Robust Stabilization of Interval Plants with Uncertain Time-Delay Using the Value Set Concept

Abstract

This paper considers the robust stabilization problem for interval plants with parametric uncertainty and uncertain time-delay based on the value set characterization of closed-loop control systems and the zero exclusion principle. Using Kharitonov’s polynomials, it is possible to establish a sufficient condition to guarantee the robust stability property. This condition allows us to solve the control synthesis problem using conditions similar to those established in the loopshaping technique and to parameterize the controllers using stable polynomials constructed from classical orthogonal polynomials.

On Robust Stability for Hurwitz Polynomials via Recurrence Relations and Linear Combinations of Orthogonal Polynomials

“…In this situation, an interesting open problem is to analyze the capabilities of the resulting system, such as closed-loop frequency response, disturbance rejection, and sensitivity [34]. It is worth noticing that properties of orthogonal polynomials have been used recently to propose a stabilization method for interval plants with uncertain time-delay by using the set value concept [30]. (2) Although the proposed Hurwitz polynomials are robustly stable for the prescribed values of the Complexity parameters, an interesting open problem is to deduce the equations of motion of the roots in terms of the parameters (i.e., their behavior in terms of the involved parameters or, equivalently, the algebraic expressions for the curves they describe on the complex plane), as well as to determine the regions of the complex plane where the roots will be located for every value of the parameters.…”

“…As a consequence, the corresponding inner product, the associated orthogonal sequence, and the second kind polynomials will depend on t. at is, P n (x, t) 􏼈 􏼉 n≥0 will be orthogonal for every value of t, and thus the sequence f n (x, t) 􏼈 􏼉 n≥1 constructed by using eorem 1 will be Hurwitz for every value of t. In other words, if we consider t as an uncertain parameter, the sequence f n (x, t) 􏼈 􏼉 n≥1 will be robustly stable. It is worth mentioning that robustly stable families constructed in such a way have been already used in the design of a compensator that robustly stabilizes an interval plant with uncertain time-delay (see [30]) and can also be used for applications in control design.…”