# Robust Stabilization of Discrete-Time Systems with

Authors: Valter J. S. Leite; Márcio F. Miranda
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

Sufficient linear matrix inequality (LMI) conditions to verify the robust stability and to design robust
state feedback gains for the class of linear discrete-time systems with time-varying delay and polytopic uncertainties are presented. The conditions are obtained through parameter-dependent Lyapunov-Krasovskii functionals and use some extra variables, which yield less conservative LMI conditions. Both problems, robust stability analysis and robust synthesis, are formulated as convex problems where all system matrices
can be affected by uncertainty. Some numerical examples are presented to illustrate the advantages of the
proposed LMI conditions.


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## Body

## 1. Introduction

Time-delay systems have received a lot of
attention from both academics and industrial engineers in the last decades. This can be verified by the great number of
papers published in this area. See, for example, [1–5] and the references therein. It
is worthwhile to mention that time delays affect both continuous and
discrete-time systems and are presented in several systems like thermal
processes, communication systems, internet dataflow, biological systems,
regenerative chatter in metal cutting, and so on [2, 4]. The presence of delay
yields, in general, performance degradation and, eventually, leads the system
to instability.There are two main classes of robust stability
analysis that have been investigated, namely, delay-dependent and
delay-independent conditions. For a system whose stability does not depend on
the time-delay value, the analysis performed through delay-dependent conditions
can be very conservative. Also, delay-independent conditions cannot be obtained
as a limit case of delay-dependent ones just by imposing the maximum delay
valuedmax→+∞, leading to a gap between these two types of delay-stability
conditions [4, page 146].An important approach used in the last years to deal
with delay in the states of the systems is the use of Lyapunov-Krasovskii
functionals. This approach has been largely used to obtain convex conditions
mainly for continuous-time systems subject to retarded states and for neutral
systems [6]. On the
other hand, discrete-time systems with state delay (DTSSD) have received
little attention compared with its continuous-time counterpart. The main
reason for this is that for precisely known
discrete-time systems with constant time-delay, it is always possible to obtain
an augmented system without delayed states [7]. This approach, however, does not seem to be suitable
for time-varying delay, delay-independent stability
characterization, and for
robust-system stabilization. Despite the fact that the digital control has increased in
importance for practical robust control applications in the last decades, there
are few papers dealing with DTSSD.
Besides this, the majority of the results available in the literature for DTSSD
are formulated for norm-bounded uncertainty employing constant quadratic
Lyapunov-Krasovskii functionals, which may lead to conservative results
[8]. This approach,
called quadratic stability, is the base of several results available in the
literature as it can be seen in [9–12].Recent results on
DTSSD can be found in [13] where the real stability
radii of time-invariant state-delayed systems have been considered. In [14], there has been presented the
descriptor approach to obtain stability and design conditions for discrete
time-delay systems. See also [15] where some nonconvex conditions were proposed to the
synthesis problem. In the context of systems with Markovian jumps, see
[3, 16, 17] and references therein.
More recently, in [18, 19], robust
stabilizing conditions have been proposed, but most of the results have been
obtained through quadratic stability approach and the state feedback gain is
designed directly from the Lyapunov-Krasovskii matrices. In [20], a dynamic controller is
designed achieving a guaranteed cost control for norm-bounded DTSSD, but the design conditions are
nonlinear. In [21],
precisely known discrete-time systems with time-varying delays in the state are
investigated and convex conditions are given to design a memoryless control
law. The design conditions presented in [21] do not seem to be suitable to deal with polytopic
systems since the matrices of the system multiply the Lyapunov-Krasovskii
functional matrices. Observe that convex extensions of these results to treat
polytopic type of uncertainty in the context of robust stability analysis and
robust stabilization are not straightforward. Delay-independent conditions for
robust stabilization of DTSSD have been considered in [22], where a
parameter-dependent functional has been used, but only for
time-invariant   delays.In this paper, new results for robust stability
analysis as well as for robust stabilization of uncertain DTSSD
with time-varying delay are proposed as convex conditions depending on
the maximum and minimum values assumed by the delay,d¯ and d¯, respectively. A Lyapunov-Krasovskii
functional similar to that employed in [21] is used. Differently from other results in the literature,
both analysis and synthesis conditions are formulated as LMI feasibility
problems that can be solved efficiently in polynomial time by specialized
numerical algorithms. The main contribution of this paper is the improvement of
the recently published results on DTSSD with time-varying delay given in [21]. Furthermore, the scope of the results
of [21] is extended to
deal with polytopic-type uncertainty affecting DTSSD
and the possibility of design memory state feedback control gain.
Parameter-dependent Lyapunov-Krasovskii functional used conjoint some extra
matrices to achieve less conservative results. The approach proposed here does
not introduce any additional dynamics and leads to a product separation between
the matrices of the system and the matrices from the functional, allowing a
convex formulation for the synthesis problem.In Section2 some
definitions and the problem formulation are provided. Then, in Section 3, the main results are presented for both
robust-stability analysis and for the synthesis of state feedback gains
assuring the robust stability of the closed-loop system. The computational
complexity of the proposed conditions and decentralized control design are
discussed. In Section 4, some examples are given to illustrate the efficiency
of the proposed conditions. In Section 5 some
conclusions are presented.
Notation.
The notation used here is quite
standard.ℝ is the set of
real numbers and ℕ is the set of
natural numbers. In and 0 denote,
respectively, the n×n identity matrix
and the null matrix of appropriate dimensions. M>0 (M<0) means that
matrix M is positive
(negative) definite. M′ is the
transpose of M. The symbol ⋆ stands for
symmetric blocks in the LMIs.


## 2. Preliminaries and Problem Formulation

Consider the following discrete-time system with a
time-varying delay in the state:(2.1)xk+1=A˜(α)xk+A˜d(α)xk−d(k)+B˜(α)uk,xk=ϕ(k),k∈[−d¯,0],where k is the sampling
time, xk∈ℝn is the state
vector, uk∈ℝn is the input
control signal, d(k) is the
time-varying delay, where its time variation is bounded as (2.2)d¯≤d(k)≤d¯,(d¯,d¯)∈ℕ×ℕwith d¯ and d¯ being the
minimum and maximum delay values, respectively. (A˜(α),A˜d(α),B˜(α))≡(A˜,A˜d,B˜)(α)∈ℝn×2n+ℓ are unknown
constant matrices belonging to a polytope 𝒫˜,(2.3)𝒫˜≡{(A˜,A˜d,B˜)(α):(A˜,A˜d,B˜)(α)=∑i=1N(A˜,A˜d,B˜)iαi,α∈Ω},(2.4)Ω≡{α:α∈ℝN,∑i=1Nαi=1,αi≥0},where the vertices (A˜i,A˜di,B˜i)≡(A˜,A˜d,B˜)i are precisely
known. In special, observe that if d¯=d¯, then the delay is uncertain, belonging to [0,d¯], but it is time invariant. Note that if d(k)=0, then (2.1) is rewritten as xk+1=(A˜(α)+A˜d(α))xk+B˜(α)uk. Also define(2.5)β=d¯−d¯+1.In this paper, the following control
law is considered:(2.6)u(k)=Kx(k)+Kdx(k−d(k)),where (K,Kd)∈ℝℓ×2n are the robust
state feedback gains that assure the robust stability of the closed-loop
system, that is, the stability of (2.1)–(2.4) with (2.6) is assured for all α∈Ω. Therefore, this uncertain closed-loop system is
given by(2.7)xk+1=A(α)xk+Ad(α)xk−d(k)with
(2.8)A(α)≡A˜(α)+B˜(α)K,Ad(α)≡A˜d(α)+B˜(α)Kd,where (A,Ad)(α)∈𝒫 with(2.9)𝒫≡{(A,Ad)(α):(A,Ad)(α)=∑i=1N(A,Ad)iαi,α∈Ω}.It is worth to mention that if
the delay d(k) is not known at
each sample time k, then it is enough to make Kd=0 in (2.6). If d(k) is known at
each sample time k, then the possibility of using K and Kd may improve the
performance of the closed-loop system (2.7).The objective of this paper is to give convex
conditions solving the following problems.Problem 1.
Givend¯ and d¯ subject to (2.2),
determine if the uncertain DTSSD given in (2.7) is robustly stable.Problem 2.
Find a pair of gains(K,Kd) such that the system (2.1)–(2.4) controlled by (2.6) is robustly stable.Remark 2.1.
Generally speaking, in the cases where the time-delay depends on a physical parameter (such as
velocity of a transport belt, the stem position of a valve, etc.) it may be possible to determine the delay value
at each sample-time. As a special case, consider the regenerative chatter in metal cutting. In this process a
cylindrical workpiece has an angular velocityω while a machine tool (lathe) translates along the axis of this
workpiece. For details, see [2, page 2]. In this case the delay depends on the velocity ω and thus, if this angular
velocity can be measured, the delay could be determined at each instant. Note, however, that a detailed study on
physical application is not the focus of the paper.


## 3. Main Results

First, sufficient LMI conditions to solve Problem1 are given. The approach used here does not introduce any dynamics and leads to
a product separation between the matrices of the system and those from the
Lyapunov-Krasovskii functional. Then, these conditions are exploited to provide
some convex synthesis results. The following theorems provide some LMI
conditions depending on the values d¯ and d¯ to determine
the robust stability of (2.7) or to design robust state feedback gains K and Kd that assure the
robust closed-loop stability.


### 3.1. Robust Stability Analysis

Theorem 3.1.
System (2.7) subject to (2.2), (2.4),
and (2.9) is robustly stable if there exist symmetric matrices 0<P(α)∈ℝn×n and 0<Q(α)∈ℝn×n, such that one of the following equivalent conditions
is verified:
(a)(3.1)Γ(α)≡[A(α)′P(α)A(α)+βQ(α)−P(α)A(α)′P(α)Ad(α)⋆Ad(α)′P(α)Ad(α)−Q(α)]<0;
(b) there exist
parameter-dependent matricesF(α)∈ℝn×n, G(α)∈ℝn×n, and H(α)∈ℝn×n,  such that(3.2)ℳ(α)≡[P(α)+F(α)′+F(α)G(α)′−F(α)A(α)H(α)′−F(α)Ad(α)⋆ϑ22(α)−A(α)′H(α)′−G(α)Ad(α)⋆⋆ϑ33(α)]<0,
where ϑ22(α) denotes −A(α)′G(α)′−G(α)A(α)+βQ(α)−P(α) and ϑ33(α) denotes −(Q(α)+H(α)Ad(α)+Ad(α)′H(α)′) with β given by (2.5).
In this case, the functional
(3.3)V(α,k)=∑v=13Vv(α,k)>0,with
(3.4)V1(α,k)=xk′P(α)xk,V2(α,k)=∑j=k−d(k)k−1xj′Q(α)xj,V3(α,k)=∑ℓ=2−d¯1−d¯∑j=k+ℓ−1k−1xj′Q(α)xj,is such that
(3.5)ΔV(α,k)<0,∀[x(k)′x(k−d(k))′]′≠0and is called a
Lyapunov-Krasovskii functional, assuring the robust stability of (2.7).Proof.
The positivity of the functional
(3.3)) is assured with the hypothesis of P(α)=P(α)′>0, Q(α)=Q(α)′>0. For (3.3)) is a Lyapunov-Krasovskii functional,
besides its positivity, it is necessary to verify (3.5) for allα∈Ω. From hereafter, the α dependency is omitted
in the expressions Vv(k), v=1,…,3, for simplicity of the notation. To calculate (3.5),
consider(3.6)ΔV1(k)=xk+1'P(α)xk+1−xk'P(α)xk,(3.7)ΔV2(k)=xk'Q(α)xk−xk−d(k)'Q(α)xk−d(k)+∑i=k+1−d(k+1)k−1xi'Q(α)xi−∑i=k+1−d(k)k−1xi'Q(α)xi,(3.8)ΔV3(k)=(d¯−d¯)xk'Q(α)xk−∑i=k+1−d¯k−d¯xi'Q(α)xi.Observe that the third term in
(3.7) can be rewritten as(3.9)Ξk≡∑i=k+1−d(k+1)k−1xi′Q(α)xi=∑i=k+1−d¯k−1xi′Q(α)xi+∑i=k+1−d(k+1)k−d¯xi′Q(α)xi≤∑i=k+1−d(k)k−1xi′Q(α)xi+∑i=k+1−d¯k−d¯xi′Q(α)xi.Using (3.9) in (3.7), one
gets(3.10)ΔV2(k)≤xk′Q(α)xk−xk−d(k)′Q(α)xk−d(k)+∑i=k+1−d¯k−d¯xi′Q(α)xi.So, taking into account (3.6),
(3.8), and (3.10) the following upper bound for (3.5) can be
obtained:(3.11)ΔV(k)≤xk+1'P(α)xk+1+xk'[βQ(α)−P(α)]xk−xk−d(k)'Q(α)xk−d(k)<0.Replacing xk+1 in (3.11) by the
right-hand side of (2.7) one gets (3.1). The equivalence between (3.1) and (3.2) can
be established as follows. First, note that (3.1) can be rewritten as
(3.12)[A(α)′P(α)Ad(α)′P(α)]P(α)−1[P(α)A(α)P(α)Ad(α)]−[P(α)−βQ(α)00Q(α)]<0which by Schur complement is
equivalent to(3.13)[−P(α)P(α)A(α)P(α)Ad(α)⋆βQ(α)−P(α)0⋆⋆−Q(α)]<0.
Therefore, the equivalence
between (a) and (b) is the same as that between (3.2) and
(3.13). So, if (3.13) is verified, then (3.2) is true for F(α)=F(α)′=−P(α), G(α)=H(α)=0. On the other hand, if (3.2) is verified, then Γ(α)=T(α)′ℳ(α)T(α) with
(3.14)T(α)=[A(α)Ad(α)In00In]completing the proof.Note that (3.7) keeps a relation with Finsler's lemma
where the slack variables F(α) and G(α) depend on the
uncertain parameter α. The conditions presented in Theorem 3.1 are of
infinite dimension if α belongs to a
continuous domain. These conditions can be numerically treated by using
different approaches such as those presented in [23, 24], where it is possible to consider the products of
matrices depending on α by means of LMI
relaxations. In case of α belonging to a
discrete, countable, and finite domain, the conditions of Theorem 3.1 state a
finite set of LMIs defined at each value of α. In this paper, the structure of matrices P(α) and Q(α) is supposed to
be linear in α [25]:
(3.15)P(α)=∑i=1NαiPi,Q(α)=∑i=1NαiQiand α∈Ω. The extra matrices are chosen to be fixed F(α)=F, G(α)=G, and H(α)=H. Although other structures may lead to a less
conservative condition for Problem 1, no improvement is expected for Problem 2 as it deals with constant state feedback gains K and Kd. Also observe that, since the conditions of Theorem 3.1
depend on the size of delay variation, d¯−d¯, and not on the delay value itself, these conditions
are delay-independent conditions.
Theorem 3.2.
System (2.7) subject to (2.2), (2.4), and (2.9) is robustly
stable if there exist symmetric matrices 0<Pi∈ℝn×n and 0<Qi∈ℝn×n,i=1,…,N, such that
(3.16)ℳi≡[Pi+F′+FG′−FAiH′−FAdi⋆βQi−Pi−Ai′G′−GAi−Ai′H′−GAdi⋆⋆−(Qi+HAdi+Adi′H′)]<0,i=1,…,N,are verified with β given by (2.5).
Beside this, (3.3)) with (3.4) and (3.15) is a Lyapunov-Krasovskii functional
for (2.7).Proof.
Observe thatℳ(α) can be obtained
by multiplying (3.16) by αi and summing it
up, that is, ℳ(α)=∑i=1Nαiℳi, α∈Ω.It is worth to mention that the parameter-dependent
structure, imposed toP(α) and Q(α), (3.15), cannot be directly used in the conditions
presented in (3.1) and (3.13). This limitation is due to the products between the
system matrices and the Lyapunov-Krasovskii candidate matrices.The approach based on quadratic stability can be
recovered from (3.1), (3.2), (3.13) (or (3.16)). In this case, it is sufficient to
impose P(α)=P,Q(α)=Q(Pi=P,Qi=Q,i=1,…,N) and replace A(α) and Ad(α) by Ai and Adi, respectively. Then it is necessary to test those
conditions for i=1,…,N. Note that all the quadratic stability conditions
obtained as described above are equivalent and the difference between them is
just the computational burden necessary to solve each one.Observe that conditions presented in Theorems3.1 and 3.2 can be used to test the robust stability of both systems (A(α),Ad(α)) and (A(α)′,Ad(α)′), since their eigenvalues are the same.


### 3.2. Robust Feedback Design

The stability analysis conditions presented in Theorem3.2 can be used to obtain a convex condition for robust synthesis of the gains K
and
Kd such that the
control law (2.6) applied in (2.1) results in a robust stable closed-loop system,
and, therefore, resulting in a solution to Problem 2.Theorem 3.3.
If there exist symmetric matrices0<Pi∈ℝn×n,0<Qi∈ℝn×n,i=1,…,N, matrices F∈ℝn×n,W∈ℝn×ℓ, and Wd∈ℝn×ℓ, such that the following LMIs(3.17)[Pi+F+F′−(FA˜i′+WB˜i′)−(FA˜di′+WdB˜i′)⋆βQi−Pi0⋆⋆−Qi]<0,i=1,…,Nare verified with β given by (2.5),
then  system given by  (2.1)–(2.4)
is robustly stabilizable with (2.6), where the
static feedback gains are given by
(3.18)K=W′(F′)−1,Kd=Wd′(F′)−1yielding a convex solution to
Problem 2. Beside this, (3.3)–(3.4) is a
Lyapunov-Krasovskii functional  that
guarantees the robust stability of the resulting
closed-loop system with P(α) and Q(α) given in (3.15).Proof.
Condition (3.17) is obtained from (3.16)
by choosing G=H=0, replacing Ai and Adi by (A˜i+B˜iK)′ and (A˜di+B˜iKd)′, respectively, and making the change of variables FK′=W and FKd′=Wd.Note that conditions presented in Theorem3.3 encompass
quadratic stability, since it is always possible to choose Pi=P and Qi=Q,i=1,…,N. Also observe that if d(k) is not
available at each sample, and therefore x(k−d(k)) cannot be used
in the feedback, then it is enough to choose Wd=0 leading to a
control law given by u(k)=Kx(k). Another relevant note is that Theorem 3.3 presents
some LMI conditions for the synthesis of state feedback gains, differently from
other approaches found in the literature where nonconvex techniques are
employed.Decentralized Control
The results of Theorem3.3 can be used to deal with
decentralized control. This can be done by imposing a diagonal structure on F=FD=block-diag{F1,…,Fκ}, W=WD=block-diag{W1,…,Wκ}, and Wd=WdD=block-diag{Wd1,…,Wdκ}, κ being the
number of subsystems. In this case, robust block-diagonal
state feedback gains given by KD=WD′(FD′)−1 and KdD=WdD′(FD′)−1 can be obtained. Beside this, no structure is imposed to the matrices
of the Lyapunov-Krasovskii functional, P(α) and Q(α), which may lead to less conservative results.Computational Complexity
The computational complexity of the conditions
presented in this paper can be determined by the number of scalar variables,𝒦, and the number of rows, ℛ, involved in the optimization problems. Theorem 3.2 presents 𝒦T2=n[3n+N(n+1)] scalar
variables and ℛT2=5Nn LMI rows and in
Theorem 3.3 it is found that 𝒦T3=n[2ℓ+n+N(n+1)] and ℛT3=3Nn. In the case of using LMI control toolbox [26], the computational
complexity is 𝒪(𝒦3ℛ) and using the
solver SeDuMi [27] the
computational complexity is 𝒪(𝒦2ℛ2.5+ℛ3.5).


## 3.1. Robust Stability Analysis

Theorem 3.1.
System (2.7) subject to (2.2), (2.4),
and (2.9) is robustly stable if there exist symmetric matrices 0<P(α)∈ℝn×n and 0<Q(α)∈ℝn×n, such that one of the following equivalent conditions
is verified:
(a)(3.1)Γ(α)≡[A(α)′P(α)A(α)+βQ(α)−P(α)A(α)′P(α)Ad(α)⋆Ad(α)′P(α)Ad(α)−Q(α)]<0;
(b) there exist
parameter-dependent matricesF(α)∈ℝn×n, G(α)∈ℝn×n, and H(α)∈ℝn×n,  such that(3.2)ℳ(α)≡[P(α)+F(α)′+F(α)G(α)′−F(α)A(α)H(α)′−F(α)Ad(α)⋆ϑ22(α)−A(α)′H(α)′−G(α)Ad(α)⋆⋆ϑ33(α)]<0,
where ϑ22(α) denotes −A(α)′G(α)′−G(α)A(α)+βQ(α)−P(α) and ϑ33(α) denotes −(Q(α)+H(α)Ad(α)+Ad(α)′H(α)′) with β given by (2.5).
In this case, the functional
(3.3)V(α,k)=∑v=13Vv(α,k)>0,with
(3.4)V1(α,k)=xk′P(α)xk,V2(α,k)=∑j=k−d(k)k−1xj′Q(α)xj,V3(α,k)=∑ℓ=2−d¯1−d¯∑j=k+ℓ−1k−1xj′Q(α)xj,is such that
(3.5)ΔV(α,k)<0,∀[x(k)′x(k−d(k))′]′≠0and is called a
Lyapunov-Krasovskii functional, assuring the robust stability of (2.7).Proof.
The positivity of the functional
(3.3)) is assured with the hypothesis of P(α)=P(α)′>0, Q(α)=Q(α)′>0. For (3.3)) is a Lyapunov-Krasovskii functional,
besides its positivity, it is necessary to verify (3.5) for allα∈Ω. From hereafter, the α dependency is omitted
in the expressions Vv(k), v=1,…,3, for simplicity of the notation. To calculate (3.5),
consider(3.6)ΔV1(k)=xk+1'P(α)xk+1−xk'P(α)xk,(3.7)ΔV2(k)=xk'Q(α)xk−xk−d(k)'Q(α)xk−d(k)+∑i=k+1−d(k+1)k−1xi'Q(α)xi−∑i=k+1−d(k)k−1xi'Q(α)xi,(3.8)ΔV3(k)=(d¯−d¯)xk'Q(α)xk−∑i=k+1−d¯k−d¯xi'Q(α)xi.Observe that the third term in
(3.7) can be rewritten as(3.9)Ξk≡∑i=k+1−d(k+1)k−1xi′Q(α)xi=∑i=k+1−d¯k−1xi′Q(α)xi+∑i=k+1−d(k+1)k−d¯xi′Q(α)xi≤∑i=k+1−d(k)k−1xi′Q(α)xi+∑i=k+1−d¯k−d¯xi′Q(α)xi.Using (3.9) in (3.7), one
gets(3.10)ΔV2(k)≤xk′Q(α)xk−xk−d(k)′Q(α)xk−d(k)+∑i=k+1−d¯k−d¯xi′Q(α)xi.So, taking into account (3.6),
(3.8), and (3.10) the following upper bound for (3.5) can be
obtained:(3.11)ΔV(k)≤xk+1'P(α)xk+1+xk'[βQ(α)−P(α)]xk−xk−d(k)'Q(α)xk−d(k)<0.Replacing xk+1 in (3.11) by the
right-hand side of (2.7) one gets (3.1). The equivalence between (3.1) and (3.2) can
be established as follows. First, note that (3.1) can be rewritten as
(3.12)[A(α)′P(α)Ad(α)′P(α)]P(α)−1[P(α)A(α)P(α)Ad(α)]−[P(α)−βQ(α)00Q(α)]<0which by Schur complement is
equivalent to(3.13)[−P(α)P(α)A(α)P(α)Ad(α)⋆βQ(α)−P(α)0⋆⋆−Q(α)]<0.
Therefore, the equivalence
between (a) and (b) is the same as that between (3.2) and
(3.13). So, if (3.13) is verified, then (3.2) is true for F(α)=F(α)′=−P(α), G(α)=H(α)=0. On the other hand, if (3.2) is verified, then Γ(α)=T(α)′ℳ(α)T(α) with
(3.14)T(α)=[A(α)Ad(α)In00In]completing the proof.Note that (3.7) keeps a relation with Finsler's lemma
where the slack variables F(α) and G(α) depend on the
uncertain parameter α. The conditions presented in Theorem 3.1 are of
infinite dimension if α belongs to a
continuous domain. These conditions can be numerically treated by using
different approaches such as those presented in [23, 24], where it is possible to consider the products of
matrices depending on α by means of LMI
relaxations. In case of α belonging to a
discrete, countable, and finite domain, the conditions of Theorem 3.1 state a
finite set of LMIs defined at each value of α. In this paper, the structure of matrices P(α) and Q(α) is supposed to
be linear in α [25]:
(3.15)P(α)=∑i=1NαiPi,Q(α)=∑i=1NαiQiand α∈Ω. The extra matrices are chosen to be fixed F(α)=F, G(α)=G, and H(α)=H. Although other structures may lead to a less
conservative condition for Problem 1, no improvement is expected for Problem 2 as it deals with constant state feedback gains K and Kd. Also observe that, since the conditions of Theorem 3.1
depend on the size of delay variation, d¯−d¯, and not on the delay value itself, these conditions
are delay-independent conditions.
Theorem 3.2.
System (2.7) subject to (2.2), (2.4), and (2.9) is robustly
stable if there exist symmetric matrices 0<Pi∈ℝn×n and 0<Qi∈ℝn×n,i=1,…,N, such that
(3.16)ℳi≡[Pi+F′+FG′−FAiH′−FAdi⋆βQi−Pi−Ai′G′−GAi−Ai′H′−GAdi⋆⋆−(Qi+HAdi+Adi′H′)]<0,i=1,…,N,are verified with β given by (2.5).
Beside this, (3.3)) with (3.4) and (3.15) is a Lyapunov-Krasovskii functional
for (2.7).Proof.
Observe thatℳ(α) can be obtained
by multiplying (3.16) by αi and summing it
up, that is, ℳ(α)=∑i=1Nαiℳi, α∈Ω.It is worth to mention that the parameter-dependent
structure, imposed toP(α) and Q(α), (3.15), cannot be directly used in the conditions
presented in (3.1) and (3.13). This limitation is due to the products between the
system matrices and the Lyapunov-Krasovskii candidate matrices.The approach based on quadratic stability can be
recovered from (3.1), (3.2), (3.13) (or (3.16)). In this case, it is sufficient to
impose P(α)=P,Q(α)=Q(Pi=P,Qi=Q,i=1,…,N) and replace A(α) and Ad(α) by Ai and Adi, respectively. Then it is necessary to test those
conditions for i=1,…,N. Note that all the quadratic stability conditions
obtained as described above are equivalent and the difference between them is
just the computational burden necessary to solve each one.Observe that conditions presented in Theorems3.1 and 3.2 can be used to test the robust stability of both systems (A(α),Ad(α)) and (A(α)′,Ad(α)′), since their eigenvalues are the same.


## 3.2. Robust Feedback Design

The stability analysis conditions presented in Theorem3.2 can be used to obtain a convex condition for robust synthesis of the gains K
and
Kd such that the
control law (2.6) applied in (2.1) results in a robust stable closed-loop system,
and, therefore, resulting in a solution to Problem 2.Theorem 3.3.
If there exist symmetric matrices0<Pi∈ℝn×n,0<Qi∈ℝn×n,i=1,…,N, matrices F∈ℝn×n,W∈ℝn×ℓ, and Wd∈ℝn×ℓ, such that the following LMIs(3.17)[Pi+F+F′−(FA˜i′+WB˜i′)−(FA˜di′+WdB˜i′)⋆βQi−Pi0⋆⋆−Qi]<0,i=1,…,Nare verified with β given by (2.5),
then  system given by  (2.1)–(2.4)
is robustly stabilizable with (2.6), where the
static feedback gains are given by
(3.18)K=W′(F′)−1,Kd=Wd′(F′)−1yielding a convex solution to
Problem 2. Beside this, (3.3)–(3.4) is a
Lyapunov-Krasovskii functional  that
guarantees the robust stability of the resulting
closed-loop system with P(α) and Q(α) given in (3.15).Proof.
Condition (3.17) is obtained from (3.16)
by choosing G=H=0, replacing Ai and Adi by (A˜i+B˜iK)′ and (A˜di+B˜iKd)′, respectively, and making the change of variables FK′=W and FKd′=Wd.Note that conditions presented in Theorem3.3 encompass
quadratic stability, since it is always possible to choose Pi=P and Qi=Q,i=1,…,N. Also observe that if d(k) is not
available at each sample, and therefore x(k−d(k)) cannot be used
in the feedback, then it is enough to choose Wd=0 leading to a
control law given by u(k)=Kx(k). Another relevant note is that Theorem 3.3 presents
some LMI conditions for the synthesis of state feedback gains, differently from
other approaches found in the literature where nonconvex techniques are
employed.Decentralized Control
The results of Theorem3.3 can be used to deal with
decentralized control. This can be done by imposing a diagonal structure on F=FD=block-diag{F1,…,Fκ}, W=WD=block-diag{W1,…,Wκ}, and Wd=WdD=block-diag{Wd1,…,Wdκ}, κ being the
number of subsystems. In this case, robust block-diagonal
state feedback gains given by KD=WD′(FD′)−1 and KdD=WdD′(FD′)−1 can be obtained. Beside this, no structure is imposed to the matrices
of the Lyapunov-Krasovskii functional, P(α) and Q(α), which may lead to less conservative results.Computational Complexity
The computational complexity of the conditions
presented in this paper can be determined by the number of scalar variables,𝒦, and the number of rows, ℛ, involved in the optimization problems. Theorem 3.2 presents 𝒦T2=n[3n+N(n+1)] scalar
variables and ℛT2=5Nn LMI rows and in
Theorem 3.3 it is found that 𝒦T3=n[2ℓ+n+N(n+1)] and ℛT3=3Nn. In the case of using LMI control toolbox [26], the computational
complexity is 𝒪(𝒦3ℛ) and using the
solver SeDuMi [27] the
computational complexity is 𝒪(𝒦2ℛ2.5+ℛ3.5).


## 4. Numerical Examples

Example 4.1.
This example shows that the
condition presented by Theorem3.2 can be less conservative than other conditions
available in the literature. Consider the
DTSSD given by (2.7),
where(4.1)A1=[0.600.350.7],Ad1=[0.100.20.1].This system has been
investigated in [18]
and its stability has been assured for 2≤d(k)≤13 by using a
Lyapunov-Krasovskii with 5 terms instead
of the 3 terms presented
here (see (3.3))-(3.4)). Using Theorem 3.2 the same delay interval is obtained, but
with a lower-computational complexity. Using [21, Theorem 3.1] that employs a
Lyapunov-Krasovskii functional similar to that used here, it is possible to
verify the stability of this system but with a narrower delay interval:  2≤d(k)≤10.
Consider now that this system is affected by an
uncertain parameter such that it can be described by a polytope (2.3) with A1 and Ad1 given above and A2=1.1A1 and Ad2=1.1Ad1. In this case, the conditions of [18, 21] are no longer applicable,
since the system is uncertain. Using Theorem 3.2, it is possible to assure the
robust stability of this system for |d(k+1)−d(k)|≤3. Thus, a more flexible analysis condition is provided
by Theorem 3.2.Example 4.2.
Consider
the discrete-time system with delayed states which is described by (2.1), where the
system matrices are given by(4.2)A˜=[0.001.00−2.00−3.00],A˜d=[0.010.100.000.10],B˜=[0.001.00].This system has been
investigated for d¯=1≤d(k)≤10=d¯ in [21] where a state feedback gain K=[2.0005,2.9051] has been
obtained by means of application of its Theorem 4.1 that has 𝒦 [21] =19 scalar
variables and ℛ [21] =16 rows. On the
other hand, the conditions of Theorem 3.3 have 𝒦T3=12 scalar
variables and ℛT3=6 rows. Thus, by
using LMI control toolbox [26], the condition
proposed in [21] is
more complex than that in Theorem 3.3. Conditions of Theorem 3.3
yield(4.3)Q=[0.15580.00030.00030.0209],P=[2.3902−0.0026−0.00260.4315],F=[−1.54410.00470.0005−0.4866]and K=[2.0000,2.9929], which is close to the gain obtained by [21] with a lower computational
cost. Therefore, this example illustrates that the conditions proposed here are
numerically more efficient than those proposed in [21].Example 4.3.
Consider
the system investigated in Example4.2 where some uncertainties have been added
as follows:(4.4)A˜(ρ)=(1+ρ)A˜,A˜d(θ)=(1+θ)A˜d,B˜(η)=(1+η)B˜with |ρ|≤0.07, |θ|≤0.1, and |η|≤0.1. These parameters lead to a polytope with 8 vertices
determined by the combination of the extreme values of ρ, θ, and η. Keeping the same delay variation interval considered
in Example 4.2, that is, d¯=1 and d¯=10, the conditions of Theorem 3.3 are used to stabilize
this uncertain system resulting in
(4.5)[P1P2P3P4P5P6P7P8]=[3.6298−0.49312.8302−0.43743.6173−0.51932.9327−0.5202−0.49310.7591−0.43740.6360−0.51930.7868−0.52020.75442.6414−0.75873.5305−0.56152.7431−0.73743.5308−0.5874−0.75870.8439−0.56150.8096−0.73740.8481−0.58740.8380],[Q1Q2Q3Q4Q5Q6Q7Q8]=[0.1976−0.01710.1292−0.03060.2015−0.02120.1515−0.0385−0.01710.0538−0.03060.0326−0.02120.0550−0.03850.03960.03820.01380.2055−0.01940.0847−0.00130.2102−0.02410.01380.0207−0.01940.0527−0.00130.0237−0.02410.0543],F=[−2.41780.54760.5056−0.7569],K=[1.96702.7170].The behavior
of the states of
the closed-loop response of this uncertain discrete-time system with
time-varying delay is shown in Figure 1. It has been simulated that the time
response of this system at each vertex of the polytope was used in the controller
design. The respective control signals are shown in Figure 2. The initial
conditions have been chosen as x(k)=[1,−1], k=−d¯,…,0, and the value of the delay, d(k), has been varied randomly as shown in Figure 2(b). In Figure
1, the stability of the uncertain
closed-loop system is illustrated, assured by the state feedback
calculated by means of Theorem 3.3, for 1≤d(k)≤10. The control effort is shown in Figure 2(a).
It is worth to mention that the results presented in [21] cannot be directly used with
polytopic uncertain systems and cannot be applied in this case. This example shows
the efficacy of the conditions proposed here applied to uncertain discrete-time
delayed systems.The behavior of the statesx1(k) (a) and x2(k), with (b)  1≤d(k)≤10 (see Figure 2(b)), K=[109670,2.7170].
(a)(b)Control signal (a) and
time-varying delay (b).
(a)(b)Example 4.4.
In this
example, it is shown how conditions of Theorem3.3 can be used to design robust
decentralized state feedback gains. Consider the system (2.1) with 4 states and 2 vertex matrices
given by(4.6)[A˜1|A˜2]=[0.900.00−0.080.030.900.00−0.080.030.000.700.05−0.030.000.900.05−0.03−0.080.05−0.291.00−0.040.03−0.100.000.03−0.030.000.950.01−0.011.00−0.20],[A˜d1|A˜d2]=[−0.100.00−0.100.03−0.100.10−0.100.06−0.10−0.100.06−0.03−0.10−0.100.03−0.03−0.100.060.010.01−0.100.060.010.000.03−0.030.000.020.03−0.030.010.02],[B˜1|B˜2]=[1000000001001000001000110000000.10].
By imposing a block diagonal
structure on F and W given by {2,2} and choosing Wd=0, it is possible to apply the conditions of Theorem 3.3
getting(4.7)K=[K100K2]withK1=[0.4873−0.87851.6425−1.7437],K2=[0.2587−0.60890.22990.4833]for d¯=1 and d¯=4. Thus, this example shows how the proposed synthesis
conditions can be used to design robust decentralized state feedback control
gains without imposing any structure on Lyapunov-Krasovskii functional
matrices, yielding full matrices Pi and Qi,i=1,2 .Example 4.5.
This last
example is presented to illustrate how the proposed conditions can be used
within time-varying systems, that is, time-varying delay systems with matricesA˜, A˜d, and B˜ depending on α(t). Consider the uncertain discrete-time system with
time-varying delay described by a polytope of matrices with vertices given
by(4.8)[A˜1|A˜2]=[1.331.260.370.741.491.460.911.14],[A˜d1|A˜d2]=[0.200.060.16−0.060.010.14−0.010.06],[B˜1|B˜2]=[0.390.110.480.32].
The open-loop system is not
robustly stable as can be noticed by the eigenvalues of A1, given by 0.0233 and 2.7667, and the eigenvalues of A2, given by −0.1514 and 1.6614. The unforced system is simulated for an initial
state given by xk=[1,−1]′,k∈[−2,0] and a constant α1=0.4, that is, A˜(α)=0.4A˜1+0.6A˜2, A˜d(α)=0.4A˜d1+0.6A˜d2, and B˜(α)=0.4B˜1+0.6B˜2. The unstable behavior of the states is shown in
Figure 3.
Using Theorem3.3 with Wd=0 and Pi=P, Qi=Q, i=1,2, it is possible to obtain K=−[3.1599,3.4971] that assures
the quadratic stability of the system for d¯=0 and d¯=19. Therefore, A˜(k)=α1(k)A˜1+α2(k)A˜2, A˜d(k)=α1(k)A˜d1+α2(k)A˜d2, and B˜(k)=α1(k)B˜1+α2(k)B˜2, α(k)∈Ω. The closed-loop system behavior is illustrated in
Figures 4 and 5. The state behavior of the closed-loop system is shown in
Figure 4(a) and the control signal is presented in the bottom part. This
system has been simulated for α1(k)=0.5(sin(k)+1) and α2(k)=1−α1(k) as seen in
Figure 5(a), with initial conditions x(k)=[1,−1]′,k=−19,…,0. The time-varying delay d(k) has been
randomly generated as indicated in Figure 5(b).
Therefore, this example shows that
conditions of Theorem3.3 can be used in the context of time-varying systems
encompassing, in this case, quadratic stability conditions.Figure 3
The behavior of the statesx1(k) (×) and x2(k) (⋅) for the unforced system with  1≤d(k)≤2 and α1=0.4.The behavior of the statesx1(k) (×) and x2(k) (⋅) on (a) and the control signal on (b).
(a)(b)α1(k) (a) and time-varying delay, d(k) (b).
(a)(b)


## 5. Conclusions

Some sufficient
convex conditions were proposed to solve two problems: the robust stability
analysis and the synthesis of robust state feedback gains for the class of
polytopic discrete-time systems with time-varying delay. The presented LMI
conditions include some extra variables and no additional dynamic in the
investigated system, thus yielding less conservative results. Some examples,
with numerical simulation, are given to demonstrate some relevant
characteristics of the proposed design methodology such as robust stabilization
using memory or memoryless state feedback gains in the control law,
decentralized control, and design for time-varying discrete-time systems with
time-varying delay. Some of these examples have been compared with other
results available in the literature.



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