Mathematical Problems in Engineering

Volume 2015, Article ID 217904, 8 pages

http://dx.doi.org/10.1155/2015/217904

## New Algorithm Based on Sign Decomposition to Verify the Robust Stability Property for a Class of Linear Time-Delay Systems

^{1}ECEN, Texas A&M University, College Station, TX 77840, USA^{2}UAT-UAM Reynosa Rodhe, 88779 Reynosa, TAMPS, Mexico^{3}FIME, Universidad Autónoma de Nuevo León, 66450 San Nicolás de los Garza, NL, Mexico

Received 22 December 2014; Accepted 16 April 2015

Academic Editor: Baocang Ding

Copyright © 2015 Iván D. Díaz-Rodríguez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The main focus of this paper is to analyze the robust stability property for a class of time-delay systems when parametric polynomic uncertainty is considered. The analysis is made by replacing the time-delay part with an auxiliary equation and then using the sign definite decomposition to deal with the polynomic parametric uncertainty. Also, it is shown that it is possible to verify the robust stability property by first obtaining the Hurwitz matrix from the characteristic equation for this class of systems and then checking the leading principal minors positivity using the sign definite decomposition. Finally, an algorithm codified in MATLAB is used to evaluate and graphically show the robust stability property. This is shown by a series of points that were calculated using the sign definite decomposition.

#### 1. Introduction

Time-delay systems arose as a result of inherent delays in system’s components and also due to the introduction of deliberated delay in the system for control purposes; see [1–3]. Over the years, time-delay systems interest and popularity have grown steadily. In particular, in the last 10 to 15 years there has been a surge in research and a proliferation of new techniques and results. Many of these new results include systems not only with time-delay analysis, but also with uncertainty in the system to be considered. For example, in [4–6] an analysis of robust stability for time-delay dynamical systems with parametric uncertainty in the mathematical model and in the time-delay is shown; in addition, the value set concept and the zero exclusion condition are used to verify the robust stability property of interval plants; see also [7, 8]. In [9, 10] the robust stabilization problem for a class of time-delay systems is considered where they involve parametric affine perturbations; in [11], new results to compute the time-delay of the hot-dip galvanizing control system are considered. In [12, 13] the authors present a robust model predictive control for systems represented by Takagi-Sugeno models and this technique was applied to the continuous stirred tank reactor (CSTR). They use Linear Matrix Inequalities (LMI) to solve the optimization problem. A technique based on a representation in the time domain of a class of* differential-difference* systems is presented in [14, 15]. Here, the authors make an application to the active suspension systems with actuator delay using the aforementioned technique. In [16, 17] new robust stability results for LTI systems with parametric uncertainty using sign definite decomposition were developed. In [18] the robust stability problem for a polynomial family was considered whose coefficients are polynomial functions of the parameters of interest. They used the sign definite decomposition for the controllers design. In [19] the robust stability positivity of a real function is considered while the real vector varies over a box. They determined Hurwitz robust stability for a polynomial family using the sign definite decomposition described in [16].

In 1981, the characteristic polynomial including a time-delay, for a linear differential-difference system, considered replacing the term by a regular polynomial . After this, it was possible to verify the asymptotic stability property for a class of time-delay systems; see [20]. Previously, a different approach was made in [21] by replacing by . But, it was found in [20] that the main problem with this substitution was that the two sets of image points were not identical for all and . In the present paper we outline a new algorithm to verify the robust stability property for a class of linear time-delay systems including a special case of polynomic parametric uncertainty, which one has not been considered in systems involving a time-delay. This is by using sign definite decomposition theory to verify the robust stability of the system using the stability conditions of Hurwitz matrix. We illustrate this using a numerical example.

This paper is organized as follows. In the preliminaries section, the Hurwitz stability criterion, a special polynomic parametric uncertainty case, and the sign definite decomposition are described. Then, the problem statement is presented. After that, the methodology and proposed algorithm are shown. An illustrative numerical example is presented to show the effectiveness of this approach. Finally, we discuss our results and future research.

#### 2. Main Contribution

As it may be seen from the section above, some of the previous results use techniques based on a representation in the time domain of* differential-difference* systems. Thus, to analyze and design them it is necessary to use the Lyapunov technique. It should be also mentioned that the uncertainty that they experiment should be represented by time functions. However, there are many applications where the uncertainty depends on variables other than time, such as resistors, capacitors, and inductors in an electrical circuit, which have parameters that are uncertain and whose uncertainty depends mainly on temperature and therefore could not be analyzed with these techniques. Also, in previous section some other results were mentioned that consider uncertainty structures like interval or linear affine and systems without delay. However, the main result of this paper is to obtain sufficient conditions to verify the robust stability property of a class of quasi-polynomials that represent the characteristic equation of* differential-difference* dynamics systems. It considers polynomic parametric uncertainty structure in the coefficients of quasi-polynomials and also interval uncertainty in the time-delay. First of all, a transformation of the delay’s operator is performed in order to get a two-variable polynomial; after this, to obtain the robust stability property, a result based on the Hurwitz matrix is applied, and then checking the leading principal minors positivity using the sign definite decomposition.

#### 3. Preliminaries

##### 3.1. Hurwitz Stability Criterion

Theorem 1 (Hurwitz stabilty). *Given a real polynomial , the polynomial is stable; that is, all its roots lie in the open left half plane (LHP) of the complex plane, if and only if, all of the leading principal minors, defined by , of the matrix are positive; see [22]:*

*3.2. Uncertainty*

*There exists a case where the precise value of the parameters of the mathematical model is unknown; however, its lower and upper bounds are known , respectively. The collection of all parameters involved in the mathematical model forms a vector of parameters which is an element of a parametric uncertainty box :For different lower and upper bound values, it is always possible to make a coordinate transformation of the physical parameters without losing their original properties. Such transformation can be , and in this case is taken in , where, for simplicity, we can name to the new coordinate . When we consider a parametric uncertainty, we have a polynomial family defined asThere exist a class of polynomials with parametric uncertainty . It is called polynomic uncertainty structure; that is, it has all of its coefficients where at least one parameter appears with power greater than one. For example, .*

*Definition 2 (see [23]). *Let be a positive convex cone in a vector space , for all , it is said that with respect to if , the interior of .

*From this point, we will consider and . This implies that .*

*Definition 3 (see [23]). * being a continuous function and a convex subset, it is said that is a nondecreasing function in , if implies .

*3.3. Sign Definite Decomposition*

*Definition 4 (see [16, 17]). * being a continuous function and a convex subset, it is said that has sign decomposition in if there exist two nondecreasing bounded functions , , such that for all . One will call those functions the positive and negative parts of the function:

*The negative and positive parts constitute a representation of the function in with a graphic representation in the plane according to Figure 1.*