Event-Triggered Reliable Control in Networked Control Systems with Probabilistic Actuator Faults

Abstract

This paper introduces a novel event-triggered scheme into networked control systems which is used to determine when to transmit the newly sampled state information to the controller. Considering the effect of the network transmission delay and probabilistic actuator fault with different failure rates, a new actuator fault model is proposed under this event-triggered scheme. Then, criteria for the exponential mean square stability (EMSS) and criteria for codesigning both the feedback and the trigger parameters are derived by using Lyapunov functional method. These criteria are obtained in the form of linear matrix inequalities. A simulation example is employed to show that our event-triggered scheme can lead to a larger release period than some existing ones.

1. Introduction

Nowadays, more and more attention has been paid to the study of stability analysis and control design of networked control systems (NCSs). NCSs have a relatively new structure where the links from sensor to controller and from controller to actuator are not connected directly, but through a network [1, 2]. The application of networks into control systems can be advantageous in terms of simplicity scalability, and cost-effectiveness. However, the introduction of a communication network can also bring about many problems, such as network-induced delay and packet dropout. Therefore, the tasks in traditional systems, such as the control problems and signal estimation problem, should be reconsidered. In recent years, the stability analysis and control design for NCS have been invested, and lots of outstanding results have been obtained [3, 4]. In these works, most are based on the periodic triggered control method, which is called a time-trigged control. In this triggering scheme, the fixed sampling period is determined under worst conditions such as external disturbances, uncertainties, and time-delays. However, in practical systems, the worst cases seldom occur. Thus, this kind of triggering method may often lead to transmitting many unnecessary signals through the network, which in turn will increase the load of network transmission and waste the network bandwidth. Hence, it is an important problem to reduce communication requirements.

Recently, event-triggered scheme for control design, advocating the use of actuation only when some function of the system state exceeds a threshold, has received considerable attention, and many important results have been reported [5–9]. Event triggering method provides a useful way of determining when the sampling action is carried out, compared with periodic sampling methods, and it has the following advantages: (1) it only samples when necessary; (2) the burden of the network communication is reduced; (3) the computation cost of the controller and the occupation of the sensor and actuator are reduced.

More specifically, a networked estimator problem for event-triggered sampling systems with packet dropouts was solved in [7]. Refrence[9] studied event design in event-triggered feedback systems, and a novel event triggering scheme was presented to ensure exponential stability of the resulting sampled-data system. Refrence [8] studied how the event triggered as well as the self-triggered control systems could be reformed in the case of discrete-time systems. The methods for design or implication of controllers in the event-triggered form based on dissipation inequalities were proposed for both linear and nonlinear systems in [6]. The work in [5] examined event-triggered data transmission in distributed networked control systems with packet loss and transmission delays.

Moreover, in NCSs, the temporary measurements failure and probabilistic distortion are usually unavoidable for a variety of reasons, for example, networked delay, sensor/actuators aging, electromagnetic interference, and zero shift, which may lead to intolerable system performance [10]. Therefore, from a safety as well as performance point of view, it is required to design a reliable controller that can tolerate actuators failures as well as networked delay. Recently, the fault model has received a lot of interest, and lots of outstanding results have been obtained [11–13]. In [11], the authors considered the problem of delay-dependent adaptive reliable 𝐻_∞ controller design against actuator faults for linear time-varying delay systems. By using a linear matrix inequality technique and an adaptive method, they established a new delay-dependent reliable 𝐻_∞ controller, which guaranteed the stability and adaptive 𝐻_∞ performance of closed-loop systems in normal and faulty cases. Considering the different failure rates of each sensor or actuator, the authors studied the reliable controller for networked control systems in [12, 13], reliable controllers were designed, and sufficient conditions for the exponentially mean square stability of NCS were obtained. Up to now, to the best of authors’ knowledge, there are no papers to deal with the reliable control for event-triggered networked control systems with probabilistic actuator faults, which still remains as a challenging problem.

In this paper, firstly, an new event-triggered scheme is introduced to networked control systems, which can reduce the burden of the network communication. Then, considering different failure rates of actuators and the measurements distortion of every actuator, a new probabilistic actuator fault model is proposed under the proposed event-triggered scheme. By using Lyapunov functional method, criteria for the exponential stability and criteria for codesigning both the feedback and the trigger parameters are derived in terms of linear matrix inequalities. A simulation example is given to show that the proposed event-triggered scheme is superior to some existing ones.

Notation. ℝ^𝑛 and ℝ^𝑛×𝑚 denote the n-dimensional Euclidean space and the set of 𝑛×𝑚 real matrices; the superscript “𝑇” stands for matrix transposition; 𝐼 is the identity matrix of appropriate dimension; ‖⋅‖ stands for the Euclidean vector norm or the induced matrix 2-norm as appropriate; the notation 𝑋>0 (resp., 𝑋≥0) for 𝑋∈ℝ^𝑛×𝑛 means that the matrix 𝑋 is real symmetric positive definite (resp., positive semi definite). When 𝑥 is a stochastic variable, 𝔼{𝑥} stands for the expectation of 𝑥. For a matrix 𝐵 and two symmetric matrices 𝐴 and 𝐶, 𝐴∗𝐵𝐶 denotes a symmetric matrix, where ∗ denotes the entries implied by symmetry.

2. System Description

In this paper, we consider the following system: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡), (1) where 𝑥(𝑡)∈ℝ^𝑛 and 𝑢(𝑡)∈ℝ^𝑚 denote the state vector and control vector, respectively; 𝐴 and 𝐵 are parameter matrices with appropriate dimensions.

Throughout this paper, we assume that the system (1) is controlled though a network.

As is well known, periodic sampling mechanism has been widely used in practical systems; however, it may often lead to transmitting many unnecessary signals through the network, which in turn increases the load of network transmission and wastes the network bandwidth. Therefore, for the control of networked control systems shown in Figure 1, in order to save network resources such as network bandwidth, it is significant to introduce an event triggered mechanism which decides whether the newly sampled state should be sent out to the controller. As is shown in Figure 1, the state are sampled regularly by the sampler of the smart sensor with period ℎ and feeds into an event generator that decides when to transmit the states to the controller via a network medium by a specified trigger condition, which will be given in sequel. The following function of network architecture in Figure 1 is expected.(1) The states are sampled at time 𝑘ℎ by sampler with a given period ℎ. The next state is at time (𝑘+1)ℎ.(2) As is shown in Figure 1, the event generator is constructed between the sensor and the controller which is used to determine whether the newly sampled state will be sent out to the controller or not by using the following judgement algorithm: 𝑥𝑘+𝑗ℎ−𝑥(𝑘ℎ)^𝑇Ω𝑥𝑘+𝑗ℎ−𝑥(𝑘ℎ)≤𝜌𝑥^𝑇𝑘+𝑗ℎΩ𝑥𝑘+𝑗ℎ. (2)(3)Under the event-triggered scheme (2), the release times are assumed to be 𝑡₀ℎ, 𝑡₁ℎ, 𝑡₂ℎ,…, where 𝑡₀ is the initial time. 𝑠_𝑖ℎ=𝑡_𝑖+1ℎ−𝑡_𝑖ℎ denotes the release period of event generator in (2). Considering the effect of the transmission delay on the network system, we suppose that the time-varying delay in the network communication is 𝜏_𝑘 and 𝜏_𝑘∈[0,𝜏), where 𝜏 is a positive real number. Therefore, the states 𝑥(𝑡₀ℎ),𝑥(𝑡₁ℎ),𝑥(𝑡₂ℎ),… will arrive at the controller side at the instants 𝑡₀ℎ+𝜏₀,𝑡₁ℎ+𝜏₁,𝑡₂ℎ+𝜏₂,…, respectively. Notice that the set of the release instants, that is, {𝑡₀,𝑡₁,𝑡₂,…} is a subset of  {0,1,2,…}. The amount of  {𝑡₀,𝑡₁,𝑡₂,…}, depends on not only the value of 𝜌, but also the variation of the state. When 𝜌=0, {𝑡₀,𝑡₁,𝑡₂,…}={0,1,2,…}, it reduces to the case with periodic release times.

Figure 1 

The structure of an event-triggered networked control system.

Based on the previous analysis, considering the effect of the transmission delay, the system model under the event generator with (2) can be described as 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢𝑡_𝑘ℎ,𝑡∈𝑡_𝑘ℎ+𝜏_𝑘,𝑡_𝑘+1ℎ+𝜏_𝑘+1. (3)

Assumption 1. The actuators in the closed-loop systems have different failure rates because of different working conditions. Furthermore, the measurements distortion of every actuator is also take into consideration.

Under Assumption 1, the control 𝑢(𝑡)=𝐾𝑥(𝑡) can be described as 𝑢^𝐹(𝑡)=Ξ𝐾𝑥(𝑡)=^𝑚_𝑖=1𝜉_𝑖𝐶_𝑖𝐾𝑥(𝑡), (4) where Ξ=diag{𝜉₁,…,𝜉_𝑚} and 𝜉_𝑖(𝑖=1,2,…,𝑚) are 𝑚 unrelated variables taking values on the interval [0,𝜃], where 𝜃≥1, the mathematical expectation and variance of 𝜉_𝑖 are 𝜇_𝑖 and 𝜎²_𝑖(𝑖=1,2,…,𝑚), and 𝐶_𝑖=diag{0,…,0_𝑖−1,1,0,…,0_𝑚−𝑖}.

Define Ξ=diag{𝜇₁,…,𝜇_𝑚}=∑^𝑚_𝑖=1𝜇_𝑖𝐶_𝑖, and obviously, 𝔼(Ξ)=Ξ, 𝔼(Ξ−Ξ)=0, 𝔼(𝜉_𝑖−𝜇_𝑖)²=𝜎²_𝑖.

Under (4), for 𝑡∈[𝑡_𝑘ℎ+𝜏_𝑘,𝑡_𝑘+1ℎ+𝜏_𝑘+1), (3) can be rewritten as 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵Ξ𝐾𝑥𝑡_𝑘ℎ. (5)

Remark 2. 𝑢(𝑡) is the control input without actuator failure, and 𝑢^𝐹(𝑡) is the control input after actuator failures occur. 𝜉_𝑖=0 and 𝜉_𝑖=1 represent the meaning of completely failure or completely normal [14]. 0<𝜉_𝑖<1 means partial failure [15, 16]. 𝜉_𝑖>1 represents the condition of data distortion [12, 13].

Remark 3. 𝜇_𝑖=1 does not represent that the 𝑖th actuator is always in good work condition, but it means that the expectation of 𝑖th actuator is 1.

For technical convenience, consider the following two cases.

Case 1. If 𝑡_𝑘ℎ+ℎ+𝜏≥𝑡_𝑘+1ℎ+𝜏_𝑘+1, where 𝜏=max𝜏_𝑘, define a function 𝜏(𝑡) as 𝜏(𝑡)=𝑡−𝑡_𝑘ℎ,𝑡∈𝑡_𝑘ℎ+𝜏_𝑘,𝑡_𝑘+1ℎ+𝜏_𝑘+1; (6) clearly, 𝑡_𝑘≤𝜏(𝑡)≤𝑡_𝑘+1−𝑡_𝑘ℎ+𝜏_𝑘+1≤ℎ+𝜏. (7)

Case 2. If 𝑡_𝑘ℎ+ℎ+𝜏<𝑡_𝑘+1ℎ+𝜏_𝑘+1, consider the following two intervals: 𝑡_𝑘ℎ+𝜏_𝑘,𝑡_𝑘ℎ+ℎ+𝜏,𝑡_𝑘ℎ+𝑖ℎ+𝜏,𝑡_𝑘ℎ+𝑖ℎ+ℎ+𝜏. (8) Since 𝜏_𝑘≤𝜏, it can be easily shown that there exists 𝑑_𝑀 such that 𝑡_𝑘ℎ+𝑑_𝑀ℎ+𝜏<𝑡_𝑘+1ℎ+𝜏_𝑘+1≤𝑡_𝑘ℎ+𝑑_𝑀ℎ+ℎ+𝜏. (9) Moreover, 𝑥(𝑡_𝑘ℎ) and 𝑡_𝑘ℎ+𝑖ℎ with 𝑖=1,2,…,𝑑_𝑀 satisfy (2). Let 𝐼₀=𝑡_𝑘ℎ+𝜏_𝑘,𝑡_𝑘ℎ+ℎ+𝜏,𝐼_𝑖=𝑡_𝑘ℎ+𝑖ℎ+𝜏,𝑡_𝑘ℎ+𝑖ℎ+ℎ+𝜏,𝐼_𝑑𝑀=𝑡_𝑘ℎ+𝑑_𝑀ℎ+𝜏,𝑡_𝑘+1ℎ+𝜏_𝑘+1, (10) where 𝑖=1,2,…,𝑑_𝑀−1. One can see that 𝑡_𝑘ℎ+𝜏_𝑘,𝑡_𝑘+1ℎ+𝜏_𝑘+1=^{𝑖=𝑑𝑀}_𝑖=0𝐼_𝑖. (11)
Define 𝜏(𝑡)=𝑡−𝑡_𝑘ℎ,𝑡∈𝐼₀,𝑡−𝑡_𝑘ℎ−𝑖ℎ,𝑡∈𝐼_𝑖,𝑖=1,2,…,𝑑_𝑀−1,𝑡−𝑡_𝑘ℎ−𝑑_𝑀ℎ,𝑡∈𝐼_𝑑𝑀. (12) then, we have 𝑡_𝑘≤𝜏(𝑡)<ℎ+𝜏,𝑡∈𝐼₀,𝑡_𝑘≤𝜏≤𝜏(𝑡)<ℎ+𝜏,𝑡∈𝐼_𝑖,𝑖=1,2,…,𝑑_𝑀𝑡−1,_𝑘≤𝜏≤𝜏(𝑡)<ℎ+𝜏,𝑡∈𝐼_𝑑𝑀, (13) where the third row in (13) holds because 𝑡_𝑘+1ℎ+𝜏_𝑘+1≤𝑡_𝑘ℎ+(𝑑_𝑀+1)ℎ+𝜏. Obviously, 0≤𝜏_𝑘≤𝜏(𝑡)≤ℎ+𝜏≜𝜏_𝑀,𝑡∈𝑡_𝑘ℎ+𝜏_𝑘,𝑡_𝑘+1ℎ+𝜏_𝑘+1. (14)
In Case 1, for 𝑡∈[𝑡_𝑘ℎ+𝜏_𝑘,𝑡_𝑘+1ℎ+𝜏_𝑘+1), define 𝑒_𝑘(𝑡)=0. In Case 2, define𝑒_𝑘(𝑡)=0,𝑡∈𝐼₀,𝑥𝑡_𝑘ℎ−𝑥𝑡_𝑘ℎ+𝑖ℎ,𝑡∈𝐼_𝑖,𝑖=1,2,…,𝑑_𝑀𝑥𝑡−1,_𝑘ℎ−𝑥𝑡_𝑘ℎ+𝑑_𝑀ℎ,𝑡∈𝐼_𝑑𝑀. (15)
From the definition of 𝑒_𝑘(𝑡) and the triggering algorithm (2), it can be easily seen that, for 𝑡∈[𝑡_𝑘ℎ+𝜏_𝑘,𝑡_𝑘+1ℎ+𝜏_𝑘+1), 𝑒^𝑇_𝑘(𝑡)Ω𝑒_𝑘(𝑡)≤𝜌𝑥^𝑇(𝑡−𝜏(𝑡))Ω𝑥(𝑡−𝜏(𝑡)). (16)
Utilizing 𝜏(𝑡) and 𝑒_𝑘(𝑡), (5) can be rewritten as 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵Ξ𝐾𝑥(𝑡−𝜏(𝑡))+𝑒_𝑘(𝑡(𝑡))=𝐴𝑥+𝐵Ξ𝐾𝑥(𝑡−𝜏(𝑡))+𝐵Ξ𝐾𝑒_𝑘(𝑡)+𝐵Ξ−Ξ𝐾𝑥(𝑡−𝜏(𝑡))+𝐵Ξ−Ξ𝐾𝑒_𝑘(𝑡), (17) where 𝑡∈[𝑡_𝑘ℎ+𝜏_𝑘,𝑡_𝑘+1ℎ+𝜏_𝑘+1).
For the system (17), we supplement the initial condition of the state 𝑥(𝑡) on [𝜏_𝑀,0] as 𝑥(𝑡)=𝜙(𝑡),𝑡∈𝜏_𝑀,0, (18) where 𝜙(𝑡) is a continuous function on [𝜏_𝑀,0].

Remark 4. If 𝜏_𝑘=0, it means that no transmission delay exists or transmission delay can be ignored, and the maximum sampling period is 𝜏_𝑀. Note that 𝜏_𝑀=ℎ+𝜏. If 𝜏_𝑘>0, the selecting sampling period ℎ<𝜏_𝑀, 𝜏=𝜏_𝑀−ℎ, is the allowable maximum transmission delay.

Remark 5. Notice that if 𝜌=0, then 𝑒_𝑘(𝑡)=0; in this situation, the model (17) reduces to a model for a networked control system with a time-triggered scheme. It means that, if 𝜌→0⁺, the dynamic of the system under event-triggered scheme will approach to the one with a time-triggered scheme.

In the following, we will introduce the following definitions and lemmas, which are needed in the next section.

Definition 6. For a given function 𝑉∶𝐶^𝑏_𝐹0([−𝜏_𝑀,0],ℝ^𝑛)×𝑆, its infinitesimal operator ℒ is defined as ℒ𝑉𝜂(𝑡)=lim_Δ→0+1Δ𝔼𝑉𝜂_𝑡+Δ∣𝜂_𝑡−𝑉𝜂_𝑡. (19)

Definition 7. System (17) is said to be exponentially of mean square stability (EMSS) if there exist constants 𝛼>0 and 𝜖>0 such that for 𝑡≥0, 𝔼‖𝑥(𝑡)‖²≤𝛼𝑒^−𝜖𝑡𝔼sup_{−𝜏𝑀≤𝑠≤0}𝜓(𝑠)². (20)

Lemma 8 (see [17]). For any vectors 𝑥,𝑦∈𝑅^𝑛 and positive definite matrix 𝑄∈𝑅^𝑛×𝑛, the following inequality holds: 2𝑥^𝑇𝑦≤𝑥^𝑇𝑄𝑥+𝑦^𝑇𝑄⁻¹𝑦. (21)

Lemma 9 (see [18]). Ξ₁, Ξ₂, and Ω are matrices with appropriate dimensions, 𝜏(𝑡) is a function of 𝑡, and 0≤𝜏(𝑡)≤𝜏_𝑀; then 𝜏(𝑡)Ξ₁+𝜏_𝑀−𝜏(𝑡)Ξ₂+Ω<0 (22) if and only if 𝜏_𝑀Ξ₁𝜏+Ω<0,_𝑀Ξ₂+Ω<0. (23)

3. Main Results

In this section, the following theorem provides the EMSS criteria for system (17) with the controller (4) under the event generator (2).

Theorem 10. For given 𝜇_𝑖, 𝜎_𝑖(𝑖=1,…,𝑚), 𝜌, and matrix 𝐾, the system described by (17) is EMSS, if there exists matrices 𝑃>0, 𝑄>0, 𝑅>0, Ω>0, and 𝑁, 𝑀 with which appropriate dimensions such that for 𝑠=1,2, Σ(𝑠)=Σ₁₁+Γ+Γ^𝑇Σ∗∗∗₂₁Σ−𝑅∗∗₃₁0Σ₃₃∗Σ₄₁(𝑠)00−𝑅<0,𝑠=1,2, (24) where Σ₁₁=𝑃𝐴+𝐴^𝑇𝐾𝑃+𝑄∗∗∗_𝑇Ξ^𝑇𝐵^𝑇𝐾𝑃𝜌Ω∗∗00−𝑄∗_𝑇Ξ^𝑇𝐵^𝑇,,Σ𝑃00−ΩΓ=𝑁−𝑁+𝑀−𝑀0₂₁=√𝜏_𝑀𝑅𝐴√𝜏_𝑀𝑅𝐵Ξ𝐾0√𝜏_𝑀𝑅𝐵,ΣΞ𝐾₃₁=,𝜎0Π00000ΠΠ=¹√𝜏_𝑀𝑅𝐵𝐶₁𝐾𝜎₂√𝜏_𝑀𝑅𝐵𝐶₂𝐾⋮𝜎_𝑚√𝜏_𝑀𝑅𝐵𝐶_𝑚𝐾,Σ₃₃=diag−𝑅,…,−𝑅_2𝑚,Σ₄₁(1)=√𝜏_𝑀𝑁^𝑇,Σ₄₁(2)=√𝜏_𝑀𝑀^𝑇. (25)

Proof. Choose the following Lyapunov functional candidate as 𝑉𝑥_𝑡=𝑉₁𝑥_𝑡+𝑉₂𝑥_𝑡+𝑉₃𝑥_𝑡, (26) where 𝑉₁𝑥_𝑡=𝑥^𝑇(𝑡)(𝑡),𝑉𝑃𝑥₂𝑥_𝑡=^𝑡_{𝑡−𝜏𝑀}𝑥^𝑇(𝑠)(𝑠)𝑉𝑄𝑥𝑑𝑠,₃𝑥_𝑡=^𝑡_{𝑡−𝜏𝑀}^𝑡_𝑠𝑥̇^𝑇(𝑣)𝑅𝑥̇(𝑣)𝑑𝑣𝑑𝑠, (27) in which 𝑃, 𝑄, and 𝑅 are symmetric positive definite matrices.
From the definition of Ξ, it can be concluded that 𝔼{Ξ−Ξ}=diag{0,…,0}, and using the infinitesimal operator (19) for 𝑉(𝑥_𝑡) and taking expectation on it, we obtain 𝔼ℒ𝑉₁𝑥_𝑡=2𝑥^𝑇(𝑡)(𝑡),𝔼𝑃Υℒ𝑉₂𝑥_𝑡=𝑥^𝑇(𝑡)𝑄𝑥(𝑡)−𝑥^𝑇((𝑡((𝑡,𝔼𝑡−𝜏))𝑄𝑥𝑡−𝜏))ℒ𝑉₃𝑥_𝑡=𝔼𝜏_𝑀𝑥̇^𝑇(𝑡)𝑅𝑥̇(𝑡)−^𝑡_{𝑡−𝜏𝑀}𝑥̇^𝑇(𝑠)𝑅𝑥̇(𝑠)𝑑𝑠, (28) where Υ(𝑡)=𝐴𝑥(𝑡)+𝐵Ξ𝐾𝑥(𝑡−𝜏(𝑡))+𝐵Ξ𝐾𝑒_𝑘(𝑡).
Notice that 𝔼𝜏_𝑀𝑥̇^𝑇(𝑡)𝑅𝑥̇(𝑡)=𝜏_𝑀Υ^𝑇(𝑡)(𝑡)+𝑅Υ_𝑚𝑖=1𝜏_𝑀𝜎²_𝑖𝑒^𝑇_𝑘(𝑡)𝐾^𝑇𝐶^𝑇_𝑖𝐵^𝑇𝑅𝐵𝐶_𝑖𝐾𝑒_𝑘(𝑡)+^𝑚_𝑖=1𝜏_𝑀𝜎²_𝑖𝑥^𝑇(𝑡−𝜏(𝑡))𝐾^𝑇𝐶^𝑇_𝑖𝐵^𝑇𝑅𝐵𝐶_𝑖×𝐾𝑥(𝑡−𝜏(𝑡)). (29)
Combining (28) and (29), we obtain 𝔼ℒ𝑉𝑥_𝑡=2𝑥^𝑇(𝑡)𝑃Υ(𝑡)+𝑥^𝑇(𝑡)𝑄𝑥(𝑡)−𝑥^𝑇(𝑡−𝜏(𝑡))𝑄𝑥(𝑡−𝜏(𝑡))+𝜏_𝑀Υ^𝑇(𝑡)(𝑡)+𝑅Υ_𝑚𝑖=1𝜏_𝑀𝜎²_𝑖𝑒^𝑇_𝑘(𝑡)𝐾^𝑇𝐶^𝑇_𝑖𝐵^𝑇𝑅𝐵𝐶_𝑖𝐾𝑒_𝑘(𝑡)+^𝑚_𝑖=1𝜏_𝑀𝜎²_𝑖𝑥^𝑇(𝑡−𝜏(𝑡))𝐾^𝑇𝐶^𝑇_𝑖𝐵^𝑇𝑅𝐵𝐶_𝑖((𝑡−×𝐾𝑥𝑡−𝜏))_{𝑡𝑡−𝜏𝑀}𝑥̇^𝑇(𝑠)𝑅𝑥̇(𝑠)𝑑𝑠+Γ₁+Γ₂, (30) where Γ₁ and Γ₂ are introduced by employing free weight matrix method [19, 20] Γ₁=2𝜁^𝑇(𝑡)𝑁𝑥(𝑡)−𝑥(𝑡−𝜏(𝑡))−^𝑡_{(𝑡)𝑡−𝜏}𝑥̇(𝑠)Γ𝑑𝑠=0,₂=2𝜂^𝑇(𝑡)𝑀𝑥(𝑡−𝜏(𝑡))−𝑥𝑡−𝜏_𝑀−^{𝑡−𝜏(𝑡)}_{𝑡−𝜏𝑀}𝑥̇(𝑠)𝑑𝑠=0, (31) where 𝑁 and 𝑀 are matrices with appropriate dimensions, and 𝜁^𝑇(𝑡)=𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−𝜏(𝑡))𝑥^𝑇𝑡−𝜏_𝑀𝑒^𝑇_𝑘(𝑡). (32)
By Lemma 8, we have −2𝜁^𝑇(𝑡)𝑁^𝑡_{(𝑡)𝑡−𝜏}𝑥̇(𝑠)𝑑𝑠≤𝜏(𝑡)𝜁^𝑇(𝑡)𝑁𝑅⁻¹𝑁^𝑇𝜁(𝑡)+^𝑡_{(𝑡)𝑡−𝜏}𝑥̇^𝑇(𝑠)𝑅𝑥̇(𝑠)𝑑𝑠,−2𝜁^𝑇(𝑡)𝑀^{𝑡−𝜏(𝑡)}_{𝑡−𝜏𝑀}𝑥̇(𝑠)𝑑𝑠≤𝜏_𝑀−𝜏(𝑡)𝜁^𝑇(𝑡)𝑁𝑅⁻¹𝑀^𝑇𝜁(𝑡)+^{𝑡−𝜏(𝑡)}_{𝑡−𝜏𝑀}𝑥̇^𝑇(𝑠)𝑅𝑥̇(𝑠)𝑑𝑠. (33)
Combining (16), substitute (33) into (30), and we obtain that 𝔼ℒ𝑉𝑥_𝑡≤2𝑥^𝑇(𝑡)𝑃Υ(𝑡)+𝑥^𝑇(𝑡)𝑄𝑥(𝑡)−𝑥^𝑇(𝑡−𝜏(𝑡))𝑄𝑥(𝑡−𝜏(𝑡))+𝜏_𝑀Υ^𝑇(𝑡)(𝑡)+𝑅Υ_𝑚𝑖=1𝜏_𝑀𝜎²_𝑖𝑒^𝑇_𝑘(𝑡)𝐾^𝑇𝐶^𝑇_𝑖𝐵^𝑇𝑅𝐵𝐶_𝑖𝐾𝑒_𝑘(𝑡)+^𝑚_𝑖=1𝜏_𝑀𝜎²_𝑖𝑥^𝑇(𝑡−𝜏(𝑡))𝐾^𝑇𝐶^𝑇_𝑖𝐵^𝑇𝑅𝐵𝐶_𝑖×𝐾𝑥(𝑡−𝜏(𝑡))+2𝜁^𝑇(𝑡)𝑁[𝑥(𝑡)−𝑥(𝑡−𝜏(𝑡))]+2𝜁^𝑇(𝑡)𝑀×𝑥(𝑡−𝜏(𝑡))−𝑥𝑡−𝜏_𝑀−𝑒^𝑇_𝑘(𝑡)Ω𝑒_𝑘(𝑡)+𝜌𝑥^𝑇((𝑡((𝑡+𝜏𝑡−𝜏))Ω𝑥𝑡−𝜏))_𝑀−𝜏(𝑡)𝜁^𝑇(𝑡)𝑀𝑅⁻¹𝑀^𝑇𝜁(𝑡)+𝜏(𝑡)𝜁^𝑇(𝑡)𝑁𝑅⁻¹𝑁^𝑇𝜁(𝑡); (34) that is, 𝔼ℒ𝑉𝑥_𝑡≤𝜁^𝑇(𝑡)Θ𝜁(𝑡), (35) where Θ=Σ₁₁+Γ+Γ^𝑇+Σ^𝑇₂₁𝑅⁻¹Σ₂₁−Σ^𝑇₃₁Σ⁻¹₃₁Σ₃₁+(𝜏_𝑀−𝜏(𝑡))𝑀𝑅⁻¹𝑀^𝑇+𝜏(𝑡)𝑁𝑅⁻¹𝑁^𝑇.
By using Schur complement and Lemma 9, we have Θ<0, if and only if the following holds: Σ₁₁+Γ+Γ^𝑇+Σ^𝑇₂₁𝑅⁻¹Σ₂₁−Σ^𝑇₃₁Σ⁻¹₃₁Σ₃₁∗Σ₄₁(𝑠)−𝑅<0,𝑠=1,2. (36)
By using Schur complement, we can obtain that (36) is equivalent to (24). Furthermore, 𝔼ℒ𝑉𝑥_𝑡≤−𝜆𝜁^𝑇(𝑡)𝜁(𝑡), (37) where 𝜆=min{𝜆_min[Θ]}. Define a new function as 𝑊𝑥_𝑡=𝑒^𝜖𝑡𝑉𝑥_𝑡. (38)
Its infinitesimal operator ℒ is given by ℒ𝑊𝑥_𝑡=𝜖𝑒^𝜖𝑡𝑉𝑥_𝑡+𝑒^𝜖𝑡ℒ𝑉𝑥_𝑡. (39)
From (39), we can obtain that 𝔼𝑊𝑥_𝑡−𝔼𝑊𝑥₀=^𝑡₀𝜖𝑒^𝜖𝑠𝔼𝑉𝑥_𝑠+^𝑡₀𝑒^𝜖𝑠𝔼ℒ𝑉𝑥_𝑠. (40)
Then using the similar method of [15], we can observe that there exists a positive number 𝛼 such that for 𝑡≥0𝔼𝑉𝑥_𝑡≤𝛼sup_{−𝜏𝑀≤𝑠≤0}𝑒^−𝜖𝑠𝔼𝜓(𝑠)². (41)
Since 𝑉(𝑥_𝑡)≥𝜆_min(𝑃)𝑥^𝑇(𝑡)𝑥(𝑡), it can be shown from (41) that, for 𝑡≥0, 𝔼𝑥^𝑇(𝑡)𝑥(𝑡)≤𝛼𝑒^−𝜖𝑡sup_{−𝜏𝑀≤𝑠≤0}𝔼𝜓(𝑠)², (42) where 𝛼=𝛼/𝜆_min(𝑃). The proof can be completed.

In the following, based on Theorem 10, we will design the feedback gain 𝐾 in (4) under the event-trigger (2).

Theorem 11. For given 𝜇_𝑖, 𝜎_𝑖(𝑖=1,…,𝑚), 𝜌 and 𝜀, system (17) with the feedback gain 𝐾=𝑌𝑋⁻¹ under the event trigger condition (2) is EMSS if there exist matrices 𝑋>0, 𝑄^∼>0, 𝑅^∼>0, Ω^∼>0, and 𝑁^∼, 𝑀^∼ with appropriate dimensions such that 𝑠=1,2Σ^∼₁₁+Γ^∼+Γ^∼𝑇∗∗∗_∼Σ₂₁−2𝜀𝑋+𝜀𝑅^2∼∗∗_∼Σ₃₁0Σ^∼₃₃∗_∼Σ₄₁(𝑠)00−2𝜀𝑋+𝜀𝑅^2∼<0,𝑠=1,2, (43) where Σ^∼₁₁=𝐴𝑋+𝑋𝐴^𝑇+𝑄^∼𝑌∗∗∗_𝑇Ξ^𝑇𝐵^𝑇𝜌Ω^∼∗∗00−𝑄^∼∗𝑌^𝑇Ξ^𝑇𝐵^𝑇00−Ω^∼_∼,Γ=𝑁^∼−𝑁^∼+𝑀^∼−𝑀^∼0,_∼Σ₂₁=√𝜏_𝑀𝐴𝑋√𝜏_𝑀𝐵Ξ𝑌0√𝜏_𝑀𝐵Ξ𝑌_∼,Σ₃₁=0Π^∼00000Π^∼,^∼𝜎Π=¹√𝜏_𝑀𝐵𝐶₁𝑌𝜎₂√𝜏_𝑀𝐵𝐶₂𝑌⋮𝜎_𝑚√𝜏_𝑀𝐵𝐶_𝑚𝑌_∼,Σ₃₃=diag−2𝜀𝑋+𝜀𝑅^2∼,…,−2𝜀𝑋+𝜀𝑅^2∼_2𝑚,_∼Σ₄₁(1)=√𝜏𝑁_𝑀^∼𝑇,Σ^∼₄₁(2)=√𝜏𝑀_𝑀^∼𝑇. (44)

Proof. By using Schur complement, we can obtain that the following is equivalent to (24): Σ₁₁+Γ+Γ^𝑇Φ∗∗∗₂₁−𝑃𝑅⁻¹Φ𝑃∗∗₃₁0Φ₃₃∗Σ₄₁(𝑠)00−𝑅<0,𝑠=1,2, (45) where Φ₂₁=√𝜏_𝑀𝑃𝐴√𝜏_𝑀𝑃𝐵Ξ𝐾0√𝜏_𝑀𝑃𝐵,ΦΞ𝐾₃₁=0Π¹00000Π¹,Π¹=𝜎₁√𝜏_𝑀𝑃𝐵𝐶₁𝐾𝜎₂√𝜏_𝑀𝑃𝐵𝐶₂𝐾⋮𝜎_𝑚√𝜏_𝑀𝑃𝐵𝐶_𝑚𝐾,Φ₃₃=diag−𝑃𝑅⁻¹𝑃,…,−𝑃𝑅⁻¹𝑃_2𝑚. (46)
Due to (𝑅−𝜀⁻¹𝑃)𝑅⁻¹(𝑅−𝜀⁻¹𝑃)≥0, we have −𝑃𝑅⁻¹𝑃≤−2𝜀𝑃+𝜀²𝑅. (47)
Substituting −𝑃𝑅⁻¹𝑃 with −2𝜀𝑃+𝜀²𝑅 into (45), we obtain Σ₁₁+Γ+Γ^𝑇Φ∗∗∗₂₁−2𝜀𝑃+𝜀²Φ𝑅∗∗₃₁0Φ₃₃∗Σ₄₁(𝑠)00−𝑅<0,𝑠=1,2, (48) where Φ₃₃=diag{−2𝜀𝑃+𝜀²𝑅,…,−2𝜀𝑃+𝜀²𝑅_2𝑚}.
Denoting 𝑋=𝑃⁻¹, 𝑄^∼=𝑋𝑄𝑋, 𝑅^∼=𝑋𝑅𝑋, 𝑁^∼=𝑋𝑁𝑋, 𝑀^∼=𝑋𝑀𝑋, Ω^∼=𝑋Ω𝑋, and 𝑌=𝐾𝑋, then pre- and postmultiplying (48) with diag{𝑋,…,𝑋_2𝑚+5,𝐼}, (43) can be obtained.

Remark 12. Theorem 11 shows that, for given 𝜌 and 𝜀, we can obtain the feedback gain 𝐾 by solving a set of LMIs in (43); on the other hand, using Theorem 11, for the preselected Ω and the feedback gain 𝐾, event-triggered parameter 𝜌 can be obtained. Therefore, we can use Theorem 11 to codesign the the feedback gain 𝐾 and the event-triggered parameter.

Remark 13. Notice that (43) includes the information transmission delay, and we can use (43) to obtain the feedback gain and the event-triggered parameter, which can be used to guarantee the required performance even though the transmission delay exists.

4. Simulation Examples

Example 14. To illustrate the effectiveness and application of the proposed method, we consider an inverted pendulum on top of a moving cart. The plants linearized state is depicted as the following system [21]: 𝑥̇(𝑡)=010000−𝑚𝑔𝑀0𝑔000100𝑙0𝑥(𝑡)+01𝑀0−1𝑢(𝑡)𝑀𝑙, (49) where 𝑥^𝑇=[𝑥₁𝑥₂𝑥₃𝑥₄]=[𝑦𝑦̇𝜃𝜃̇] is the system state, 𝑦 is the carts position, and 𝜃 is the pendulum bob’s angle with respect to the vertical. 𝑀=10kg is the cart mass, 𝑚=1kg is the mass of the pendulum bob, 𝑙=3m is the length of the pendulum arm, and 𝑔=10m/s² is gravitational acceleration. The initial state is the vector 𝑥₀=[0.9800.20].
In the following, we will give two cases. Case 1 is used to show how the upper bound of 𝜏_𝑀 varies along the values of 𝜌, under given feedback gain 𝐾 as (50), when there is no failure in the actuator. In Case 2, we consider that the actuators have probabilistic failure rates; firstly, we give an example to design both the feedback and the trigger parameters, and the upper bound of 𝜏_𝑀 and the release interval for 𝑡∈[𝑜,40] are also derived; secondly, we suppose that the feedback gain 𝐾 is given as (50) and study how the upper bound of 𝜏_𝑀 varies along the values of 𝜌.

Case 1. When Ξ=1, that is, there is no failure in the actuator, let 𝐾=212378210 (50) which is the same as the one in [22].
For given 𝜎₁=0, 𝜀=0.53, and 𝜌=0.2, by using Theorem 10, we can have 𝜏_𝑀=0.1270 and Ω=0.00010.00030.01120.00620.00030.00230.06660.03760.01120.06662.11001.16620.00620.03761.16620.6510. (51) Also, we can obtain Table 1 and Figure 2, which describe the upper bound of 𝜏_𝑀 varying along the values of 𝜌 and the state response of (17).
With feedback gain 𝐾 as (50), from Table 1, we can see that when 𝜌=0.53 and 𝜎=0.2, the upper bound of 𝜏_𝑀 is 0.1270. Suppose that 𝜏_𝑘=0; since 𝜏_𝑀=ℎ+𝜏, it can be known that the maximum sampling period is 0.1270. Moreover, under our event-trigger scheme, the maximum release period is 2.3. It can be seen that under the same conditions, our scheme can provide a larger release interval than some existing ones in [21, 22].

Table 1 

The computation results of the upper bound 𝜏_𝑀 for given Ξ=1, 𝜎=0 and 𝜀=0.53.

Figure 2 

The state response of (49) with the feedback gain (50) for given Ξ=1, 𝜎1=0, 𝜀=0.53, and 𝜌=0.2.

Case 2. When Ξ=0.6 and 𝜎₁=0.02, that is, the actuators have probabilistic failure rate, setting 𝜀=0.53 and 𝜌=0.2, by using Theorem 11, we can obtain the feedback gain 𝐾 of (4) as follows: ,.𝐾=5.069623.2085594.9876331.9409Ω=162.181772.523−13.8855−11.951972.5232525.4131−67.738−32.7170−13.8855−67.738252.8001−70.5906−11.9519−32.7170−70.5906132.2203 (52)
For given Ξ=0.6, 𝜎=0.02, and 𝜀=0.53, under different values of 𝜌, the upper bound of 𝜏_𝑀 and the release interval for 𝑡∈[0,40] are given in Table 2. Figures 3 and 4, respectively, represent the release instants and release interval and the probabilistic actuator failures, when Ξ=0.6, 𝜎=0.02, 𝜀=0.53, 𝜌=0.1, and 𝜏_𝑀=0.2399.
When 𝐾 is given as (50), let Ξ=0.6, Ξ=1, 𝜎₁=0, and 𝜀=0.53, and the upper bound of 𝜏_𝑀 varies along the values of 𝜌, which is shown in Table 3.

Table 2 

Some computation results for given Ξ=0.6, 𝜎=0.02, and 𝜀=0.53.

Table 3 

The computation results of the upper bound 𝜏_𝑀 for given Ξ=0.6, 𝜎=0.02, and 𝜀=0.53.

Figure 3 

The release instants and release interval for given Ξ=0.6, 𝜎=0.02, and 𝜀=0.53.

Figure 4 

The probabilistic actuator failures.

Remark 15. For given 𝜌 and 𝜀, we can obtain the upper bound of 𝜏_𝑀. we can see that the larger 𝜌, the larger average release period; Thus, the load of network communication delay is reduced and the transmission delay is decreased. This fact can be illustrated by Tables 1, 2 and 3.

5. Conclusion

In order to save the communication network bandwidth, a novel event triggering scheme is used to determine when to transmit the sampled state information. Under this event-triggered scheme, this paper considers networked systems with probabilistic actuators faults. In terms of different failure rates and the measurements distortion of every actuator, a new probabilistic actuator fault model for event-triggered networked control systems is proposed. By using Lyapunov functional method, criteria for the EMSS and criteria for codesigning both the feedback and the trigger parameters are derived in the form of linear matrix inequalities. A simulation example is given to illustrate that our event-triggered scheme can lead to a larger release period than some existing works.

Acknowledgments

This work is partly supported by the National Natural Science Foundation of China (No. 11226240), the Natural Science Foundation of Jiangsu Province of China (no. BK2012469), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (no. 12 KJD120001), the Young Teacher Supporting Foundation of Shanghai (no. ZZGJD12036), and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).