Adaptive Fault-Delay Accommodation for a Class of State-Delay Systems With Actuator Faults

Shengjuan Huang and Chunrong Li

Abstract—Fault and delay accommodating simultaneously for a class of linear systems subject to state delays, actuator faults and disturbances is investigated in this work. A matrix norm minimization technique is applied to minimize the norms of coefficient matrix on time delay terms of the system in consideration. Compared with the matrix inequality scaling technique, the minimization technique can relax substantially the obtained stability conditions for state delay systems, especially, when the coefficient matrices of time delay terms have a large order of magnitudes.An output feedback adaptive fault-delay tolerant controller (AFDTC) is designed subsequently to stabilize the plant with state delays and actuator faults. Compared with the conventional fault tolerant controller (FTC), the designed output feedback AFDTC can be updated on-line to compensate the effect of both faults and delays on systems. Simulation results under two numerical examples exhibit the effectiveness and merits of the proposed method.

I. Introduction

IT is known that delay phenomena are commonly encountered in a number of real systems such as mechanics,physics, biology, medicine, and engineering systems [1]. Aircraft stabilization, manual control, models of lasers, neural networks, nuclear reactors, ship stabilization, and systems with loss less transmission lines often suffer from time delays[2]. Time delays are frequently the source of instability generating oscillations in systems. For example, as described in [3],the input and output data will be inevitably randomly missed partially due to the time delays and packet loss from the networked-induce. How to cope with the negative effect of time delays on systems efficiently is a challenging problem. For time delay systems, the robust control [4], [5], robustH∞control [6]–[8], fault tolerant control (FTC) [9]–[17], other stability analysis and stabilization [18]–[22] have been widely studied in past decades.

In recent years, some time delay system research teams are studying how to derive stability conditions with less conservativeness by optimizing the Lyapunov-Krasovskii functional, so as to obtain a large admissible upper bound of time delay, see e.g., [23]–[30]. In [1], to obtain stability conditions with less conservativeness for the delay fuzzy systems with actuator faults, a novel integral inequality technique has been introduced. A time delay decomposition technique has been applied to study the upper bound of time delay for time delay systems in [26]. The convex quadratic function approach has been presented in [25]. Lyapunov-Krasovskii functional combined with the Wirtinger-based integral inequality approach are investigated in [28] and [29].However, few of the existing literature consider how to design an effective control mechanism to compensate time delay terms in the system model. The obtained stability conditions for state-delay systems will be conservative when the coefficient matrices of time delay terms have a large order of magnitudes. In this case, the obtained stability conditions cannot guarantee a larger admissible upper bound of time delay. This research gap stimulates us the current work to cope with the fault-delay accommodation for a class of linear systems with state delays and actuator faults.

On the other hand, the research of FTC for various control systems is still enduring. FTC can provide a strong guarantee for the security and reliability of a control system, and has attracted many scholars’ unremitting research. FTC is usually divided into passive FTC and active FTC, while active FTC is the research mainstream. We have witnessed many outstanding achievements about FTC for linear/nonlinear control systems, see e.g., [31]–[45]. In the last decade, faulttolerant control and fault diagnosis for time-delay systems have attracted the attention of some fault-tolerant control research teams. In our previous work [1], the problem of dynamic output feedback fault tolerant control for a class of delay fuzzy systems with actuator faults has been studied. A delay-dependent adaptive reliableH∞controller has been designed against actuator faults for linear time-varying delay systems in [31]. A fault accommodation approach was proposed in [42], where a class of linear systems subject to uncertainties and time-varying state delays were considered.The fault tolerant synchronization problem was discussed for complex interconnected neural networks in [37], where time delays and sensor faults were both considered. However, in the mentioned literature, the matrix inequality scaling technique is the main tool to deal with time delay terms, and the controller’s reasonable compensation for time delay terms was ignored. The controller with the accommodation of simultaneous fault and time delay for delay control systems has not been fully investigated. This research gap further stimulates us the work in the present.

In mind of these motivations, the present work is concerned with the fault and delay accommodating simultaneously for a class of linear systems subject to state delays and actuator faults. A matrix norm minimization technique is a key tool to minimize the norms of coefficient matrix on time delay terms of the system in consideration. Then, an adaptive fault-delay accommodation based output feedback controller is designed to stabilize the state delay system with actuator faults. The designed controller can be updated on-line to compensate the effect of both faults and delays on systems effectively.Simulation results exhibit the effectiveness and merits of the proposed method. The following summarizes the main contributions of the present work:

1) The effect of faults and state delays on the linear systems in consideration can be compensated simultaneously on-line through the designed output feedback adaptive fault-delay tolerant controller (AFDTC). However, in the literature, see e.g., [1], [2], [27], [42], time delay terms are usually removed by using the inequality scaling technique, while the compensation for the effect of delay terms on systems was ignored.

2) The matrix norm minimization technique is applied to minimize the norms of coefficient matrix on time delay terms of the system in consideration. In this case, the obtained stability conditions for the closed-loop state delay system will be less conservative compared with those in the existing literature such as [1], [24], [31], [41], [42], in particular, when the coefficient matrices of time delay terms have a large order of magnitudes.

The organization of this paper is as follows. Section II gives the considered state delay system model subject to actuator faults, where a matrix norm minimization technique is applied to minimize the norms of coefficient matrix on time delay terms of the system. An output feedback AFDTC is designed,and some sufficient stability conditions to guarantee theH∞stability of the closed-loop state delay system are given in Section III, then simulation results are exhibited in Section IV.The work is concluded in Section V.

II. Problem Formulation

Consider the following linear time-invariant (LTI) system subject to state delays, actuator faults and disturbances represented as

wherex(t)∈Rnis the system state vector and unmeasurable;u(t)∈Rmis the control input andf(t)∈Rmis the additive actuator fault signal and unknown;w(t)∈Rlrepresents the exogenous disturbance belonging toL2[0,∞) ; whilez(t)∈Rqrepresents the control output andy(t)∈Rpis the measured output in disturbances and delays free; τ is a constant time delay of states and ?(t) is an initial continuous vector-valued function defined on the interval [?τ,0].A,Aτ,B,C,D,E,Eτ, andHare constant matrices with appropriate dimensions. Assume that (A,B) is controllable.

Before starting the design procedure, the models described in (1)–(4) need to satisfy the following assumptions:

Assumption 1:The additive actuator faultf(t) satisfies:

Assumption 2:The control matrixBin (1) satisfiesrank(B)=m.

Assumption 3:rank(CB)=rank(B)=m.

Assumptions 1–3 exhibit some general conditions, which can be seen in the literature, see, e.g., [1], [33], [44]. As described in [33], the derivatives of the faults are energybounded, i.e.,f˙(t)∈L2[0,∞), which are more general compared with those in [34], [35].

In the past research for state delay systems, the state delay terms in systems, such asAτx(t?τ) in system (1), are in general included in the derivatives of the designed Lyapunov-Krasovskii functional, where the termis usually estimated aswithP>0 andQ>0. However, the estimation might lead to the conservativeness of the obtained stability conditions whenAτhas a large order of magnitudes. Therefore, an accommodation mechanism is to be introduced to adjust the coefficient matrix of state delay term. In this paper, a matrix norm minimization technique will be applied to achieve this requirement. The system in (1) can be rewritten as

with

where ?τ(t) represents the combination of fault and delay to be compensated on-line, while the accommodation matrixKτis to be designed and satisfies

Remark 1:It can be seen that in system (5), the coefficient matrix of state delay has been adjusted asAτ?BKτ. Under the minimum condition in (7), the adjusted coefficient matrix of state delay will have a smaller order of magnitudes, which implies thatIn this case, the obtained stability conditions will be less conservative than the conventional ones that are obtained based on only the inequality scaling technique.

In particular, in system (5), ifAτandBis matched, i.e.,there exists a matrixKτsuch thatAτ=BKτ, then (5) reduces to

In this case, it is obvious that the obtained stability conditions will be less conservative, which can guarantee a large admissible upper bound of time delay.

The following work will be concentrated on the designing of an adaptive fault-delay tolerant output feedback controller to stabilize the state delay system in (5). Sufficient conditions to guarantee theH∞stability of the closed-loop state delay system are to be given subsequently.

III. Adaptive Fault-delay Accommodation Based Controllers Design

Based on the system in (5), where the coefficient matrix of state delay has been accommodated, in the following, the adaptive output feedback control and the stability analysis for the closed-loop system are to be solved.

A. Output Feedback AFDTC

Firstly, based on the key source of measured output in (3),an adaptive fault-delay tolerant controller (AFDTC) for the plant in (5) with (2) and (4) is to be designed. The control strategy is constructed as

with the adaptive law

Remark 2:The output derivatives here used to construct the adaptive law in (10), in fact, are frequently applied to design adaptive fault estimators (or adaptive laws) in the literature,see e.g., [44], [45]. It should be noted that the output derivatives are of no real significance for practical systems and seldom used in adaptive control, especially, when the states of systems are known, see e.g., [17], [37]. However, the designed adaptive fault estimator in the literature or adaptive law in (10) is only applied to the stability analysis but not for the practical applications. For example, based on (10), in practical applications, the estimate of ?τ(t) can be computed by

Under the designed output feedback based AFDTC in (9)with the adaptive law in (10), one has the following overall closed-loop state delay system described by

Consequently, the adaptive law can be inferred as

Remark 3:The designed output feedback based AFDTC in(9) can be updated on-line by the adaptive law in (10) to compensate simultaneously the effect of state delays and faults on the system. However, it cannot be achieved by using the conventional AFTC in the existing literature, see e.g., [1],[42].

Design the tuning gainMas

Then, (13) is simplified as

wi th

Denote

Then, one has

B. Stability Analysis and AFDTC Gains Design

Under the designed AFDTC strategy, in the following,sufficient conditions to ensure theH∞stability of the closedloop state delay system are to be given.

Lemma 1:For given positive scalars: γ1and γ2, if there exist matricesK,Kτ, Γ,Ma,P>0, andQ>0 such that the optimization problem

is solved, where

Proof:Set

whereP>0. The derivative ofV(t) in (22) with respect to timetis inferred as

Then, if the matrix inequality in (19) holds, it follows that

where

It is obvious from (22) thatx(t)=0 , ?t∈[?τ,0], when ?(t)=0 , ande?(t)=0 , which implies thatX(t)=0, and

Therefore, from (24) one has

SinceV(T)>0, the prescribedH∞performance in (21) can be verified.

On the other hand, if (19) holds, then one haswhenandw(t)=0, based on the time derivatives ofV(t) in(22) along the solution of (16). Therefore, the asymptotic stability of (16) (and (12)) with state delay follows immediately. So far, the proof is completed.

Under the sameV(t), by using the conventional design methods,the stability conditions in Lemma 1 will reduce to, w here

It will be conservative since thatThis further implies that, under the same Lyapunov-Krasovskii functional such as in [1], [27]–[29] and so on, one can obtain less conservative stability conditions for the closed-loop system with the designed AFDTC and the obtained upper bounds of time delays will be larger than the existing ones. In view of our research goal lying in fault-delay accommodation based output feedback control for the system, this work is omitted.

Remark 4:In the proof of Lemma 1, one can see that the termis hard to be decoupled, that is, ideal feasible solutions for the derived matrix inequality will be very difficult to be found. In fact, it is one of the difficulties of output feedback control and also an interesting problem for us to study and overcome in the future.

Theorem 1:For given positive scalars: γ1and γ2, if there exist matricesK,Kτ, Γ0>0 ,Mˉa, andQ>0 such that the optimization problem

Proof:Following similar steps as in the proof of Lemma 1,the proof can be derived withand the variable changing Γ=Γ0Φ,Ma=Γ0Mˉaand the Schur complement. The details are omitted here.

Remark 5:It should be noted that by using the conventional design methods, the LMI condition can be obtained as

whereare to be designed.

Remark 6:In fact, Theorem 1 just gives a feasible method of converting the matrix inequalities in Lemma 1 into LMIs,and there might be other approaches such as the linear transformation method. However, due to the existence of the matrix inequality (30), it will be a challenge to find another ideal method of converting the matrix inequalities in Lemma 1 into LMIs. On the other hand, in Theorem 1, ∥Aτ∥ has been minimized as ∥Aτ?BKτ∥, which can guarantee the state of the closed-loop system to fluctuate in a small boundary.

IV. Simulation Results

In order to validate the proposed AFDTC strategy, two state-delay models are provided in this section: one is a chemical reactor model, the other is a delay system with a large order of magnitudes in the coefficient matrix of state delay terms.

A. Example 1

A two-stage chemical reactor model subject to delayed recycle streams [46] is considered. The state-space model is exhibited as:

and the control output and measured output are described as

where the model parameters can be seen in Table I,f(t) andw(t)represent actuator faults and input disturbances, respectively. The parameters in the model are assumed to be:q1=q2=1,a=b=1,V1=V2=1,R1=0.5,R2=2,G=1.

TABLE I Model Parameters

For simulation purpose, the actuator fault and disturbance are created as in [42], while the state delay τ =5. Set γ1=0.25 and γ2=0.1, solving the the optimization problem in (28), one has δmin=6.0035×10?4, and a set of feasible solutions for the LMIs in (29) and (30) can be obtained, and the control gains and adaptive law gains are designed as:K=4.476,Kτ=[0.5,2], Γ =9.3215,M=?60.4739.

However, under the given scalars above, there is no feasible solution by solving the LMI in (31). Reset γ1=0.45 and γ2=0.65, a set of feasible solutions for the LMI in (31) can be obtained, and the control gains and adaptive law gains are designed as:K=6.7365, Γ=1.8802,M=?16.3486. It demonstrates in practice that under the minimum of norm for the coefficient of state delay, one can obtain smallerH∞performance indexes by the proposed theorem compared with the conventional ones.

Fig. 1(a) shows the response curves of fault-delay termθτ(t)and its estimate by the designed adaptive law in (10). It can be seen that the fault delay θτ(t) can be depicted with a good accuracy. It means that the designed AFDTC can compensate on-line the effect of fault and time delay on the plant effectively. However, by using the conventional AFTC approach, the fault cannot be better estimated by the designed adaptive law due to the effect of state delay in the plant, and Fig. 1(b) exhibits the case.

The trajectories of measured output and control output under our proposed AFDTC and the conventional AFTC are further shown in Fig. 2, respectively. From the figure, it can be seen that the measured output and control output can both fluctuate in a smaller boundary under the designed AFDTC compared with the conventional AFTC.

Fig. 1. Estimation results. (a) Response curves of fault-delay and its estimate by AFDTC. (b) Response curves of fault and its estimate by AFTC.

Fig. 2. Output results under AFDTC and AFTC respectively. (a) Response curves of measured outputs. (b) Response curves of control outputs.

B. Example 2

To further exhibit the designed AFDTC strategy, we consider the following linear system with state delays and actuator faults, where the coefficient matrix of state delay is of a large order of magnitudes. The model is described as

and the control output and measured output in this paper are described as

wheref(t) andw(t) represent actuator faults and input disturbances, respectively. In this example, the norm of the coefficient matrix (Aτ) of state delay is 1 29.5630 (| |Aτ||2). Now, set γ1=γ2=0.1, solving the the optimization problem in (28),one has δmin=4.0853, and a set of feasible solutions for the LMIs in (29) and (30) as follows:

Then, the adaptive law gains can be obtained as

On the other hand, the coefficient matrix (Aτ) of state delay has been accommodated as

with the norm | |Aτ?BKτ||2=16.6897.

However, without using the proposed delay accommodation method, there is no feasible solution by solving the LMI in(31) with the givenH∞performance indexes as γ1=γ2=0.1.We need to reset the scalars as γ1=0.2 , and γ2=0.7, then we can obtain a set of feasible solutions for the LMI in (31), and the control gain and adaptive law gains can be designed as

It further exhibits in practice that the proposed stability conditions under the accommodation of fault and delay are less conservative compared with the conventional ones.

For simulation purpose, the state delay is assumed that τ=2, while the actuator faults and input disturbance are created as

Run the simulation for the example under the proposed AFDTC and the conventional AFTC, respectively. The simulation results are shown in Figs. 3–6.

Fig. 3 shows the response curves of fault-delay terms θτi(t),i=1,2, and their estimates by the proposed AFDTC with the adaptive law in (10). It can be seen that the fault-delays have been also estimated with a good accuracy. It means that the designed AFDTC can compensate effectively the effect of faults and state delays on the system in consideration.However, this cannot be achieved by the conventional AFTC.

Fig. 4 exhibits the response curves of fault termsfi(t),i=1,2, and their estimates by the conventional AFTC. It can be seen clearly that the faults has not been estimated with a good precision under the conventional AFTC in this example.

Fig. 3. Estimation results by AFDTC. (a) Response curves of θ τ1(t) and its estimate. (b) Response curves of θ τ2(t) and its estimate.

Fig. 4. Estimation results under AFTC. (a) Response curves of f1(t) and its estimate. (b) Response curves of f2(t) and its estimate.

Fig. 5 further exhibits the response trajectories of measured output under our proposed AFDTC (Fig. 5(a)) and the conventional AFTC (Fig. 5(b)), respectively. Compared with the conventional AFTC, one can see that the proposed AFDTC can guarantee the measured outputs fluctuate in a smaller boundary. It further shows that the designed AFDTC can compensate on-line the effect from both faults and state delays on systems effectively. Fig. 6 illustrates as well this case, where the simulation results forx1(t) (Fig. 6(a)) andz(t)(Fig. 6(b)) are shown under our AFDTC and the conventional AFTC, respectively.

V. Conclusion

Fig. 5. Measured output results. (a) Response curves of y (t) under AFDTC.(b) Response curves of y (t) under AFTC.

Fig. 6. Simulation results under AFDTC and AFTC respectively. (a) Response curves of x1-states. (b) Response curves of control outputs.

This present work has studied the strategy of AFDTC for a class of linear systems subject to simultaneous state delays and actuator faults. A matrix norm minimization technique has been applied to cope with the designing of the adaptive output feedback controller. Based on the control strategy, the faults and state delays can be simultaneously compensated effectively on-line so as to stabilize the state delay system.Under the matrix norm minimization technique, relaxed stability conditions for the delay closed-loop system have been derived. Finally, the proposed AFDTC is applied to run a chemical reactor model and a delay system with a large order of magnitudes in the coefficient matrix of state delay terms,respectively, so as to show its effectiveness and merits. The future work will be concentrated on the study of upper bounds of delays or/and convergent output feedback FTC for delay systems.