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Stabilization of Discrete-Time Control Systems with Multiple State Delays
Advances in Difference Equations volume 2009, Article number: 240707 (2009)
Abstract
We give sufficient conditions for the exponential stabilizability of a class of perturbed time-varying difference equations with multiple delays and slowly varying coefficients. Under appropiate growth conditions on the perturbations, combined with the "freezing" technique, we establish explicit conditions for global feedback exponential stabilizability.
1. Introduction
Let us consider a discrete-time control system described by the following equationin:


where denotes the
dimensional space of complex column vectors,
is a given integer,
is the state,
(
) is the input,
is the set of nonnegative integers. Hence forward,
is the Euclidean norm;
and
are variable matrices of compatible dimensions,
is a variable
matrix such that

and is a given vector-valued function, that is,
The stabilizability question consists on finding a feedback control law for keeping the closed-loop system

asymptotically stable in the Lyapunov sense.
The stabilization of control systems is one of the most important properties of the systems and has been studied widely by many reseachers in control theory; (see, e.g., [1–11]) and the references therein. It is recognized that the Lyapunov function method serves as a main technique to reduce a given complicated system into a relatively simpler system and provides useful applications to control theory, but finding Lyapunov functions is still a difficult task (see, e.g., [1–3, 12, 13]). By contrast, many methods different from Lyapunov functions have been successfully applied to establish stabilizability results for discrete-time equations. For example, to the linear system

if the evolution operator generated by
is stable, then the delay control system (1.1)-(1.2) is asymptotically stabilizable under appropiate conditions on
(see [4, 8, 14]). For infinite-dimensional control systems, the study of stabilizabilization is more complicated and requires sophisticated techniques from semigroup theory.
The concept of stabilizability has been developed and successfully applied in different settings (see, e.g., [9, 15, 16]). For example, finite- and infinite-dimensional discrete-time control systems have been studied extensively (see, e.g., [2, 5, 6, 10, 17–20]).
The stabilizability conditions obtained in this paper are derived by using the "freezing" technique (see, e.g., [21–23]) for perturbed systems of difference equations with slowly varying coefficients and do not involve either Lyapunov functions or stability assumptions on the associated evolution operator . With more precision, the freezing technique can be described as follows. If
is any fixed integer, then we can think of the autonomous system

as a particular case of the system (1.1), with its time dependence "frozen" at time Thus, in this paper it is shown that if each frozen system is exponentially stabilizable and the rate of change of the coefficients of system (1.1) is small enough, then the nonautonomous system (1.1)-(1.2) is indeed exponentially stabilizable.
The purpose of this paper is to establish sufficient conditions for the global exponential feedback stabilizability of perturbed control systems with both time-varying and time-delayed states.
Our main contributions are as follows. By applying the "freezing" technique to the control system (1.1)-(1.2), we derive explicit stabilizability conditions, provided that the coefficients are slowly varying. Applications of the main results to control systems with many delays and nonlinear perturbations will also be established in this paper. This technique will allow us to avoid constructing the Lyapunov functions in some situations. For instance, it is worth noting that Niamsup and Phat [2] established sufficient stabilizability conditions for the zero solution of a discrete-time control system with many delays, under exponential growth assumptions on the corresponding transition matrix. By contrast, our approach does not involve any stability assumption on the transition matrix.
The paper is organized as follows. In Section 2 we introduce notations, definition, and some preliminary results. In Section 3, we give new sufficient conditions for the global exponential stabilizability of discrete-time systems with time-delayed states. Finally, as an application, we consider the global stabilization of the nonlinear control systems.
2. Preliminaries
In this paper we will use the following control law:

where is a variable
matrix.
To formulate our results, let us introduce the following notation. Let be a constant
matrix and let
denote the eigenvalues of
, including their multiplicities. Put

where is the Hilbert-Schmidt (Frobenius) norm of
; that is,
The following relation

is true, and will be useful to obtain some estimates in this work.
Theorem 2 A (11,Theorem 3.7).
For any -matrix
, the inequality

holds for every nonnegative integer , where
is the spectral radius of
, and
.
Remark 2.1.
In general, the problem of obtaining a precise estimate for the norm of matrix-valued and operator-valued functions has been regularly discussed in the literature, for example, see Gel'fond and Shilov [24] and Daleckii and Krein [25].
The following concepts of stability will be used in formulating the main results of the paper (see, e.g., [26]).
Definition 2.2.
The zero solution of system (1.4)–(1.2) is stable if for every and every
there is a number
(depending on
and
) such that every solution
of the system with
for all
, satisfies the condition

Definition 2.3.
The zero solution of (1.4) is globally exponentially stable if there are constants and
such that

for any solution of (1.4) with the initial conditions (1.2).
Definition 2.4.
The pair is said to be stabilizable for each
if there is a matrix
such that all the eigenvalues of the matrix
are located inside the unit disk for every fixed
Namely,

Remark 2.5.
The control is a feedback control of the system.
Definition 2.6.
System (1.1) is said to be globally exponentially stabilizable (at ) by means of the feedback law (2.1) if there is a variable matrix
such that the zero solution of (1.4) is globally exponentially stable.
3. Main Results
Now, we are ready to establish the main results of the paper, which will be valid for the system (1.1)-(1.2) with slowly varying coefficients.
Consider in the equation

subject to the initial conditions (1.2), where is a given integer and
is a variable
matrix.
Proposition 3.1.
Suppose that
-
(a)
-
(b)
(3.2)
-
(c)
Then the zero solution of system (3.1)–(1.2) is globally exponentially stable. Moreover, any solution of (3.1) satisfies the inequality

where

Proof.
Rewrite (3.1) in the form

with a fixed nonnegative integer . The variation of constants formula yields

Taking , we have

Hence,

Thus,

Hence,

From this inequality we obtain

But, the right-hand side of this inequality does not depend on . Thus, it follows that

This proves the global stability of the zero solution of (3.1)–(1.2).
To establish the global exponential stability of (3.1)–(1.2), we take the function

with small enough, where
is a solution of (3.1).
Substituting (3.13) in (3.1), we obtain

where

Applying the above reasoning to (3.14), according to inequality (3.3), it follows that is a bounded function. Consequently, relation (3.13) implies the global exponential stability of the zero solution of system (3.1)–(1.2).
Computing the quantities and
, defined by

is not an easy task. However, in this section we will improve the estimates to these formulae.
Proposition 3.2.
Assume that (a) and (b) hold, and in addition

where is the spectral radius of
for each
If

then the zero solution of system (3.1)–(1.2) is globally exponentially stable.
Proof.
Let us turn now to inequality (3.3). Firstly we will prove the inequality

where

Consider

By Theorem A, we have

But

Hence,

Proceeding in a similar way, we obtain

These relations yield inequality (3.19). Consequently,

where

Relation (3.26) proves the global stability of the zero solution of system (3.1)–(1.2). Establishing the exponential stability of this equation is enough to apply the same arguments of the Proposition 3.1.
Theorem 3.3.
Under the assumption (a), let be stabilizable for each fixed
with respect to a matrix function
satisfying the following conditions:
-
(i)
-
(ii)
and
-
(iii)
If,

then system (1.1)-(1.2) is globally exponentially stabilizable by means of the feedback law (2.1).
Proof.
Rewrite (1.4) in the form

where
According to (i), (ii), and (iii), the conditions (b) and (3.17) hold. Furthermore, condition (3.28) assures the existence of a matrix function such that condition (3.18) is fulfilled. Thus, from Proposition 3.2, the result follows.
Put

where the minimum is taken over all matrices
satisfying (i), (ii), and (iii).
Corollary 3.4.
Suppose that (a) holds, and the pair is stabilizable for each fixed
If

then the system (1.1)-(1.2) is globally exponentially stabilizable by means of the feedback law (2.1).
Now, consider in the discrete-time control system

subject to the same initial conditions (1.2), where and
are constant matrices. In addition, one assumes that the pair
is stabilizable, that is, there is a constant matrix
such that all the eigenvalues of
are located inside the unit disk. Hence,
In this case,
and
Thus,

Hence, Theorem 3.3 implies the following corollary.
Corollary 3.5.
Let be a stabilizable pair of constant matrices, with respect to a constant matrix
satisfying the condition

Then system (3.32)-(1.2), under condition (a), is globally exponentially stabilizable by means of the feedback law (2.1).
Example 3.6.
Consider the control system in :

where and
, subject to the initial conditions

where is a given function with values in
,
are positive scalar-valued bounded sequences with the property

and are positive scalar-valued sequences with

In the present case, the pair is controllable. Take

Then

where and
By inequality

it follows that

Assume that

Since and
are constants, by (3.37) we have
Hence, according to (3.28),

If and
are small enough such that for some
and
we have
then by Theorem 3.3, system (3.35)-(3.36), under conditions (3.37) and (3.38), is globally exponentially stabilizable.
In the same way, Theorem 3.3 can be extended to the discrete-time control system with multiple delays


where (
) are variable
matrices,
Denote

Theorem 3.7.
Let be stabilizable for each
with respect to a matrix function
satisfying the conditions (i), (ii), and (iii). In addition, assume that

If

then system (3.45)-(3.46) is globally exponentially stabilizable by means of the feedback law (2.1). Moreover, any solution of (3.45)-(3.46) satisfies the inequality

As an application, one consider, the stabilization of the nonlinear discrete-time control system


where   
is a given nonlinear function satisfying

for some positive numbers and
One recalls that nonlinear control system (3.51)-(3.52) is stabilizable by a feedback control where
is a matrix, if the closed-loop system

is asymptotically stable.
Theorem 3.8.
Under (3.53), let be stabilizable for each
with respect to a matrix function
satisfying conditions (i), (ii), and (iii). In addition, assume that

If

then system (3.51)-(3.52) is globally exponentially stabilizable by means of the feedback law (2.1).
Proof.
Rewrite (3.54) in the form

where
Thus, by reasoning as in Theorem 3.3, and using the estimates established in Proposition 3.2, the result follows.
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Acknowledgment
This research was supported by Fondecyt Chile under Grant no. 1.070.980.
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Rigoberto, M. Stabilization of Discrete-Time Control Systems with Multiple State Delays. Adv Differ Equ 2009, 240707 (2009). https://doi.org/10.1155/2009/240707
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DOI: https://doi.org/10.1155/2009/240707