Adaptive Fault-TolerantH∞Controller Design for Networked Systems with Actuator Saturation

In this paper, an indirect adaptive fault-tolerant controller design method is proposed for networked systems in the presence of actuator saturation. Based on the on-line estimation of eventual faults, the parameters of controller are being updated automatically to compensate the fault effects on systems. The designs are given in linear matrix inequalities (LMIs) approach, which can guarantee the disturbance tolerance level and adaptive performances of networked systems in the cases of actuator saturation and actuator failures. An example is given to illustrate the efficiency of the design method.


Introduction
With the rapid developments in network technologies, more and more communication networks are used in control systems; especially, the stability analysis for systems with time delays have become an active research area.However, networked control systems with actuator saturation and time delays are often encountered in many practical systems such as electrical heaters and long transmission lines in pneumatic, hydraulic, and rolling mill systems.Since the existence of time delay and actuator saturation in a physical system often induces instability of poor performance, research on time delay systems with actuator saturation is a topic of great practical and theoretical importance.If the saturation and time delay are ignored in system analysis and design, the performance of the overall system can be degenerated.More seriously, saturation and time delay can cause instability of the overall system.Therefore, over the last several decades, many researchers have considered various control problems of disturbance rejection for linear systems subject to actuator saturation [1][2][3][4][5][6][7][8][9][10][11].Papers [4,5] carried out the  2 gain analysis and minimization.Although there are plenty of papers that are devoted to dealing with different problems for systems with actuator saturation, the main difference and difficulty lie in their treatment of saturation nonlinearity.In paper [2], authors gave a method for maximization of an ellipsoid which is invariant under input saturation, but persistent disturbances.The works of [1,3,[6][7][8] consider the situation where disturbance are bounded in energy.The works of [1,6,7] formulated and solved the problem of stability analysis and design.In [9,10], authors presented LMI-based methods for regional stability and performance of linear antiwindup compensators for linear control systems.[12] presents a method for the analysis and control design of linear systems in the presence of actuator saturation and  2 disturbances.During the last few years, problems about actuator saturation have been extended to many other fields of automatic control, such as singular systems [13], systems with parameters uncertainty [14], Markovian jump systems [15], decentralized control systems [16], and Hamiltonian systems [17].
Time delays are frequently encountered in almost all networked systems.Since the existence of a delay in a physical system often induces instability of poor performance, research on time-delay systems is a topic of great practical and theoretical importance.During the last decade, the control problem of systems with time delay has received considerable attention.The main methods can be classified into two types: delay-independent ones and delay-dependent ones.
On the other hand, fault tolerant has become a hot research area because of its importance in practical engineering [18][19][20][21][22][23][24][25][26][27][28].And the design approach can be broadly classified into two types: Passive approach and Active approach.A passive fault-tolerant controller commonly has a simple structure and is easily implemented [18][19][20][21][22].The system performances in normal and fault modes can be optimized.Some of these active fault-tolerant control methods may readjust controller parameters or change controller structure to compensate the fault effects on systems.Some of these methods include a strategy involving a fast subsystem for fault detection and isolation (FDI) and a supervisory system that chooses the corresponding controller for a particular type of fault.Most of the results in adaptive fault-tolerant control are based on model reference adaptive control (MRAC) [29][30][31], but the disturbance attenuation performances of systems have not been addressed yet within the MRAC framework.Paper [32] considered the problem of adaptive reliable controller via state feedback and dynamic output feedback, respectively, for linear time-delay systems against actuator faults.However, when actuator saturation problem is considered, the methods of [32] cannot be used.
As we all know, actuator faults and saturation always happen at the same time for networked systems.However, noting all above results, there is no work that deals with this problem.There are only a few papers that considered the problems about systems with actuator saturation and faults [13,33,34].Motivated by the above observations, this paper studies the problem of designing adaptive fault-tolerant  ∞ controllers for networked systems with actuator saturation.The designs are developed in the framework of LMIs approach, which can guarantee the disturbance tolerance level and adaptive  ∞ performances of networked systems in the cases of actuator saturation and actuator failures.The difference between this paper and some existing results is that in this paper the fault tolerant and saturation are considered at same time for networked systems.
The remainder of this paper is organized as follows.Section 2 introduces notation to be used in the paper, and problem statement is given in it.An adaptive fault-tolerant  ∞ controller design method is described for networked system in Section 3. In Section 4, an example is given to illustrate the efficiency of the design method.The paper will be concluded in Section 5.

Problem Statement and Preliminaries
In this paper, the following LTI plant will be considered: where () ∈   and   is the plant state at time  defined by   () = ( + ),  ∈ [−ℎ, 0], sat() ∈   is the saturated control input, () ∈   is the regulated output, and () ∈   is an exogenous disturbance in  2 [0, ∞], respectively.,  1 ,  1 ,  2 , , and , are known constant matrices of appropriate dimensions.For simplicity only, we take single delay ().The results of this paper can be easily applied to the case of multiple delays.
The following case for time-varying delay () is considered.That is, () is differentiable funcion where  is an upper bound on the derivative of ().
In this paper, we formulate the fault-tolerant control problem by using the following model form which is considered in [19,22]: where where For convenience, the following uniform actuator fault model is exploited: where  is described by  = diag[ 1 ,  2 , . . .,   ].The following definitions and lemmas will be used in the sequel.
Definition 1 (see [33]).Consider the following system: where  is parameter vector and ρ() is a time-varying parameter vector to be chosen.Let  > 0 be a given constant, then system ( 7) is said to be with an adaptive  ∞ performance index no larger than  if for any  > 0, there exists a ρ() such that the following inequality holds: Definition 2. For a matrix   ∈  × , denote the th row of   as   , and define Lemma 3 (see [32]).If there exists a symmetric matrix Θ with and Θ 11 , Θ 22 ∈  × such that the following inequalities hold: then inequality holds for all   ∈ [    ], where  =   ∈  × and Let D be a set of  ×  diagonal matrices whose diagonal elements are either 1 or 0. There are 2  elements in D, and one denotes its elements as   ,  ∈ I[0, 2  − 1], where for Then, one has the following.
For a networked system, the performance of closedloop system can be measured by the  2 gain.However, this gain cannot be well defined for closed-loop system, since a sufficiently large disturbance may lead to unstable closedloop system.For this reason, we need to consider a class of disturbances whose energy is bounded by a given value; that is, In this paper, we will consider the following problems.The first question is, what is the maximal value of  such that the state will be bounded for all  ∈ M  for systems with time delay?This question can be referred to as disturbance tolerance level.The system performance can be measured by the restricted  2 gain over M  .In this paper,  2 gain and M  will be considered at same time for networked system.
(I) The trajectories of the closed-loop system that start from the origin will remain inside the domain   () for every  ∈ M  .
From Theorem 10, we have the following algorithm to optimize the adaptive  ∞ performance in normal and fault cases and the disturbance tolerance level  with considering time delay.Algorithm 11.Suppose that   and   denote the adaptive  ∞ performance indexes for the normal case and fault cases of the closed-loop system (30), respectively.Let  denote the disturbance tolerance level.Then,   ,   are minimized, and  is maximized if the following optimization problem is solvable: 35) , ( 36) , (37) , (38) where , and , , and  are weighting coefficients.
Remark 12. Theorem 10 prevents a condition for the existence of an adaptive fault tolerant  ∞ controller.In Theorem 10, if set   = 0,   = 0,   = 0, and   = 0,  ∈ I[1, ], the condition of Theorem 10 is reduced to fixed gains condition.By the following example, we can get that the adaptive controller can guarantee better effect.

Examples
Example 13.Consider the system of the form (1) with and the following two possible fault modes.
Mathematical Problems in Engineering Fault mode 1: both of the two actuators are normal; that is  1 1 =  1 2 = 0. Fault mode 2: the first actuator is outage, and the second actuator may be normal or loss of effectiveness, described by  2 1 = 1, 0 ≤  2 2 ≤ , where  = 0.5 denotes the maximal loss of effectiveness for the second actuator.
During the following simulation, fault case is considered as follows.At 0 second, the first actuator is outage.Here, we choose  1 =  2 = 100.
Firstly, we consider the  ∞ performance.The disturbance is given as Figures 1 and 2 show the responses curves of the first state in normal and fault case, respectively.Then, we consider the disturb tolerance problem.The disturbance is given as Figures 3 and 4 show the responses curves of the states in normal case.

Conclusions
In this paper, an adaptive fault-tolerant  ∞ controllers design method was given for networked systems with actuator  saturation.The designs were proposed in LMIs approach, which could guarantee the disturbance tolerance ability and adaptive  ∞ performances of networked systems in the cases of actuator saturation and actuator failures.An example, has been given to illustrate the efficiency of the design method.where Furthermore, by (20) it follows that where where It is obvious from the requirement of 0 <  1 and the fact that in (A.13) −( 3 +   3 ) must be negative and  is nonsingular. Defining . .,    ],  ∈ I[1, ].Considering the lower and upper bounds    and    , the following set can be defined:   = {    = diag [

Figure 1 :
Figure 1: Response curve of the first state in normal case with the adaptive controller (solid) and the fixed gain controller (dashed).

Figure 2 :
Figure 2: Response curve of the first state in fault case with the adaptive controller (solid) and the fixed gain controller (dashed).

Figure 3 :
Figure 3: Responses curves of the states with adaptive controller in normal case.

Figure 4 :
Figure 4: Responses curves of the states with fixed gain controller in normal case.