Event-Triggered Faults Tolerant Control for Stochastic Systems with Time Delays

This paper is concerned with the state-feedback controller design for stochastic networked control systems (NCSs) with random actuator failures and transmission delays. Firstly, an event-triggered scheme is introduced to optimize the performance of the stochastic NCSs. Secondly, stochastic NCSs under event-triggered scheme are modeled as stochastic time-delay systems. Thirdly, some less conservative delay-dependent stability criteria in terms of linearmatrix inequalities for the codesign of both the controller gain and the trigger parameters are obtained by using delay-decomposition technique and convex combination approach. Finally, a numerical example is provided to show the less sampled data transmission and less conservatism of the proposed theory.


Introduction
Networked control systems (NCSs) are control systems wherein the control loops are closed through a certain digital communication network, which have been widely used in many fields, for example, vehicle industry, robot, and unmanned aerial vehicles, because of their benefits, for example, low installation and maintenance costs, high reliability, increased system flexibility, and decreased wiring.However, inserting a network to a control loop also brings a series of network-induced constraints, for example, time delays, packet losses, and competition of multiple nodes accessing network.The network-induced constraints will deteriorate the systems performance or even destabilize the system [1,2].Therefore, NCSs with network-induced constraints have gained considerable research interests and numerous results have been published [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21].
So far, most of the analysis and synthesis for NCSs utilize time-triggered communications scheme, for example, stabilization [3,4], filtering [5,6], and tracking control [7].It is well known that network-induced constraints are mainly caused by the limited network bandwidth.Moreover, it is very significant to mitigate the networked load to increase the lifespan of the battery of the nodes in wireless NCSs [8].Therefore, to reduce the unnecessary waste of computation and transmitting sample data not only has theoretical importance but also has practical significance.In general, NCSs using event-triggered communication scheme can considerably reduce the network resource occupancy while maintaining the control performance compared to NCSs using time-triggered communication scheme [9].So, event-triggered control for NCSs has received considerable attention in the past decade because of energy conservation and optimal performance, especially in the wireless NCSs [8], for example, event-triggered state-feedback control [10,11], event-triggered guaranteed cost control [12], event-triggered dynamic output feedback control [13], event-triggered output tracking control [14], and event-triggered fuzzy filtering [15].Considering the event-triggered control for NCS with stochastic perturbations [22][23][24][25], limited work has been reported in the open literatures.In [16], event-triggered stabilization for networked stochastic systems with multiplicative noise and network-induced delays has been investigated.However, the time delays in state and the effect of fault are not taken into account.On the one hand, fault tolerant control for NCSs has attracted great attention over the past decades because actuator faults caused by actuator aging or actuator in a hostile environment will have a great threat to the security and the reliability of NCSs [17][18][19][20][21]. On the other hand, systems with delays are ubiquitous, for example, 2 Scientific Programming NCSs, biology systems, and hydraulic rolling mill systems [2,[26][27][28].To the best of the authors' knowledge, there is no result reported in the open literatures on event-triggered faults tolerant control for stochastic systems with time delays.This motivates the study presented in the paper.
In the paper, we deal with the state-feedback controller design for stochastic NCSs with random actuator failures and transmission delays.The main works of this paper are as follows: (1) a triggered scheme is introduced to optimize the performance of stochastic NCSs; (2) stochastic NCSs under an event-triggered scheme are modeled as stochastic timedelays systems; (3) some less conservative stability criteria for the codesign of both the controller gain and the trigger parameters are obtained by using delay-decomposition technique and convex combination approach; (4) a practice example on event-triggered control for an unstable batch reactor is provided to show the merits of the proposed theory.
Notation.The notation used throughout the paper is fairly standard.  and  −1 denote the transpose and the inverse of matrix , respectively.For a square matrix , Sym{} is defined as +  .The symmetric term in a matrix is denoted by * ; for example, The notation  =   > 0 ( =   ≥ 0) means that the matrix  is real symmetric positive definite (positive semidefinite).(Ω, ϝ, {ϝ  } ≥0 , P) is a complete probability space and {⋅} stands for the expectation operator with respect to the probability measure.diag{⋅ ⋅ ⋅ } stands for a block diagonal matrix.Matrices, if not explicitly stated, are assumed to have appropriate dimensions.

Problem Formulation
Consider a linear system with state delay and Itô process, which is represented by the following stochastic differential equation: where () ∈ R  and () ∈ R  denote the state vector and the control input of the system, respectively,   is a constant delay, ,   , , , and   are some constant matrices with appropriate dimensions, () is a scalar standard Brownian Motion defined on a complete probability space (Ω, ϝ, {ϝ  } ≥0 , P) with a natural filtration {ϝ  } ≥0 and satisfies {()} = 0 and {() 2 } = , and () is a continuous vector valued initial function defined on [−  , 0].
The stochastic system controlled by a network is described in Figure 1.
To simplify the exposition, we make the following assumptions.
Assumption 1.All the system states variables of the controlled plant are available for a state-feedback control.The sensor is time-triggered with a constant sampling period ℎ.The sampling sequence is described by the set (ℎ),  = 1, 2, . . ., ∞.
Assumption 2. The sampled data is transmitted in a single packet, and the packet losses do not occur in transmission.
Whether the packets should be transmitted or not over network is determined by an event-triggered communication scheme.The successfully transmitted packet sequence is described by the set (  ℎ),  ∈ {1, 2, . . ., ∞}.
Assumption 3. The controllers and the actuators are eventtriggered.The control input at the actuator is generated by a zero-order-holder (ZOH) with the holding time  ∈ [  ℎ +    ,  +1 ℎ+  +1 ], where    is the communication delay.Before the first control signal reaches the plant, the control input () = 0.
The following state-feedback control law is employed for system (1): Considering the effect of the transmission delay, (2) can be rewritten as In order to improve the performance of the NCSs, the following event-triggered communication scheme is introduced to reduce the energy cost and the adverse effect of the transmission delay on performance of the NCSs: where  = 0, 1, . . .,  +1 −   − 1, 0 ≤  < 1, and Ω 1 and Ω 2 are the positive definite weighting matrices to be designed.

Remark 4.
Only the sampled data satisfying inequality (4) will be transmitted since the transmitter in Figure 1 has a logic function to determine whether one should be transmitted or not; when  = 0, inequality (4) holds for all the sampled state; hence, it shrinks to the periodic release case in [17,18].Obviously, the amount of transferred data can be reduced by set  ̸ = 0 in (4).If the amount of transferred data is reduced, the networked load will be reduced.Then, the energy cost used to transmit the unnecessary sampled data to the controller will be saved, and the adverse effect of the transmission delay on performance of the NCSs will be reduced since the transmission delay is mainly caused by the network traffic congestion.Remark 5. A similar step to the one used in [16], two different weighting matrices Ω 1 and Ω 2 are introduced in event-triggering communication scheme (4) to obtain less conservative results.We further illustrate this in the following numerical example.
For a detailed timing analysis, we divide the holding interval of ZOH  ∈ [  ℎ +    ,  +1 ℎ +   +1 ] into the following subsets:  Then, we have The event-triggered algorithm (4) can be rewritten as Considering the effect of the stochastic actuator failure, the relationship between the control input () and the real actuator output û() is described as follows: where where  ≥ 1 and Λ  are the random variables which describe the relationship between the real executive amount of th actuator and the control input for th actuator.Λ  = 0 represents that the th actuator is completely fails, Λ  = 1 represents that the th actuator works normally, 0 < Λ  < 1 and Λ  > 1 represent that the real actuator output is smaller and greater than the control input, respectively.The expectation and variance of Λ  are Λ  and  2  , respectively. Define Based on the above analysis, (3) can be rewritten as Substituting ( 12) into (1), the resultant closed-loop stochastic system with time delays can be expressed as follows:

Main Results
This paper aims to develop a novel state-feedback controller for event-triggered stochastic NCSs with state delay, networked induce delay, and stochastic actuator failure.Before presenting the sufficient stabilization condition, we introduce the following lemmas, which are indispensable for the proofs in the sequel.
In the following, we will show that stochastic system (21) satisfies event-triggered algorithm (4).Integrate on both sides of inequality (31) from 0 to , and according to (0) = 0, () ≥ 0, we can learn that if inequalities (17) are feasible, then This completes the proof.
The sufficient condition that stochastic system (13) with time-delay satisfying condition ( 7) is stochastically asymptotically stable is given in Theorem 8 with coupled matrix variables, but the controller gain matrix cannot be obtained directly.So, based on Theorem 8, we next prove a new theorem for the event-triggered controller design in terms of linear matrix inequalities through congruent transformation.
Remark 10.It is worth mentioning that Theorem 9 is also applicable to the event-triggered faults tolerant control for system (1) without state-delay and stochastic perturbations.
The proof follows a similar line to Theorems 8 and 9. Now we formally present this result as the following corollary.

A Numerical Example
In this section, a numerical example is used to illustrate the effectiveness and improvement of the proposed approach in this paper.Consider a simplified model of an unstable batch reactor with parameters: The eigenvalues of  are 1.9910, 0.0635, −5.0566, and −8.6659; it is obvious that the unforced system (47) is unstable.The unstable system is controlled under a network which is described in Figure 1, and the event-triggered communication scheme ( 4) is introduced to optimize the performance of the NCSs.Considering the effect of the transmission delay, it is assumed that the lower bound of the network-induced delay  1 = 0.01 s.The purpose of the design is to make this upper bound,  2 , as large as possible and the sampled data transmitted as low as possible so that the system can work under actuator failures conditions.Two cases of probability distributions are considered.
Case 1. Actuator failures Λ  are Bernoulli distributed white sequences with parameters: It is not difficult to verify that when  = 0 and Ω 1 ̸ = Ω 2 .We conduct the simulation studies with different Λ  .The maximal allowable delays  2 and fault tolerant controller parameter  from Corollary 11 and those in [23,24] are listed in Table 1  (50) Case 2.2.We have the following: when  = 0 and Ω 1 ̸ = Ω 2 .Table 2 lists the maximal allowable delays  2 and fault tolerant controller  obtained from Corollary 11 and those in [17,18].From Table 2, it can be concluded that our results are less conservative than the results from [17,18].
In the following, under actuators failures Λ  in Case 2.2 with Λ 1 = 0.2, Λ 2 = 0.3,  1 = 0.2, and  2 = 0.3, we will demonstrate that the fault tolerant state-feedback controller also optimizes the performance of the NCSs, that is, considerable reduction of the network resource occupancy: (1) When  = 0.1 and Ω 1 ̸ = Ω 2 , applying Corollary 11, we can solve the maximum  2 = 0.23s, and the Remark 12. From Tables 1 and 2, it can be concluded that when  = 0, that is, in the case of time-triggered communication scheme, our results are less conservative than existing ones.Moreover, when  = 0.1, the maximal allowable delay  2 obtained from Corollary 11 under condition Ω 1 ̸ = Ω 2 ( 2 = 0.23 s) is bigger than the case of Ω 1 = Ω 2 ( 2 = 0.15 s); that is, two different weighting matrices Ω 1 and Ω 2 are introduced in (4) which guaranteed less conservative results.
Select the sampling period ℎ = 0.10 s.According to (7), we can see that the allowable maximum transmission delay can be 0.05 s.The state response of system (47) with controller  in (53) and Ω in (54) is shown in Figure 2. Figure 3 plots the release instants and release interval.
From Figures 2 and 3, we can see that system (47) remains asymptotically stable under the feedback controller  in (53) and trigger matrix Ω in (54); the simulation results for  ∈ [0, 30] show that only 58 sampled signals need to be transmitted to the controller through the transmitter, which takes 19.33% of the whole sample signals.Thus, the fault tolerant state-feedback controller presented in this paper also optimizes the performance of the NCSs by reducing the networked load.
Remark 13.The simulation results show that the amount of transferred data is reduced by introducing the event-triggering communication scheme.In addition, the maximum   value,  2 , in this paper is bigger than the value,  2 , in [17,18].Therefore, the proposed controller design method in the paper not only is less conservative but also has less transferred data than [17,18].

Conclusion
The fault tolerant state-feedback controller design problem for stochastic NCSs has been investigated in the paper.Improved state-feedback controller design method with less conservation and less sampled data transmission has been proposed by employing event-triggered communication scheme, the delay-decomposition technique, and convex combination approach.A practical numerical example for the stabilization of batch reactor has been given to demonstrate

Figure 1 :
Figure 1: A framework of stochastic NCSs under an event-triggered communication scheme.

Case 2 . 1 .
We have the following:

Figure 2 :
Figure 2: The state responses of system (47) under the feedback controller  in (53) and trigger matrix Ω in (54).
Event-based release instants and release interval

Figure 3 :
Figure 3: The release instants and release interval under the feedback controller  in (53) and trigger matrix Ω in (54).

Table 1 :
The maximal allowable delays  2 and fault tolerant controller  for different Λ  in Case 1.

Table 2 :
The maximal allowable delays  2 and fault tolerant controller  for different Λ  in Case 2. When  = 0.1 and Ω 1 = Ω 2 , applying Corollary 11, we can solve the maximum  2 = 0.15 s, and the corresponding controller  and the trigger parameter Ω are given by