# Stability Guaranteed Active Fault-Tolerant  Control of Networked Control Systems

Authors: Shanbin Li; Dominique Sauter; Christophe Aubrun; Joseph Yamé
Journal: Journal of Control Science and Engineering
 (2008)

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## Abstract

The stability guaranteed active fault-tolerant control against actuators failures and plant uncertainties in networked control systems (NCSs) is addressed. A detailed design procedure is formulated as a convex optimization
problem which can be efficiently solved by existing software. An illustrative example
is given to show the efficiency of the proposed method for network-based
control for uncertain systems.


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## Body

## 1. Introduction

Fault-tolerant
control (FTC) techniques against actuator faults can be classified into two
groups [1]: passive
and active approaches. In passive FTC systems, a single controller with fixed
structure/parameters is used to deal with all possible failure scenarios which
are assumed to be known a priori. Consequently, the passive controller is
usually conservative. Furthermore, if a failure out of those considered in the
design occurs, the stability and performance of the closed-loop system might
not be guaranteed. Such potential limitations of passive approaches provide a
strong motivation for the development of methods and strategies for active FTC
(AFTC) systems.In contrast to passive FTC systems, AFTC techniques
rely on a real-time fault detection and isolation (FDI) scheme and a controller
reconfiguration mechanism. Such techniques allow a flexibility to select different controllers according to different component failures, and therefore
better performance of the closed-loop system can be expected. However, this
holds true when the FDI process does not make an incorrect or delayed decision
[2]. Some preliminary
results have been obtained on AFTC which is immune to imperfect FDI
process [3, 4]. In [5], the latter issue is further
discussed in a classical setting (i.e., point-to-point control) by using the
guaranteed cost control approach and online controller switching in order to
ensure stability of the closed-loop system at all times. The aim of this paper
is to extend the results in [5] to uncertain plants controlled over digital communication networks. In such networks, the information transfer from sensors to controllers and from controllers to actuators is not instantaneous but suffers communication delays. Such communication delays can be highly variable due to their strong dependence on variable network conditions such as congestion and channel quality. These network-induced delays may impact
adversely on the stability and performance of the control system [6, 8]. Networked control systems
(NCSs) are now pervasive (see, e.g., the recent special issue of the
proceedings of the IEEE [7]), and such systems are long-running real-time systems which should function in a
correct manner even in the presence of failures. This makes the issue of fault
tolerant control in NCS an important one and entails designing strategies to
cope with some of the fundamental problems introduced by the network such as
bandwidth limitations, quantization and sampling effects, message scheduling
and communication delays. Motivated by the above considerations, we address the
problem of fault tolerant control in NCSs with time-varying delays.
Specifically, we extend the results of [5] for the stabilization of a plant, subject to model uncertainties and actuator faults, which is controlled over a communication network
that induces time-varying but bounded delays.The outline of the paper is as follows. In Section2, a network-based control model for an uncertain plant
subject to actuator failures is proposed and the guaranteed cost control
problem is formulated. In Section 3, the detailed procedure for designing the NCSs-based
fault-tolerant controller is given. Section 4 presents a design example to illustrate the benefit
of the proposed FTC design procedure. Finally, conclusions are given in Section 5. The proof of the main theorem of the paper is
reported in the appendix.


## 2. Problem Statement

Figure1 shows the basic networked control architecture which is considered in this paper and which
consists of a single uncertain plant, with few sensors and actuators,
controlled by a digital controller in a centralized structure.Figure 1
Networked control system with actuator failures.The delays induced by the network in the closed-loop
control system are modeled as time-varying quantitiesτ(k)=τksc arising from
the communication delays between sensors and controllers at time k. Without loss of generality, we assume that there are
no transmission time-delays between the controllers and the actuators. The
actuators might be subject to faults during the system operation. Thus, taking
into account the potential failures of actuators, the interconnection of the
uncertain discrete-time plant and a discrete-time controller through the
digital communication link as depicted in Figure 1 can be described by the following dynamical and state-delayed feedback equations: (1)x(k+1)=(A+DΔ(k)E)x(k)+Bℒu(k),(2)x(0)=x0,(3)u(k)=Kx(k−τ(k)), where x(k)∈ℛn is the state of
the uncertain plant, u(k)∈ℛm is the control
input, A,B,D,E are all real
constant matrices, and matrix K is the
controller gain matrix to be designed. The time-varying matrix Δ(k) represents
norm-bounded parameter uncertainties and satisfies the bound Δ(k)TΔ(k)≤I where I denotes the
identity matrix with appropriate dimension. The constant matrices D and E characterize
the structure of these parameter uncertainties. The fault indicator matrix ℒ is given by (4)ℒ=diag{l1,…,lm} with lj∈{0,1} for j=1,2,…,m, where lj=1 means that the jth actuator is
in healthy state, whereas the jth actuator is
meant to experience a total failure when lj=0. Having a finite number of actuators, the set of
possible related failure modes is also finite and, by abuse of notation, we
denote this set by (5)ℒ={ℒ1,ℒ2,…ℒN}with N=2m−1. Each failure mode ℒi,(i=1,2…,N) is therefore an
element of the set ℒ. We also view ℒi as a matrix,
that is, as a particular pattern of matrix ℒ in (4) depending on the
values of lj(j=1,2,…m). Throughout, when ℒ is invoked as a
matrix, it will mean that matrix ℒ varies over the
set of matrices in (5). Note that the faulty mode ℒi in the NCS
architecture of Figure 1 is estimated by the FDI unit. In order to ensure that
system (1)
should remain controllable, we assume that the set ℒ excludes the
element diag{0,0,…,0}, that is, at least one actuator should be healthy.We further assume that the time-varying delaysτ(k) lie between the
following positive integer bounds τm and τM, that is,(6)τm≤τ(k)≤τM. Given positive definite symmetric matrices Q1 and Q2, we consider the quadratic cost function (7)J=∑k=0∞[xT(k)Q1x(k)+uT(k)Q2u(k)], and with respect to this cost function, we define the guaranteed cost
controller in the event of actuator failures as follows.Definition 1.
If there exists a control lawu(k) and a positive scalar J∗ such that for
all admissible uncertainties Δ(k) and all failure modes ℒi∈ℒ the closed-loop
system (1)–(3) is stable with cost function (7) satisfying J≤J∗, then J∗ is said to be a
guaranteed cost and u(k) a guaranteed cost
controller for the uncertain system (1).In Section3, we will proceed through two main steps
to design a cost guaranteed active fault tolerant control in the NCS framework. These steps are(i)
construct a
fault-tolerant controller (i.e., a robust controller), with structure as given
by (3), which
achieves the smallest possible value for J∗ under all
admissible plant uncertainties and all actuator failure modes;(ii)
redesign that
part of the above controller associated to only one fault-free actuator
in order to improve the robust performance without loss of the stability
property of the design in step (i); step (ii) repeats for allm actuators and results
in a bank of m controllers. It follows from inequality m≤N=2m−1, that the cardinality of the bank of controllers
(which is equal to the number of actuators) is less than the cardinality of the
set ℒ of faulty
modes. For each faulty mode ℒi, the controller to be switched on should be the best
as ranked with respect to a closed-loop performance index. In this paper, we
will not address the switching and reconfiguration
mechanisms; we focus on the design of the bank of m controllers.


## 3. AFTC Design for NCS

### 3.1. Robust Stability

In this subsection,
we establish a sufficient condition for the existence of a guaranteed cost
network-based controller for the uncertain plant (1). Note that the
control law (3) applied to plant (1) results in the
following system: (8)x(k+1)=A1x(k)+BℒKx(k−τ(k)), where A1=A+DΔ(k)E. The cost function associated to system (8) is therefore (9)J=∑k=0∞xeT(k)Qxe(k), where xeT(k)=[xT(k),xT(k−τ(k))] and Q=diag{Q1,KTQ2K}. Under the assumptions made in Section 2, we can state the following result.Theorem 1.
If there exist a gain matrixK, a scalar ϵ>0, symmetric positive-definite matrices P1∈ℛn×n, R∈ℛn×n, S∈ℛn×n, and matrices P2∈ℛn×n, P3∈ℛn×n, W∈ℛ2n×2n, M∈ℛ2n×n such that the
following matrix inequalities are satisfied: (10)[ΓPT​​[0BℒK]​−​M[ET0​​]∗−R​+​KTQ2K0∗∗−ϵI]<0,(11)[​​WM∗S]≥0 with (12)Γ=PT[0IA−I−I]+[0IA−I−I]TP+ϵPT[000DDT]P+[μR+Q100P1+τMS]+τMW+[M0]+[M0]T,μ=1+(τM−τm),P=[P10P2P3], then system (8) is asymptotically stable and the cost function
(9) satisfies
the inequality (13)J≤xT(0)P1x(0)+∑l=−τM−1xT(l)Rx(l)+∑θ=−τM+10∑l=−1+θ−1yT(l)Sy(l)+∑θ=−τM+2−τM+1∑l=θ−1−1xT(l)Rx(l), where y(l)=x(l+1)−x(l).Proof.
See the appendix.Remark 2.
The “∗” sign represents blocks that are readily inferred by symmetry.Remark 3.
Note that
the upper bound in (13) depends on the initial condition of system (8). To remove the
dependence on the initial condition, we suppose that the initial state of
system (8)
might be arbitrary but belongs to the set 𝒮={x(l)∈ℛn:x(l)=Uv,vTv≤1,l=−τM,−τM+1,…,−τm}, where U is a given
matrix. Inequality (13) leads to (14)J≤λmax(UTP1U)+ρ1λmax(UTRU)+ρ2λmax(UTSU), where λmax(⋅) denotes the
maximum eigenvalue of matrix (⋅), ρ1=μ(τM+τm)/2, and ρ2=2τM(τM+1).


### 3.2. Step (i): Controller Design

Now, we derive
the guaranteed cost controller in terms of the feasible solutions to a set of
linear matrix inequalities.Using Sherman-Morrison matrix inversion formula, we
have(15)P−1=[P1−10−P3−1P2P1−1P3−1]. In the sequel, we will denote X=P1−1,Y=P3−1, and Z=−P3−1P2P1−1. We further
restrict M to the
following case in order to obtain a linear matrix inequality (LMI) (see, e.g.,
[9]): (16)M=δPT[0BℒK], where δ is a scalar
parameter. Pre- and postmultiply (10) by diag{(P−1)T,P1−1,I} and diag{P−1,P1−1,I}, respectively; also pre- and postmultiply (11) by diag{(P−1)T,P1−1} and diag{P−1,P1−1} and denote (17)L=P1−1RP1−1,F=KP1−1,S¯=S−1,(P−1)TWP−1=[W¯1W¯2∗W¯3]. Applying the Schur complement and expanding the block matrices, we obtain the
following result under the assumptions made in Section 2.Theorem 2.
Suppose that for a prescribed
scalarδ, there exist a scalar ϵ>0, matrices X>0,Y,Z,F,L>0,S¯>0,W¯1,W¯2,W¯3, such that the following matrix inequalities are
satisfied: (18)[Ψ1Ψ20Ψ41*Ψ3(1−δ)BℒFΨ42**−LΨ43***Ψ5]<0,(19)[W¯1W¯20∗W¯3δBℒF∗∗XS¯−1X]≥0, where (20)Ψ1=Z+ZT+μL+τMW¯1,Ψ2=Y+X(A−I)T−ZT+τMW¯2+δ(BℒF)T,Ψ3=−Y−YT+τMW¯3+ϵDDT,[Ψ41Ψ42Ψ43]=[XETτMZT0XZT0τMYT00YT00FT00],Ψ5=diag{−ϵI,−τMS¯,−Q2−1,−Q1−1,−X}. Then, the control law (21)u(k)=FX−1x(k−τ(k)) is a guaranteed cost networked control law for system (1) and the
corresponding cost function satisfies (22)J≤λmax(UTX−1U)+ρ1λmax(UTX−1LX−1U)+ρ2λmax(UTS¯−1U), where ρ1=μ(τM+τm)/2 and ρ2=2τM(τM+1).Remark 4.
From (22), we establish the following inequalities: (23)[−αIUT∗−X]<0,[−βIUT∗−XL−1X]<0,[−γIUT∗−S¯]<0, where α,β, and γ are scalars to
be determined. It is worth noting that condition (23) is not an LMI
because of the term −XL−1X. This is also the case for condition (19) which is not an
LMI because of the term XS¯−1X. Note that for any matrix X>0 we have (24)XS¯−1X≥2X−S¯,XL−1X≥2X−L.Given a prescribed
scalarδ, the design problem of the optimal guaranteed cost
controller can be formulated therefore as the following optimization problem: (25)OP1:minϵ,X,Y,Z,F,L,S¯,W¯1,W¯2,W¯3(α+ρ1β+ρ2γ)s.t.{(i)inequality(18),(ii)[W¯1W¯20∗W¯3δBℒF∗∗2X−S¯]≥0,(iii)[−αIUT∗−X]<0,[−βIUT∗−2X+L]<0,[−γIUT∗−S¯]<0. Clearly, the above optimization problem (25) is a convex
optimization problem which can be effectively solved by existing LMI software
[10]. Thus, the
minimization of α+ρ1β+ρ2γ implies the
minimization of the cost in (9). By applying a simple one-dimensional search over δ for a certain τM, a global optimum cost can be found.


### 3.3. Robust Stability with at Least a Fault-Free Actuator

Based on the
controller designed in Theorem2, let us assume that actuator i is fault-free,
then we can redesign the i th row of
controller gain matrix K to improve the
robust performance for the system against actuator failures. We can rewrite the
overall control system as (26)x(k+1)=A1x(k)+(Bi¯ℒi¯Ki¯+biki)x(k−τ(k)), where A1=A+DΔ(k)E, matrix Ki¯ is obtained by
deleting the ith row from K, Bi¯ is obtained by
deleting the ith column from B, and ℒi¯ is obtained by
deleting ith row and ith column from ℒ. The cost function associated to system (26) reads as (27)J=∑k=0∞xeT(k)Qxe(k) with xeT(k)=[xT(k),xT(k−τ(k))], Q=diag{Q1,kiTQ2iki+Ki¯TQ2i¯Ki¯}, where Q2i¯ is obtained by
deleting the ith row and ith column from Q2. With regard to system (26) where Ki¯ is assumed to
be known, we have the following result.Theorem 3.
If there exist a gain matrixki, a scalar ϵ>0, symmetric positive-definite matrices P1∈ℛn×n, R∈ℛn×n, S∈ℛn×n, and matrices P2∈ℛn×n, P3∈ℛn×n, W∈ℛ2n×2n, M∈ℛ2n×n such that the
following matrix inequalities are satisfied: (28)[ΓPT​​[​​0Bi¯ℒi¯Ki¯+biki]−M[ET0]∗−R​+kiTQ2iki+Ki¯TQ2i¯Ki¯0∗∗−ϵI]<0,(29)[​​WM∗S​​]≥0, then, system (26) is asymptotically
stable and the cost function (27) satisfies inequality (13).Proof.
The proof is similar to the proof
of Theorem1.


### 3.4. Step (ii): Controller Redesign

Proceeding as
in step (i), we restrictM to the
following case in order to obtain an LMI: (30)M=δPT[0biki], where δ is a scalar
parameter. Pre- and postmultiply (28) with diag{(P−1)T,P1−1,I} and diag{P−1,P1−1,I}, respectively; also pre- and postmultiply (29) with diag{(P−1)T,P1−1} and diag{P−1,P1−1}, respectively, and denote (31)L=P1−1RP1−1,F*=kiP1−1,S¯=S−1,(P−1)TWP−1=[W¯1W¯2∗W¯3]. The Schur complement trick leads to the following controller redesign result.Theorem 4.
Suppose that for a prescribed scalarδ, there exist a scalar ϵ>0, matrices X>0,Y,Z,F*,L>0,S¯>0,W¯1,W¯2,W¯3, such that the following matrix inequalities are
satisfied: (32)[Ψ˜1Ψ˜20Ψ˜41*Ψ˜3Bi¯ℒi¯Ki¯X+(1−δ)biF*Ψ˜42**−LΨ˜43***Ψ˜5]<0,(33)[W¯1W¯20∗W¯3δbiF*∗∗XS¯−1X]≥0, where (34)Ψ˜1=Z+ZT+μL+τMW¯1,Ψ˜2=Y+X(A−I)T−ZT+τMW¯2+δ(biF*)T,Ψ˜3=−Y−YT+τMW¯3+ϵDDT,[Ψ˜41Ψ˜42Ψ˜43]=[XETτMZT00XZT0τMYT000YT00(F*)TXKi¯T00],Ψ˜5=diag{−ϵI,−τMS¯,−Q2i−1,−Q2i¯−1,−Q1−1,−X}. Then, the ith control law (35)ui(k)=F*X−1x(k−τ(k)) is a guaranteed cost networked control law of system (26) and the
corresponding cost function satisfies (36)J≤λmax(UTX−1U)+ρ1λmax(UTX−1LX−1U)+ρ2λmax(UTS¯−1U),where ρ1=μ(τM+τm)/2andρ2=2τM(τM+1).Given a prescribed
scalarδ, the redesign problem of the optimal guaranteed cost
controller can be formulated as the following convex optimization problem: (37)OP2:minϵ,X,Y,Z,F*,L,S¯,W¯1,W¯2,W¯3(α+ρ1β+ρ2γ)s.t.{(i)inequality(32),(ii)[W¯1W¯20∗W¯3δbiF*∗∗2X−S¯]≥0,(iii)[−αIUT∗−X]<0,[−βIUT∗−2X+L]<0,[−γIUT∗−S¯]<0.


## 3.1. Robust Stability

In this subsection,
we establish a sufficient condition for the existence of a guaranteed cost
network-based controller for the uncertain plant (1). Note that the
control law (3) applied to plant (1) results in the
following system: (8)x(k+1)=A1x(k)+BℒKx(k−τ(k)), where A1=A+DΔ(k)E. The cost function associated to system (8) is therefore (9)J=∑k=0∞xeT(k)Qxe(k), where xeT(k)=[xT(k),xT(k−τ(k))] and Q=diag{Q1,KTQ2K}. Under the assumptions made in Section 2, we can state the following result.Theorem 1.
If there exist a gain matrixK, a scalar ϵ>0, symmetric positive-definite matrices P1∈ℛn×n, R∈ℛn×n, S∈ℛn×n, and matrices P2∈ℛn×n, P3∈ℛn×n, W∈ℛ2n×2n, M∈ℛ2n×n such that the
following matrix inequalities are satisfied: (10)[ΓPT​​[0BℒK]​−​M[ET0​​]∗−R​+​KTQ2K0∗∗−ϵI]<0,(11)[​​WM∗S]≥0 with (12)Γ=PT[0IA−I−I]+[0IA−I−I]TP+ϵPT[000DDT]P+[μR+Q100P1+τMS]+τMW+[M0]+[M0]T,μ=1+(τM−τm),P=[P10P2P3], then system (8) is asymptotically stable and the cost function
(9) satisfies
the inequality (13)J≤xT(0)P1x(0)+∑l=−τM−1xT(l)Rx(l)+∑θ=−τM+10∑l=−1+θ−1yT(l)Sy(l)+∑θ=−τM+2−τM+1∑l=θ−1−1xT(l)Rx(l), where y(l)=x(l+1)−x(l).Proof.
See the appendix.Remark 2.
The “∗” sign represents blocks that are readily inferred by symmetry.Remark 3.
Note that
the upper bound in (13) depends on the initial condition of system (8). To remove the
dependence on the initial condition, we suppose that the initial state of
system (8)
might be arbitrary but belongs to the set 𝒮={x(l)∈ℛn:x(l)=Uv,vTv≤1,l=−τM,−τM+1,…,−τm}, where U is a given
matrix. Inequality (13) leads to (14)J≤λmax(UTP1U)+ρ1λmax(UTRU)+ρ2λmax(UTSU), where λmax(⋅) denotes the
maximum eigenvalue of matrix (⋅), ρ1=μ(τM+τm)/2, and ρ2=2τM(τM+1).


## 3.2. Step (i): Controller Design

Now, we derive
the guaranteed cost controller in terms of the feasible solutions to a set of
linear matrix inequalities.Using Sherman-Morrison matrix inversion formula, we
have(15)P−1=[P1−10−P3−1P2P1−1P3−1]. In the sequel, we will denote X=P1−1,Y=P3−1, and Z=−P3−1P2P1−1. We further
restrict M to the
following case in order to obtain a linear matrix inequality (LMI) (see, e.g.,
[9]): (16)M=δPT[0BℒK], where δ is a scalar
parameter. Pre- and postmultiply (10) by diag{(P−1)T,P1−1,I} and diag{P−1,P1−1,I}, respectively; also pre- and postmultiply (11) by diag{(P−1)T,P1−1} and diag{P−1,P1−1} and denote (17)L=P1−1RP1−1,F=KP1−1,S¯=S−1,(P−1)TWP−1=[W¯1W¯2∗W¯3]. Applying the Schur complement and expanding the block matrices, we obtain the
following result under the assumptions made in Section 2.Theorem 2.
Suppose that for a prescribed
scalarδ, there exist a scalar ϵ>0, matrices X>0,Y,Z,F,L>0,S¯>0,W¯1,W¯2,W¯3, such that the following matrix inequalities are
satisfied: (18)[Ψ1Ψ20Ψ41*Ψ3(1−δ)BℒFΨ42**−LΨ43***Ψ5]<0,(19)[W¯1W¯20∗W¯3δBℒF∗∗XS¯−1X]≥0, where (20)Ψ1=Z+ZT+μL+τMW¯1,Ψ2=Y+X(A−I)T−ZT+τMW¯2+δ(BℒF)T,Ψ3=−Y−YT+τMW¯3+ϵDDT,[Ψ41Ψ42Ψ43]=[XETτMZT0XZT0τMYT00YT00FT00],Ψ5=diag{−ϵI,−τMS¯,−Q2−1,−Q1−1,−X}. Then, the control law (21)u(k)=FX−1x(k−τ(k)) is a guaranteed cost networked control law for system (1) and the
corresponding cost function satisfies (22)J≤λmax(UTX−1U)+ρ1λmax(UTX−1LX−1U)+ρ2λmax(UTS¯−1U), where ρ1=μ(τM+τm)/2 and ρ2=2τM(τM+1).Remark 4.
From (22), we establish the following inequalities: (23)[−αIUT∗−X]<0,[−βIUT∗−XL−1X]<0,[−γIUT∗−S¯]<0, where α,β, and γ are scalars to
be determined. It is worth noting that condition (23) is not an LMI
because of the term −XL−1X. This is also the case for condition (19) which is not an
LMI because of the term XS¯−1X. Note that for any matrix X>0 we have (24)XS¯−1X≥2X−S¯,XL−1X≥2X−L.Given a prescribed
scalarδ, the design problem of the optimal guaranteed cost
controller can be formulated therefore as the following optimization problem: (25)OP1:minϵ,X,Y,Z,F,L,S¯,W¯1,W¯2,W¯3(α+ρ1β+ρ2γ)s.t.{(i)inequality(18),(ii)[W¯1W¯20∗W¯3δBℒF∗∗2X−S¯]≥0,(iii)[−αIUT∗−X]<0,[−βIUT∗−2X+L]<0,[−γIUT∗−S¯]<0. Clearly, the above optimization problem (25) is a convex
optimization problem which can be effectively solved by existing LMI software
[10]. Thus, the
minimization of α+ρ1β+ρ2γ implies the
minimization of the cost in (9). By applying a simple one-dimensional search over δ for a certain τM, a global optimum cost can be found.


## 3.3. Robust Stability with at Least a Fault-Free Actuator

Based on the
controller designed in Theorem2, let us assume that actuator i is fault-free,
then we can redesign the i th row of
controller gain matrix K to improve the
robust performance for the system against actuator failures. We can rewrite the
overall control system as (26)x(k+1)=A1x(k)+(Bi¯ℒi¯Ki¯+biki)x(k−τ(k)), where A1=A+DΔ(k)E, matrix Ki¯ is obtained by
deleting the ith row from K, Bi¯ is obtained by
deleting the ith column from B, and ℒi¯ is obtained by
deleting ith row and ith column from ℒ. The cost function associated to system (26) reads as (27)J=∑k=0∞xeT(k)Qxe(k) with xeT(k)=[xT(k),xT(k−τ(k))], Q=diag{Q1,kiTQ2iki+Ki¯TQ2i¯Ki¯}, where Q2i¯ is obtained by
deleting the ith row and ith column from Q2. With regard to system (26) where Ki¯ is assumed to
be known, we have the following result.Theorem 3.
If there exist a gain matrixki, a scalar ϵ>0, symmetric positive-definite matrices P1∈ℛn×n, R∈ℛn×n, S∈ℛn×n, and matrices P2∈ℛn×n, P3∈ℛn×n, W∈ℛ2n×2n, M∈ℛ2n×n such that the
following matrix inequalities are satisfied: (28)[ΓPT​​[​​0Bi¯ℒi¯Ki¯+biki]−M[ET0]∗−R​+kiTQ2iki+Ki¯TQ2i¯Ki¯0∗∗−ϵI]<0,(29)[​​WM∗S​​]≥0, then, system (26) is asymptotically
stable and the cost function (27) satisfies inequality (13).Proof.
The proof is similar to the proof
of Theorem1.


## 3.4. Step (ii): Controller Redesign

Proceeding as
in step (i), we restrictM to the
following case in order to obtain an LMI: (30)M=δPT[0biki], where δ is a scalar
parameter. Pre- and postmultiply (28) with diag{(P−1)T,P1−1,I} and diag{P−1,P1−1,I}, respectively; also pre- and postmultiply (29) with diag{(P−1)T,P1−1} and diag{P−1,P1−1}, respectively, and denote (31)L=P1−1RP1−1,F*=kiP1−1,S¯=S−1,(P−1)TWP−1=[W¯1W¯2∗W¯3]. The Schur complement trick leads to the following controller redesign result.Theorem 4.
Suppose that for a prescribed scalarδ, there exist a scalar ϵ>0, matrices X>0,Y,Z,F*,L>0,S¯>0,W¯1,W¯2,W¯3, such that the following matrix inequalities are
satisfied: (32)[Ψ˜1Ψ˜20Ψ˜41*Ψ˜3Bi¯ℒi¯Ki¯X+(1−δ)biF*Ψ˜42**−LΨ˜43***Ψ˜5]<0,(33)[W¯1W¯20∗W¯3δbiF*∗∗XS¯−1X]≥0, where (34)Ψ˜1=Z+ZT+μL+τMW¯1,Ψ˜2=Y+X(A−I)T−ZT+τMW¯2+δ(biF*)T,Ψ˜3=−Y−YT+τMW¯3+ϵDDT,[Ψ˜41Ψ˜42Ψ˜43]=[XETτMZT00XZT0τMYT000YT00(F*)TXKi¯T00],Ψ˜5=diag{−ϵI,−τMS¯,−Q2i−1,−Q2i¯−1,−Q1−1,−X}. Then, the ith control law (35)ui(k)=F*X−1x(k−τ(k)) is a guaranteed cost networked control law of system (26) and the
corresponding cost function satisfies (36)J≤λmax(UTX−1U)+ρ1λmax(UTX−1LX−1U)+ρ2λmax(UTS¯−1U),where ρ1=μ(τM+τm)/2andρ2=2τM(τM+1).Given a prescribed
scalarδ, the redesign problem of the optimal guaranteed cost
controller can be formulated as the following convex optimization problem: (37)OP2:minϵ,X,Y,Z,F*,L,S¯,W¯1,W¯2,W¯3(α+ρ1β+ρ2γ)s.t.{(i)inequality(32),(ii)[W¯1W¯20∗W¯3δbiF*∗∗2X−S¯]≥0,(iii)[−αIUT∗−X]<0,[−βIUT∗−2X+L]<0,[−γIUT∗−S¯]<0.


## 4. Illustrative Example

The dynamics are described by the following matrices:(38)A=[0.900.20.5],B=[0.20.10−0.1],D=[00.10.10],E=[0.100.1−0.1], and the design parameters are chosen as (39)Q1=[1001],Q2=[0.1000.1],U=[1001]. When τm=1,τM=2, and δ=1, by OP1 (25), the cost is obtained as J1=61.6653 and the
fault-tolerant controller can be designed for step (i): (40)[k1k2]=[−0.0812×10−5−0.1333×10−5−0.1865×10−5−0.3060×10−5]. In step (ii), by OP2 (37), the cost and the feedback gains are redesigned as (41)J2=39.0026,k1*=[−0.8776−0.2857],J3=49.9616,k2*=[−0.6494−0.4161]. As a result, the two controllers are determined as follows: (42)#1:[k1*k2]=[−0.8776−0.2857−0.1865×10−5−0.3060×10−5],#2:[k1k2*]=[−0.0812×10−5−0.1333×10−5−0.6494−0.4161]. Figure 2 reports the simulation for two failures scenarios of the actuators. For this simulation, the time-varying norm-bounded uncertain
matrix Δ(k) is taken as (43)Δ(k)=(sin(πk)00cos(πk)), which clearly satisfies the bound Δ(k)TΔ(k)≤I. The left column of Figure 2 is related to failures of actuator 1 and the right
column is related to failures of actuator 2. Note also that for these two
failures scenarios the system has been disturbed by step disturbances entering additively in the state equations. This is illustrated in the simulation where the step
disturbance 1 shown in Figure 2(a) disturbs the state variables at time instant 35
and disappears at time instant 40. The step disturbance 2 shown in Figure 2(b) disturbs the system at time instant 5 and
disappears at time instant 10. In Figure 2(c), the solid line represents the failure of actuator
1 which occurs at time instant 15 and disappears at time instant 35, occurs
again at time instant 55 and disappears at time instant 65. In Figure 2(d), the solid line represents the failure of actuator
2 which occurs at time instant 35 and disappears at time instant 45, occurs
again at time instant 65 and disappears at time instant 80. The delay of fault
detection is assumed to be 3 time steps, which is represented by dot-dashed
lines as shown in Figures 2(c) and 2(d). Under the above simulation setting, the state
responses are shown in Figures 2(e) and 2(f):Disturbance, actuator failures, and state
response.
(a)(b)(c)(d)(e)(f)(i)
the dotted line
represents the state response for controller-switching sequence N°1 : #2 is the initial
controller, and #1 is switched-on
at time instant 38, then #2 at time instant
48, #1 at time instant
68;(ii)
the solid line
represents the state response for controller-switching sequence N°2 : #1 is the initial
controller, and #2 is switched-on
at time instant 38, then #1 at time instant
48, #2 at time instant
68;(iii)
the dot-dashed
line represents the state response under the fault tolerant control (i.e.,
robust control) of step (i). The trace of
matrices (x∗)(x∗)T/(simulation time) is used as a
performance measure for comparison, where x∗ represents the
state trajectory in the different schemes and the simulation time is 80
seconds. After computation, we obtain for the above three control schemes the
traces Tr1=0.0279,Tr2=0.0338,Tr3=0.0499, which means that Tr1<Tr2<Tr3. We conclude that the proposed method for sequence N°1 is the best control scheme. We
also observe that for all possible switching sequences with controllers in the
designed controllers bank, the proposed active FTC is able to guarantee at
least the closed-loop stability of the overall system.


## 5. Conclusion

In this paper,
the stability guaranteed active fault tolerant control against actuators
failure in networked control system with time-varying but bounded delays has
been addressed. Plants with norm-bounded parameter uncertainty have been
considered, where the uncertainty may appear in the state matrix. Modelling
network-induced delays as communication delays between sensors and actuators,
linear memoryless state feedback fault-tolerant controllers have been developed
through LMI-based methods. A simulation example has been presented to show the
potentials of the proposed method for fault-tolerant control in networked
control systems.



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*Source: 189064-2008-03-06.xml*