Integrity Design for Networked Control Systems with Actuator Failures and Data Packet Dropouts

Abstract

The integrity design problem of fault tolerant control for networked control system (NCS) with actuator failures and data packet dropouts is investigated. The data packet dropouts in both sensor-controller (S-C) and controller-actuator (C-A) links are described by two switches, which can be modeled as a discrete event system with known rate. After introducing the matrix of actuator failure, the closed-loop NCS is developed, which can be viewed as asynchronous dynamical systems (ADSs). Then, the sufficiency of exponential stability for the NCS is obtained based on the theory of ADSs. The output feedback controllers that can guarantee system stability are also proposed. Finally, two numerical examples are given to demonstrate the validity of our proposed approach.

1. Introduction

Along with the rapid development of communication networks, a great amount of effort has been made on fault-tolerant control (FTC) problems for networked control systems (NCSs) recently. NCSs are distributed systems, which are comprised of controlled plants actuators, sensors, and controllers. The essential feature of NCSs is that the information is exchanged through some form of communication networks (as in [1–5]). The use of a shared network to connect spatially distributed devices results in flexible architectures and generally reduces installation and maintenance costs. Consequently, NCSs have been widely applied to many complex control systems, for example, unmanned aerial vehicles, avionics industries, remote surgery, and rapid transit trains.

However, the insertion of networks also brings some new issues, such as network-induced delay (as in [1, 6–12]) and data packet dropout (as in [1, 13–15]), which make NCSs more vulnerable to faults than conventional systems. As we know, research on FTC strives to make the system stable and retain acceptable performance under the system faults. An important part of FTC is the one specializing in actuator faults, FTC techniques dealing with actuator faults are relevant for practical application and have already been the subject of many publications (as in [16–19]). Therefore, a suitable architecture for FTC of NCSs must take the dynamical behavior of network into consideration (as in [8–12, 14, 15, 20–23]).

A wide range of research has recently been reported dealing with problems related to the FTC for NCSs with network-induced delay (as in [8–11]). As compared to the plentiful works on FTC for NCSs with network-induced delay, only a few attention has been paid to the study of FTC for NCSs with data packet dropout (as in [14, 15, 23]). As we know, packet dropouts can be modeled either as stochastic or deterministic phenomena, such as packet dropouts in both S-C link and C-A link are characterized by Bernoulli process in [14]. Finite-state Markovian process is used to model correlated packet dropouts in [23]. Deterministic models for data dropouts have also been proposed, specified in terms of time averages as in [1, 15], such as packet dropouts in the S-C link modeled by ADSs, discussed in [15].

From the above description, considering data packet dropout in both S-C and C-A links, the problem of FTC for NCSs with actuator failures is still a challenging problem. Thus, this paper is devoted to study FTC for NCSs with actuator failures and packet dropout in both S-C and C-A links. According to the method in [1], data packet dropouts in both S-C and C-A links are described by two independent switches, which can be modeled as a discrete event system with known rate. After introducing the matrix of actuator failures, the model of closed-loop NCSs with actuator failures is addressed. It should be mentioned that the closed-loop NCSs can be viewed as ADSs. Then, based on the theory of ADSs, the sufficiency of exponential stability for such NCSs is derived and the output feedback controllers that can guarantee system stability are presented. Finally, two numerical examples are provided to show the effectiveness of the present approach.

The rest of the paper is organized as follows. Section 2 gives the formulation of the problem, and the NCSs with data packet dropout is modeled as a discrete event system. The integrity design of FTC for the NCSs is addressed in Section 3. The output-feedback controllers are also presented in Section 3. Two numerical examples to testify the effectiveness of the proposed method are shown in Section 4. Section 5 makes the concluding remarks.

2. Problem Formulation

The LTI discrete-time system is considered as follows: 𝑥_𝑘+1=𝐴𝑥_𝑘+𝐵𝑢_𝑘,𝑦_𝑘=𝐶𝑥_𝑘, (1) where 𝑥_𝑘∈𝐑^𝑛, 𝑦_𝑘∈𝐑^𝑚, and 𝑢_𝑘∈𝐑^𝑝 denote the state vector, the measurable output vector, and the control input vector, respectively. 𝐴,𝐵, and𝐶 are known matrices with appropriate dimensions.

Networks can be viewed as unreliable data transmission paths, where packet collision and network node failure occasionally occur. When there is a packet collision, instead of repeated retransmission attempts, it might be advantageous to drop the old packet and transmit a new one. Hence, in this study, we focus on the phenomenon of data packet dropout. Figure 1 shows a typical closed loop of NCSs with data packet dropout. Some assumptions in this paper are as follows. (1)The controller and the actuator are event-driven.(2)Data are single-packet transmission, and the dropout packet rate is known.

Figure 1 

Schematic description of NCSs with random data packet dropouts.

From Figure 2, it can been seen that data packet dropouts in the network can be treated as switches that close at a certain rate. When these switches are closed (position 𝑆₁,𝑆₃), data packets containing 𝑦_𝑘, 𝑢_𝑘 are transmitted successfully, whereas when they are open (position 𝑆₂,𝑆₄), the outputs of these switches are held at the previous values 𝑦_𝑘−1,𝑢_𝑘−1, stored in the buff, and the current data packet is missing. Thus, the dynamic model of this network with data packet dropout is given by 𝑆₁:𝑦_𝑘=𝑦_𝑘,𝑆₂:𝑦_𝑘=𝑦_𝑘−1,𝑆₃:𝑢_𝑘=−𝐾_𝜃𝑘𝑦_𝑘,𝑆₄:𝑢_𝑘=𝑢_𝑘−1, (2) where 𝜃_𝑘 is a four-state time-homogenous Markov chain which takes values in index set Υ={1,2,3,4}; the rate at which the event 𝜃_𝑘=𝑗 occurs is defined by the following: 𝑝_𝑖𝑗=Pr𝜃_𝑘+1=𝑗∣𝜃_𝑘=𝑖_4𝑗=1,𝑝_𝑖𝑗=1,∀𝑖,𝑗∈Υ,𝑝_𝑖𝑗≥0. (3)

Figure 2 

Four-state discrete event system.

From the above discussion, data packet dropouts in the network can be viewed as two independent switches; that is, we can obtain the network model (2) by analyzing the features of these switches. Meanwhile, it is worth noting that the network model (2) can also be a discrete event system, which contains four events as follows. Event 1: 𝑆₁𝑆₃, no data packets are lost in the whole loop.  Event 2: 𝑆₂𝑆₃, data packets are lost in the S-C link.  Event 3: 𝑆₁𝑆₄, data packets are lost in the C-A link.  Event 4: 𝑆₂𝑆₄, data packets’s loss exist in both the S-C and the C-A link.

Actually, sensor failures model can be transformed into actuator failures model by changing format of system with faults. So, in this paper, we only discuss actuator failures model. Usually, when a system is under actuator failures, a matrix 𝐿=diag{𝑙₁,𝑙₂…,𝑙_𝑝} will be introduced into this system between its coefficient matrix and the feedback control matrix to describe actuator failures. We have 𝑙_𝑗=∈()1,noactuatorfault,0,the𝑗thactuatorhasfault,0,1,the𝑗thactuatorhaspartialfault, (4) where 𝑙_𝑗=1 denotes that the 𝑗th actuator is in the normal situation, 𝑙_𝑗=0 denotes that the 𝑗th actuator is in the full fault situation, and 𝑙_𝑗∈(0,1) denotes that the 𝑗th actuator is in partial fault situation.

Define a new augmented vector 𝑧_𝑘=[𝑥^𝑇_𝑘,𝑦^𝑇_𝑘−1,𝑢^𝑇_𝑘−1]^𝑇; then the closed-loop NCS model with actuator failures will be 𝑧_𝑘+1=Φ_𝜃𝑘𝑧_𝑘, (5) where Φ₁=𝐴−𝐵𝐿𝐾₁𝐶00𝐶00−𝐿𝐾₁,Φ𝐶00²=,Φ𝐴0𝐵𝐶0000𝐼₃=𝐴−𝐵𝐿𝐾₃00𝐼00−𝐿𝐾₃0,Φ₄=.𝐴0𝐵0𝐼000𝐼 (6)

Up till now, the state space expressions for different discrete events of the closed-loop NCS are acquired.

3. Integrity Design of FTC for the NCS

In this section, the integrity design of FTC for the NCS (5) is discussed. As we know, the integrity design is that system can keep asymptotically stable under some sensors failures or actuators failures and is one of the important means for the FTC. Meanwhile, because system (5) can also be viewed as ADSs, which are systems that incorporate continuous and discrete dynamics, the continuous dynamics are governed by differential or difference equations, whereas the discrete dynamics are governed by finite automata that are driven asynchronously by external discrete events with fixed rates [24, 25]. Then before studying the problem of integrity design, an important lemma about ADSs should be introduced.

Lemma 1 (as in [24]). Given an ADS as follows: 𝑥(𝑘+1)=𝑓_𝑠(𝑥(𝑘)),𝑠=1,2,…,𝑁. (7)
Suppose a Lyapunov function 𝑉∶𝑅_𝑛→𝑅₊ and 𝛽₁‖𝑥‖²≤𝑉(𝑥)≤𝛽₂‖𝑥‖², (8) where 𝛽_1,2>0. The ADS (7) is exponentially stable with the decay rate 𝛿 if there exist the scalars 𝛼₁,𝛼₂,…,𝛼_𝑀>0, satisfying 𝛼^𝑟1₁𝛼^𝑟2₂⋯𝛼^𝑟𝑀_𝑀𝑉(𝑥((𝑥(𝑘≤𝛼>𝛼>1,𝑘+1))−𝑉))_𝑠⁻²₁𝛼_𝑠⁻²₂⋯𝛼_𝑠⁻²_𝑀𝑠−1𝑉(𝑥(𝑘)), (9) where 𝑀_𝑠 is the number of events to each discrete state, 𝑀 is the number of events in total, and 𝑟_1,…,𝑀 are corresponding to each event rate.

Remark 2. Note that the 𝛼_𝑠𝑗,𝑗=1,…,𝑀_𝑠 belongs to the set {𝛼₁,𝛼₂,…,𝛼_𝑀}. And Lemma 1 implies that the Lyapunov function 𝑉(𝑥) does not have to decrease monotonically at some rate 𝛿 along all trajectories, but rather it should decrease at a rate 𝛿 on the average.

Theorem 3. If 𝑟₁,𝑟₂ represents the rate of events 𝑆₁,𝑆₃, respectively and 𝑟₁̂,𝑟₂̂,𝑟₃̂,𝑟₄̂ are the rates of events 𝑆₁𝑆₃, 𝑆₁𝑆₄, 𝑆₂𝑆₃, 𝑆₂𝑆₄ in NCS (5), then 𝑟₁̂,𝑟₂̂,𝑟₃̂,𝑟₄̂ can be described as 𝑟₁̂=𝑟₁𝑟₂,𝑟₂̂=𝑟₁1−𝑟₂,𝑟₃̂=1−𝑟₁𝑟₂,𝑟₄̂=1−𝑟₁1−𝑟₂. (10)

Proof. Since the switches in Figure 1 are independent, then 𝑆₁,𝑆₂,𝑆₃,𝑆₄ are also independent events. According to the probability theory, the probabilities of these four independent events can be obtained as follows: Pr𝑆₁𝑆₃=Pr𝑆₁Pr𝑆₃=𝑟₁𝑟₂,Pr𝑆₁𝑆₄=Pr𝑆₁Pr𝑆₄=𝑟₁1−𝑟₂,Pr𝑆₂𝑆₃=Pr𝑆₂Pr𝑆₃=1−𝑟₁𝑟₂,Pr𝑆₂𝑆₄=Pr𝑆₂Pr𝑆₄=1−𝑟₁1−𝑟₂. (11)
Then, Theorem 3 can be proved.

Theorem 4. The NCS (5) is exponentially stable with the decay rate 𝛿 if there exist 𝛿>0, 𝛼_𝑖>0,𝑖=1,2,3,4, positive definite matrices 𝑃, 𝑄, 𝑅, and matrices 𝐾₁, 𝐾₃, satisfying the following: 𝛼^𝑟1𝑟2₁𝛼^{𝑟1(1−𝑟2)}₂𝛼^{(1−𝑟1)𝑟2}₃𝛼^{(1−𝑟1)(1−𝑟2)}₄>𝛿>1,−𝛼₁⁻²𝑃⁻¹00𝑃⁻¹𝐴^𝑇−𝑃⁻¹𝐶^𝑇𝐾^𝑇₁𝐿^𝑇𝐵^𝑇𝑃⁻¹𝐶^𝑇𝑃⁻¹𝐶^𝑇𝐾^𝑇₁𝐿^𝑇0−𝛼₁⁻²𝑄⁻¹000000−𝛼₁⁻²𝑅⁻¹000𝐴𝑃⁻¹−𝐵𝐿𝐾₁𝐶𝑃⁻¹00−𝑃⁻¹00𝐶𝑃⁻¹000−𝑄⁻¹0𝐿𝐾₁𝐶𝑃⁻¹0000−𝑅⁻¹<0,−𝛼₂⁻²𝑃⁻¹00𝑃⁻¹𝐴^𝑇𝑃⁻¹𝐶^𝑇00−𝛼₂⁻²𝑄⁻¹0000001−𝛼₂⁻²𝑅⁻¹𝑅⁻¹𝐵^𝑇0𝑅⁻¹𝐴𝑃⁻¹0𝐵𝑅⁻¹−𝑃⁻¹00𝐶𝑃⁻¹000−𝑄⁻¹000𝑅⁻¹00−𝑅⁻¹<0,−𝛼₃⁻²𝑃⁻¹00𝑃⁻¹𝐴^𝑇0001−𝛼₃⁻²𝑄⁻¹0−𝑄⁻¹𝐾^𝑇₃𝐿^𝑇𝐵^𝑇𝑄⁻¹−𝑄⁻¹𝐾^𝑇₃𝐿^𝑇00𝛼₃⁻²𝑅⁻¹000𝐴𝑃⁻¹−𝐵𝐿𝐾₃𝑄⁻¹0−𝑃⁻¹000𝑄⁻¹00−𝑄⁻¹00−𝐿𝐾₃𝑄⁻¹000−𝑅⁻¹<0,−𝛼₄⁻²𝑃⁻¹00𝑃⁻¹𝐴^𝑇0001−𝛼₄⁻²𝑄⁻¹00𝑄⁻¹0001−𝛼₄⁻²𝑅⁻¹𝑅⁻¹𝐵^𝑇0𝑅⁻¹𝐴𝑃⁻¹0𝐵𝑅⁻¹−𝑃⁻¹000𝑄⁻¹00−𝑄⁻¹000𝑅⁻¹00−𝑅⁻¹<0. (12)(13)(14)(15)(16)

Proof. From Lemma 1 and (10), we can get (12) directly.
Next, for nonzero 𝑥_𝑘,𝑦_𝑘−1,𝑢_𝑘−1, choose the Lyapunov function as follows: 𝑉=𝑥^𝑇_𝑘𝑃𝑥_𝑘+𝑦^𝑇_𝑘−1𝑄𝑦_𝑘−1+𝑢^𝑇_𝑘−1𝑅𝑢_𝑘−1. (17)
For simplicity, let 𝑥_𝑘=𝑥,𝑦_𝑘−1=𝑦,  and 𝑢_𝑘−1=𝑢. Then, each discrete event of NCS will be discussed, respectively.
Event 1. If event 𝑆₁𝑆₃∶𝑦_𝑘=𝑦_𝑘,𝑢_𝑘=−𝐿𝐾₁𝑦_𝑘 occurs, it follows that 𝑉𝑧_𝑘+1−𝛼₁⁻²𝑉𝑧_𝑘=𝑥^𝑇𝐴^𝑇𝑃𝐴𝑥−𝑥^𝑇𝐴^𝑇𝑃𝐵𝐿𝐾₁𝐶𝑥−𝑥^𝑇𝐶^𝑇𝐾^𝑇₁𝐿^𝑇𝐵^𝑇𝑃𝐴𝑥+𝑥^𝑇𝐶^𝑇𝑄𝐶𝑥+𝑥^𝑇𝐶^𝑇𝐾^𝑇₁𝐿^𝑇𝐵^𝑇𝑃𝐵𝐿𝐾₁𝐶𝑥+𝑥^𝑇𝐶^𝑇𝐾^𝑇₁𝐿^𝑇𝑅𝐿𝐾₁𝐶𝑥−𝛼₁⁻²𝑥^𝑇𝑃𝑥−𝛼₁⁻²𝑦^𝑇𝑄𝑦−𝛼₁⁻²𝑢^𝑇𝑅𝑢. (18)
So, we have 𝑉𝑧_𝑘+1−𝛼₁⁻²𝑉𝑧_𝑘=𝑥^𝑇𝑦^𝑇𝑢^𝑇𝐻₁𝑥𝑦𝑢, (19) where

𝐻₁=Π₁₁000−𝛼₁⁻²𝑄000−𝛼₁⁻²𝑅,Π₁₁=𝐴^𝑇𝑃𝐴+𝐴^𝑇𝐶^𝑇𝑄𝐶𝐴−𝐶^𝑇𝐾^𝑇₁𝐿^𝑇𝐵^𝑇𝑃𝐴−𝐴^𝑇𝑃𝐵𝐿𝐾₁𝐶+𝐶^𝑇𝐾^𝑇₁𝐿^𝑇𝐵^𝑇𝑃𝐵𝐿𝐾₁𝐶+𝐶^𝑇𝐾^𝑇₁𝐿^𝑇𝑅𝐿𝐾₁𝐶−𝛼₁⁻²𝑃. (20)

By Schur complement, 𝐻₁<0 is equivalent to −𝛼₁⁻²𝑃000−𝛼₁⁻²𝑄000−𝛼₁⁻²𝑅+𝐴^𝑇−𝐶^𝑇𝐾^𝑇₁𝐿^𝑇𝐵^𝑇𝐶^𝑇𝐶^𝑇𝐾^𝑇₁𝐿^𝑇×000000−𝑃⁻¹000−𝑄⁻¹000−𝑅⁻¹⁻¹𝐴−𝐵𝐿𝐾₁𝐶00𝐶00𝐿𝐾₁𝐶00<0. (21)

Then, −𝛼₁⁻²𝑃00𝐴^𝑇−𝐶^𝑇𝐾^𝑇₁𝐿^𝑇𝐵^𝑇𝐶^𝑇𝐶^𝑇𝐾^𝑇₁𝐿^𝑇0−𝛼₁⁻²𝑄000000−𝛼₁⁻²𝑅000𝐴−𝐵𝐿𝐾₁𝐶00−𝑃⁻¹00𝐶000−𝑄⁻¹0𝐿𝐾₁𝐶0000−𝑅⁻¹<0. (22)

Next, after pre- and postmultiplying inequality (22) by diag{𝑃⁻¹,𝑄⁻¹,𝑅⁻¹,𝐼,𝐼,𝐼}, we have (13). Therefore, the following inequality holds: 𝑉𝑧_𝑘+1−𝛼₁⁻²𝑉𝑧_𝑘<0. (23)

Then, according to the Lemma 1, the NCS (5) in discrete events 𝜃_𝑘=1 is exponentially stable.

Event 2. If event 𝑆₂𝑆₃∶𝑦_𝑘=𝑦_𝑘−1,𝑢_𝑘=−𝐿𝐾₃𝑦_𝑘 occurs, then we get 𝑉(𝑥(𝑘+1))−𝛼₃⁻²𝑉(𝑥(𝑘))=𝑥^𝑇𝐴^𝑇𝑃𝐴𝑥−𝑥^𝑇𝐴^𝑇𝑃𝐵𝐿𝐾₃𝑦−𝑦^𝑇𝐾^𝑇₃𝐿^𝑇𝐵^𝑇𝑃𝐴𝑥+𝑦^𝑇𝐾^𝑇₃𝐿^𝑇𝐵^𝑇𝑃𝐵𝐿𝐾₃𝑦+𝑦^𝑇𝑄𝑦+𝑦^𝑇𝐾^𝑇₃𝐿^𝑇𝑅𝐿𝐾₃𝑦−𝛼₃⁻²𝑥^𝑇𝑃𝑥−𝛼₃⁻²𝑦^𝑇𝑄𝑦−𝛼₃⁻²𝑢^𝑇𝑅𝑢. (24)

So, we have 𝑉𝑧_𝑘+1−𝛼₃⁻²𝑉𝑧_𝑘=𝑥^𝑇𝑦^𝑇𝑢^𝑇𝐻₃𝑥𝑦𝑢, (25) where 𝐻₃=𝐴^𝑇𝑃𝐴−𝛼₃⁻²𝑃−𝐴^𝑇𝑃𝐵𝐿𝐾₃0−𝐾^𝑇₃𝐿^𝑇𝐵^𝑇𝑃𝐴Π₃₃0000, (26) where Π₃₃=𝐾^𝑇₃𝐿^𝑇𝐵^𝑇𝑃𝐵𝐿𝐾₃+𝑄+𝐾^𝑇₃𝐿^𝑇𝑅𝐿𝐾₃−𝛼₂⁻²𝑄.

Applying Schur complement theory, 𝐻₃<0 is equivalent to −𝛼₃⁻²𝑃00𝐴^𝑇0001−𝛼₃⁻²𝑄0−𝐾^𝑇₃𝐿^𝑇𝐵^𝑇𝐼−𝐾^𝑇₃𝐿^𝑇𝑅00𝛼₂⁻²𝑅000𝐴−𝐵𝐿𝐾₃0−𝑃⁻¹000𝐼00−𝑄⁻¹00−𝐿𝐾₃000−𝑅⁻¹<0. (27)

Next, after pre- and postmultiplying inequality (27) by diag{𝑃⁻¹,𝑄⁻¹,𝑅⁻¹,𝐼,𝐼,𝐼}, we have (15). Therefore, the inequality 𝑉(𝑧_𝑘+1)−𝛼₃⁻²𝑉(𝑧_𝑘)<0 holds.

If event 3 and event 4 occur, that is, 𝑆₁𝑆₄:𝑦_𝑘=𝑦_𝑘,𝑢_𝑘=𝑢_𝑘−1 and 𝑆₂𝑆₄:𝑦(𝑘)=𝑦_𝑘−1,𝑢_𝑘=𝑢_𝑘−1, then we get 𝑉(𝑧(𝑘+1))−𝛼₂⁻²𝑉(𝑧(𝑘))=𝑥^𝑇𝐴^𝑇𝑃𝐴𝑥+𝑥^𝑇𝐴^𝑇𝑃𝐵𝑢+𝑢^𝑇𝐵^𝑇𝑃𝐴𝑥+𝑢^𝑇𝐵^𝑇𝑢+𝑢𝑃𝐵_𝑇𝑅𝑢+𝑥^𝑇𝐶^𝑇𝑄𝐶𝑥−𝛼₂⁻²𝑥^𝑇𝑃𝑥−𝛼₂⁻²𝑦^𝑇𝑄𝑦−𝛼₂⁻²𝑢^𝑇𝑅𝑉(𝑥(𝑢,𝑘+1))−𝛼₄⁻²𝑉(𝑥(𝑘))=𝑥^𝑇𝐴^𝑇𝑃𝐴𝑥+𝑥^𝑇𝐴^𝑇𝑃𝐵𝑢+𝑢^𝑇𝐵^𝑇𝑃𝐴𝑥+𝑢^𝑇𝐵^𝑇𝑢+𝑢𝑃𝐵_𝑇𝑅𝑢+𝑦^𝑇𝑄𝑦−𝛼₄⁻²𝑥^𝑇𝑃𝑥−𝛼₄⁻²𝑢^𝑇𝑅𝑢−𝛼₄⁻²𝑦^𝑇𝑄𝐻𝑦,₂=𝐴^𝑇𝑃𝐴+𝐶^𝑇𝑄𝐶−𝛼₂⁻²𝑃0𝐴^𝑇𝑃𝐵0−𝛼₂⁻²𝐵𝑄0_𝑇𝑃𝐴0𝐵^𝑇𝑃𝐵+𝑅−𝛼₂⁻²𝑅,𝐻₄=𝐴^𝑇𝑃𝐴−𝛼₄⁻²𝑃0𝐴^𝑇𝑃𝐵0𝑄−𝛼₄⁻²𝐵𝑄0_𝑇𝑃𝐴0𝐵^𝑇𝑃𝐵+𝑅−𝛼₄⁻²𝑅. (28)

Similar to the above process, (14) and (16) can be obtained. Then, the following inequalities hold: 𝑉𝑥_𝑘+1−𝛼₂⁻²𝑉𝑥_𝑘𝑉𝑥<0,_𝑘+1−𝛼₄⁻²𝑉𝑥_𝑘<0. (29)

At last, form Lemma 1 and Theorem 3, the conclusion that the NCS (5) is exponentially stable with the decay rate 𝛿 is made.

Remark 5. From Theorem 4, it can be found that the NCS (5) is exponentially stable. According to the characteristic of trivial solution of system (5), we can get that the exponential stability of NCS (5) is equivalent to asymptotic stability, which corresponds with the requirement of integrity design for FTC.

4. Numerical Examples

In this section, two numerical examples are given to illustrate the performance of the proposed approach.

Example 1. Consider the following linear model of a helicopter (as in [19]): 𝑥̇(𝑡)=𝐴𝑥(𝑡)+(𝑡),𝑦(𝑡)=𝐵𝑢𝐶𝑥(𝑡), (30) where 𝑥=[𝑉_ℎ,𝑉_𝑣,𝑞,𝜃]^𝑇 is system state, 𝑢=[𝛿_𝑐,𝛿_𝑙]^𝑇 is control input, 𝑉_ℎ is horizontal velocity, 𝑉 is vertical velocity, 𝑞 is pitch rate, 𝜃 is angle of pitch, 𝛿_𝑐 is blade control, and the matrices of model (30) are 𝐴=,−0.03660.02710.0188−0.15550.0482−1.010.0024−4.02080.10020.3681−0.7071.4200010𝐵=,0.44220.17613.5446−7.5922−5.524.4900𝐶=.1000011000100001 (31)
Assume that the sampling period is 𝑇=0.1s and the initial state 𝑥₀=[25050108]^𝑇, then the matrices of zero-order holding system (30) are given by ,.𝐴=0.99640.0025960.001039−0.015960.0045130.9037−0.01879−0.38340.0097620.033880.93830.13030.00049220.0017410.096771.007𝐵=0.044240.016870.3407−0.7249−0.52780.4214−0.026780.02151,𝐶=1000010000100111 (32)
Suppose the rate of packet dropout is 30% in the S-C link and the rate of packet dropout is 20% in the C-A link; then, we have 𝑟₁=0.7  and  𝑟₂=0.8.

Denote actuator failure matrix 𝐿 as follows: 𝐿=1001. (33)

From Figure 3, it can be seen that the system is unstable when the data packet dropout and actuator faults happen. Then, using the proposed approach in Theorem 4 and taking 𝛼₁=3.1641, 𝛼₂=0.3461,  and  𝛼₃=0.6121, 𝛼₄=0.6188, the controllers can be obtained as follows: 𝐾₁=,𝐾0.61290.3626−1.480−2.5049−0.3064−1.06510.25900.0393₃=.−1.1034−0.11490.84541.0289−0.13821.5734−0.1126−0.574 (34)

Figure 3 

Output of the system with data packet dropout and actuator fault.

Besides, from Figure 4, we can see that controllers can guarantee system stability when the data packet dropout and actuator faults happen. Therefore, the simulation result of Example 1 demonstrates the effectiveness of our presented method.

(a)   Event 1

(b) Event 3

Figure 4 

Output of the closed-loop NCS with data packet dropout and actuator fault.

Example 2. Consider the discrete system as follows (as in [13]): 𝑥()=𝑥(𝑘)+𝑢(𝑘),𝑦(𝑘)=𝑥(𝑘),𝑘+10.550.03500.620.0060.240.1000.1 (35) with the sampling period 𝑇=0.3s.
When data packet dropouts occur in the communication channels, a controller presented in [13] can stabilize the closed-loop NCS. But, when the closed-loop NCS with actuator failure matrix 𝐿=[10], the suitable controller and 𝛼_𝑖 cannot be found using method in [13]. Comparing with method in [13], for given 𝑟₁=0.6, 𝑟₂=0.7, 𝛼₁=1.574, 𝛼₂=0.913, 𝛼₃=0.705,  and  𝛼₄=0.72, the suitable output feedback controllers can be found by the proposed approach in this paper:

𝐾₁=,𝐾0.01510.488000₃=.0.31710.004900 (36)

Figure 5 shows the step responses of NCS (5) in events 1 and 3. The case without data packet dropout and actuator failure is displayed in Figure 5(a) as the solid lines, controllers can switch immediately between two different control loops, and the system is exponentially stable. When an actuator fault occurs, one of the trajectories becomes zero as shown in Figure 5(a) by the star parts. Similarly, Figure 5(b) shows that the proposed controllers can guarantee the stability even though both the data packet dropout and actuator failure happen. From a comparison between Figures 5(a) and 5(b), we can see that system performance in event 3 is not as good as that in event 1 because of the data packet dropout. Hence, the simulation results of Example 2 imply that the desired goal is well achieved.

(a)   Event 1

(b) Event 3

Figure 5 

Step responses under different discrete events.

5. Conclusion

The problem of FTC for NCSs with actuator failures and data packet dropout in both S-C and C-A links is discussed in this paper. These data packet dropouts are described by two independent switches, which can be modeled as a discrete event system with known rate. Introducing the matrix of actuator failures, the model of NCS with actuator failures is addressed as ADSs. Then, based on the theory of ADSs, the sufficiency of exponential stability for such NCS is derived and the output feedback controllers guaranteed system performance are presented. Finally, two numerical examples are exploited to show the effectiveness of the proposed method.

Acknowledgments

The authors would like to acknowledge the National Natural Science Foundation of P. R. China under Grant no. 61174044, the Independent Innovation Fund of Shandong University (2012JC005), and the Shandong Province Natural Science Foundation under Grant no. ZR2010FM016. The authors also wish to thank the reviewers for their valuable suggestions.