Controllability Robustness of Linear Interval Systems with/without State Delay and with Unstructured Parametric Uncertainties

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Volume 2013, Article ID 346103,10pages http://dx.doi.org/10.1155/2013/346103

Research Article

Controllability Robustness of Linear Interval

Systems with/without State Delay and with Unstructured

Parametric Uncertainties

Shinn-Horng Chen

1

and Jyh-Horng Chou

1,2,3

1_{Department of Mechanical/Electrical Engineering, National Kaohsiung University of Applied Sciences, 415 Chien-Kung Road,}

Kaohsiung 807, Taiwan

2_{Institute of Electrical Engineering and Department of Mechanical and Automation Engineering,}

National Kaohsiung First University of Science and Technology, 1 University Road, Yanchao, Kaohsiung 824, Taiwan

3_{Department of Healthcare Administration and Medical Informatics, Kaohsiung Medical University, 100 Shi-Chuan 1st Road,}

Kaohsiung 807, Taiwan

Correspondence should be addressed to Jyh-Horng Chou; choujh@nkfust.edu.tw Received 29 April 2013; Accepted 18 June 2013

Academic Editor: Elena Braverman

Copyright © 2013 S.-H. Chen and J.-H. Chou. 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.

The robust controllability problem for the linear interval systems with/without state delay and with unstructured parametric uncertainties is studied in this paper. The rank preservation problem is converted to the nonsingularity analysis problem of the minors of the matrix under discussion. Based on some essential properties of matrix measures, two new sufficient algebraically elegant criteria for the robust controllability of linear interval systems with/without state delay and with unstructured parametric uncertainties are established. Two numerical examples are given to illustrate the applications of the proposed sufficient algebraic criteria, where one example is also presented to show that the proposed sufficient condition for the linear interval systems having no state delay and no unstructured parametric uncertainties can obtain less conservative results than the existing ones reported recently in the literature.

1. Introduction

It is well known that time delay effect may occur naturally because of the inherent characteristics of some system com-ponents or part of the control process [1,2]. In addition, the controllability is of particular importance in control theory and plays an important role in dynamic control systems [3,4]. Then, the controllability problem of continuous linear time delay systems has been studied by some researchers (see, e.g., [2, 5–15]). On the other hand, the problems of con-trolling objects whose models contain interval uncertainties arise from the control theory, differential games, operations research, and other areas of engineering and natural sciences [16]. However, the results reported in the literature [2,5–15] cannot be applied to solve the robust controllability problems of the linear interval systems with state delay.

For the time delay systems, there are two cases considered in the literature: (i) delay in state and (ii) delay in control input. The authors of this paper have studied the control-lability problem of the uncertain/interval system with delay

in control input [17–19], whereas the controllability problem

of the interval system with delay in state is considered in this paper. Here it should be noticed that the controllability problem of the continuous linear systems with both paramet-ric uncertainties and delay in state has been considered by Chen and Chou [20]. The same mathematical means as that used by Chen et al. [17,18] and Chen and Chou [19] is used in this paper, but the rationale, formulation, and concept of analyzing controllability for the delay in state case are very different from those for the delay in control input case. On the other hand, here it should be also noticed that, in the works of Chen et al. [18] and Chen and Chou [19], all the

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2 Abstract and Applied Analysis elements in the interval system matrix and in the interval

input matrices, respectively, are assumed to vary with both synchronous direction and same magnitude. So, the results of Chen et al. [18] and Chen and Chou [19] cannot be used to cover all matrices in the interval system.

Recently, the robustness issues of interval multiple-input-multiple-output (MIMO) systems without state delay have been studied by many researchers (see, e.g., [16, 17,21–31] and references therein). But, till now, only a few researchers studied the controllability issue of the interval MIMO systems without state delay [16,19–25,31]. The approaches proposed by Zhirabok [24] and Ashchepkov [16,25] need to consider the solvability of dynamic systems. Most notably, the methods proposed by Cheng and Zhang [21], Ahn et al. [22], Chen et al. [23] as well as Chen and Chou [19,20,31] give algebraically elegant derivations. However, the interval matrices consid-ered by Cheng and Zhang [21] must satisfy the sign-invariant condition, and all the interval matrices considered by Chen and Chou [31] must have the same variations.

On the other hand, it is well known that an approximate system model is always used in practice, and sometimes the approximation error should be covered by introducing both structured (elemental) and unstructured (norm-bounded) uncertainties in control system analysis and design [32]. That is, it is not unusual that at times we have to deal with a system simultaneously consisting of two parts: one part has only the structured parameter perturbations and the other part has the unstructured parameter uncertainties. Here it should be noticed that the system with structured uncertainties may be viewed as a special case of the interval system [33–35]. To the authors’ best knowledge, the robust controllability problem of linear interval systems with/without state delay and with unstructured parametric uncertainties has not been studied in the literature.

The purpose of this paper is to study the robust controlla-bility problem of linear interval MIMO systems with/without state delay and with unstructured parametric uncertainties. Based on some essential properties of matrix measures, two new sufficient algebraic criteria are proposed to guarantee the controllability robustness of linear interval MIMO systems with/without state delay and with unstructured parametric uncertainties. The proposed approach gives the algebraically elegant derivations. Two numerical examples are given in this paper to illustrate the applications of the proposed sufficient algebraic criteria. And, for the linear interval systems without both state delay and unstructured parametric uncertainties, the result is also given to compare with those results obtained from the existing methods reported in the literature.

2. Linear Interval Systems with Both State

Delay and Unstructured Uncertainties

Let𝐷 = {𝑑_𝑖𝑗} and 𝐷 = {𝑑_𝑖𝑗} be real 𝛼 × 𝛽 matrices satisfying 𝐷 ≤ 𝐷, that is, 𝑑_𝑖𝑗 ≤ 𝑑_𝑖𝑗, 𝑖 = 1, 2, . . . , 𝛼 and 𝑗 = 1, 2, . . . , 𝛽. The set of matrices[𝐷, 𝐷] = {𝐷; 𝐷 ≤ 𝐷 ≤ 𝐷} is called an interval matrix. Consider a linear interval MIMO system with both state delay and unstructured parametric uncertainties as the following form:

̇𝑥 (𝑡) = 𝐴𝑥 (𝑡) + ̃𝐴𝑥 (𝑡) + 𝐵𝑥 (𝑡 − 𝜏)

+ ̃𝐵𝑥 (𝑡 − 𝜏) + 𝐶𝑢 (𝑡) + ̃𝐶𝑢 (𝑡) , (1) where𝑥(𝑡) ∈ 𝑅𝑛is the system state vector,𝑢(𝑡) ∈ 𝑅𝑚is the control input vector,𝜏 > 0 denotes the time delay, 𝐴 ∈ [𝐴, 𝐴], 𝐵 ∈ [𝐵, 𝐵], and 𝐶 ∈ [𝐶, 𝐶] are, respectively, the 𝑛 × 𝑛, 𝑛 × 𝑛, and𝑛 × 𝑚 interval matrices, and the unstructured parametric matrices ̃𝐴, ̃𝐵, and ̃𝐶 are assumed to be bounded, that is,

󵄩󵄩󵄩󵄩

󵄩𝐴󵄩󵄩󵄩󵄩󵄩 ≤ 𝛽̃ 1, 󵄩󵄩󵄩󵄩_󵄩̃𝐵󵄩󵄩󵄩󵄩󵄩 ≤ 𝛽2, 󵄩󵄩󵄩󵄩_󵄩𝐶󵄩󵄩󵄩󵄩󵄩 ≤ 𝛽̃ 3, (2)

where𝛽₁,𝛽₂, and𝛽₃are nonnegative real constant numbers, and ‖ ⋅ ‖ denotes any matrix norm. Let ̂Β_𝑎 be the Banach space of real𝑛-vector-valued continuous functions defined on the interval[𝑡₀− 𝜏, 𝑡₀] with the uniform norm; that is, if Φ ∈ ̂Β𝑎, we have‖Φ‖ = max𝑡∈[𝑡0−𝜏,𝑡0]|Φ(𝑡)|. The initial

func-tion space is assumed to be ̂Β_𝑎, the space of continuous func-tions mapping[𝑡₀− 𝜏, 𝑡₀] into 𝑅𝑛, and the𝑅𝑚-valued control function 𝑢(𝑡) is measurable and bounded on every finite time interval [6]. The system in (1), called the linear inter-val MIMO system with both state delay and unstructured parametric uncertainties, is said to be controllable if each combination( ̂𝐴, ̂𝐵, ̂𝐶) is controllable, where ̂𝐴 = 𝐴 + ̃𝐴, ̂𝐵 = 𝐵 + ̃𝐵, ̂𝐶 = 𝐶 + ̃𝐶, 𝐴 ∈ [𝐴, 𝐴], 𝐵 ∈ [𝐵, 𝐵], and 𝐶 ∈ [𝐶, 𝐶].

For an interval matrix[𝐷, 𝐷] and for 𝑑_𝑖𝑗−𝑑_0𝑖𝑗≤ 𝜀_𝑖𝑗≤ 𝑑_𝑖𝑗− 𝑑_0𝑖𝑗, the 𝛼 × 𝛽 matrix 𝐷 = 𝐷₀+ ∑𝛼_𝑖=1∑𝛽_𝑗=1𝜀_𝑖𝑗𝐷_𝑖𝑗denotes that it varies between𝐷 and 𝐷, in which 𝐷 = [𝑑_𝑖𝑗] and 𝐷 = [𝑑_𝑖𝑗] are, respectively, the lower bound and upper bound matrices of interval matrix,𝐷_𝑖𝑗is an𝛼 × 𝛽 constant matrix with 1 in the ijth entry and 0 elsewhere, and𝐷₀ = [𝑑_0𝑖𝑗] ∈ [𝐷, 𝐷] is any given constant matrix. Then, the interval matrices[𝐴, 𝐴], [𝐵, 𝐵], and [𝐶, 𝐶] can be written as

[𝐴, 𝐴] = 𝐴0+ 𝑛 ∑ 𝑖=1 𝑛 ∑ 𝑗=1 𝜀𝑎𝑖𝑗𝐴𝑖𝑗, [𝐵, 𝐵] = 𝐵₀+∑𝑛 𝑖=1 𝑛 ∑ 𝑗=1 𝜀_𝑏𝑖𝑗𝐵_𝑖𝑗, [𝐶, 𝐶] = 𝐶₀+∑𝑛 𝑗=1 𝑚 ∑ 𝑘=1 𝜀_𝑐𝑗𝑘𝐶_𝑗𝑘, (3) where𝐴 = [𝑎_𝑖𝑗], 𝐴 = [𝑎_𝑖𝑗], 𝐵 = [𝑏_𝑖𝑗], 𝐵 = [𝑏_𝑖𝑗], 𝐶 = [𝑐_𝑖𝑗], 𝐶 = [𝑐𝑖𝑗], 𝐴𝑖𝑗, 𝐵𝑖𝑗, and 𝐶𝑗𝑘 are, respectively, 𝑛 × 𝑛, 𝑛 × 𝑛, and

𝑛 × 𝑚 constant matrices with 1 in the ijth or jkth entry and 0 elsewhere (for𝑖, 𝑗 = 1, 2, . . . , 𝑛 and 𝑘 = 1, 2, . . . , 𝑚), 𝑎_𝑖𝑗−𝑎_0𝑖𝑗≤ 𝜀_𝑎𝑖𝑗 ≤ 𝑎_𝑖𝑗− 𝑎_0𝑖𝑗,𝑏_𝑖𝑗− 𝑏_0𝑖𝑗 ≤ 𝜀_𝑏𝑖𝑗 ≤ 𝑏_𝑖𝑗− 𝑏_0𝑖𝑗, and𝑐_𝑗𝑘− 𝑐_0𝑗𝑘 ≤ 𝜀_𝑐𝑗𝑘≤ 𝑐_𝑗𝑘− 𝑐_0𝑗𝑘, and𝐴₀= [𝑎_0𝑖𝑗] ∈ [𝐴, 𝐴], 𝐵₀= [𝑏_0𝑖𝑗] ∈ [𝐵, 𝐵], and𝐶₀= [𝑐_0𝑗𝑘] ∈ [𝐶, 𝐶] are, respectively, any given 𝑛 × 𝑛, 𝑛 × 𝑛, and 𝑛 × 𝑚 constant matrices with that the combination (𝐴₀, 𝐵₀, 𝐶₀) is controllable.

Before we investigate the property of robust controllabil-ity for the linear interval system with both state delay and un-structured parametric uncertainties of (1), the following def-initions and lemmas need to be introduced first.

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(x)𝜀_𝑎21𝜙₂₁+ 𝜀_𝑎32𝜙₃₂+ 𝜀_𝑏31𝜑₃₁+ 𝜀_𝑐22𝜃₂₂+ 𝛽₁+ 𝛽₂+ 𝛽₃≤ 0.86400 < 1, for 𝜀_𝑎21∈ [−0.6, 0] , 𝜀_𝑎32∈ [0, 0.4] , 𝜀_𝑏31∈ [0, 1] , 𝜀_𝑐22 ∈ [−0.01, 0] ; (39j) (xi)𝜀_𝑎21𝜙₂₁+ 𝜀_𝑎32𝜙₃₂+ 𝜀_𝑏31𝜑₃₁+ 𝜀_𝑐22𝜃₂₂+ 𝛽₁+ 𝛽₂+ 𝛽₃≤ 0.75766 < 1, for 𝜀_𝑎21∈ [−0.6, 0] , 𝜀_𝑎32∈ [0, 0.4] , 𝜀_𝑏31∈ [−0.5, 0] , 𝜀_𝑐22∈ [−0.01, 0] ; (39k) (xii)𝜀_𝑎21𝜙₂₁+ 𝜀_𝑎32𝜙₃₂+ 𝜀_𝑏31𝜑₃₁+ 𝜀_𝑐22𝜃₂₂+ 𝛽₁+ 𝛽₂+ 𝛽₃≤ 0.73766 < 1, for 𝜀_𝑎21∈ [−0.6, 0] , 𝜀_𝑎32∈ [0, 0.4] , 𝜀_𝑏31 ∈ [−0.5, 0] , 𝜀_𝑐22∈ [0, 1.6] ; (39l) (xiii)𝜀_𝑎21𝜙₂₁+ 𝜀_𝑎32𝜙₃₂+ 𝜀_𝑏31𝜑₃₁+ 𝜀_𝑐22𝜃₂₂+ 𝛽₁+ 𝛽₂+ 𝛽₃≤ 0.77189 < 1, for 𝜀_𝑎21∈ [−0.6, 0] , 𝜀_𝑎32∈ [−0.4, 0] , 𝜀_𝑏31∈ [0, 1] , 𝜀_𝑐22∈ [0, 1.6] ; (39m) (xiv)𝜀_𝑎21𝜙₂₁+ 𝜀_𝑎32𝜙₃₂+ 𝜀_𝑏31𝜑₃₁+ 𝜀_𝑐22𝜃₂₂+ 𝛽₁+ 𝛽₂+ 𝛽₃≤ 0.79189 < 1, for 𝜀_𝑎21∈ [−0.6, 0] , 𝜀_𝑎32∈ [−0.4, 0] , 𝜀_𝑏31∈ [0, 1] , 𝜀_𝑐22∈ [−0.01, 0] ; (39n) (xv)𝜀_𝑎21𝜙₂₁+ 𝜀_𝑎32𝜙₃₂+ 𝜀_𝑏31𝜑₃₁+ 𝜀_𝑐22𝜃₂₂+ 𝛽₁+ 𝛽₂+ 𝛽₃≤ 0.68554 < 1, for 𝜀_𝑎21∈ [−0.6, 0] , 𝜀_𝑎32∈ [−0.4, 0] , 𝜀_𝑏31∈ [−0.5, 0] , 𝜀_𝑐22∈ [−0.01, 0] ; (39o) (xvi)𝜀_𝑎21𝜙₂₁+ 𝜀_𝑎32𝜙₃₂+ 𝜀_𝑏31𝜑₃₁+ 𝜀_𝑐22𝜃₂₂+ 𝛽₁+ 𝛽₂+ 𝛽₃≤ 0.66554 < 1, for 𝜀_𝑎21∈ [−0.6, 0] , 𝜀_𝑎32∈ [−0.4, 0] , 𝜀_𝑏31 ∈ [−0.5, 0] , 𝜀_𝑐22∈ [0, 1.6] . (39p) Hence, from the results obtained above, we can conclude that, for any𝜏 > 0, the linear interval system with both state delay and unstructured parametric uncertainties is robustly controllable in sense of Weiss [6].

5. Conclusions

The robust controllability problem for the linear interval MIMO system with/without state delay and with unstruc-tured parametric uncertainties has been investigated. The

rank preservation problem for robust controllability of the linear interval system with/without state delay and with unstructured parametric uncertainties is converted to the nonsingularity analysis problem. Based on some essential properties of matrix measures, two new sufficient algebra-ically elegant criteria for the robust controllability of linear in-terval MIMO systems with/without state delay and with un-structured parametric uncertainties have been established. Two numerical examples have been given to illustrate the applications of the proposed sufficient algebraic criteria. It has also been shown that the proposed sufficient criterion for linear interval systems having no state delay and no unstruc-tured parametric uncertainties can obtain less conservative results than the existing sufficient criteria given by Cheng and Zhang [21], Ahn et al. [22], Chen et al. [23], and Chen and Chou [19,20,31].

Acknowledgment

This work was in part supported by the National Science Council, Taiwan, under Grants nos. NSC101-2221-E151-031 and NSC101-2221-E-151-076.

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Journal of Hindawi Publishing Corporation

Function Spaces

Abstract and Applied Analysis Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014 International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporation http://www.hindawi.com Volume 2014

The Scientific

World Journal

Discrete Dynamics in Nature and Society

Discrete Mathematics

Journal of

Controllability Robustness of Linear Interval Systems with/without State Delay and with Unstructured Parametric Uncertainties

Research Article