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

Authors: Wei Guan
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

In this paper, an indirect adaptive fault-tolerantH∞ 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 H∞ 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.


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## Body

## 1. 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 andtime 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 [1–11]. Papers [4, 5] carried out the L2 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, 6–8] 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 L2 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 [18–28]. 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 [18–22]. 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) [29–31], 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 H∞ 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 H∞ 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. Section2 introduces notation to be used in the paper, and problem statement is given in it. An adaptive fault-tolerant H∞ 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.


## 2. Problem Statement and Preliminaries

In this paper, the following LTI plant will be considered:(1)x˙(t)=Ax(t)+A1x(t-τ(t))+B1ω(t)+B2σ(u),z(t)=Cx(t)+Dsat(u),x(t)=ϕ(t)t∈[-h,0],
where x(t)∈Rn and xt is the plant state at time t defined by xt(s)=x(t+s), s∈[-h,0], sat(u)∈Rm is the saturated control input, z(t)∈Rs is the regulated output, and ω(t)∈Rd is an exogenous disturbance in L2[0,∞], respectively. A,  A1,  B1, B2, C, and D, are known constant matrices of appropriate dimensions. For simplicity only, we take single delay τ(t). The results of this paper can be easily applied to the case of multiple delays.The following case fortime-varying delayτ(t) is considered. That is, τ(t) is differentiable funcion
(2)0≤τ(t)≤h,τ˙(t)≤d<1,satisfying∀t≥0,
where d is an upper bound on the derivative of τ(t).In this paper, we formulate the fault-tolerant control problem by using the following model form which is considered in [19, 22]:
(3)ujqF(t)=(1-ρjq)σ(uj(t)),0≤ρ_jq≤ρjq≤ρ¯jq,j∈I[1,m],q∈I[1,L],
where ujqF(t) represents the signal from the jth actuator which has failed in the qth fault mode, ρjq is an unknown constant, the L is the total fault modes, and ρ_jq and ρ¯jq represent the lower and upper bounds of ρjq, respectively. Denote that
(4)uqF(t)=[u1qF(t),u2qF(t),…,umqF(t)]T=(I-ρq)σ(u(t)),
where ρq=diag[ρ1q,ρ2q,…,ρmq],  q∈I[1,L]. Considering the lower and upper bounds ρ_jq and ρ¯jq, the following set can be defined:
(5)Nρq={ρqρq=diag[ρ1q,ρ2q,…,ρmq],ρjq=ρ_jqorρjq=ρ¯jq}.For convenience, the following uniform actuator fault model is exploited:(6)uF(t)=(I-ρ)sat(u(t)),ρ∈{ρ1⋯ρL},
where ρ is described by ρ=diag[ρ1,ρ2,…,ρm]. The following definitions and lemmas will be used in the sequel.Definition 1 (see [33]).
Consider the following system:(7)x˙(t)=Aa(ρ^(t),ρ)x(t)+Ba(ρ^(t),ρ)ω(t),z(t)=Ca(ρ^(t),ρ)x(t),x(0)=0,
where ρ is parameter vector and ρ^(t) is a time-varying parameter vector to be chosen. Let r>0 be a given constant, then system (7) is said to be with an adaptive H∞ performance index no larger than r if for any ε>0, there exists a ρ^(t) such that the following inequality holds:
(8)∫0∞‍zT(t)z(t)dt≤r2∫0∞‍ωT(t)ω(t)dt+ε.Definition 2.
For a matrixCcl∈Rm×n, denote the jth row of Ccl as Cclj, and define
(9)℘(Ccl)={x∈Rn:|Ccljx|≤1,j∈I[1,m]}.Lemma 3 (see [32]).
If there exists a symmetric matrixΘ with
(10)Θ=[Θ11Θ12Θ12TΘ22]
and Θ11,  Θ22∈RNn×Nn such that the following inequalities hold:
(11)Θ22jj≤0,j∈I[1,N],Θ11+Θ12Δ(δ)+(Θ12Δ(δ))T+Δ(δ)Θ22Δ(δ)≥0,δ∈Δv,[QEETF]+UTU+GTΘG<0,ρ∈{ρ1⋯ρL},ρq∈Nρq,
then inequality
(12)L(δ)=Q+∑j=1N‍δjEj+(∑j=1N‍δjEj)T+∑j=1N‍∑p=1N‍δjδpFjp+(U0+∑j=1N‍δjUj)T×(U0+∑j=1N‍δjUj)<0
holds for all δj∈[δ_jδ¯j], where Q=QT∈Rn×n and Fpj=FpjT∈Rn×n,Ej∈Rn×n(13)Δ(δ)=diag[δ1In×n⋯δNIn×n],E=[E1E2···EN],U=[U0U1···UN],F=[F11⋯F1N⋯⋯⋯FN1⋯FNN],G=[[In×n⋯In×n]00INn×Nn].
LetD be a set of m×m diagonal matrices whose diagonal elements are either 1 or 0. There are 2m elements in D, and one denotes its elements as Di, i∈I[0,2m-1], where for i=z12m-1+Z22m-2+⋯+zm with zj∈{0,1}, the diagonal elements of Di are {1-z1,1-z2,…,1-zm}. Denote Di-=I-Di. It is easy to see that Di-∈D. Then, one has the following.Lemma 4 (see [35]).
For two vectorsu, v∈Rm. Suppose that ∥v∥∞≤1. Then,
(14)sat(u)∈co{Diu+Di-v:i∈[0,2m-1]},
where co denotes the convex hull.For a networked system, the performance of closed-loop system can be measured by theL2 gain. However, this gain cannot be well defined for closed-loop system, since a sufficiently large disturbance may lead to unstable closed-loop system. For this reason, we need to consider a class of disturbances whose energy is bounded by a given value; that is,
(15)𝔐δ:={ω:R+⟶Rd:∫0∞‍ωT(t)ω(t)dt≤δ}.In this paper, we will consider the following problems. The first question is, what is the maximal value ofδ such that the state will be bounded for all ω∈𝔐δ for systems with time delay? This question can be referred to as disturbance tolerance level. The system performance can be measured by the restricted L2 gain over 𝔐δ. In this paper, L2 gain and 𝔐δ will be considered at same time for networked system.


## 3. Main Results

The dynamics with actuator faults (6) and saturation are described by
(16)x˙(t)=Ax(t)+A1x(t-τ(t))+B1ω(t)+B2(I-ρ)sat(u(t)),z(t)=Cx(t)+D(I-ρ)sat(u(t)),x(t)=ϕ(t)t∈[-h,0].
Rewrite (16) as
(17)x˙(t)=y(t),y(t)=(A+A1)x(t)+B1ω(t)+B2(I-ρ)sat(u(t))-A1∫t-τ(t)t‍y(s)ds,z(t)=Cx(t)+D(I-ρ)sat(u(t)).The controller structure is given as(18)u(t)=K(ρ^(t))x(t)=(K0+Ka(ρ^(t))+Kb(ρ^(t)))x(t),
where ρ^(t) is used to estimate ρ, Ka(ρ^(t))=∑j=1m‍Kajρ^j(t) and Kb(ρ^(t))=∑j=1m‍Kbjρ^j(t).By Lemma3, the following equality is given, with x∈℘(H(ρ^)):
(19)sat(K(ρ^(t))x(t))=∑i=02m-1‍ηi[DiK(ρ^(t))+Di-H(ρ^(t))]x(t),
for some scalars 0≤ηi≤1, i∈I[0,2m-1], such that ∑i=02m-1‍ηi=1, and the following equality holds:
(20)(I-ρ)sat(u(t))=∑i=02m-1‍ηi[(I-ρ)DiK0+DiKa(ρ)-ρDiKa(ρ^)+(I-ρ^(t))DiKb(ρ^(t))+DiKa(ρ~(t))+ρ~DiKb(ρ^(t))+(I-ρ)Di-H0+Di-Ha(ρ)-ρDi-Ha(ρ^)+(I-ρ^(t))Di-Hb(ρ^(t))+Di-Ha(ρ~(t))+ρ~Di-Hb(ρ^(t))]x(t),
where ρ~(t)=ρ^(t)-ρ. Denote Bj=[0⋯bj⋯0] with B=[b1⋯bm], and
(21)Δρ^={ρ^=(ρ^1⋯ρ^m):ρ^j∈{minq{ρ_jq},maxq{ρ¯jq}},q∈I[1,L]minq{ρ_jq}}.Definition 5.
LetP1∈Rn×n be a positive-definite matrix. Denote
(22)ε(P1,δ)={x∈Rn:xTP1x≤δ},ε*(P1,δ)={x∈Rn:xTP1x+∑j=1m‍ρ~j2(t)lj≤δ}.
Assuming that lj>0 is given, we denote δ*=δ+max{∑j=1m‍(ρ~j2(t)/lj)}.The following lemma provides a method for choosing ofηi's, which are Lipschitzian functions in x and ρ^ and thus are useful in controller design method.Lemma 6 (see [11]).
Letx∈℘(H(ρ^)). For each j∈I[1,m],
(23)λj(x(t),ρ^(t))={1,ifK(ρ^(t))jx(t)=H(ρ^(t))jx(t)σ(K(ρ^(t))jx(t))-H(ρ^(t))jx(t)(K(ρ^(t))j-H(ρ^(t))j)x(t),otherwise,
and for each i∈I[0,2m-1], let zj∈{0,1} such that i=z12m-1+z22m-2+⋯+zm, and define
(24)ηi(x(t),ρ^(t))=∏j=1m[zj(1-λj(x,ρ^))+(1-zj)λj(x,ρ^)].
Then,ηi's are functions Lipschitz in x and ρ^, such that, ∑i=02m-1‍ηi=1, 0≤ηi≤1, i∈I[0,2m-1]. Moreover, they satisfy relation (19).
By using the functionsηi(x(t),ρ^(t))’s and controller (18), plant (16) can be written in a quasi-LPV form as follows:
(25)y(t)=(A+A1)x(t)+B1ω(t)+B2∑i=02m-1‍ηi[+Hb(ρ^(t)))(I-ρ)Di(K0+Ka(ρ^(t))+Kb(ρ^(t)))+(I-ρ)Di-(Hb(ρ^(t))H0+Ha(ρ^(t))x(t)+Hb(ρ^(t)))]-A1∫t-τ(t)t‍y(s)ds.Lemma 7 (see [36]).
For anya∈Rn, b∈R2n, Z0∈R2n×n, R∈Rn×n,Y∈Rn×2n, and Z∈R2n×2n, the following holds:
(26)-2bTFa≤[ab]T[RY-Z0TYT-Z0Z][ab],
where  [RYYTZ]≥0.Definition 8.
Firstly, for system (25), consider the following Lyapunov-Krasovskii functional
(27)V=V1+V2+V3+∑j=1m‍ρ~j2(t)lj,
where
(28)V1=x-T(t)EPx-(t),V2=∫h0‍∫t+θt‍yT(s)Ry(s)dsdθ,V3=∫t-τ(t)t‍xT(s)Sx(s)ds,E=[In×n000],P=[P10P2P3],P1=P1T>0.
Then, the following set can be given:(29)MV(δ)={x(t)∈Rn:V=V1+V2+V3+∑j=1m‍ρ~j2(t)lj≤δ}.Remark 9.
By Definitions5 and 8, we can draw the conclusion that MV(δ)⊂ε*(P1,δ)⊂ε(P1,δ).
By Lemma6, we analyze the auxiliary LPV system as follows, of which the closed-loop system comprising of (25) and (18) is a special case, for all x(t)∈MV(δ)⊂℘(H(ρ^)):
(30)x˙(t)=A(η)x(t)+B1ω,η∈Γ,
where η=[η0,η1,…,η2m-1], and
(31)Γ={η∈R2m:∑i=02m-1‍ηi=1,0≤ηi≤1,i∈I[0,2m-1]},A(η)=A+A1+B2∑i=02m-1‍ηi[(I-ρ)DiK0+DiKa(ρ)-ρDiKa(ρ^)+(I-ρ^(t))DiKb(ρ^(t))+DiKa(ρ~(t))+ρ~DiKb(ρ^(t))+(I-ρ)Di-H0+Di-Ha(ρ)-ρDi-Ha(ρ^)+(I-ρ^(t))Di-Hb(ρ^(t))+Di-Ha(ρ~(t))+ρ~Di-Hb(ρ^(t))]-A1∫t-τ(t)t‍y(s)ds.Theorem 10.
Letrf>0, rn>0, d,h>0 and δ>0 be given constants, then the following two conditions are satisfied.(I)
The trajectories of the closed-loop system that start from the origin will remain inside the domainMV(δ) for every ω∈𝔐δ.(II)
In normal case, that is,ρ=0, for x(0)=0,
(32)∫0∞‍zT(t)z(t)dt≤rn2∫0∞‍ωT(t)ω(t)dt+rn2∑j=1m‍ρ~j2(0)lj,and in actuator failures cases, that is, ρ∈{ρ1⋯ρL}, for x(0)=0,
(33)∫0∞‍zT(t)z(t)dt≤rf2∫0∞‍ωT(t)ω(t)dt+rf2∑j=1m‍ρ~j2(0)lj,
where ρ~(t)=diag{ρ~1(t)⋯ρ~m(t)}, ρ~j(t)=ρ^j(t)-ρj if there exist matrices Q1>0,  Q2,  Q3,  S-,  R-,  Z-1,  Z-2,  Z-3,  O-0,  O-aj,  O-bj,  Y-0,  Y-aj,  Y-bj,  j∈I[1,m] and symmetric matrixes Θi, i∈I[0,2m-1], with
(34)Θi=[Θ11iΘ12iΘ12iTΘ22i]
and Θ11i, Θ22i∈Rmn×mn such that the following inequalities hold for all Di∈D and MV(δ)⊂℘(H(ρ^)); that is, |H(ρ^)jx|≤1 for all x∈MV(δ),j∈I[1,m]:
(35)Θ22jji≤0,j∈I[1,m],i∈I[0,2m-1]Θ11i+Θ12iΔ(ρ^)+(Θ12iΔ(ρ^))T+Δ(ρ^)Θ22iΔ(ρ^)≥0,ρ^∈Δρ^(36)[MiV0iT[0B10][0A1(I-ε)S-0][Q100][hQ2ThQ3T0]*-rn2I0000**-I000***-(1-d)S-00****-S-0*****-hR-]ρ=0,(37)[MiV0iT[0B10][0A1(I-ε)S-0][Q100][hQ2ThQ3T0]*-rf2I0000**-I000***-(1-d)S-00****-S-0*****-hR-]<0,ρ∈{ρ1⋯ρL},ρq∈Nρq(38)[R-0R-εA1T*Z-1Z-2**Z-3]≥0,
where
(39)Mi=[N0iUiUiTΩi]+GTΘiG,N0i=[Q2+Q2T+hZ-1T1i*-Q3-Q3T+hZ-3],Ui=[U1iU2i⋯Umi],V0i=[V00iV01i⋯V0mi],Ωi=[Ωijp],j,p=1⋯m,T1i=Q3-Q2T+Q1(AT+εA1T)+hZ-2+(I-ρ)DiY-0T+(I-ρ)Di-O-0T,V00i=[CQ1+D(I-ρ)DiY-0+D(I-ρ)Di-O-00],V0ji=[D(I-ρ)Di(Y-aj+Y-bj)+D(I-ρ)Di-(O-aj+O-bj)0],Uji=[0-Y-ajTDiρB2T+Y-bjTDiB2T-O-ajTDiρB2T+O-bjTDiB2T00],G=[[I⋮I]00I],Ωijp=[0-B2jDiY-bp-Y-bjTDiB2pT-B2jDiO-bp-O-bjTDiB2pT00],Y-a(ρ)=∑j=1m‍Y-ajρj,
and also ρ^j(t) is determined according to the adaptive law
(40)ρ^˙j=Proj[minq{ρ_jq}maxq{ρ¯jq}{Lj}={0,ifρ^j=minq{ρ_jq},Lj≤0orρ^j=maxq{ρ¯jq},Lj≥0Lj,otherwise,
where
(41)Lj=-ljx-T(t)[00Λji0]x-(t),Λji=B2j(∑i=02m-1‍ηiDi)Kb(ρ^)+B2(∑i=02m-1‍ηiDi)Kaj+B2j(∑i=02m-1‍ηiDi-)Hb(ρ^)+B2(∑i=02m-1‍ηiDi-)Haj,Q=[Q10Q2Q3],
where Kaj=Y-ajQ1-1, Kbj=Y-bjQ1-1, Haj=O-ajQ1-1, and Hbj=O-bjQ1-1·lj>0  (j∈I[1,m]) are the adaptive law gains which can be given according to practical applications. Then, the following controller can be given:
(42)K(ρ^)=Y-0Q1-1+∑j=1m‍ρ^jY-ajQ1-1+∑j=1m‍ρ^jY-bjQ1-1.Proof.
See appendix.From Theorem10, we have the following algorithm to optimize the adaptive H∞ performance in normal and fault cases and the disturbance tolerance level δ with considering time delay.Algorithm 11.
Suppose thatrn and rf denote the adaptive H∞ performance indexes for the normal case and fault cases of the closed-loop system (30), respectively. Let δ denote the disturbance tolerance level. Then, rn, rf are minimized, and δ is maximized if the following optimization problem is solvable:
(43)minη=αηn+βηf+γηδs.t.(a)(35),(36),(37),(38)(b)MV(δ)⊂℘(H(ρ^)),
where ηn=rn2, ηf=rf2, ηδ=1/δ*=1/(δ+max{∑j=1m‍(ρ~j2(t)/lj)}), and α, β, and γ are weighting coefficients.
By Definition2, condition (b) cannot be shown as LMIs directly. However, obviously, MV(δ)⊂ε*(P1,δ*)⊂ε(P1,δ*), which implies that (b) can be replaced with (b1) as follows:
(44)(b1)ε(P1,δ*)⊂℘(H(ρ^)).
Equation (44) is equivalent to
(45)δ*h(ρ^)jP1-1h(ρ^)jT≤1⟺[1δ*h(ρ^)jP1-1*P1-1]≥0,
for all j∈I[1,m], where h(ρ^)j is the jth row of H(ρ^). Then, it can be drawn that (45) is equivalent to the following inequalities:
(46)(b2)[-ηδ-O-0s*-Q1]+∑j=1m‍ρ^j[0-O-ajs-O-bjs*0]≤0,
where ρ^∈Δρ^ and O-ajs is the sth row of O-aj,  s∈I[1,m].Remark 12.
Theorem10 prevents a condition for the existence of an adaptive fault tolerant H∞ controller. In Theorem 10, if set Yaj=0, Ybj=0, Oaj=0, and Obj=0, j∈I[1,m], 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.


## 4.  Examples

Example 13.
Consider the system of the form (1) with
(47)A=[32340],A1=[-100-2],B1=[1010],B2=[400040],C=[400000]T,D=[00.50001]T,τ(t)=1+sin(t)4,
and the following two possible fault modes.Fault mode 1: both of the two actuators are normal; that isρ11=ρ21=0. Fault mode 2: the first actuator is outage, and the second actuator may be normal or loss of effectiveness, described by ρ12=1, 0≤ρ22≤a, where a=0.5 denotes the maximal loss of effectiveness for the second actuator.Letα=10,  β=1, and γ=10, and let the optimal indexes with fixed controller gains are ηn=0.1963, ηf=9.8933, ηδ=20.5385, and η=217.2408. By solving (43), the optimal indexes can be given as ηn=0.5881, ηf=9.1236, ηδ=9.6701, and  η=111.7048. For getting smaller number for every optimal index, we may revise α=110,β=0.2, and γ=0.5. Then, the indexes can be drawn that ηn=0.1676, ηf=7.1242, and ηδ=18.6399. For illustrating the efficiency of the design method, the following simulations is given.During the following simulation, fault case is considered as follows. At0 second, the first actuator is outage. Here, we choose l1=l2=100.Firstly, we consider theH∞ performance. The disturbance is given as
(48)ω1(t)=ω2(t)={cos(t),4.2≤t≤6.90,otherwise.
Figures 1 and 2 show the responses curves of the first state in normal and fault case, respectively.Figure 1
Response curve of the first state in normal case with the adaptive controller (solid) and the fixed gain controller (dashed).Figure 2
Response curve of the first state in fault case with the adaptive controller (solid) and the fixed gain controller (dashed).Then, we consider the disturb tolerance problem. The disturbance is given as(49)ω1(t)=ω2(t)={27.2,4≤t≤50,otherwise.
Figures 3 and 4 show the responses curves of the states in normal case.Figure 3
Responses curves of the states with adaptive controller in normal case.Figure 4
Responses curves of the states with fixed gain controller in normal case.


## 5. Conclusions

In this paper, an adaptive fault-tolerantH∞ 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 H∞ 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.



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