𝐡=,𝐷0=⎑⎒⎒⎒⎒⎣0⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦⎑⎒⎒⎒⎒⎣1⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝐢0.50.2,𝐻=0.50.50=,𝐢112,𝐢=0.100𝑑=0.100,𝑁=0.100,𝐷1=0.1.(24) Choose the same parameters as [22], packet loss rate is 𝑝=0.1, delay constant is 𝑑=3. With the method in [22], the optimal performance 𝛾 is 3.3339. Using the proposed method in this paper, the optimal performance 𝛾 is 1.3614, which means the smaller performance has been obtained.

Next, let the packet loss rate and fault probability 𝑝=0.3, delay constant is 𝑑=2, under the initial condition π‘₯(0)=[βˆ’0.200.5]𝑇,𝛾=1.6845, and 𝛿=2.0753, the fault-tolerant controller parameters are designed by Theorem 7 as follows.𝐴𝑓=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝐡0.13572.4132βˆ’0.9861βˆ’2.1036βˆ’1.0003βˆ’0.0584βˆ’1.5372βˆ’0.4739βˆ’2.9271𝑓=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βˆ’0.06970.7264βˆ’0.0983,𝐢𝑓=.βˆ’0.1016βˆ’0.07120.2598(25)

From Figure 2, it can be seen that state responses under the designed controller can stabilize the NCS with data packet loss, probabilistic actuator failures, and/or sensor failures, which can illustrate the effectiveness of the proposed method.

734758.fig.002
Figure 2: State responses under the fault-tolerant controller.

6. Conclusion

Motivated by robust FTC problem over networks, a new stochastic NCS model with fault is addressed, which includes the state-delay, model uncertainty, disturbance, random packet dropout, probabilistic sensors failures, and actuators failures. The random packet dropout in both S-C link and C-A link, the sensor failure and the actuator failure are described as a binary random variable. The sufficient condition for asymptotical mean-square stability of the NCS has been derived and the closed-loop NCS satisfies 𝐻∞ performance constraints. Finally, by solving a linear matrix inequality, the fault tolerant controller is designed.

Acknowledgments

The authors would like to acknowledge the National Natural Science Foundation of China under Grant (61174044), and the Shandong Province Natural Science Foundation under Grant (ZR2010FM016). The authors also wish to thank the reviewers for their valuable suggestions.

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