Stabilization of Linear Sampled-Data Systems by a Time-Delay Feedback Control

We consider one-dimensional, time-invariant sampled-data linear systems with constant feedback gain, an arbitrary fixed time delay, which is amultiple of the sampling period and a zero-order hold for reconstructing the sampled signal of the system in the feedback control. We obtain sufficient conditions on the coefficients of the characteristic polynomial associated with the system. We get these conditions by finding both lower and upper bounds on the coefficients. These conditions let us give both an estimation of the maximum value of the sampling period and an interval on the controller gain that guarantees the stabilization of the system.


Introduction
The sampled-data systems are particular cases of a general type of systems called networked control systems, that have an important value in applications see Hespanha et  When in networked control systems it is satisfied that the plant outputs and the control inputs are delivered at the same time, then we obtain a sampled-data system.In this paper, we focus our attention on sampled-data systems.These systems have widely been studied due to their importance in engineering applications see Åstr öm and Wittenmark 11 , Chen and Francis 12 , Franklin et al. 13 , and Kolmanovskii and Myshkis 14 .

Mathematical Problems in Engineering
A sampled-data linear system with fixed time delay in the feedback is a continuous plant such that the feedback control of the closed loop system is discrete and has a delay r, namely, ẋ Ax t Bu k−r t , 1.1 Another idea is to propose a control depending on a parameter and then prove that the control stabilizes the system when is small enough.This idea was developed by Yong and Arapostathis 19 .Since the existence has been proved for these last authors, now we focus on estimating an interval for .In order to reduce the difficulty of the problem, we will restrict our study to the one-dimensional sampled-data systems.From 1.6 we obtain the following difference equation: By making the change of variable J k − N, the difference equation 1.8 becomes a homogeneous difference equation of order N 1, namely, This homogeneous difference equation of order N 1 can be rewritten as the following system of N 1 difference equations of order one.Indeed let

1.10
Using 1.9 , we obtain the following system of difference equations: which in matrix form becomes where To give stability conditions of the system of difference equations, we first obtain the characteristic polynomial of the matrix A: Thus the problem of stabilizing system 1.3 is equivalent to giving conditions on the coefficients of the characteristic polynomial 1.14 so that this polynomial is Schur stable.The problem of characterizing the stability region of 1.12 or equivalently 1.14 is considered an interesting problem 1 although it is known that it is very difficult 9 .Our objective in this paper is to find information about the stability region, which is explained below.System 1.1 has been studied, and necessary and sufficient conditions on A, B for the r-stabilization of the system have been obtained see Yong and Arapostathis 19 , but they are not easily verifiable.For the one-dimensional case 1.3 , their result is the following.
for a sufficiently small .However in a design problem we need to say how to find such an , or to obtain an estimation of the maximum sampling interval for which the stability is guaranteed, that is very important see Hespanha et al. 1 .In this paper, we find a 0 such that the polynomial That is, we get an estimation of the largest max with the property that the polynomial 1.15 is Schur stable for 0 < < max .Some general results about the stability for retarded differential equations with piecewise constant delays were obtained by Cooke and Wiener 21 .Problems 1.1 and 1.3 for continuous-time systems were studied by Yong 22,23 with an analogous approach.

Main result
Consider a polynomial P z a n z n a n−1 z n−1 a 0 such that −n/ n − 1 < a n−1 /a n < −1.Our objective is to give values of the coefficient a 0 such that P z is Schur stable.The result is the following.Choose a 0 −a n−1 a n − 1 , then P z is Schur stable if satisfies the inequality 0 < < 3n/ 2n − 1 3 n − 1 / 2n − 1 a n−1 /a n .We begin by establishing the result when the degree of P z is two in fact, we have here necessary and sufficient conditions .
Proof.P z is Schur stable if and only if its coefficients satisfy 24 the following:

2.3
To prove this last part, we define Then Since the coefficient of 2 is positive, g < 0 if and only if We have, it holds that.On the other hand, The arbitrary degree proof depends on the following lemma and several technical propositions that can be checked in the appendix.

Lemma 2.2. Fix an arbitrary integer
where Hence to prove the lemma, it is sufficient to show that A straightforward calculation shows that inequality 2.9 holds if and

6 Mathematical Problems in Engineering
We will split the analysis into the following two cases: We analyze 2.11 .By Proposition A.2, the first inequality in 2.11 is satisfied if and only if a n a n 1 2 .

2.13
Since 1 1/2 a n /a n 1 > 0, it follows that By straightforward calculations, and since > 0, it must satisfy a n a n 1 2 .

2.16
For the second inequality in 2.11 , we use Proposition A.3.So a 2 n 1 − a 2 0 a 0 a n > 0 if and only if

2.17
Since − n 1 /n < a n /a n 1 < −1, we have the following two inequalities:

2.18
Now, since we are interested in > 0, it must satisfy

2.19
By straightforward calculations, it follows that We now prove the main result for an arbitrary degree.
Theorem 2.3 fix an arbitrary integer n ≥ 2 .Let P z a n z n a n−1 z n−1 a 0 be a polynomial such that Proof.We make induction over n.The case n 2 is part of Theorem 2.1.Now suppose that the theorem holds for n ≥ 2, and let P z a n 1 z n 1 a n z n a 0 be a polynomial of degree n 1 such that

2.24
If we define the polynomials Q and R as in lemma, then replacing P and Q in the polynomial R, we obtain 2.26 By 2.24 , −a 0 a n a 2 n a n a n 1 − a n a n 1 or equivalently

2.28
Comparing this with 2.26 , we see that

2.29
Moreover by induction hypothesis must satisfy the condition

2.30
Substituting , A n−1 and A n into 2.30 , we obtain

2.31
The first inequality in 2.31 is equivalent to

2.32
And by Proposition A.5 this holds if and only if 0 < < 2 a n a n 1 .

2.33
Now we will analyze the second inequality in 2.31 which is equivalent to

2.34
By Proposition A.6, inequality 2.34 is obtained if and only if where c a n /a n 1 and H n c

2.39
We now analyze the right-hand side of 2.39 .Let

2.40
By Proposition A.9, it holds that F c is increasing and convex, F − n 1 /n 0 and F − n 1 /n 3n/ 2n 1 .
We now get the equation of the tangent line of the function F at the point c − n 1 /n.To do this, we use fact hat F − n 1 /n 0 and F − n 1 /n 3n/ 2n 1 .So that the equation of the tangent line passing through the point from which Theorem 2.3 follows.
Remark 2.4.Note that the inequality − N 1 /N < −A d < −1 implies that the number a in 1.3 must be positive since A d e ah with h > 0 and then: −e ah < −1 is satisfied if and only if a > 0.

Mathematical Problems in Engineering
The next corollary is a consequence of our results.
Corollary 2.5.Suppose that the system 1.3 has a proportional control 1.4 with delay r Nh and suppose that a, b > 0. If the sampling period and the gain of the controller satisfy

2.42
then the sampled-data system is stabilizable.

Example
We consider the sampled-data system where the values of the parameters are a 1, b 1, N 4, and r 4h.The difference equation 1.8 is ε k 1 e h ε k e h − 1 Kε k − 4 and the characteristic polynomial 1.12 associated with the system is P λ λ 5 − e h λ 4 e h − 1 which is Schur stable for 0 < < 5 − 4e h /3 by Theorem 2.1.Furthermore by Corollary 2.5 the maximum sampling period is h < ln 5/4 and the interval for the gain of the controller is

3.2
Now for h 0.22, the interval of the gain that guaranties the stabilization of the system is −1.02 < K < −1.

Appendix
In what follows we prove several inequalities.
Proof.Replacing the value of a 0 , we see that The roots of the equation h 0 are 1 2 a n /a n 1 and 2 a n /a n 1 .Since the coefficient of 2 is positive, then h < 0 if and only if a n /a n 1 < < 2 a n /a n 1 .But since a n /a n 1 < 0, we obtain 0 < < 2 a n /a n 1 .
Proof.Substituting a 0 into the first inequality, we see that a n a n 1 − a n a n 1 , then Since the coefficient of 2 is negative, Proof.Replacing a 0 into the inequality a 2 n 1 − a 2 0 a 0 a n > 0, we get That is, But this is true because the discriminant −4n 4 − 4n 3 15n 2 16n 4 of this quadratic function is negative for n ≥ 3 and the coefficient of c 2 is positive.
Therefore the first inequality is satisfied if and only if 0 < < 2 a n /a n 1 .
Proposition A.6.If a 0 −a n a n 1 − 1 , then where c a n /a n 1 and H n c Proof.The inequality Replacing a 0 into the last inequality, we see that this is equivalent to say that 0 < f , where and the coefficient of 2 is negative, it holds that for all n ≥ 1, and the result follows.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: al. 1 , Hespanha et al. 2 , Hikichi et al. 3 , Meng et al. 4 , Naghshtabrizi and Hespanha 5 Ögren et al. 6 , Seiler and Sengupta 7 and Shirmohammadi and Woo 8 .The networked control systems can be studied either from the approach of control theory or communication theory see Hespanha et al. 1 .Among the reported papers in control theory that have researched about networked control systems it is worth to mention the works by Zhang, Tipsuwan, and Hespanha see Zhang et al. 9 , Tipsuwan and Chow 10 , and Hespanha et al. 1 .

First
Round of ReviewsMarch 1, 2009 16notes the integer part of α, A is an n × n matrix, B ∈ R n , r ∈ R, and h is the interval between the successive sample instants t k and t k 1 .If h is a constant, it is called the sampling period and t k kh.Recommendable references about time-delay systems are the books by Hale and Verduyn Lunel 15 , and Kolmanovskii and Myshkis 14 .On the other hand, the theory about n-dimensional sampled-data control systems can be studied in the books by Åstr öm and Wittenmark 11 or Chen and Francis 12 .In relation with the study of sampled-data systems and the problem of proving the existence of a stabilizing control, it is worth to mention the work by Fridman et al.16, which is based on solving a linear matrix inequality.The application of this approach has been very successful in subsequent works seeFridman et al. 17 , and Mirkin 18 .
Note that not depending on 2.12 , we get that|A n | > |A 0 | if 2.21 is satisfied, so we can omit the analysis of 2.12 .
a 0 a n is a Schur stable polynomial if and only if P is Schur stable 25 .The inequality |a n 1 | > |a 0 | was proved in the lemma., A n−1 a n 1 a n and A 0 −a 0 a n and since the inequality |A n | > |A 0 | is satisfied which was proved in the lemma , then by induction hypothesis the polynomial a 1 2 c 2 .