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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.

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, [

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
value

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 [

Recent results on
DTSSD can be found in [

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,

In Section

The notation used here is quite
standard.

Consider the following discrete-time system with a
time-varying delay in the state:

In this paper, the following control
law is considered:

The objective of this paper is to give convex conditions solving the following problems.

Given

Find a pair of gains

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 velocity

First, sufficient LMI conditions to solve Problem

System (

(a)

(b) there exist
parameter-dependent matrices

The positivity of the functional
(

Note that (

System (

Observe that

It is worth to mention that the parameter-dependent
structure, imposed to

The approach based on quadratic stability can be
recovered from (

Observe that conditions presented in Theorems

The stability analysis conditions presented in Theorem

If there exist symmetric matrices

Condition (

Note that conditions presented in Theorem

The results of Theorem

The computational complexity of the conditions
presented in this paper can be determined by the number of scalar variables,

This example shows that the
condition presented by Theorem

Consider now that this system is affected by an
uncertain parameter such that it can be described by a polytope (

Consider
the discrete-time system with delayed states which is described by (

Consider
the system investigated in Example

The behavior of the states

Control signal (a) and time-varying delay (b).

In this
example, it is shown how conditions of Theorem

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 matrices

Using Theorem

Therefore, this example shows that
conditions of Theorem

The behavior of the states

The behavior of the states

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.

This paper has
been supported by the Brazilian agencies CNPq (485496/2006-2) and FAPEMIG
(TEC