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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.

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 [

In 1981, the characteristic polynomial including a time-delay, for a linear differential-difference system, considered replacing the term

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.

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

Given a real polynomial

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

Let

From this point, we will consider

The negative and positive parts

Rectangle containing the function

It will be called minimum and maximum euclidean vertex

Since the negative

In order to define the positivity or negativity of a function using the

The determinant of the matrix

Let

The main interest of this research is to analyze the robust stability property of difference-differential dynamical systems which are characterized by polynomic parametric uncertainty and time-delay of the form:

The robust stability property is determined by the analysis of the characteristic equation (

A polynomial

The roots of this associated polynomial have an important relation with the roots of the quasi-polynomial. This relation is presented in the following theorem.

Suppose that

With this transformation we can get the relation between

The Hurwitz matrix is

The Hurwitz matrix being (

According to the

Let

Let

Let

Let

Let

Let

In this subsection we describe the steps to follow to analyze the robust stability property of the system including time-delay and polynomic parametric uncertainty. The algorithm is the following.

Consider the time-delay system described in (

Determine the characteristic equation

Define the Hurwitz matrix

For each element in the Hurwitz matrix, perform a separation in positive and negative part. This is given in (

With respect to the

Determine the points

If the points

If the points

If we have at least one point

If the points

Consider a first order system:

Sign definite decomposition of the Hurwitz matrix for the example.

We can see in Figure

If we analyze a small variation in the time-delay with values of

Sign definite decomposition of the determinant of the Hurwitz matrix for the example with a little variation in the time-delay.

In this research it was shown that the robust stability property of linear dynamical systems, which have polynomic uncertain parameters and time-delay, can be verified by the application of an algorithm based on the method of sign definite decomposition. The positivity of the determinant of the Hurwitz matrix is verified by checking the positivity of all leading principal minors of the matrix in terms of

The authors declare that there is no conflict of interests regarding the publication of this paper.

The first author wants to thank Consejo Nacional de Ciencia and Tecnologia (CONACYT) for the support to study abroad by the scholarship no. 308935. Also, the second author wants to thank to the Autonomous University of Tamaulipas and the Fondo Mixto de Fomento a la Investigacion Cientifica and Tecnologica-Gobierno del Estado de Tamaulipas for the partial financial support granted through the Project Agreement 185932.