# Robust Adaptive Fault-Tolerant Control of Stochastic Systems with Modeling Uncertainties and Actuator Failures

Authors: Wenchuan Cai; Lingling Fan; Yongduan Song
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

This paper deals with the problem of fault-tolerant control (FTC) of uncertain stochastic systems subject to modeling uncertainties and actuator failures. A robust adaptive fault-tolerant controller design method based on stochastic Lyapunov theory is developed to accommodate the negative impact on system performance arising from uncertain system parameters and external disturbances as well as actuation faults. There is no need for on-line fault detection and diagnosis (FDD) unit in the proposed FTC scheme, which not only simplifies the design process but also makes the implementation inexpensive. Numerical examples are provided to validate and illustrate the benefits of the proposed control method.


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## Body

## 1. Introduction

Stability analysis and control design of stochastic systems have received increasing attention during the past decades. Under the framework of Itô equations together with the notion of mean-square stability, some interesting results have been obtained in terms of generalized algebraic Riccati equations, linear matrix inequality (LMI), or spectra of some operators (see, for instance, [1–4] and the references cited therein).However, to our knowledge, very few works have dealt with the stabilization of general stochastic systems where actuator failures, parameter uncertainties, and state-dependent disturbances are involved simultaneously. This motivates us to investigate the reliable control problem of stochastic systems, aiming at maintaining an acceptable performance for the closed-loop systems in the presence of actuator failures and modeling uncertainties.Actuator failures can cause severe performance deterioration of control systems, or even system instability, leading to catastrophic accidents. Fault-tolerant control (FTC) has been viewed as one of the most promising methods to increase system safety and reliability and has thus received considerable attention from control and system engineering research community [5–17]. Most existing FTC methods can be broadly classified as active FTC and passive FTC. The active FTC requires a fault detection and diagnosis (FDD) mechanism to detect and identify the faults in real time, and a mechanism to reconfigure the controller according to the on-line fault information from the FDD [9–17]. The main idea of the passive FTC approach is to design a single controller that is robust against faults and uncertainties. In contrast to the passive approach, active methods utilize control reconfiguration to adjust controllers in real time so that the impacts of the failures can be compensated and the stability as well as the acceptable performance of the system can be maintained. Remarkable progress have been made in the area of actuator accommodation control with various effective design methods developed such as linear quadratic [18], multiple model designs [19–21], model following [2], FDD-dependent designs [22–24], and sliding mode control-based designs [10, 25].It is noted that, by blending adaptive control into FTC, the resultant control scheme turns out to be effective in reconfigurable control of systems with actuator failures [9, 26–30]. However, it is noted that few of the aforementioned works address the fault-tolerant control problem of stochastic systems.In this research, we will consider robust adaptive FTC for uncertain stochastic systems subject to actuator faults. The system under consideration involves parameter uncertainties and state-dependent disturbances. Moreover, there involves actuation faults that are assumed to be unpredictable during the system operation. We are interested in developing an FTC control scheme without the need for FDD. The developed FTC scheme is user friendly in the fact that no complicated computation is involved in its design and implementation.The remaining part of the paper is organized as follows. In Section2, the control problem is formulated. The design and analysis of the proposed control schemes are given in Section 3. Numerical simulations are conducted to demonstrate various features of the proposed control method and the results are presented in Section 4. Finally, the paper is closed with some concluding comments in Section 5.Notation. The notations in this paper are quite standard. Rn and Rn×m denote, respectively, the n dimensional Euclidean space and the set of all n×m real matrices. The superscript “T” denotes the transpose and the notation X≥Y (resp., X>Y) where X and Y are symmetric matrices, which means that X-Y is positive semidefinite (resp., positive definite). I is the identity matrix with compatible dimension. |·| is the Euclidean norm in Rn. If A is a matrix, denote by ∥A∥ its operator norm; that is, ∥A∥=sup⁡{|Ax|:|x|=1}=λmax⁡1/2(ATA), where λmax⁡(·) [resp., λmin⁡(·)] means the largest (resp., smallest) eigenvalues of A. Moreover, (Ω,F,P) is probability space with Ω the sample space, F the σ-algebra of subsets of the sample space, and P the probability measure. Ξ{·} stands for the mathematical expectation operator with respect to the given probability measure P. L2 and L∞ denote the spaces of square-integrable vector and bounded vector functions over [0,∞), respectively.


## 2. Problem Statement

Consider the stabilization problem of the following uncertain stochastic systems subject to actuator faults and external disturbances:(1)dx(t)=[(A+ΔA(t))x(t)+B(ua(t)+f(x(t)))]dt+(C+ΔC(t))x(t)dω(t),
where x(t)∈Rn is state, ua(t)∈Rm is actual control input, f(x,t)∈Rm is unknown external disturbances.Here,ω(t) is a one-dimensional Brownian motion defined on the probability space (Ω,F,P) with Ξ{ω(t)}=0 and Ξ{ω2(t)}=1.A; B and C are known real constant matrices with appropriate dimensions. Without loss of generality, it is assumed that the pair (A,B) is controllable. ΔA(t),ΔC(t), and C denote parameter uncertainties and satisfy
(2)ΔA(t)=BF1(t)ΔC(t)=BF2(t)C=BF3,
where F3 is known constant matrix, F1(t) and F2(t) are unknown time-varying matrix satisfying ∥F1(t)∥≤aF1<∞ and (∥F2(t)∥+∥F3∥)2≤aF2<∞.Remark 1.
It is observed that the parameter uncertainty structure as in (2) is more relaxed than the most existing methods. The parameter uncertainty structure which has been widely used in the problems of robust control and robust filtering of uncertain systems is assumed to be (ΔA(t)TΔC(t)T)T=(E1TE2T)TF(t)H, where E1,E2, and H are known constant matrices and F(t) is an known time-varying matrix satisfying FT(t)F(t)<I (see, for instance, [31–34]). Obviously, the structure herein which only needs the existence of the upper bound of F(t) is easier to be satisfied.To formulate the fault-tolerant control problem, the fault model must be established first. In system (1), the types of faults under consideration include loss of effectiveness, stuck, or combination of all. The actual control input ua(t) able to impact the system and the designed control input u(t) designed are not the same in general. In this paper, the relationship between them will be adopted. Consider
(3)ua(t)=Δ(·)u(t)+κ(t),
where Δ(·)=diag⁡{δi(t)} is a diagonal matrix with δi(t)(i=1,2,…,m) being the unknown and time-varying scalar function called actuator efficiency factor, or “health indicator.” For every fault mode, δ_i and δ-i represent the lower and upper bounds of δi, respectively. Note that, when δ_i=δ-=1, there is no fault for the ith actuator ui. When δ_i=δ-=0, the ith actuator ui is outage. When 0<δ_i≤δ-<1, the type of actuator is loss of effectiveness. κ(t) denotes a vector function reflecting the portion of the control action produced by the actuator that is completely out of control.The type of actuator failures considered in this work is listed in  Table1.Table 1
Representations of typical actuator failures.
Type of actuator failures
δ
i
(
t
)
κ
(
t
)
Healthy actuator
1
0
Loss of effectiveness only
0
<
δ
i
(
t
)
≤
1
0
Loss of effectiveness and partially out of control
0
<
δ
i
(
t
)
≤
1
Time-varying
Loss of effectiveness and partially jammed
0
<
δ
i
(
t
)
≤
1
ConstantIn order for the system to admit a feasible FTC, the following assumptions are imposed.Assumption 2.
The unparametrizable stuck-actuator fault and external disturbance are piecewise continuous bounded functions; that is, there exist unknown positive constantsaκ and af such that
(4)∥κ(t)∥≤aκ<∞,∥f(·)∥≤afψf(·)<∞.Assumption 3.
For the system under consideration, there exist some constantsα>0 and β>0 such that for all possible actuator faults, the following relation holds:
(5)α∥BTPx∥2≤β∥BTPxΔ(·)∥2,
where
(6)Δ(·)=diag⁡{δi(t)},δi(t)∈(0,1](i=1,2,…,m).Remark 4.
Assumption2 confines the vector κ(t) and external disturbances are bounded. Assumption 3, slightly less restrictive, sets constraint on the actuation faults, which a feasible FTC is able to deal with. Clearly, such condition is well justified if all the actuators with faults are still functional (i.e., δi(t)≠0), whereas the too extreme faults in that all the actuators completely fail to work (i.e., δi(t)=0) make the assumption invalid, which, if not impossible, is significantly challenging to develop a globally stable control for the stochastic system (1); thus it is not considered in this work.Remark 5.
Since(A,B) is controllable, one can choose N0 properly such that A-=A-BN0 is Hurwitz. Namely, for given Q=QT>0, there exists a symmetric and positive definite P such that the following matrix inequality is established:
(7)A-TP+PA-+ρI<-Q,
where ρ=∥BTPB∥aF2. Note that we can find the proper P very easily because (7) is much simpler than those complex LMIs. The method in frame of linear matrix inequality is well used in many existing works [8, 13, 31, 33].In the end of this section, the following important lemma is given, which will be used for the development of our result.Lemma 6 (see [35]).
The trivial solution of the stochastic differential equation(8)dx(t)=a(x,t)dt+b(x,t)dω,
with a(x,t) and b(x,t) sufficiently differentiable maps, is globally asymptotically stable in probability, if there exists a positive definite, radially unbounded, twice continuously differentiable function V(x(t),t) such that the infinitesimal generator is
(9)L[V(x(t),t)]=∂V∂t+(∂V∂x)Ta(x,t)+12b(x(t),t)T∂2V∂x2b(x(t),t)<0.


## 3. Fault-Tolerant Control Design

To show the idea of this work explicitly, several fault-tolerant control schemes are developed under different conditions in this section. At the beginning, a robust fault-tolerant control method is presented.


### 3.1. Robust Fault-Tolerant Control

In this section, a robust fault-tolerant control of the form(10a)u(t)=-N0x+N(t)
is proposed, where N0 is chosen such that A-BN0 is Hurwitz, and N(t) is generated by
(10b)N(t)=-aλmφ(·)BTPx∥BTPx∥,
with 0<λm≤α/β being a constant, where λm represents the lower bound of the health indicator matrix Δ(·); that is, 0<λm≤λmin⁡(Δ) and α>0, β>0 are suitable constants such that
(10c)α∥BTPx∥2≤β∥BTPxΔ(·)∥2,(10d)φ(·)=1+∥N0x∥+∥φf(x)∥+∥x∥,a=max⁡{1,aN,af,aF1}.Theorem 7.
Under Assumptions2 and 3, the FTC as given in ((10a), (10b), (10c), and (10d)) exponentially stabilizes (in mean square) the stochastic system described by (1), for all admissible uncertainties as well as all actuator failures corresponding to (3).Proof.
When the system is subject to the actuator failure as described in (3), its dynamic behavior becomes
(11)dx(t)=[(Δ(·)u(t)+κ(t)+f(x(t)))(A+ΔA(t))x(t)+B(Δ(·)u(t)+κ(t)+f(x(t)))]dt+(C+ΔC(t))x(t)dω(t).
With the proposed control ((10a), (10b), (10c), (10d)), one has
(12)dx(t)=[(Δ(·)u(t)+κ(t)+f(x(t)))(A+ΔA(t))x(t)+B(Δ(·)u(t)+κ(t)+f(x(t)))]dt+(C+ΔC(t))x(t)dω(t)=[B(Δ(·)(-N0x+N(t))+κ(t)+f(x(t)))(A+ΔA(t))x(t)+B(Δ(·)(-N0x+N(t))+κ(t)+f(x(t)))]dt+(C+ΔC(t))x(t)dω(t)=(A-BN0)x(t)dt+B[Δ(·)N(t)+Z(t)]dt+(C+ΔC(t))x(t)dω(t),
where
(13)Z(·)=(I-Δ(·))N0x(t)+κ(t)+f(·)+F1(t)x(t),
which is bounded as
(14)∥Z(·)∥≤∥N0x∥+∥κ(·)∥+∥f(·)∥+∥F1(t)x(t)∥≤a(1+∥N0x∥+∥ψ(x)∥+∥x∥),
based on Assumption 2, where a=max⁡{1,aN,af,aF1} and φ(·)=1+∥N0x∥+∥φf(x)∥+∥x∥. Thus, it is not difficult to get
(15)(BTPx)TZ≤aφ(·)∥BTPx∥.
Consider the following Lyapunov function candidate:
(16)V(x(t),t)=xT(t)Px(t).
Then, by Itô’s formula, the infinitesimal generator of (12) is
(17)L[V(x(t),t)]=xT(t)P(A-BN0)x(t)+((A-BN0)x(t))TPx(t)+2(BTPx)T[-Δ(·)aλmφ(·)BTPx∥BTPx∥]+2(BTPx)TZ+xT(t)(C+ΔC(t))TP(C+ΔC(t))x(t).
Note that the last term of (17) cannot be combined with Z(·); thus the adaptive updating law cannot be used to compensate its effect as usual. To establish the robust stability of the closed-loop system (12), we need to have the following development. From the fact that (∥F2(t)∥+∥F3∥)2≤aF2<∞ and using (2), it is seen that the last term of (17) can be expressed as
(18)xT(t)(C+ΔC(t))TP(C+ΔC(t))x(t)=xT(t)[B(F2(t)+F3)TPB(F2(t)+F3)]x(t)≤xT(t)[∥BTPB∥aF2]x(t);
from (10c), it holds that
(19)-(BTPx)TΔ(·)(BTPx)≤-αβ∥BTPx∥2≤-λm∥BTPx∥2.
and by defining ρ=∥BTPB∥aF2, the inequality (17) can be shown to satisfy
(20)L[V(x(t),t)]≤xT(t)(A-TP+PA-)x(t)-2aλmφ(·)∥BTPx∥(BTPx)TΔ(·)(BTPx)+aφ(·)∥BTPx∥+xT(t)∥BTPB∥∥F2∥2x(t)≤xT(t)(A-TP+PA-+ρI)x(t)≤-12xTQx<0forx(t)≠0,
where matrixes P and Q are chosen properly to satisfy A-TP+PA-+ρI≤-Q. Therefore, it is confirmed from Lemma 6 that the closed-loop system (11) is asymptotically mean square stable in probability despite faulty actuators with the proposed FTC.Remark 8.
Note that if proper constantsα and β can be obtained in advance, the proposed control ((10a), (10b), (10c), and (10d)) achieved exponential stability in mean square for the stochastic system under Assumptions 2 and 3. However, it is a little difficult to select such α and β to ensure λm≤α/β, since λm the lower bound of the eigenvalue of the health indicator matrix is not available in general. In view of this, a more feasible method is developed in the next subsection.


### 3.2. Robust Adaptive Fault-Tolerant Control

In order to develop a control scheme that is not only robust but also adaptive yet fault-tolerant, we modify the previous one to get(21a)u(t)=-N0x+N^(t),
where N0>0 is chosen such that A-BN0 is Hurwitz and N^(t) is on-line updated by
(21b)N^(t)=a^(t)φ(x)BTPx∥BTPx∥,
with
(21c)φ(x)=1+∥N0x∥+∥φf(x)∥+∥x∥,(21d)a^˙(t)=-γφ(x)∥BTPx∥,γ>0.Theorem 9.
Consider the uncertain stochastic system (11) under Assumptions 2 and 3. If the robust adaptive fault-tolerant controller ((21a), (21b), (21c), and (21d)) is implemented, the closed-loop system is ensured to be asymptotically stable.Proof.
Substituting the proposed control ((21a), (21b), (21c), and (21d)) into the stochastic system (11), we obtain the closed-loop system dynamics as follows:
(22)dx(t)=[(A+ΔA(t))x(t)+B(Δ(·)u(t)+κ(t)+f(x(t)))]dt+(C+ΔC(t))x(t)dω(t)=[(Δ(·)(-N0x+N^(t))+κ(t)+f(x(t)))(A+ΔA(t))x(t)+B(Δ(·)(-N0x+N^(t))+κ(t)+f(x(t)))]dt+(C+ΔC(t))x(t)dω(t)=(A-BN0)x(t)dt+B[Δ(·)N^(t)+Ζ(t)]dt+(C+ΔC(t))x(t)dω(t).
Consider the following Lyapunov function candidate:(23)V(x(t),t)=xTPx+1λmγ(a-a^λm)2,
where γ>0 is a constant related to adaptation rate chosen by the designer and λm>0 is constant defined as before. Upon using the control scheme with the adaptive algorithm, it is not difficult to show that
(24)L[V(x(t),t)]=xT(t)P(A-BN0)x(t)+((A-BN0)x(t))TPx(t)+2(a-a^λm)(-a^˙γ-1)+xT(t)(C+ΔC(t))TP(C+ΔC(t))x(t)=[A-x+B(Δ(·)N^(t)x+Z(·))]TPx+xTP[A-x+B(Δ(·)N^(t)x+Z(·))]+2(a-a^λm)(-a^˙γ-1)+xT(t)(C+ΔC(t))TP(C+ΔC(t))x(t)=xT(A-TP+PA-)x+2xTPB(Δ(·)N^(t)x+Z(·))+2(a-a^λm)(-a^˙γ-1)+xT(t)(C+ΔC(t))TP(C+ΔC(t))x(t).
Then
(25)L[V(x(t),t)]=xT(A-TP+PA-)x+2xTPB{Δ(·)[-a^φ(x)(BTPx)∥BTPx∥]+Z(·)}+2(a-λma^)(-γ-1a^˙)+xT(t)(C+ΔC(t))TP(C+ΔC(t))x(t).
In light of the definition ofλm, it is true that (BTPx)TΔ(·)(BTPx)≥λm∥BTPx∥2; thus the second term in (25) can be rewritten as
(26)2xTPB{Δ(·)[-a^φ(x)(BTP)∥BTPx∥]x+Z(·)}=-2a^φ(x)∥BTPx∥(BTPx)TΔ(·)(BTPx)+2(BTPx)TZ(·)≤2(a-λma^)φ(x)∥BTPx∥.
The factxT(t)(C+ΔC(t))TP(C+ΔC(t))x(t)≤xT(t)[∥BTPB∥aF2]x(t)≤xT(t)(ρI)x(t) leads (25) to
(27)L[V(x(t),t)]≤xT(t)(A-TP+PA-)x(t)+2(a-λma^)φ(x)∥BTPx∥+2(a-λma^)(-γ-1a^˙)+xT(t)(∥BTPB∥∥F2∥2)x(t)≤xT(t)(A-TP+PA-+ρI)x(t).
Using the updating law (21d) and choosing the proper matrixes P and Q to ensure that the matric inequality is established, one obtains from (27) that
(28)L[V(x(t),t)]≤-12xTQx<0forx(t)≠0.
Consequently, according to Lemma6, it can be obtained that the closed-loop system (11) is globally asymptotically stable in probability in presence of actuator failures.Remark 10.
Note that in designing and implementing the first robust fault-tolerant control method we need to predetermine the parametersa and λm. This might present analytical and technical difficulty in practice. The second robust adaptive FTC scheme, which does not need the analytic computation of the parameters a and λm, circumvents this shortcoming. Although the existence of λm>0 is used in stability analysis, none of them are used in the control algorithm.Remark 11.
It is seen that the proposed control is independent of explicit information on faults and disturbances. As with most variable structure control methods, when the states get closer to zero, the control scheme might experience chattering, which can be easily avoided by replacingz/∥z∥ with z/(∥z∥+ς), where ς is a small number, as commonly adopted in the literature. Also to prevent the estimate a^ from drifting, (21d) can be modified as(29a)
a
^
˙
(
t
)
=
-
σ
a
^
+
γ
φ
(
x
)
2
∥
B
T
P
x
∥
2
φ
(
x
)
∥
B
T
P
x
∥
+
ς
,
γ
>
0
,
σ
>
0
.In this case, we have the following ultimately uniformly bounded (UUB) stabilization result.Theorem 12.
Consider the uncertain stochastic system (11). Let the Assumptions 2 and 3 hold. If the following robust adaptive control is applied:
(29b)u(t)=-N0x+N^(t),
where N0>0 is chosen such that A-BN0 is Hurwitz, and N^(t) is generated by
(29c)N^(t)=a^(t)φ(x)2BTPx∥BTPx∥φ(x)+ςand
a
^
is updated by(29a), then the closed-loop system(11)is ensured to UUB stable.Proof.
The result can be established by using the method similar to that as in [15].Remark 13.
Since the robust FTC with the fixed gain may bring more conservatives, a new robust adaptive FTC is further addressed in the next subsection. By means of the on-line estimation of effectiveness values of faulty actuators, the robust adaptive FTC gain is adaptively updated to compensate the effects of actuator faults.


### 3.3. Improved Robust Adaptive Fault-Tolerant Control

Consider that the elements of the actuator efficiency factorΔ(·) are constants. A robust and adaptive control scheme integrated with on-line fault estimation is designed as(30a)u(t)=-Δ^(t)-1N0x+N^(t),
where Δ^(t)=diag⁡{δ^1(t),δ^2(t),…,δ^m(t)}, δ^i(t) is the estimated values of effectiveness for ith actuator, and the updating law for δ^i(t)  (i=1,2,…,m) is given as
(30b)δ^˙i(t)=Pr[δ_i,δ-i]{0,ifδ^i=δ_i,Ui≤0,orδ^i=δ-i,Ui≥0Ui,otherwise,
where Ui=ηix(t)T(PB)iΔ^(·)-1N0ix(t),ηi>0 is the adaptive law gain to be chosen according to practical applications. Here, Mi and Mi denote the ith row and ith column of a matrix M, respectively.N
0
>
0 is chosen such that A-BN0 is Hurwitz, and N^(t) is on-line updated by
(30c)N^(t)=a^(t)ψ(x)BTPx∥BTPx∥,
with
(30d)ψ(x)=1+∥φf(x)∥+∥x∥,(30e)a^˙(t)=-γψ(x)∥BTPx∥,γ>0.Remark 14.
It is noted from (30b) that Pr{·} is a projection operator [28], which projects the estimate δ^i into the interval [δ_i,δ-i] so as to satisfy the assumption on the bound of effectiveness values in (3). Because this updating law can ensure the estimated values δ^i(t) are not zero, the control signal u(t) will take effect on the plant.Theorem 15.
For the uncertain stochastic system (11), the robust adaptive fault-tolerant controller given as ((30a), (30b), (30c), (30d), and (30e)) can ensure that the state will asymptotically tend to zero.Proof.
Substituting ((30a), (30b), (30c), (30d), and (30e)) into the stochastic system (11), we obtain the closed-loop system equation as follows:
(31)dx(t)=[(Δ(·)u(t)+κ(t)+f(x(t)))(A+ΔA(t))x(t)+B(Δ(·)u(t)+κ(t)+f(x(t)))]dt+(C+ΔC(t))x(t)dω(t)=[(-Δ^(·)-1N0x+N^(t))(A+ΔA(t))x(t)+B(Δ(·)(-Δ^(·)-1N0x+N^(t))+κ(t)+f(x(t))(-Δ^(·)-1N0x+N^(t)))]dt+(C+ΔC(t))x(t)dω(t)=(A-BN0)x(t)dt+B[(I-Δ(·)Δ^(·)-1)N0x+Δ(·)N^(t)+Z(·)]dt+(C+ΔC(t))x(t)dω(t),
where Z(·)=κ(t)+f(·)+F1(t)x(t), which is bounded by
(32)∥Z(·)∥≤∥κ(·)∥+∥f(·)∥+∥F1(t)x(t)∥≤a(1+∥φ(x)∥+∥x∥)
under Assumption 2.
Consider the following Lyapunov function candidate(33)V(x(t),t)=xTPx+1λmγ(a-a^λm)2+∑i=1mηi-1δ~i2(t),
where γ>0 and η>0 are constants related to adaptation rate chosen by the designer and λm>0 is constant defined as before. Upon using the control scheme with the adaptive algorithm, it is not difficult to show that the infinitesimal operator
(34)L[V(x(t),t)]=xT(t)P(A-BN0)x(t)+((A-BN0)x(t))TPx(t)+2(a-a^λm)(-a^˙γ-1)+2∑i=1mηi-1δ~i(t)δ^˙i(t)=[+Δ(·)N^(t)x+Ζ(·))TA-x+B(-δ~δ^-1N0x+Δ(·)N^(t)x+Z(·))T]Px+xTP[A-x+B(-δ~δ^-1N0x+Δ(·)N^(t)x+Z(·))]+2(a-a^λm)(-a^˙γ-1)+2∑i=1mηi-1δ~i(t)δ^˙i(t)=xT(A-TP+PA-)x+2xTPBδ~δ^-1N0x+2xTPB(Δ(·)N^(t)x+Z(·))+2(a-a^λm)(-a^˙γ-1)+2∑i=1mηi-1δ~i(t)δ^˙i(t).
Considering that PBδ~δ^-1=∑i=1mδ~(PB)iδ^-1 and the adaptive law (30b), we have
(35)-2xTPBδ~δ^-1N0x+2∑i=1mηi-1δ~i(t)δ^˙i(t)≤0.
Then, L[V(x(t),t)] becomes
(36)L[V(x(t),t)]=xT(A-TP+PA-)x+2xTPB{Δ(·)[-a^ψ(x)(BTPx)∥BTPx∥]+Z(·)}+2(a-λma^)(-γ-1a^˙)+xT(t)(C+ΔC(t))TP(C+ΔC(t))x(t),
in which the second term in (36) can be rewritten as
(37)2xTPB{Δ(·)[-a^ψ(x)(BTP)∥BTPx∥]x+Z(·)}=-2a^ψ(x)∥BTPx∥(BTPx)TΔ(·)(BTPx)+2(BTPx)TZ(·)≤2(a-λma^)ψ(x)∥BTPx∥;
by using (19) it is true that (BTPx)TΔ(·)(BTPx)≥λm∥BTPx∥2. Thus by using the fact that xT(t)(C+ΔC(t))TP(C+ΔC(t))x(t)≤xT(t)(ρI)x(t) and the updating law (30e), the function L[V(x(t),t)] eventually is bounded as
(38)L[V(x(t),t)]≤xT(A-TP+PA-+ρI)x+2(a-λma^)ψ(x)∥BTPx∥+2(a-λma^)(-γ-1a^˙)≤-12xTQx<0forx(t)≠0,
as long as proper matrixes P and Q are select to ensure (7). Therefore, it can be obtained from Lemma 6 that the state of the stochastic system is asymptotically stable in probability and the estimation parameters (a-λma^) and δ~i are bounded.


## 3.1. Robust Fault-Tolerant Control

In this section, a robust fault-tolerant control of the form(10a)u(t)=-N0x+N(t)
is proposed, where N0 is chosen such that A-BN0 is Hurwitz, and N(t) is generated by
(10b)N(t)=-aλmφ(·)BTPx∥BTPx∥,
with 0<λm≤α/β being a constant, where λm represents the lower bound of the health indicator matrix Δ(·); that is, 0<λm≤λmin⁡(Δ) and α>0, β>0 are suitable constants such that
(10c)α∥BTPx∥2≤β∥BTPxΔ(·)∥2,(10d)φ(·)=1+∥N0x∥+∥φf(x)∥+∥x∥,a=max⁡{1,aN,af,aF1}.Theorem 7.
Under Assumptions2 and 3, the FTC as given in ((10a), (10b), (10c), and (10d)) exponentially stabilizes (in mean square) the stochastic system described by (1), for all admissible uncertainties as well as all actuator failures corresponding to (3).Proof.
When the system is subject to the actuator failure as described in (3), its dynamic behavior becomes
(11)dx(t)=[(Δ(·)u(t)+κ(t)+f(x(t)))(A+ΔA(t))x(t)+B(Δ(·)u(t)+κ(t)+f(x(t)))]dt+(C+ΔC(t))x(t)dω(t).
With the proposed control ((10a), (10b), (10c), (10d)), one has
(12)dx(t)=[(Δ(·)u(t)+κ(t)+f(x(t)))(A+ΔA(t))x(t)+B(Δ(·)u(t)+κ(t)+f(x(t)))]dt+(C+ΔC(t))x(t)dω(t)=[B(Δ(·)(-N0x+N(t))+κ(t)+f(x(t)))(A+ΔA(t))x(t)+B(Δ(·)(-N0x+N(t))+κ(t)+f(x(t)))]dt+(C+ΔC(t))x(t)dω(t)=(A-BN0)x(t)dt+B[Δ(·)N(t)+Z(t)]dt+(C+ΔC(t))x(t)dω(t),
where
(13)Z(·)=(I-Δ(·))N0x(t)+κ(t)+f(·)+F1(t)x(t),
which is bounded as
(14)∥Z(·)∥≤∥N0x∥+∥κ(·)∥+∥f(·)∥+∥F1(t)x(t)∥≤a(1+∥N0x∥+∥ψ(x)∥+∥x∥),
based on Assumption 2, where a=max⁡{1,aN,af,aF1} and φ(·)=1+∥N0x∥+∥φf(x)∥+∥x∥. Thus, it is not difficult to get
(15)(BTPx)TZ≤aφ(·)∥BTPx∥.
Consider the following Lyapunov function candidate:
(16)V(x(t),t)=xT(t)Px(t).
Then, by Itô’s formula, the infinitesimal generator of (12) is
(17)L[V(x(t),t)]=xT(t)P(A-BN0)x(t)+((A-BN0)x(t))TPx(t)+2(BTPx)T[-Δ(·)aλmφ(·)BTPx∥BTPx∥]+2(BTPx)TZ+xT(t)(C+ΔC(t))TP(C+ΔC(t))x(t).
Note that the last term of (17) cannot be combined with Z(·); thus the adaptive updating law cannot be used to compensate its effect as usual. To establish the robust stability of the closed-loop system (12), we need to have the following development. From the fact that (∥F2(t)∥+∥F3∥)2≤aF2<∞ and using (2), it is seen that the last term of (17) can be expressed as
(18)xT(t)(C+ΔC(t))TP(C+ΔC(t))x(t)=xT(t)[B(F2(t)+F3)TPB(F2(t)+F3)]x(t)≤xT(t)[∥BTPB∥aF2]x(t);
from (10c), it holds that
(19)-(BTPx)TΔ(·)(BTPx)≤-αβ∥BTPx∥2≤-λm∥BTPx∥2.
and by defining ρ=∥BTPB∥aF2, the inequality (17) can be shown to satisfy
(20)L[V(x(t),t)]≤xT(t)(A-TP+PA-)x(t)-2aλmφ(·)∥BTPx∥(BTPx)TΔ(·)(BTPx)+aφ(·)∥BTPx∥+xT(t)∥BTPB∥∥F2∥2x(t)≤xT(t)(A-TP+PA-+ρI)x(t)≤-12xTQx<0forx(t)≠0,
where matrixes P and Q are chosen properly to satisfy A-TP+PA-+ρI≤-Q. Therefore, it is confirmed from Lemma 6 that the closed-loop system (11) is asymptotically mean square stable in probability despite faulty actuators with the proposed FTC.Remark 8.
Note that if proper constantsα and β can be obtained in advance, the proposed control ((10a), (10b), (10c), and (10d)) achieved exponential stability in mean square for the stochastic system under Assumptions 2 and 3. However, it is a little difficult to select such α and β to ensure λm≤α/β, since λm the lower bound of the eigenvalue of the health indicator matrix is not available in general. In view of this, a more feasible method is developed in the next subsection.


## 3.2. Robust Adaptive Fault-Tolerant Control

In order to develop a control scheme that is not only robust but also adaptive yet fault-tolerant, we modify the previous one to get(21a)u(t)=-N0x+N^(t),
where N0>0 is chosen such that A-BN0 is Hurwitz and N^(t) is on-line updated by
(21b)N^(t)=a^(t)φ(x)BTPx∥BTPx∥,
with
(21c)φ(x)=1+∥N0x∥+∥φf(x)∥+∥x∥,(21d)a^˙(t)=-γφ(x)∥BTPx∥,γ>0.Theorem 9.
Consider the uncertain stochastic system (11) under Assumptions 2 and 3. If the robust adaptive fault-tolerant controller ((21a), (21b), (21c), and (21d)) is implemented, the closed-loop system is ensured to be asymptotically stable.Proof.
Substituting the proposed control ((21a), (21b), (21c), and (21d)) into the stochastic system (11), we obtain the closed-loop system dynamics as follows:
(22)dx(t)=[(A+ΔA(t))x(t)+B(Δ(·)u(t)+κ(t)+f(x(t)))]dt+(C+ΔC(t))x(t)dω(t)=[(Δ(·)(-N0x+N^(t))+κ(t)+f(x(t)))(A+ΔA(t))x(t)+B(Δ(·)(-N0x+N^(t))+κ(t)+f(x(t)))]dt+(C+ΔC(t))x(t)dω(t)=(A-BN0)x(t)dt+B[Δ(·)N^(t)+Ζ(t)]dt+(C+ΔC(t))x(t)dω(t).
Consider the following Lyapunov function candidate:(23)V(x(t),t)=xTPx+1λmγ(a-a^λm)2,
where γ>0 is a constant related to adaptation rate chosen by the designer and λm>0 is constant defined as before. Upon using the control scheme with the adaptive algorithm, it is not difficult to show that
(24)L[V(x(t),t)]=xT(t)P(A-BN0)x(t)+((A-BN0)x(t))TPx(t)+2(a-a^λm)(-a^˙γ-1)+xT(t)(C+ΔC(t))TP(C+ΔC(t))x(t)=[A-x+B(Δ(·)N^(t)x+Z(·))]TPx+xTP[A-x+B(Δ(·)N^(t)x+Z(·))]+2(a-a^λm)(-a^˙γ-1)+xT(t)(C+ΔC(t))TP(C+ΔC(t))x(t)=xT(A-TP+PA-)x+2xTPB(Δ(·)N^(t)x+Z(·))+2(a-a^λm)(-a^˙γ-1)+xT(t)(C+ΔC(t))TP(C+ΔC(t))x(t).
Then
(25)L[V(x(t),t)]=xT(A-TP+PA-)x+2xTPB{Δ(·)[-a^φ(x)(BTPx)∥BTPx∥]+Z(·)}+2(a-λma^)(-γ-1a^˙)+xT(t)(C+ΔC(t))TP(C+ΔC(t))x(t).
In light of the definition ofλm, it is true that (BTPx)TΔ(·)(BTPx)≥λm∥BTPx∥2; thus the second term in (25) can be rewritten as
(26)2xTPB{Δ(·)[-a^φ(x)(BTP)∥BTPx∥]x+Z(·)}=-2a^φ(x)∥BTPx∥(BTPx)TΔ(·)(BTPx)+2(BTPx)TZ(·)≤2(a-λma^)φ(x)∥BTPx∥.
The factxT(t)(C+ΔC(t))TP(C+ΔC(t))x(t)≤xT(t)[∥BTPB∥aF2]x(t)≤xT(t)(ρI)x(t) leads (25) to
(27)L[V(x(t),t)]≤xT(t)(A-TP+PA-)x(t)+2(a-λma^)φ(x)∥BTPx∥+2(a-λma^)(-γ-1a^˙)+xT(t)(∥BTPB∥∥F2∥2)x(t)≤xT(t)(A-TP+PA-+ρI)x(t).
Using the updating law (21d) and choosing the proper matrixes P and Q to ensure that the matric inequality is established, one obtains from (27) that
(28)L[V(x(t),t)]≤-12xTQx<0forx(t)≠0.
Consequently, according to Lemma6, it can be obtained that the closed-loop system (11) is globally asymptotically stable in probability in presence of actuator failures.Remark 10.
Note that in designing and implementing the first robust fault-tolerant control method we need to predetermine the parametersa and λm. This might present analytical and technical difficulty in practice. The second robust adaptive FTC scheme, which does not need the analytic computation of the parameters a and λm, circumvents this shortcoming. Although the existence of λm>0 is used in stability analysis, none of them are used in the control algorithm.Remark 11.
It is seen that the proposed control is independent of explicit information on faults and disturbances. As with most variable structure control methods, when the states get closer to zero, the control scheme might experience chattering, which can be easily avoided by replacingz/∥z∥ with z/(∥z∥+ς), where ς is a small number, as commonly adopted in the literature. Also to prevent the estimate a^ from drifting, (21d) can be modified as(29a)
a
^
˙
(
t
)
=
-
σ
a
^
+
γ
φ
(
x
)
2
∥
B
T
P
x
∥
2
φ
(
x
)
∥
B
T
P
x
∥
+
ς
,
γ
>
0
,
σ
>
0
.In this case, we have the following ultimately uniformly bounded (UUB) stabilization result.Theorem 12.
Consider the uncertain stochastic system (11). Let the Assumptions 2 and 3 hold. If the following robust adaptive control is applied:
(29b)u(t)=-N0x+N^(t),
where N0>0 is chosen such that A-BN0 is Hurwitz, and N^(t) is generated by
(29c)N^(t)=a^(t)φ(x)2BTPx∥BTPx∥φ(x)+ςand
a
^
is updated by(29a), then the closed-loop system(11)is ensured to UUB stable.Proof.
The result can be established by using the method similar to that as in [15].Remark 13.
Since the robust FTC with the fixed gain may bring more conservatives, a new robust adaptive FTC is further addressed in the next subsection. By means of the on-line estimation of effectiveness values of faulty actuators, the robust adaptive FTC gain is adaptively updated to compensate the effects of actuator faults.


## 3.3. Improved Robust Adaptive Fault-Tolerant Control

Consider that the elements of the actuator efficiency factorΔ(·) are constants. A robust and adaptive control scheme integrated with on-line fault estimation is designed as(30a)u(t)=-Δ^(t)-1N0x+N^(t),
where Δ^(t)=diag⁡{δ^1(t),δ^2(t),…,δ^m(t)}, δ^i(t) is the estimated values of effectiveness for ith actuator, and the updating law for δ^i(t)  (i=1,2,…,m) is given as
(30b)δ^˙i(t)=Pr[δ_i,δ-i]{0,ifδ^i=δ_i,Ui≤0,orδ^i=δ-i,Ui≥0Ui,otherwise,
where Ui=ηix(t)T(PB)iΔ^(·)-1N0ix(t),ηi>0 is the adaptive law gain to be chosen according to practical applications. Here, Mi and Mi denote the ith row and ith column of a matrix M, respectively.N
0
>
0 is chosen such that A-BN0 is Hurwitz, and N^(t) is on-line updated by
(30c)N^(t)=a^(t)ψ(x)BTPx∥BTPx∥,
with
(30d)ψ(x)=1+∥φf(x)∥+∥x∥,(30e)a^˙(t)=-γψ(x)∥BTPx∥,γ>0.Remark 14.
It is noted from (30b) that Pr{·} is a projection operator [28], which projects the estimate δ^i into the interval [δ_i,δ-i] so as to satisfy the assumption on the bound of effectiveness values in (3). Because this updating law can ensure the estimated values δ^i(t) are not zero, the control signal u(t) will take effect on the plant.Theorem 15.
For the uncertain stochastic system (11), the robust adaptive fault-tolerant controller given as ((30a), (30b), (30c), (30d), and (30e)) can ensure that the state will asymptotically tend to zero.Proof.
Substituting ((30a), (30b), (30c), (30d), and (30e)) into the stochastic system (11), we obtain the closed-loop system equation as follows:
(31)dx(t)=[(Δ(·)u(t)+κ(t)+f(x(t)))(A+ΔA(t))x(t)+B(Δ(·)u(t)+κ(t)+f(x(t)))]dt+(C+ΔC(t))x(t)dω(t)=[(-Δ^(·)-1N0x+N^(t))(A+ΔA(t))x(t)+B(Δ(·)(-Δ^(·)-1N0x+N^(t))+κ(t)+f(x(t))(-Δ^(·)-1N0x+N^(t)))]dt+(C+ΔC(t))x(t)dω(t)=(A-BN0)x(t)dt+B[(I-Δ(·)Δ^(·)-1)N0x+Δ(·)N^(t)+Z(·)]dt+(C+ΔC(t))x(t)dω(t),
where Z(·)=κ(t)+f(·)+F1(t)x(t), which is bounded by
(32)∥Z(·)∥≤∥κ(·)∥+∥f(·)∥+∥F1(t)x(t)∥≤a(1+∥φ(x)∥+∥x∥)
under Assumption 2.
Consider the following Lyapunov function candidate(33)V(x(t),t)=xTPx+1λmγ(a-a^λm)2+∑i=1mηi-1δ~i2(t),
where γ>0 and η>0 are constants related to adaptation rate chosen by the designer and λm>0 is constant defined as before. Upon using the control scheme with the adaptive algorithm, it is not difficult to show that the infinitesimal operator
(34)L[V(x(t),t)]=xT(t)P(A-BN0)x(t)+((A-BN0)x(t))TPx(t)+2(a-a^λm)(-a^˙γ-1)+2∑i=1mηi-1δ~i(t)δ^˙i(t)=[+Δ(·)N^(t)x+Ζ(·))TA-x+B(-δ~δ^-1N0x+Δ(·)N^(t)x+Z(·))T]Px+xTP[A-x+B(-δ~δ^-1N0x+Δ(·)N^(t)x+Z(·))]+2(a-a^λm)(-a^˙γ-1)+2∑i=1mηi-1δ~i(t)δ^˙i(t)=xT(A-TP+PA-)x+2xTPBδ~δ^-1N0x+2xTPB(Δ(·)N^(t)x+Z(·))+2(a-a^λm)(-a^˙γ-1)+2∑i=1mηi-1δ~i(t)δ^˙i(t).
Considering that PBδ~δ^-1=∑i=1mδ~(PB)iδ^-1 and the adaptive law (30b), we have
(35)-2xTPBδ~δ^-1N0x+2∑i=1mηi-1δ~i(t)δ^˙i(t)≤0.
Then, L[V(x(t),t)] becomes
(36)L[V(x(t),t)]=xT(A-TP+PA-)x+2xTPB{Δ(·)[-a^ψ(x)(BTPx)∥BTPx∥]+Z(·)}+2(a-λma^)(-γ-1a^˙)+xT(t)(C+ΔC(t))TP(C+ΔC(t))x(t),
in which the second term in (36) can be rewritten as
(37)2xTPB{Δ(·)[-a^ψ(x)(BTP)∥BTPx∥]x+Z(·)}=-2a^ψ(x)∥BTPx∥(BTPx)TΔ(·)(BTPx)+2(BTPx)TZ(·)≤2(a-λma^)ψ(x)∥BTPx∥;
by using (19) it is true that (BTPx)TΔ(·)(BTPx)≥λm∥BTPx∥2. Thus by using the fact that xT(t)(C+ΔC(t))TP(C+ΔC(t))x(t)≤xT(t)(ρI)x(t) and the updating law (30e), the function L[V(x(t),t)] eventually is bounded as
(38)L[V(x(t),t)]≤xT(A-TP+PA-+ρI)x+2(a-λma^)ψ(x)∥BTPx∥+2(a-λma^)(-γ-1a^˙)≤-12xTQx<0forx(t)≠0,
as long as proper matrixes P and Q are select to ensure (7). Therefore, it can be obtained from Lemma 6 that the state of the stochastic system is asymptotically stable in probability and the estimation parameters (a-λma^) and δ~i are bounded.


## 4. Numerical Simulation

Two examples are used to demonstrate the features of the proposed control scheme.Example 1.
Consider the uncertain stochastic system (11) with
(1)A=[0.2-0.20-0.2-0.60.30.2-0.4-0.2],B=[-0.20.21-1.70.6-0.7],C=[-0.040.20.07-0.030.10.040.04-0.2-0.07],ΔA=[0.02sin(t)0.04cos⁡2(t)0.04cos⁡(2t)0.020.04sin(2t)cos⁡(t)0.040.03cos⁡(t)0.060.06sin(t)],ΔC=[-0.007cos⁡(t)0.0140.014sin(2t)0.004cos⁡2(t)0.008sin2(t)0.008-0.007-0.014sin(t)cos⁡(2t)-0.014sin(t)cos⁡(3t)],f(x(t))=(sin⁡(x1(t))sin(x2(t))2x1(t)cos⁡(x2(t))).
It is seen that the uncertaintiesΔA and ΔC are complex to be described by the form of (ΔA(t)TΔC(t)T)T=(E1TE2T)TF(t)H. But the form of (2) is easy to satisfy. The external disturbance f(·) is state-dependent and unknown. For the simulation, the initial conditions are x(0)=[1,2.5,3] and a^(0)=0.
The actuator efficiency variables for each of the two control channels simulated are as illustrated in Figure1, where two of the actuators suffer from the failure as shown in the figure.
The scenario simulated is that the system operates normally at the beginning, and the disturbances always exist during the system operation. After 4 seconds of the operation some faults in actuators occur: the first and the second actuators encounter severe failures in the fact that both channels lose their effectiveness by over 50% at some time and the faults are fast time-varying for some period.
The objective in this work is to design a reliable robust adaptive fault-tolerant controller such that the closed-loop system is asymptotically stable in probability despite the presence of actuator faults. In applying the control scheme ((29a), (29b), and (29c)), one can easily determine all the control parameters:
(40)N0=[-1.09120.32100.06958.0245-3.80700.0239],γ=5,σ=0.08,ε=0.001,φ(x)=1+∥N0x∥+∥φf(x)∥+∥x∥.
The simulation results in terms of stabilization of the three states are presented in Figure2. It can be seen that the states x1, x2, and x3 can converge to a small neighborhood around zero. Figure 3 shows the control signals of the two inputs. The estimated parameter a^(t) is shown in Figure 4. The results confirm the theoretical prediction.Figure 1
Profile of the time-varying actuator efficiency variable (δ1 (solid), δ2 (dot)).Figure 2
The curve ofx(t) with the proposed control scheme ((29a), (29b), and (29c)).Figure 3
The curve ofu(t) with the proposed control scheme ((29a), (29b), and (29c)).Figure 4
Updating ofa^(t) with the proposed control scheme ((29a), (29b), and (29c)).Example 2.
The second simulation is made for robust adaptive fault-tolerant controller ((30a), (30b), (30c), (30d), and (30e)). It is assumed that at t=4s, the first actuator u1 is still normal and the second actuator u2 is faulty with δ2=0.5. The simulations are shown in Figures 5, 6, 7, and 8. Also it should be pointed out from [36] that the estimated value δ^i(t)(i=1,2) can converge but may not converge to its true value δi(t). And in our controller design procedure, only the estimated value δ^i(t) is needed to construct adaptive controller and whether δ^i(t) can converge to its true values or not is not necessary.
From Figure5, the FTC scheme ((30a), (30b), (30c), (30d), and (30e)) makes the curves relatively smooth via the adaptive estimate δ^i(t) of efficiency value. The simulation results confirm that the robust adaptive FTC can achieve a good performance on dealing with the reliable control problem of stochastic systems in presence of actuator failures, parameter uncertainty, and state-dependent disturbance.Figure 5
System responses under the control of the proposed FTC ((30a), (30b), (30c), (30d), and (30e)).Figure 6
The curve ofu(t) with the proposed control scheme ((30a), (30b), (30c), (30d), and (30e)).Figure 7
Updating ofa^(t) with the proposed control scheme ((30a), (30b), (30c), (30d), and (30e)).Figure 8
The curve of the estimateΔ^(t) with η=2 (δ^1 (solid), δ^2 (dot)).


## 5. Conclusion

In this paper, the problem of robust adaptive FTC for stochastic systems with faulty actuators has been considered. By blending adaptive control into robust FTC, the proposed control method is able to accommodate actuation faults and modeling uncertainties concurrently. Both theoretical analysis and numerical simulations validate the benefits and effectiveness of the proposed approach.



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*Source: 450521-2014-03-13.xml*