Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 6142848, 12 pages

http://dx.doi.org/10.1155/2016/6142848

## Specific -Admissibility and Design Issues for Uncertain Descriptor Systems with Parametric Uncertainty in the Derivative Matrix

Department of Computer Science, University of Taipei, Taipei 100, Taiwan

Received 30 October 2015; Revised 8 January 2016; Accepted 10 January 2016

Academic Editor: Yan-Jun Liu

Copyright © 2016 Chih-Peng Huang. 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

Stability analysis issues and controller synthesis for descriptor systems with parametric uncertainty in the derivative matrix are discussed in this paper. The proposed descriptor system can extend the system’s modeling extent of physical and engineering systems from the traditional state-space model. First, based on the extended -stability definitions for the descriptor model, necessary and sufficient admissibility and -admissibility conditions for the unforced nominal descriptor system are derived and formulated by compact forms with strict linear matrix inequality (LMI) manner. In contrast, existing results need to involve nonstrict LMIs, which cannot be evaluated by current LMI solvers and need some extra treatments. Deducing from the obtained distinct results, the roust admissibility and -admissibility of the descriptor system with uncertainties in both the derivative matrix and the system’s matrices thus can be coped. Furthermore, by involving a proportional and derivative state feedback (PDSF) control law, we further address the controller design for the resulting closed-loop systems. Since all the proposed criteria are explicitly expressed in terms of the strict LMIs, we can use applicable LMI solvers for evaluating the feasible solutions. Finally, the efficiency and practicability of the proposed approach are demonstrated by two illustrative examples.

#### 1. Introduction

For capably integrating dynamic behaviors and algebraic dependent constraints into a single system’s model, descriptor systems have revealed more applicable fields and research topics than the conventional state-space models especially, such as electrical networks, power systems, robotic systems, chemical processes, and social economic system [1–4]. They also become a powerful technique, named by a descriptor system approach, to solve many concerned control problems via specific augmented forms [5, 6]. Descriptor systems also are named as generalized state-space systems, differential-algebraic systems, singular systems, implicit systems, or semistate systems. But, the arising stability issues are more intractable than the conventional state-space ones. Besides the fundamental stability, we need extra treatment to assure the regularity and impulse immunity simultaneously (see, e.g., [7–13], and the references therein).

Furthermore, considering the tolerable disturbance of realistic systems, mathematical modeling needs to embrace parameter’s uncertainties against rounding errors, varying operation circumstance, components’ aging, and so on. Some of them also can be transferred as nonlinear systems with dead zone input and the corresponding stability issues then can be efficiently treated by adaptive control strategy [14–16]. For considering descriptor systems with uncertainties, even if the nominal descriptor system is admissible, the stability, regularity, or impulse immunity can easily be destroyed by the arising uncontrollable uncertainties. Some works [8, 17–19] dealt with the robust admissibility and robust stabilization for the uncertain descriptor systems with a constant derivative matrix . But, for mathematical modeling of a physical system, if there exists parametric perturbation within its inner structures or behaviors, they should be meaningfully represented as parametric uncertainties directly into both the system matrix and the derivative matrix. Thus, few works [20, 21] have set about addressing the uncertain descriptor systems with the perturbed derivative matrices, where the presented uncertain derivative matrices had to meet some specific forms. Nevertheless, for considering the system’s performance, the system’s transient behaviors can be metricized by decay rate, damping, settle time, and so on. These characterized performance indexes mainly are dominated by the poles’ clustering location of the state-space model. Thus, for the regular state-space model, poles’ locations in various geometric regions, for example, shift half planes, vertical/horizontal strips, sectors, parabolic regions, and any intersection thereof, were deeply discussed in many works [22–28]. And the descriptor systems with specific poles’ location also have been discussed very recently [29–31]. However, the stability region analysis for the uncertain descriptor systems with the uncertain derivative matrix has drawn little attention in the past.

Furthermore, if the derivative of the state (or the output ) is accessible during the control process, a proportional and derivative state feedback (PDSF) (or proportional and derivative output feedback (PDOF)) controller can be equipped to stabilize the closed-loop descriptor systems [32–37]. But, some results are based on the matrix decomposition, and they cannot be extended to embrace uncertain parameters. To date, robust normalization and stabilization for uncertain descriptor systems were addressed [36]. According to an augmented form approach associated with a PDSF (or (PDOF)) controller, they presented some design criteria for the resulting closed-loop descriptor system with norm-bounded uncertainties in the derivative matrix and the system’s matrices simultaneously.

In this study, we investigate the robust -admissibility and the controller design for the descriptor system with the existing uncertainties in both the derivative matrix and the system matrices simultaneously. Compact admissibility and -admissibility conditions for the nominal descriptor system, which can be directly expressed by one strict linear matrix inequality (LMI), are first proposed. In contrast, some existing results involve additional nonstrict LMIs [38] or [12] in their results, where they cannot be directly evaluated by current LMI solvers and need some extra treatments [39, 40]. Based on the proposed explicit forms, we thus can treat the admissibility and -admissibility for the system with uncertainties in both the derivative matrix and the system matrix. In addition, a PDSF control law is involved to normalize and stabilize the closed-loop descriptor models. Since all the proposed criteria are explicitly expressed by strict LMIs’ forms, we can handily evaluate them via existing LMI tools [40, 41] for verification. The remainder content is organized as follows. The concerned problem and some preliminary results are described in Section 2. The robust admissibility and -admissibility issues are addressed in Section 3. In Section 4, an augmented form approach associated with the PDSF control is further investigated for the controller design. In Section 5, two illustrative examples are given to demonstrate the applicability and validity of the proposed method. Some conclusions are drawn in Section 6.

#### 2. Problem Formulation and Preliminaries

Consider an uncertain descriptor system described aswhere is the descriptor vector, is the control input, and the perturbed derivative matrix may be singular, that is, , and belongs to a polytopic set defined aswhere is given a priori. and stand for the nominal system matrices with appropriate dimensions and and represent the parameter uncertainties bounded bywhere , , and are constant matrices with compatible dimensions and satisfies

*Remark 1. *For system’s model (1), the presented uncertainties in the derivative matrix and the system matrices are formulated by difference forms. For mathematical modeling of a physical system, if there exists parametric perturbation within its inner structures or behaviors, the mainly parametric uncertainties usually can be cast into system matrices and the remainder few uncertainties thus are cast into the derivative matrix. So, we can reasonably formulate the system’s matrices with whole uncertainties by the norm bound in (3) and the derivative matrix with individual uncertainties by the polytopic form in (2).

*Definition 2 (see [12, 42]). *(i) The pair () is said to be regular if is not identically zero.

(ii) The pair () is said to be impulse free if .

(iii) The descriptor system is said to be admissible if it is regular and impulse free, and lie in the open left half plane, where denotes all roots of .

(iv) The descriptor system is said to be -admissible if it is regular and impulse free and lie in a specific stability region within the open left half plane.

Lemma 3. *The pair () is regular and impulse free if and only if the pair , , , with , is regular and impulse free.*

*Proof. *By replacing the variable of with in Definition 2, according to items (i) and (ii) the proof can be directly attained.

Lemma 4 (see [43]). *Given a symmetric matrix and matrices and of appropriate dimensions, thenfor all satisfying , if and only if there exists a scalar such that*

Denote a specific poles’ region in Figure 1. The complex plane is separated into two open-half planes and by the line that intersects the real axis at and the imaginary axis at and makes an angle with respect to the positive imaginary axis, where is defined to be positive with a counterclockwise sense and [24]. For the regular state-space system , a condition of poles’ location in the region is presented in advance.