# Robust Fault Tolerant Control for a Class of Time-Delay Systems with Multiple Disturbances

Authors: Songyin Cao; Jianzhong Qiao
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

A robust fault tolerant control (FTC) approach is addressed for a class of nonlinear systems with
time delay, actuator faults, and multiple disturbances. The first part of the multiple disturbances is supposed to be an
uncertain modeled disturbance and the second one represents a norm-bounded variable. First, a composite observer
is designed to estimate the uncertain modeled disturbance and actuator fault simultaneously. Then, an FTC strategy
consisting of disturbance observer based control (DOBC), fault accommodation, and a mixedH2/H∞ controller is constructed to reconfigure the considered systems with disturbance rejection and attenuation performance. Finally, simulations for a flight control system are given to show the efficiency of the proposed approach.


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## Body

## 1. Introduction

To reduce the influence of model uncertainties and system disturbances, there are several control approaches focusing on nonlinear systems with unknown disturbances (see the survey paper [1] and references therein). The methodologies can be mainly classified into disturbance attenuation methods (such as H∞ and H2 control) and disturbance rejection approaches (such as output regulation theory, disturbance observer based control). The disturbance attenuation approaches have conservativeness for bounded stochastic disturbance. The disturbance rejection methods are established based on model-matching conditions. It has been shown that multiple disturbances exist in most practical systems. The idea of disturbance observer based control (DOBC) is to construct an observer to estimate and compensate some external disturbances [2]. A composite control scheme combining DOBC and PD (proportional derivative) control for flexible spacecraft attitude control was proposed in the presence of model uncertainty, elastic vibration, and external disturbances [3]. For nonlinear systems with multiple disturbances, it has been seen that the H∞ and variable structure control have been integrated with DOBC in [4, 5]. In [6], a composite DOBC and adaptive control approach were proposed for a class of nonlinear systems with multiple disturbances. By constructing a disturbance compensation gain vector in the composite control law, a nonlinear robust DOBC was proposed to attenuate the mismatched disturbances and the influence of parameter variations from output channels [7].In order to increase the reliability and safety of practical engineering, the issues of fault diagnosis and fault tolerant control (FTC) have become an attractive topic and have been paid much attention in recent years (see [8–13] and references therein). It is difficult to accommodate faults if the disturbances and faults exist simultaneously in the controlled systems. In [14], an optimal fault tolerant control approach was proposed for the nonlinear systems, where generalized H∞ optimization was applied to estimate the fault and attenuate the disturbances. In [15], a robust observer was proposed to simultaneously estimate system states, faults, and their finite time derivatives and attenuate disturbances; then an FTC approach was designed based on their estimations. For systems with modeled disturbance, a fault diagnosis approach based on disturbance observer was firstly proposed in [16] with disturbance rejection performance. For the nonlinear system with multiple disturbances, [17] addressed a fault tolerant control approach with disturbance rejection and attenuation performances. It is well known that time delay frequently occurs in many practical systems, such as manufacturing systems, telecommunication, and economic systems. Therefore, the problem of fault accommodation for time-delay systems has been a hot topic in the control field. Many important results have been reported in the literature (see [18–21] and references therein). In [18], an adaptive fuzzy fault accommodation control approach was proposed for nonlinear time-delay system. In [19], a fault accommodation was addressed for time-varying delay system using adaptive fault diagnosis observer. [20] dealt with fault tolerant guaranteed cost controller design problem for linear time-delay system against actuator faults.In this paper, FTC problem is discussed for a class of time-delay systems with actuator fault and multiple disturbances. The first part of multiple disturbances is modeled disturbance formulated by an exogenous system and the second one is norm bounded uncertain variable. A composite observer is designed to estimate the modeled disturbance and time-varying fault. Then, an FTC scheme is addressed with disturbance rejection and attenuation performance by combining fault accommodation and DOBC with a robustH2/H∞ controller.


## 2. Model Description

In this paper, we consider the following nonlinear system with time-varying faults, time-delay, and multiple disturbances simultaneously:(1)x˙(t)=Ax(t)+Adx(t-τ)+Gg(x(t))+J[u(t)+F(t)]+J1d1(t)+J2d2(t)y(t)=Cx(t),
where x(t)∈Rn is system state, u(t) is control input, and y(t)∈Rm represents output variable. F(t) is time-varying actuator fault to be diagnosed. A,Ad,C,G,J,J1, and J2 represent coefficient matrices of the system with suitable dimensions. τ is a known constant delay. The modeled external disturbance d1(t) is supposed to be generated by a linear exogenous system described by
(2)ω˙(t)=Wω(t)+J3δ(t),d1(t)=Vω(t),
where ω(t) is state variable, W∈Rp×p, and V and J3 are known parameter matrices of the exogenous system. δ(t) is additional disturbance which results from perturbations and uncertainties in the exogenous system. The disturbances d2(t) and δ(t) are supposed to have the bounded H2 norm.For a known matrixU1,g(x(t)) is a known nonlinear vector function that is supposed to satisfy g(0)=0 and the following norm condition:
(3)∥g(x1(t))-g(x2(t))∥≤∥U1(x1(t)-x2(t))∥
for any x1(t) and x2(t).The following assumptions are required so that the considered problem can be well-posed in this paper.Assumption 1.
(
A
,
J
) is controllable; (W,J1V) is observable.Assumption 2.
rank
(
J
,
J
1
)
=
rank
(
J
).Remark 1.
In practical engineering, the exogenous model (2) can represent many kinds of disturbances including harmonic disturbance signal caused by vibration, unknown constant load in the motor, inertial sensor drift represented by first-order Gaussian Markov process, and so on. Compared with the previous works [17, 19, 20], both the time-delay and multiple disturbances are simultaneously considered in this paper. Furthermore, the modeled disturbance d1(t) and control input are assumed in different channels, while in [17] are in the same channel.


## 3. Robust Fault Tolerant Controller Design

### 3.1. Disturbance Observer

In order to reject the modeled external disturbance, disturbance observer should be designed in this subsection. In this paper, we only consider the case of available states. The disturbance observer is formulated as(4)ω^(t)=ξ(t)-Lx(t),d^1(t)=Vω^(t),
where ξ(t) is auxiliary variable generated by
(5)ξ˙(t)=(W+LJ1V)[ξ(t)-Lx(t)]+L×[Ax(t)+Adx(t-τ)+Gg(x(t))+Ju(t)+Jufc(t)]ω^(t) is estimation of ω(t), d^1(t) is estimation of modeled disturbance d1(t), matrix L is the disturbance observer gain to be determined later, and ufc(t)=F^(t) is compensation term to be designed in fault diagnosis observer, where F^(t) is denoted as an estimation of fault F(t).By definingeω(t)=ω(t)-ω^(t) and eF(t)=F(t)-F^(t), estimation error system can be obtained from (1), (2), (4), and (5) to show the following:
(6)e˙ω(t)=(W+LJ1V)eω(t)+LJeF(t)+LJ2d2(t)+J3δ(t).In the following subsection, we will construct a fault diagnosis observer with disturbance estimation so that the modeled disturbance can be rejected and fault can be diagnosed.


### 3.2. Fault Diagnosis Observer

The following fault diagnosis observer is constructed to diagnose the time-varying actuator fault:(7)F^(t)=η(t)-Kx(t),η˙(t)=KJ[η(t)-Kx(t)]+K[Ax(t)+Adx(t-τ)+Gg(x(t))+Ju(t)+J1udc(t)],
where F^(t) is estimation of F(t). The disturbance observer based control term udc(t)=d^1(t) is applied to reject modeled disturbance d1(t) by its estimation from disturbance observer. K is the fault diagnosis observer gain to be determined later.The fault estimation error system yields(8)e˙F(t)=F˙(t)-F^˙(t)=F˙(t)+KJeF(t)+KJ1Veω(t)+KJ2d2(t).In the next subsection, a composite fault tolerant controller should be determined for reconfiguring the systems with disturbance rejection and attenuation performance.


### 3.3. Composite Fault Tolerant Controller

In this section, the object is to construct a control approach to guarantee that the system (1) is stable in the presence (or absence) of faults and multiple disturbances simultaneously for the considered time-delay system. The structure of composite fault tolerant controller is formulated as
(9)u(t)=usc(t)-ufc(t)-J*J1udc(t),
where ufc(t)=F^(t),udc(t)=d^1(t), and usc(t)=Sx(t), S is the state feedback controller gain to be determined later. Substituting (9) into (1), it can be seen that
(10)x˙(t)=(A+JS)x(t)+Adx(t-τ)+Gg(x(t))+JeF(t)+J1d1(t)-JJ*J1d^1(t)+J2d2(t).From Assumption2, it can be seen that the vector space spanned by the columns of J1 is a subset of the space spanned by the column vectors of J [19]; that is, span(J1)⊂span(J), which is equivalent to the existence of J, such that
(11)J1-JJ*J1=0.Then, it can be concluded that(12)x˙(t)=(A+JS)x(t)+Adx(t-τ)+Gg(x(t))+JeF(t)+J1Veω(t)+J2d2(t).Combing estimation error equations (6) and (8) with (12) yields(13)x-˙(t)=A-x-(t)+A-dx-(t-τ)+G-g(x-(t))+J-d-(t),z2(t)=C-2x-(t)+C-d2x-(t-τ),z∞(t)=C-∞x-(t)+C-d∞x-(t-τ)+Dd-(t),
where
(14)x-(t)=[x(t)eω(t)eF(t)],A-=[A+JSJ1VJ0W+LJ1VLJ0KJ1VKJ],G-=[G00],A-d=[Ad00000000],J-=[J200LJ2J30KJ20I],d-(t)=[d2(t)δ(t)F˙(t)],g(x-(t))=g(x(t))z2(t) is H2 reference output,
(15)z2(t)=C21x(t)+C22[eωT(t)eFT(t)]T+Cd2x(t-τ)z∞(t) is H∞ reference output,
(16)z∞(t)=C∞1x(t)+C∞2[eωT(t)eFT(t)]T+Cd∞x(t-τ)+Dd-(t),
where C21,C22,Cd2,C∞1,C∞2,Cd∞, and D are the selected weighting matrices.Definition 2.
For constantsγ1>0,γ2>0,andγ3>0, the H∞ performance is denoted as follows:
(17)J∞=∥z∞(t)∥2-γ12∥d2(t)∥2-γ22∥δ(t)∥2-γ32∥F˙(t)∥2-δ(P1,Q),
where
(18)δ(P1,Q)=ϕT(0)P1ϕ(0)+∫-d0ϕT(τ)Qϕ(τ)dτ.Definition 3.
TheH2 performance measure for (13) is defined as J2=∥z2(t)∥2.Remark 4.
Compared with [17], H2/H∞ mixed multiobjective optimization technique is used for the composite system (13). In the proposed approach, the modeled disturbance and fault are rejected by their estimations, while H∞ performance is adopted to attenuate norm bounded uncertain disturbances and H2 performance index is applied to optimize estimation error.
At this stage, the objective is to findK,L, and S such that system (13) is stable. The following result provides a design method based on convex optimization technology [22].Theorem 5.
If for the parameterλ>0,ri(i=1,…,4), matrices C21,C22,Cd2,C∞1,C∞2,Cd∞, and D, there exist matrices P0>0,P2>0,Q0>0,R0, and R2 and constants γ1>0,γ2>0, and  γ3>0, if the following LMI-based optimization problem holds:
(19)min{r1γ1+r2γ2+r3γ3+r4ϕT(0)P0-1ϕ(0)}
subject to
(20)[Ξ1AdQ0GV-Ξ15P0U1TP0C∞1TP0C21TP0*-Q00000Cd∞TCd2T0**-1λI000000***Ξ4Ξ450C∞2TC22T0****Ξ50DT00*****-λI000******-I00*******-I0********-Q0]<0,
where
(21)Ξ1=sym(AP0+JR0),Ξ15=[J200],Ξ4=sym(P2W-+R2V-),Ξ45=[R2J2P2J-3P2J-1],Ξ5=[-γ12I000-γ22I000-γ32I],W-=[W000],V-=[J1VJ],J-1=[0I],J-3=[J30],C-=[C1C2],
then with gains S=R0P0-1 and  L-=[LK]=P2-1R2, error system (13) is stable and satisfies J∞<0 and J2≤δ(P1,Q). The symmetric terms in a symmetric matrix are denoted by *. The symbol sym() represents sym(Θ):=Θ+ΘT.Proof.
Consider the following Lyapunov function:(22)Π(t)=xT(t)P1x(t)+∫t-τt‍xT(τ)Qx(τ)dτ+e-T(t)P2e-(t)+λ∫0t‍[∥Ux(τ)∥2-∥g(τ)∥2]dτ.
It is verified thatΠ(t)≥0 holds for all arguments. Along with the trajectories of (13), it can be shown that(23)Π˙(t)=2xT(t)P1x˙(t)+xT(t)Qx(t)-x(t-τ)TQx(t-τ)+2e-T(t)P2e-˙(t)+λ[∥Ux(t)∥2-∥g(x(t))∥2]=2xT(t)P1[(A+JS)x(t)+Adx(t-τ)+Gg(x(t))+J1Veω(t)+JeF(t)+J2d2(t)Gg(x(t))(x(t))]+xT(t)Qx(t)-x(t-τ)TQx(t-τ)+λxT(t)UTUx(t)-λ∥g(x(t))∥2+2e-T(t)×P2[(W-+L-V-)e-(t)+L-J2d2(t)+J-3δ(t)+J-1F˙(t)].
In the absence ofd-(t) (i.e., d-(t)=0), it can be seen that
(24)V˙(t)=sT(t)[Φ-ζζT]s(t),
where
(25)sT(t)=[xT(t)xT(t-τ)gT(x(t))e-T(t)],ζT=[C21Cd20C22],Φ=[Φ11P1Ad+C21TCd2P1GP1V-+C21TC22*-Q+Cd2TCd200**-λI0***Φ44],Φ11=sym(P1(A+JS))+λUTU+Q+C21TC21,Φ44=sym(P2(W+L-V-))+C22TC22.
From (20), it can be seen that(26)[Ξ1+P0Q0-1P0+1λP0U0TU0P0Q0AdGV-Ξ15P0C∞1TP0C21T*-Q0000Cd∞TCd2T**-1λI0000***Ξ4Ξ45C∞2TC22T****Ξ500*****-I0******-I]<0.
From the first, second, third, fourth, and seventh columns and rows of the left matrix in inequality (26) it can be verified that
(27)Φ0=[Ξ1+P0Q0-1P0+1λP0U0TU0P0Q0AdGV-P0C21T*-Q000Cd2T**-1λI00***Ξ4C22T****-I]<0.
DefiningP0=P1-1,Q-1=Q0, premultiplied and postmultiplied simultaneously by diag{P0-1,Q0-1,I,I,I}, it can be seen by using Schur complement formula that Φ0<0 leads to Φ<0. When Φ<0 holds, we have
(28)V˙(t)<-sT(t)ζζTs(t)=-z2Tz2≤0.
It follows that (13) is asymptotically stable in the absence of the exogenous input d-(t). Next, we consider the performances to be optimized for (13). Consider two auxiliary functions as follows:
(29)J0=z2T(t)z2(t)+V˙(t),J1=z∞T(t)z∞(t)-γ12d2(t)Td2(t)-γ22δ(t)Tδ(t)-γ32F˙(t)TF˙(t)+V˙(t).
Following the definition of theH2 performance, we only consider the case in the absence of d-(t). It can be verified that J0≤sT(t)Φs(t).
In the presence ofd-(t), it can be seen that J1=qT(t)Ψq(t), where
(30)qT(t)=[xT(t)xT(t-τ)gT(x(t))e-T(t)d-T(t)],Ψ=[Ψ11P1AdP1GP1V-Ψ15*-Q000**-1λI00***Ψ44Ψ45****Ξ5]+[C∞1TCd∞T0C∞2T0][C∞1TCd∞T0C∞2T0]T,
where
(31)Ψ11=sym(P1A+P1JS)+1λUTU+Q,Ψ15=[P1J200],Ψ44=sym(P2W-+P2L-V-),Ψ45=[P2L-J2P2J-3P2J-1].
Denote(32)Ψ0=Ψ+[C21Cd20C220]T[C21Cd20C220].
It can be seen by using Schur complement formula that (20) leads to Ψ0<0, and then Ψ<0 holds. It can be verified that both J0<0 and J1<0 hold. This completes the proof.


## 3.1. Disturbance Observer

In order to reject the modeled external disturbance, disturbance observer should be designed in this subsection. In this paper, we only consider the case of available states. The disturbance observer is formulated as(4)ω^(t)=ξ(t)-Lx(t),d^1(t)=Vω^(t),
where ξ(t) is auxiliary variable generated by
(5)ξ˙(t)=(W+LJ1V)[ξ(t)-Lx(t)]+L×[Ax(t)+Adx(t-τ)+Gg(x(t))+Ju(t)+Jufc(t)]ω^(t) is estimation of ω(t), d^1(t) is estimation of modeled disturbance d1(t), matrix L is the disturbance observer gain to be determined later, and ufc(t)=F^(t) is compensation term to be designed in fault diagnosis observer, where F^(t) is denoted as an estimation of fault F(t).By definingeω(t)=ω(t)-ω^(t) and eF(t)=F(t)-F^(t), estimation error system can be obtained from (1), (2), (4), and (5) to show the following:
(6)e˙ω(t)=(W+LJ1V)eω(t)+LJeF(t)+LJ2d2(t)+J3δ(t).In the following subsection, we will construct a fault diagnosis observer with disturbance estimation so that the modeled disturbance can be rejected and fault can be diagnosed.


## 3.2. Fault Diagnosis Observer

The following fault diagnosis observer is constructed to diagnose the time-varying actuator fault:(7)F^(t)=η(t)-Kx(t),η˙(t)=KJ[η(t)-Kx(t)]+K[Ax(t)+Adx(t-τ)+Gg(x(t))+Ju(t)+J1udc(t)],
where F^(t) is estimation of F(t). The disturbance observer based control term udc(t)=d^1(t) is applied to reject modeled disturbance d1(t) by its estimation from disturbance observer. K is the fault diagnosis observer gain to be determined later.The fault estimation error system yields(8)e˙F(t)=F˙(t)-F^˙(t)=F˙(t)+KJeF(t)+KJ1Veω(t)+KJ2d2(t).In the next subsection, a composite fault tolerant controller should be determined for reconfiguring the systems with disturbance rejection and attenuation performance.


## 3.3. Composite Fault Tolerant Controller

In this section, the object is to construct a control approach to guarantee that the system (1) is stable in the presence (or absence) of faults and multiple disturbances simultaneously for the considered time-delay system. The structure of composite fault tolerant controller is formulated as
(9)u(t)=usc(t)-ufc(t)-J*J1udc(t),
where ufc(t)=F^(t),udc(t)=d^1(t), and usc(t)=Sx(t), S is the state feedback controller gain to be determined later. Substituting (9) into (1), it can be seen that
(10)x˙(t)=(A+JS)x(t)+Adx(t-τ)+Gg(x(t))+JeF(t)+J1d1(t)-JJ*J1d^1(t)+J2d2(t).From Assumption2, it can be seen that the vector space spanned by the columns of J1 is a subset of the space spanned by the column vectors of J [19]; that is, span(J1)⊂span(J), which is equivalent to the existence of J, such that
(11)J1-JJ*J1=0.Then, it can be concluded that(12)x˙(t)=(A+JS)x(t)+Adx(t-τ)+Gg(x(t))+JeF(t)+J1Veω(t)+J2d2(t).Combing estimation error equations (6) and (8) with (12) yields(13)x-˙(t)=A-x-(t)+A-dx-(t-τ)+G-g(x-(t))+J-d-(t),z2(t)=C-2x-(t)+C-d2x-(t-τ),z∞(t)=C-∞x-(t)+C-d∞x-(t-τ)+Dd-(t),
where
(14)x-(t)=[x(t)eω(t)eF(t)],A-=[A+JSJ1VJ0W+LJ1VLJ0KJ1VKJ],G-=[G00],A-d=[Ad00000000],J-=[J200LJ2J30KJ20I],d-(t)=[d2(t)δ(t)F˙(t)],g(x-(t))=g(x(t))z2(t) is H2 reference output,
(15)z2(t)=C21x(t)+C22[eωT(t)eFT(t)]T+Cd2x(t-τ)z∞(t) is H∞ reference output,
(16)z∞(t)=C∞1x(t)+C∞2[eωT(t)eFT(t)]T+Cd∞x(t-τ)+Dd-(t),
where C21,C22,Cd2,C∞1,C∞2,Cd∞, and D are the selected weighting matrices.Definition 2.
For constantsγ1>0,γ2>0,andγ3>0, the H∞ performance is denoted as follows:
(17)J∞=∥z∞(t)∥2-γ12∥d2(t)∥2-γ22∥δ(t)∥2-γ32∥F˙(t)∥2-δ(P1,Q),
where
(18)δ(P1,Q)=ϕT(0)P1ϕ(0)+∫-d0ϕT(τ)Qϕ(τ)dτ.Definition 3.
TheH2 performance measure for (13) is defined as J2=∥z2(t)∥2.Remark 4.
Compared with [17], H2/H∞ mixed multiobjective optimization technique is used for the composite system (13). In the proposed approach, the modeled disturbance and fault are rejected by their estimations, while H∞ performance is adopted to attenuate norm bounded uncertain disturbances and H2 performance index is applied to optimize estimation error.
At this stage, the objective is to findK,L, and S such that system (13) is stable. The following result provides a design method based on convex optimization technology [22].Theorem 5.
If for the parameterλ>0,ri(i=1,…,4), matrices C21,C22,Cd2,C∞1,C∞2,Cd∞, and D, there exist matrices P0>0,P2>0,Q0>0,R0, and R2 and constants γ1>0,γ2>0, and  γ3>0, if the following LMI-based optimization problem holds:
(19)min{r1γ1+r2γ2+r3γ3+r4ϕT(0)P0-1ϕ(0)}
subject to
(20)[Ξ1AdQ0GV-Ξ15P0U1TP0C∞1TP0C21TP0*-Q00000Cd∞TCd2T0**-1λI000000***Ξ4Ξ450C∞2TC22T0****Ξ50DT00*****-λI000******-I00*******-I0********-Q0]<0,
where
(21)Ξ1=sym(AP0+JR0),Ξ15=[J200],Ξ4=sym(P2W-+R2V-),Ξ45=[R2J2P2J-3P2J-1],Ξ5=[-γ12I000-γ22I000-γ32I],W-=[W000],V-=[J1VJ],J-1=[0I],J-3=[J30],C-=[C1C2],
then with gains S=R0P0-1 and  L-=[LK]=P2-1R2, error system (13) is stable and satisfies J∞<0 and J2≤δ(P1,Q). The symmetric terms in a symmetric matrix are denoted by *. The symbol sym() represents sym(Θ):=Θ+ΘT.Proof.
Consider the following Lyapunov function:(22)Π(t)=xT(t)P1x(t)+∫t-τt‍xT(τ)Qx(τ)dτ+e-T(t)P2e-(t)+λ∫0t‍[∥Ux(τ)∥2-∥g(τ)∥2]dτ.
It is verified thatΠ(t)≥0 holds for all arguments. Along with the trajectories of (13), it can be shown that(23)Π˙(t)=2xT(t)P1x˙(t)+xT(t)Qx(t)-x(t-τ)TQx(t-τ)+2e-T(t)P2e-˙(t)+λ[∥Ux(t)∥2-∥g(x(t))∥2]=2xT(t)P1[(A+JS)x(t)+Adx(t-τ)+Gg(x(t))+J1Veω(t)+JeF(t)+J2d2(t)Gg(x(t))(x(t))]+xT(t)Qx(t)-x(t-τ)TQx(t-τ)+λxT(t)UTUx(t)-λ∥g(x(t))∥2+2e-T(t)×P2[(W-+L-V-)e-(t)+L-J2d2(t)+J-3δ(t)+J-1F˙(t)].
In the absence ofd-(t) (i.e., d-(t)=0), it can be seen that
(24)V˙(t)=sT(t)[Φ-ζζT]s(t),
where
(25)sT(t)=[xT(t)xT(t-τ)gT(x(t))e-T(t)],ζT=[C21Cd20C22],Φ=[Φ11P1Ad+C21TCd2P1GP1V-+C21TC22*-Q+Cd2TCd200**-λI0***Φ44],Φ11=sym(P1(A+JS))+λUTU+Q+C21TC21,Φ44=sym(P2(W+L-V-))+C22TC22.
From (20), it can be seen that(26)[Ξ1+P0Q0-1P0+1λP0U0TU0P0Q0AdGV-Ξ15P0C∞1TP0C21T*-Q0000Cd∞TCd2T**-1λI0000***Ξ4Ξ45C∞2TC22T****Ξ500*****-I0******-I]<0.
From the first, second, third, fourth, and seventh columns and rows of the left matrix in inequality (26) it can be verified that
(27)Φ0=[Ξ1+P0Q0-1P0+1λP0U0TU0P0Q0AdGV-P0C21T*-Q000Cd2T**-1λI00***Ξ4C22T****-I]<0.
DefiningP0=P1-1,Q-1=Q0, premultiplied and postmultiplied simultaneously by diag{P0-1,Q0-1,I,I,I}, it can be seen by using Schur complement formula that Φ0<0 leads to Φ<0. When Φ<0 holds, we have
(28)V˙(t)<-sT(t)ζζTs(t)=-z2Tz2≤0.
It follows that (13) is asymptotically stable in the absence of the exogenous input d-(t). Next, we consider the performances to be optimized for (13). Consider two auxiliary functions as follows:
(29)J0=z2T(t)z2(t)+V˙(t),J1=z∞T(t)z∞(t)-γ12d2(t)Td2(t)-γ22δ(t)Tδ(t)-γ32F˙(t)TF˙(t)+V˙(t).
Following the definition of theH2 performance, we only consider the case in the absence of d-(t). It can be verified that J0≤sT(t)Φs(t).
In the presence ofd-(t), it can be seen that J1=qT(t)Ψq(t), where
(30)qT(t)=[xT(t)xT(t-τ)gT(x(t))e-T(t)d-T(t)],Ψ=[Ψ11P1AdP1GP1V-Ψ15*-Q000**-1λI00***Ψ44Ψ45****Ξ5]+[C∞1TCd∞T0C∞2T0][C∞1TCd∞T0C∞2T0]T,
where
(31)Ψ11=sym(P1A+P1JS)+1λUTU+Q,Ψ15=[P1J200],Ψ44=sym(P2W-+P2L-V-),Ψ45=[P2L-J2P2J-3P2J-1].
Denote(32)Ψ0=Ψ+[C21Cd20C220]T[C21Cd20C220].
It can be seen by using Schur complement formula that (20) leads to Ψ0<0, and then Ψ<0 holds. It can be verified that both J0<0 and J1<0 hold. This completes the proof.


## 4. Simulation Examples

In this section, we consider the longitudinal dynamics of A4D aircraft at a flight condition of 15000 ft altitude and 0.9 Mach given in [2]. The longitudinal dynamics can be denoted as
(33)x˙(t)=Ax(t)+Gg(x(t))+J(u(t)+F(t))+J1d1(t)+J2d2(t),
where x(t) are measurable by using sensors technique. The coefficient matrices of aircraft model are given by
(34)A=[-0.060532.37032.2-0.00014-1.47510-0.0111-34.72-2.79300010],J=[0-0.1064-33.80],J1=[0-0.0532-16.90000],J2=[0.10-30.1].It is supposed that(35)G=[00500]T,g(x(t))=sin(2π5t)x2(t).
then, the matrix U can be selected as U=diag{0,1,0,0} and the norm condition (3) can be satisfied. Ad=0.5×A,τ=2. Periodic disturbance d1(t) caused by rotating aerial propeller is assumed to be an unknown harmonic disturbance described by (2) with
(36)W=[05-50],E=[250],H3=[0.10.1],δ(t) is the additional disturbance signal resulting from the perturbations and uncertainties in the exogenous system (2) and satisfies 2-norm boundedness. In simulation, we select δ(t) as the random signal with upper 2-norm bound 1. Wind gust and system noises d2(t) can also be considered as the random signal with upper 2-norm bounded.The initial values of the states are supposed to bexT(0)=[2-232]. For the reference output, it is denoted that
(37)C∞1=[1111],C∞2=[111],Cd∞=[0.10.10.10.1],C21=[0.10.10.10.1],C22=[0.10.10.1],Cd2=[0.010.010.010.01].Forλ=1,γ1=1,γ2=1, and γ3=1, it can be solved via LMI related to (20) that the gain of fault diagnosis observer (7) is
(38)K=[20.929910.02311.355320.9682],
the gain of disturbance observer (4) is
(39)L=[-0.2584-0.5402-0.0142-0.26102.32460.75230.15262.3269],
and the gain of state feedback controller is
(40)S=[89.6396471.067074.2419465.7941].When the disturbance observer is constructed based on (4) and (5), the estimation error of exogenous disturbances is shown in Figure 1. The actuator bias fault is supposed to occur at 15th second as F=4. The estimation and its error of fault with system disturbances are demonstrated in Figure 2, where the solid line represents the real fault signal, the dash-dotted line is its estimation, and the dash line denotes its estimation error. In Figure 3, the state response signals of the control system are illustrated. It can be seen that the proposed fault tolerant controller has a good control ability for configuring fault and rejecting and attenuating disturbances simultaneously. In Figure 4, the ramp fault is assumed to occur at 15th second with slope 0.1. From Figure 4, it is shown that the time-varying fault can also be well estimated.Figure 1
Disturbances estimation error in disturbance observer.Figure 2
Bias fault, its estimation, and error.Figure 3
The responses of state variable.Figure 4
Ramp fault, its estimation, and error.


## 5. Conclusion

In this paper, a robust antidisturbance fault tolerant control problem is investigated for nonlinear time delay systems with faults and multiple disturbances. There are the following features of the proposed algorithm compared with the previous results. First, the multiple disturbances and state time-delay are considered simultaneously in this paper. Second, an FTC scheme is addressed with disturbance rejection and attenuation performance by combining fault accommodation and DOBC with a mixedH2/H∞ controller, with which the fault can be accommodated and the disturbances can be rejected and attenuated simultaneously. Finally, simulation for a flight control system is given to show the efficiency of the proposed approach.



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*Source: 372957-2013-06-15.xml*