Guaranteed Cost Fault-Tolerant Control for Networked Control Systems with Sensor Faults

1School of Mechanical Engineering, Suzhou University of Science and Technology, Suzhou 215009, China 2School of Electronical and Electronic Engineering, East China Jiaotong University, Nanchang 330013, China 3Department of Mechanics and Engineering Science, Peking University, Beijing 100871, China 4School of Mechanical and Electronic Engineering, Jingdezhen Ceramic Institute, Jingdezhen 333001, China


Introduction
Feedback control systems wherein the control loops are closed through a real-time network are called networked control systems (NCS) [1].Due to their suitable and flexible structure, NCS is frequently encountered in practice for such fields as information technology, life science, and aeronautical and space technologies.However, there exist not only induced delay, data packet loss, and sequence disordering in NCS, but also actuators or sensors faults, which could cause negative impact on the performance of the system and even lead to system instability.Recently, the fault-tolerant control of NCS has become a new popular issue in the control field [2][3][4][5][6][7][8][9].
A fault-tolerant control algorithm for networked control systems is proposed based on Lyapunov stability theorem by Zheng and Fang [2].Qu et al. have devised a faulttolerant robust control for a class of nonlinear uncertain systems with possible sensor faults considered and developed a robust measure to identify the stability-and performancevulnerable failures [3].The faults of each sensor or actuator were taken as occurring randomly by Tian et al. [5], and their failure rates are governed by two sets of unrelated random variables satisfying certain probabilistic distribution.A sufficient condition is given by Zhang et al. [7], which could guarantee the stability of NCS with sensor failures or actuator failures and guarantee robustness to parameter uncertainties, but the issue of guaranteed cost is not discussed.A robust fault-tolerant control based on the integrity control theory when actuator faults occur is discussed by Zhang et al. [9].Wang et al. [10] investigated the issue of integrity against actuator faults for NCS under variable-period sampling, in which the existence conditions of guaranteed cost faultstolerant control law are testified in terms of Lyapunov stability theory, but not referring to the effects of uncertain parameters.
Almost all literatures above consider the faults in some special cases.However, in practical application, because of large scale and complicated structure of NCS, the faults could vary from time to time when the system works in a poor environment.It is of great importance to explore a reasonable control method to guarantee the performance of NCS when Based on the network transmission environment and sampling theory, the networked control systems are firstly modeled as a discrete-time closed-loop system by considering the time-varying transmission delay and sensor faults simultaneously.The model of NCS is related to the boundary values of the sensor faults.Using Lyapunov stability theory, a sufficient condition is given, which can render the closedloop NCS asymptotically stable and can guarantee it to meet the requirements of performance indicator (the upper bound of cost function).Based on LMI, the method of designing guaranteed cost fault-tolerant controller of NCS with timevarying faults is proposed in this paper.

Modeling of Networked Control Systems with Sensor Faults
A typical structure of NCS is shown in Figure 1.
In Figure 1,  sc represents the transmission delay from sensor to the controller, while  ca represents that from controller to the actuator.When the controller is static, the induced-delay time of system can be lumped as  =  sc +  ca .
A linear control plant is described by state equation as follows: where  ∈   ,  ∈   , and  ∈   represent state, input, and output vectors separately, while   ,   ,   , and   are matrices with appropriate dimensions.
In order to facilitate the model, some rational assumptions for networked control systems are introduced as follows.
(A1) Single data package is transmitted.The packet loss and sequence disorder are not taken into consideration during the transmission process.
(A2) Uncertain time delay exists during the data transmission process.But time delay is bounded, and the maximum time delay does not exceed one sampling period; namely,  ∈ [0, ], where  is the sampling period.
(A3) Sensor is clock driving; controller and actuator are all event driving.
Based on the above assumptions (see Figure 2), within a sampling period, the system input is not a constant value but is a piecewise constant.In a cycle, system input can be described as where   is the th cycle sampling time and   is the th cycle delay.
The discrete-time model of system (1) can be obtained as follows: where , and  = ∫  0       .Moreover, according to the Jordan canonical form, matrix  0 can be described as follows: where 0 ≤   ≤  ( = 1, 2, 3),  1 +  2 +  3 = ; _  is Jordan block that is a diagonal matrix consisting of the different eigenvalues with a total of  2 , denoted by  1 ,  2 , . . .,   2 , while J is Jordan block for the repeated eigenvalues with a total of  3 , denoted by  0 ; matrix Λ is the transformational matrix of Jordan canonical form for matrix  0 .
Correspondingly, the matrix  1 with the variable   can be equivalently calculated as where Here a set of real numbers which are not equal to zero are denoted by ,  1 ,  2 , . . .,   2 ,  0 .

Object input
Actuator input
Moreover, the boundaries of faults usually can be measured in the practical work.The following definition is presented.
Definition 1.The upper bound of fault matrices is defined as follows: while the lower bound of fault matrices is defined as follows: That is to say, () ∈ [  ,   ], which is time-varying.The mean value of matrices in Definition 1 can also be obtained as furthermore, the following matrices are introduced: Obviously, we have Based on (16), we have − × ≤ () ≤  × .Based on (16), the following can be obtained: Naturally, we have  =  0 ( + ) . ( The model of closed-loop systems with sensor faults can be obtained as Remark 2. The NCS with time-varying delay and sensor faults is modeled as a closed-loop system (18) with the uncertain parameter (  ) and time-varying parameter (); unlike the previous models as [5], this model is related to the boundary values of the faults   and   .Moreover, according to the expression of matrix  0 , we undoubtedly know it is invertible.

The Design of Guaranteed Cost Fault-Tolerant Control
For the system model (18) established above, the cost function is given as follows: where  and  are symmetric positive definite matrices.
Theorem 5. Given symmetric positive definite matrices ,  and gain matrix , if symmetric positive definite matrices  and  exist, as well as a scalar  > 0, satisfying then the NCS (18) is asymptotically stable, where * represents the symmetry blocks of matrix.
Theorem 7. Given symmetric positive definite matrices  and , as well as a set of constants  1 and  2 , if symmetric positive definite matrices ,  and matrix  exist, as well as scalar  > 0, satisfying the following LMI: then the NCS ( 18) is asymptotically stable with control gain  =  −1  0 −1 , and for all allowable uncertainties of system, its performance indicator defined as (19) satisfies inequality (30).

Simulations
Consider the model of inverted pendulum device as follows: Obviously,   (  )(  ) <  is satisfied.(43) Considering the sensor faults that may occur, the boundaries of faults are given in Table 1.
By making use of LMI toolbox in MATLAB to solve the linear matrix inequality (33), guaranteed cost fault-tolerant controller parameters of NCS can be obtained: It assumes the initial state of the system is (0) = (−1) = [2 − 1.2 1.5 − 1]  ; the comprehensive performance indicator can be obtained  0 = 16.3685.The time-varying transmission delays produced in network are shown in Figure 3.When all sensors are normal, namely,  = , the state responses of NCS are shown in Figure 4, from which we know the system gets steady at 40 s.When sensor faults vary in the scope of boundary values in Table 1, the state responses of NCS are shown in Figure 5, from which we can see the transition time of state response obviously becomes longer than that in Figure 4 because of the effects of time-varying sensor faults, but the system is still asymptotically stable and gets steady at 80 s.So, the performance of NCS can be well maintained by the guaranteed cost fault-tolerant controller, which demonstrates the effectiveness and feasibility of the approach proposed in this paper.

Conclusions
When time-varying sensor faults occur, guaranteed cost fault-tolerant control problem of networked control systems is studied in this paper.Using Lyapunov stability theory and linear matrix inequality (LMI) approach, a sufficient condition is given, which can render the closed-loop NCS with sensor faults asymptotically stable and can guarantee it to meet the requirements of performance indicator.And based on LMI, the method of designing guaranteed cost faulttolerant controller is proposed.Moreover, the feasibility and effectiveness of this method have been demonstrated by a simulation example.The next research task will be analyzing the time-varying actuator faults to improve the performance of NCS further.

Figure 2 :
Figure 2: Timing diagram of signals transmitting in NCS.

Figure 3 :
Figure 3: The distribution of delays in NCS.

Figure 4 :
Figure 4: State response of NCS without sensor faults.

1 Figure 5 :
Figure 5: State response of NCS with time-varying sensor faults.

Table 1 :
The boundaries of sensor faults.