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

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.

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:𝑥_𝑘+1=(𝐴+Δ𝐴)𝑦𝑥_𝑘𝑘+𝐴_𝑑𝑥_𝑘−𝑑+𝐵𝑢_𝑘+𝐷₀𝜔_𝑘,𝑧_𝑘=𝐶₀𝑥_𝑘,=𝐶𝑥_𝑘+𝐶_𝑑𝑥_𝑘−𝑑+𝐷₁𝜔_𝑘,(1) where 𝑥_𝑘∈ℝ^𝑛,𝑦_𝑘∈ℝ^𝑚,𝑢_𝑘∈ℝ^𝑝,𝑧_𝑘∈ℝ^𝑞 denote the state, the sensor measurement, the control input, and the controlled output, respectively. 𝜔_𝑘∈ℝ^𝑟 is the disturbance input belonging to 𝑙₂[0,∞).𝐴,𝐴_𝑑,𝐵,𝐶₀,𝐶,𝐶_𝑑,𝐷₀,𝐷₁, are known real constant matrices with appropriate dimensions. 𝑑>0 is a known delay. Δ𝐴 denotes the model uncertainty, which satisfies Δ𝐴=𝐻𝐹_𝑘𝑁,𝐹^𝑇𝑘𝐹_𝑘<𝐼,𝐹_𝑘 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 𝜏_𝑘 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 𝜃_𝑘 is an identically distributed (i.i.d.) process. 𝜃_𝑘=1 means that there is no packet dropout, and the sensors and the actuators are reliable; 𝜃_𝑘=0 means packet is lost, and the sensors and the actuators have failures. The probability distribution of 𝜃_𝑘 is 𝑃{𝜃_𝑘=0}=𝑝,𝑃{𝜃_𝑘=1}=1−𝑝, where 𝑝∈(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 𝑦_𝑘 will be updated to 𝑦_𝑘 as follows:𝑦_𝑘=𝜃_𝑘𝑦_𝑘+1−𝜃_𝑘𝑦_𝑘−1.(2)

Considering the channel from the controller to the actuator, the control output 𝑢_𝑘 will be updated as follows:𝑢_𝑘=𝜃_𝑘𝑢_𝑘+1−𝜃_𝑘𝑢_𝑘−1.(3)

Remark 3. From expressions (2) and (3), when 𝜃_𝑘=0 at time 𝑘, the sensor measurements and control information at time 𝑘 are missing. The last available measurement 𝑦_𝑘−1 and controller output 𝑢_𝑘−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 𝜃_𝑘 also represents sensor failures and actuator failures, the dynamic fault-tolerant controller is designed𝑥̂_𝑘+1=𝑢_𝑘𝐴_𝑓𝑥̂_𝑘+𝐵_𝑓𝑦_𝑘,=𝐶_𝑓𝑥̂_𝑘.(4)

Define 𝜂_𝑘=[𝑥^𝑇𝑘,𝑥̂^𝑇𝑘]^𝑇,𝑑_𝑘=[𝑢^𝑇𝑘−1,𝑦^𝑇𝑘−1]^𝑇, the stochastic NCS with probabilistic sensor failures and actuator failures𝜂_𝑘+1=𝐴𝜂_𝑘+𝜃_𝑘−𝑝𝐴₂𝜂_𝑘+𝐴_𝑑𝑍𝜂_𝑘−𝑑𝑧_𝑘+𝜃_𝑘−𝑝𝐵₁𝑑_𝑘+𝐵₂𝑑_𝑘+𝐷₁𝜔_𝑘,=𝐶𝜂_𝑘+𝐶_𝑑𝑍𝜂_𝑘−𝑑+𝐷₁𝜔_𝑘,(5) where𝐴=𝐴₁+𝐻𝐹_𝑘𝑁=⎡⎢⎢⎣𝐴𝑝𝐵𝐶^𝑓𝑝𝐵_𝑓𝐶₀𝐴_𝑓⎤⎥⎥⎦+⎡⎢⎢⎣𝐻0⎤⎥⎥⎦𝐹_𝑘𝐴𝑁0₂,=⎡⎢⎢⎣0𝐵𝐶^𝑓𝐵_𝑓𝐶₀0⎤⎥⎥⎦,𝐴^𝑑=⎡⎢⎢⎣𝐴^𝑑𝐵₁0⎤⎥⎥⎦,=⎡⎢⎢⎣−0𝐵−𝐵0_𝑓⎤⎥⎥⎦,𝐵₂=⎡⎢⎢⎣(1−0𝑝)𝐵(1−0𝑝)𝐵_𝑓𝐶=𝐶0,𝐷₁⎤⎥⎥⎦,=⎡⎢⎢⎣𝐷⁰0⎤⎥⎥⎦,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 𝑧_𝑘 satisfies ^∞_𝑘=0𝔼‖‖𝑧_𝑘‖‖²<𝛾^2∞_𝑘=0𝔼‖‖𝜔_𝑘‖‖²+𝛿^2∞_𝑘=0𝔼‖‖𝑑_𝑘‖‖²(7) for all nonzero 𝑑_𝑘,𝜔_𝑘, 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 𝑊=𝑊^𝑇<0,𝐻,𝑁 be matrices of appropriate dimensions, with 𝐹 satisfying 𝐹^𝑇𝐹≤𝐼, then 𝑊+𝐻𝐹𝑁+𝑁^𝑇𝐹^𝑇𝐻^𝑇<0 holds, if and only if there exists a 𝜀>0 such that 𝑊+𝜀𝐻𝐻^𝑇+𝜀⁻¹𝑁^𝑇𝑁<0.

Definition 5. The NCS with faults given by (5) with 𝜔_𝑘=0,𝑑_𝑘=0, is asymptotically mean-square stable, if for any initial state, lim_𝑘→∞𝔼{‖𝑧_𝑘‖²}=0 holds.

Theorem 6. Given 𝛾>0,𝛿>0, the system (5) is asymptotically mean-square stable, and the output 𝑧_𝑘 satisfies (7), if there exist matrices 𝑃=𝑃^𝑇>0, and 𝑄=𝑄^𝑇>0 satisfying ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−∗𝑃−0𝐼𝑃𝐶𝐴𝑃𝐶𝐴_𝑑𝑑𝑃𝐷₁0𝜎𝑃𝐵₁𝑃𝐵₂0∗∗−𝑃0𝐷0₁𝜎𝐴0_𝑇2000𝑃00𝑍^𝑇∗∗∗∗∗∗−∗𝑄−𝛾0₂0000∗∗∗∗∗∗∗∗∗∗𝐼−∗0𝑃−𝛿00₂0000∗∗∗∗∗∗∗𝐼−𝛿0₂0∗∗∗∗∗∗∗∗𝐼−𝑄0₋₁⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0.(8)

Proof. Let Θ_𝑘=[𝜂^𝑇𝑘,𝜂^𝑇𝑘−1,…,𝜂^{𝑇𝑘−𝑑}]^𝑇, for all nonzero 𝜂_𝑘, consider the Lyapunov function 𝑉_𝑘(Θ_𝑘)=𝑉_1𝑘+𝑉_2𝑘 with 𝑉_1𝑘=𝜂^𝑇𝑘𝑃𝜂_𝑘,𝑉_2𝑘=^𝑘−1_𝑖=_𝑘−𝑑𝜂^𝑇𝑖𝑍^𝑇𝑄𝑍𝜂_𝑖.(9) The difference of 𝑉_𝑘 is Δ𝑉_1𝑘==𝜂𝔼_𝑇𝑘𝑉_1𝑘Θ_𝑘+1−𝑉_1𝑘Θ_𝑘𝐴^𝑇𝑃𝐴𝜂_𝑘+𝜂^𝑇𝑘𝐴^𝑇𝑃𝐴_𝑑𝑍𝜂_𝑘−𝑑+𝜂^𝑇𝑘𝐴^𝑇𝑃𝐵₂𝑑_𝑘+𝜂^𝑇𝑘𝐴^𝑇𝑃𝐷₁𝜔_𝑘+𝔼𝜃_𝑘−𝑝²𝜂^𝑇𝑘𝐴^𝑇2𝑃𝐴₂𝜂_𝑘+𝔼𝜃_𝑘−𝑝²𝜂^𝑇𝑘𝐴^𝑇2𝑃𝐵₁𝑑_𝑘+𝜂^{𝑇𝑘−𝑑}𝑍^𝑇𝐴^𝑇𝑑𝑃𝐴𝜂_𝑘+𝜂^{𝑇𝑘−𝑑}𝑍^𝑇𝐴^𝑇𝑑𝑃𝐴_𝑑𝑍𝜂_𝑘−𝑑+𝜂^{𝑇𝑘−𝑑}𝑍^𝑇𝐴^𝑇𝑑𝑃𝐵₂𝑑_𝑘+𝜂^{𝑇𝑘−𝑑}𝑍^𝑇𝐴^𝑇𝑑𝑃𝐷₁𝜔_𝑘+𝔼𝜃_𝑘−𝑝²𝑑^𝑇𝑘𝐵^𝑇1𝑃𝐴₂𝜂_𝑘+𝔼𝜃_𝑘−𝑝²𝑑^𝑇𝑘𝐵^𝑇1𝑃𝐵₁𝑑_𝑘+𝑑^𝑇𝑘𝐵^𝑇2𝑃𝐴𝜂_𝑘+𝑑^𝑇𝑘𝐵^𝑇2𝑃𝐴_𝑑𝑍𝜂_𝑘−𝑑+𝑑^𝑇𝑘𝐵^𝑇2𝑃𝐵₂𝑑_𝑘+𝑑^𝑇𝑘𝐵^𝑇2𝑃𝐷₁𝜔_𝑘+𝜔^𝑇𝑘𝐷^𝑇1𝑃𝐴𝜂_𝑘+𝜔^𝑇𝑘𝐷^𝑇1𝑃𝐴_𝑑𝑍𝜂_𝑘−𝑑+𝜔^𝑇𝑘𝐷^𝑇1𝑃𝐵₂𝑑_𝑘+𝜔^𝑇𝑘𝐷^𝑇1𝑃𝐷₁𝜔_𝑘−𝜂^𝑇𝑘𝑃𝜂_𝑘Δ𝑉_2𝑘==𝜂𝔼_𝑇𝑘𝑉_2𝑘Θ_𝑘+1−𝑉_2𝑘Θ_𝑘𝑍^𝑇𝑄𝑍𝜂_𝑘−𝜂^{𝑇𝑘−𝑑}𝑍^𝑇𝑄𝑍𝜂_𝑘−𝑑.(10)
For 𝔼{𝜃_𝑘−𝑝}=0, and denote 𝔼{(𝜃_𝑘−𝑝)²}=𝜎², we have Δ𝑉_𝑘=⎡⎢⎢⎢⎢⎢⎢⎣𝜂^𝑘𝑍𝜂_𝑘−𝑑𝑑_𝑘𝜔_𝑘⎤⎥⎥⎥⎥⎥⎥⎦^𝑇⎡⎢⎢⎢⎢⎢⎢⎣Φ₁₁Φ₁₂Φ₁₃Φ₁₄∗Φ₂₂Φ₂₃Φ₂₄∗∗Φ₃₃Φ₃₄∗∗∗Φ₄₄⎤⎥⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎢⎣𝜂_𝑘𝑍𝜂_𝑘−𝑑𝑑_𝑘𝜔_𝑘<𝜁_𝑇𝑘⎤⎥⎥⎥⎥⎥⎥⎦Φ_𝑖𝑗𝜁_𝑘(𝑖,𝑗=1,…,4),(11) where 𝜎𝜁_𝑘2=[𝜂^𝑇𝑘,𝜂^{𝑇𝑘−𝑑}𝑍^𝑇,𝑑^𝑇𝑘,𝜔^𝑇𝑘]^𝑇,Φ₁₁=𝐴^𝑇𝑃𝐴+𝑍^𝑇𝑄𝑍+𝜎²𝐴^𝑇2𝑃𝐴₂−𝑃,Φ₁₂=𝐴^𝑇𝑃𝐴_𝑑,Φ₁₃=𝐴^𝑇𝑃𝐵₂+𝜎₂𝐴^𝑇2𝑃𝐵₁,Φ₁₄=𝐴^𝑇𝑃𝐷₁,Φ₂₂=𝐴^𝑇𝑑𝑃𝐴_𝑑−𝑄,Φ₂₃=𝐴^𝑇𝑑𝑃𝐵₂,Φ₂₄=𝐴^𝑇𝑑𝑃𝐷₁,Φ₃₃=𝐵^𝑇2𝑃𝐵₂+𝐵^𝑇1𝑃𝐵₁,Φ₃₄=𝐵^𝑇2𝑃𝐷₁,Φ₄₄=𝐷^𝑇1𝑃𝐷₁,∗ is implicitly defined by the fact that the matrix is symmetric.
When 𝜔_𝑘=0,𝑑_𝑘=0, (11) is rewritten as Δ𝑉_{𝑘,𝜔𝑘=𝑑𝑘=0}=⎡⎢⎢⎣𝐴^𝑇𝑃𝐴+𝑍^𝑇𝑄𝑍+𝜎²𝐴^𝑇2𝑃𝐴₂−𝑃𝐴^𝑇𝑃𝐴_𝑑=Π.𝐴^𝑇𝑑𝑃𝐴𝐴^𝑇𝑑𝑃𝐴_𝑑−𝑄⎤⎥⎥⎦(12)
From (8) and the Schur complement theorem, Π<0 is arrived. Therefore, for all nonzero 𝜂_𝑘, we have Δ𝑉_𝑘<0, then the NCS (5) with sensor and actuator fault is asymptotically mean-square stable.
Next, For any nonzero 𝜔_𝑘,𝑑_𝑘, it follows from (5), (8), and (11) that 𝔼−𝑉𝛾₂Θ_𝑘+1−𝔼𝑉Θ_𝑘+𝔼𝑧^𝑇𝑘𝑧_𝑘≤𝔼𝔼𝜁_𝑇𝑘𝜔^𝑇𝑘𝜔_𝑘−𝛿²𝔼𝑑^𝑇𝑘𝑑_𝑘Φ_𝑖𝑗𝜁_𝑘+𝜂^𝑇𝑘𝐶^𝑇𝐶𝜂_𝑘+𝜂^𝑇𝑘𝐶^𝑇𝐶_𝑑𝑍𝜂_𝑘−𝑑+𝜂^𝑇𝑘𝐶^𝑇𝐷₁𝜔_𝑘+𝜂^{𝑇𝑘−𝑑}𝑍^𝑇𝐶^𝑇𝑑𝐶𝜂_𝑘+𝜂^{𝑇𝑘−𝑑}𝑍^𝑇𝐶^𝑇𝑑𝐶_𝑑𝑍𝜂_𝑘−𝑑+𝜂^{𝑇𝑘−𝑑}𝑍^𝑇𝐶^𝑇𝑑𝐷₁𝜔_𝑘+𝜔^𝑇𝑘𝐷^𝑇1𝐶𝜂_𝑘+𝜔^𝑇𝑘𝐷^𝑇1𝐶_𝑑𝑍𝜂_𝑘−𝑑+𝜔^𝑇𝑘𝐷^𝑇1𝐷₁𝜔_𝑘−𝛾²𝜔^𝑇𝑘𝜔_𝑘−𝛿²𝑑^𝑇𝑘𝑑_𝑘.(13)
Then, we have 𝔼−𝑉𝛾₂Θ_𝑘+1−𝔼𝑉Θ_𝑘+𝔼𝑧^𝑇𝑘𝑧_𝑘𝔼𝜔^𝑇𝑘𝜔_𝑘−𝛿²𝔼𝑑^𝑇𝑘𝑑_𝑘<0.(14) Now, summing (14) from 0 to ∞ with respect to 𝑘 yields ^∞_𝑘=0𝔼𝑧^𝑇𝑘𝑧_𝑘<𝛾^2∞_𝑘=0𝔼𝜔^𝑇𝑘𝜔_𝑘+𝛿^2∞_𝑘=0+𝔼𝑉₀𝔼𝑑^𝑇𝑘𝑑_𝑘−𝔼𝑉_∞.(15) Since the system (5) is asymptotically mean-square stable, we can get that the following inequality: ^∞_𝑘=0𝔼𝑧^𝑇𝑘𝑧_𝑘<𝛾^2∞_𝑘=0𝔼𝜔^𝑇𝑘𝜔_𝑘+𝛿^2∞_𝑘=0𝔼𝑑^𝑇𝑘𝑑_𝑘(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 𝑧_𝑘 satisfies the 𝐻_∞ constraints (7), if there exist real scalars 𝜀>0, and matrices 𝑆=𝑆^𝑇>0,𝑄=𝑄^𝑇>0, and 𝑅=𝑅^𝑇>0, and real matrices 𝑊₁,𝑊₂, and 𝑊₃, such that the following inequality holds: Ω_𝑖𝑗<0,𝑖,𝑗=1,…,16,(17) where (Ω1−₁₁𝑝=)𝐵−,Ω𝑆⁻¹_1,15,Ω₁₂=−𝐼,Ω₁₄=𝐴+𝑝𝐵𝑊₃,Ω₁₅=𝐴,Ω₁₆=𝐴_𝑑,Ω₁₇=𝐷₀,Ω_1,10=−𝜎𝐵,Ω_1,12=𝑅𝐷₀=𝐻,Ω₂₁=−𝐼,Ω₂₂=−𝑅,Ω₂₄=𝑅𝐴+𝑝𝑊₂𝐶₀+𝑊₁,Ω₂₅=𝑅𝐴+𝑝𝑊₂𝐶₀,Ω₂₆=𝑅𝐴_𝑑,Ω₂₇=𝐶,Ω,₃₅Ω_2,10=−𝜎𝑅𝐵,Ω_2,11=−𝜎𝑊₂,Ω_2,12=(1−𝑝)𝑅𝐵,Ω_2,13=(1−𝑝)𝑊₂,Ω_2,15=𝑅𝐻,Ω₃₃=−𝐼,Ω₃₄=𝜎𝑊_𝑇3=𝐶,Ω₃₆=𝐶_𝑑,Ω₃₇=𝐷₁,Ω₄₁=Ω^𝑇14,Ω₄₂=Ω^𝑇24,Ω₄₃=𝐶^𝑇,Ω₄₄=−𝑆,Ω₄₅=−𝑆,Ω₄₈=−𝑆,Ω𝐵^𝑇₅₅,Ω₄₉=𝜎𝑊^𝑇3𝐵^𝑇𝑅+𝜎𝐶^𝑇0𝑊^𝑇2,Ω_4,14=𝐼,Ω_4,16=𝜖𝑁^𝑇,Ω₅₁=𝐴^𝑇,Ω₅₂=Ω^𝑇25,Ω₅₃=𝐶^𝑇,Ω₅₄=−𝑄,Ω₇₁=−𝑅,Ω₅₉=𝜎𝐶^𝑇0𝑊^𝑇2,Ω_5,14=𝑄,Ω_5,16=𝜖𝑁^𝑇,Ω₆₁=𝐴^𝑇𝑑,Ω₆₁=𝐴^𝑇𝑑𝑅,Ω₆₃=𝐶^𝑇𝑑,Ω₆₆=Ω_𝑇49=𝐷^𝑇0,Ω₇₂=𝐷^𝑇0𝑅,Ω₇₃=𝐷^𝑇1,Ω₇₇=−𝛾²𝐼,Ω₈₄=Ω^𝑇48,Ω₈₈=−𝑆,Ω₈₉=−𝑆,Ω₉₄=−𝛿₂,Ω₉₅=Ω^𝑇59,Ω₉₉=−𝑅,Ω_10,1=−𝜎𝐵^𝑇,Ω_10,2=−𝜎𝐵^𝑇𝑅,Ω_10,10=Ω_11,11=Ω_12,12=Ω_13,13=𝐻_𝑇𝐼,Ω_11,2=Ω^𝑇2,11,Ω_12,1=Ω^𝑇12,1,Ω_12,2=Ω^𝑇2,12,Ω_13,2=Ω^𝑇2,13,Ω_14,4=𝐼,Ω_14,5=𝑄^𝑇,Ω_14,14=−𝐼,Ω_15,1=,Ω_15,2=𝐻^𝑇𝑅,Ω_15,15=Ω_16,16=−𝜀𝐼,Ω_16,4=Ω_16,5=𝜖𝑁, the rest of matrix entries are zero.
The fault-tolerant controller parameters are 𝐴𝐵^𝑓_𝑓=𝑋⁻¹¹²𝑊¹−𝑝𝑅𝐵𝑊₃𝑆⁻¹𝑌^𝑇12⁻¹,=𝑋⁻¹¹²𝑊₂,𝐶_𝑓=𝑊₃𝑆⁻¹𝑌^𝑇12⁻¹.(18)

Proof. The system (5) is a parameter-dependent system. By Lemma 4, (8) is rewritten as⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−∗𝑃−0𝐼𝑃𝐶𝐴₁𝑃𝐶𝐴_𝑑𝑑𝑃𝐷₁0𝜎𝑃𝐵₁𝑃𝐵₂0𝑃𝐻0∗∗−𝑃0𝐷0₁𝜎𝐴0_𝑇200000𝑃00𝑍^𝑇0𝜖𝑁^𝑇∗∗∗∗∗∗−∗𝑄−𝛾0₂000000∗∗∗∗∗∗∗∗∗∗𝐼−∗0𝑃−𝛿00₂00000000∗∗∗∗∗∗∗𝐼−𝛿0₂000∗∗∗∗∗∗∗∗𝐼−𝑄0₋₁00∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗−∗0𝜖𝐼−00𝜖𝐼⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0.(19)Next, partition 𝑃 and 𝑃⁻¹ as 𝑃=⎡⎢⎢⎣𝑅𝑋¹²𝑋^𝑇12𝑋₂₂⎤⎥⎥⎦,𝑃₋₁=⎡⎢⎢⎣𝑆⁻¹𝑌₁₂𝑌^𝑇12𝑌₂₂⎤⎥⎥⎦.(20) Define 𝑇₁=⎡⎢⎢⎣𝑆⁻¹𝑌_𝑇12𝐼0⎤⎥⎥⎦,𝑇²=⎡⎢⎢⎣𝐼0𝑋𝑅_𝑇12⎤⎥⎥⎦.(21) Obviously, we have 𝑃𝑇₁=𝑇₂,𝑇^𝑇1𝑃𝑇₁=𝑇^𝑇1𝑇₂. Performing the congruence transformation diag{𝑇^𝑇1,𝐼,𝑇^𝑇1,𝐼,𝐼,𝑇^𝑇1𝐼,𝐼,𝑇^𝑇1,𝐼,𝐼} to (19), we obtain the following: Λ_𝑖,𝑗<0,𝑖,𝑗=1,…,16,(22) where (Λ1₁₁−=𝑝)𝐵−,𝑆Λ⁻¹,_1,15Λ₁₂=−𝐼,Λ₁₄=𝐴𝑆⁻¹+𝑝𝐵𝐶_𝑓𝑌^𝑇12,Λ₁₅=𝐴,Λ₁₆=𝐴_𝑑,Λ₁₇=𝐷₀,Λ_1,10=−𝜎𝐵,Λ_1,12=𝑅𝐴+𝑝𝑋₁₂=𝐻,Λ₂₁=−𝐼,Λ₂₂=−𝑅,Λ₂₄=(𝑅𝐴+𝑝𝑋₁₂𝐵_𝑓𝐶₀)𝑆⁻¹+(𝑝𝑅𝐵𝐶_𝑓+𝑋₁₂𝐴_𝑓)𝑌^𝑇12,Λ₂₅=(1−𝑝)𝑋₁₂𝐵_𝑓𝐶₀,Λ₂₆=𝑅𝐴_𝑑,Λ₂₇=𝑅𝐷₀,Λ_2,10=−𝜎𝑅𝐵,Λ_2,11=−𝜎𝑋₁₂𝐵_𝑓,Λ_2,12=(1−𝑝)𝑅𝐵,Λ_2,13=𝑆₋₁𝐵_𝑓,Λ_2,15=𝑅𝐻,Λ₃₃=−𝐼,Λ₃₄=𝐶𝑆⁻¹,Λ₃₅=𝐶,Λ₃₆=𝐶_𝑑,Λ₃₇=𝐷₁,Λ₄₁=Λ^𝑇41,Λ₄₂=Λ^𝑇42,Λ₄₃=𝜖𝑆𝐶^𝑇₋₁,Λ₄₄=−𝑆⁻¹,Λ₄₅=−𝐼,Λ₄₈=𝜎𝑌₁₂𝐶^𝑇𝑓𝐵^𝑇,Λ₄₉=𝜎𝑌₁₂𝐶^𝑇𝑓𝐵^𝑇𝑅+𝜎𝑆⁻¹𝐶^𝑇0𝐵^𝑇𝑓𝑋^𝑇12,Λ_4,14=𝑆⁻¹,Λ_4,16=𝜖𝑁_𝑇𝑁^𝑇,Λ₅₁=𝐴^𝑇,Λ₅₂=Λ^𝑇25,Λ₅₃=𝐶^𝑇,Λ₅₄=−𝑆,Λ₅₅=−𝑅,Λ₅₉=𝜎𝐶^𝑇0𝐵^𝑇𝑓𝑋^𝑇12,Λ_5,12=𝐼,Λ_5,16=Λ_𝑇48,Λ₆₁=𝐴^𝑇𝑑,Λ₆₂=𝐴^𝑇𝑑𝑅,Λ₆₃=𝐶^𝑇𝑑,Λ₆₆=−𝑄,Λ₇₁=𝐷^𝑇0,Λ₇₂=𝐷^𝑇0𝑅,Λ₇₃=𝐷^𝑇1,Λ₇₇=−𝛾²𝐼,Λ₈₄=−𝜎𝑋,Λ₈₈₁₂=−𝑆⁻¹,Λ₈₉=−𝐼,Λ₉₄=Λ^𝑇49,Λ₉₅=Λ^𝑇59,Λ₉₉=−𝑅,Λ_10,1=−𝜎𝐵^𝑇,Λ_10,2=−𝜎𝐵^𝑇𝑅,Λ_11,2=Λ_11,11𝐵_𝑓,Λ_12,1=(1−𝑝)𝐵^𝑇,Λ_12,2=(1−𝑝)𝐵^𝑇𝑅,Λ_13,2=(1−𝑝)𝑋₁₂𝐵_𝑓,Λ_14,4=𝑆⁻¹,Λ_14,5=𝐼,Λ_10,10=𝜖𝑁𝑆₋₁=Λ_12,12=Λ_13,13=−𝛿²𝐼,Λ_14,14=−𝑄⁻¹,Λ_15,15=Λ_16,16=−𝜖𝐼,Λ_15,1=𝐻^𝑇,Λ_15,2=𝐻^𝑇𝑅,Λ_16,4=,Λ_16,5=𝜖𝑁, 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, min_{𝜖,𝑆,𝑅,𝑄>0,𝑊1,𝑊2,𝑊3}𝛾,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, 𝐼−𝑅𝑆⁻¹=𝑋₁₂𝑌^𝑇12<0, hence, the square and nonsingular matrices 𝑋₁₂ and 𝑌₁₂ 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:𝐴=𝐵⎡⎢⎢⎢⎢⎣−=000.5⎡⎢⎢⎢⎢⎣.3012⎤⎥⎥⎥⎥⎦0,00.6−00𝐷0..72.₀3⎤⎥⎥⎥⎥⎦,𝐴_𝑑=⎡⎢⎢⎢⎢⎣−000.1000.1000.2⎤⎥⎥⎥⎥⎦,𝐶=⎡⎢⎢⎢⎢⎣000..52⎤⎥⎥⎥⎥⎦,𝐻=⎡⎢⎢⎢⎢⎣001..55₀⎤⎥⎥⎥⎥⎦,𝐶=112,𝐶=0.100_𝑑,=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 𝑥(0)=[−0.200.5]^𝑇,𝛾=1.6845, and 𝛿=2.0753, the fault-tolerant controller parameters are designed by Theorem 7 as follows.𝐵𝐴_𝑓=⎡⎢⎢⎢⎢⎣−−021.1..15303537762−−210.4..04107303239−−−020...09952887641_𝑓1⎤⎥⎥⎥⎥⎦,=⎡⎢⎢⎢⎢⎣−−000.7..00269698473⎤⎥⎥⎥⎥⎦,𝐶_𝑓=−0.1016−0.07120.2598.(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.

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.