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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Definition 1 (see [6]). The system ̇𝑥(𝑡) = 𝐿𝑥(𝑡) + 𝑀𝑥(𝑡 − 𝜏) + 𝑁𝑢(𝑡) with 𝑡 > 𝑡₀and any𝜏 > 0 is controllable to the origin from time𝑡₀if for eachΦ ∈ ̂Β_𝑎, there exists a finite time𝑡₁> 𝑡₀and an admissible input𝑢(𝑡) defined on [𝑡₀, 𝑡₁] such that 𝑥(𝑡₁, 𝑡₀, Φ, 𝑢) = 0, where 𝑥(𝑡₁, 𝑡₀, Φ, 𝑢) denotes a solution to ̇𝑥(𝑡) = 𝐿𝑥(𝑡) + 𝑀𝑥(𝑡 − 𝜏) + 𝑁𝑢(𝑡) at time 𝑡1corresponding

to initial time𝑡₀, initial functionΦ ∈ ̂Β_𝑎, and input𝑢(𝑡), in which L, M, and N are, respectively, the𝑛 × 𝑛, 𝑛 × 𝑛, and 𝑛 × 𝑚 matrices.

Definition 2 (see [36]). The measure of an 𝑛 × 𝑛 complex matrix𝑊 is defined as

𝜇 (𝑊) ≡ lim

𝜃 → 0

(󵄩󵄩󵄩󵄩󵄩𝐼 + 𝜃𝑊󵄩󵄩󵄩󵄩󵄩 − 1)

𝜃 , (4)

where‖ ⋅ ‖ is the induced matrix norm on the 𝑛 × 𝑛 complex matrix.

Lemma 3 (see [7,8]). If the system ̇𝑥(𝑡) = (𝐿+𝑀)𝑥(𝑡)+𝑁𝑢(𝑡)

with𝑡 > 𝑡₀is controllable, then the linear time delay system

̇𝑥(𝑡) = 𝐿𝑥(𝑡) + 𝑀𝑥(𝑡 − 𝜏) + 𝑁𝑢(𝑡) with 𝑡 > 𝑡0is controllable in sense of Weiss [6] for any𝜏 > 0.

Lemma 4. For any 𝜏 > 0, the linear time delay system ̇𝑥(𝑡) =

𝐿𝑥(𝑡) + 𝑀𝑥(𝑡 − 𝜏) + 𝑁𝑢(𝑡) with 𝑡 > 𝑡0is controllable in sense of Weiss [6] if the following𝑛2× 𝑛(𝑛 + 𝑚 − 1) controllability matrix 𝐸 = [ [ [ [ [ [ [ [ [ [ 𝐼_𝑛 0 ⋅ ⋅ ⋅ 0 0 ⋅ ⋅ ⋅ 0 𝑁 − (𝐿 + 𝑀) 𝐼𝑛 ⋅ ⋅ ⋅ 0 0 ⋅ ⋅ ⋅ 𝑁 0 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 0 0 ⋅ ⋅ ⋅ 𝐼_𝑛 0 ⋅ ⋅ ⋅ 0 0 0 0 ⋅ ⋅ ⋅ − (𝐿 + 𝑀) 𝑁 ⋅ ⋅ ⋅ 0 0 ] ] ] ] ] ] ] ] ] ] (5)

has rank𝑛2, where𝐿, 𝑀 ∈ 𝑅𝑛×𝑛,𝑁 ∈ 𝑅𝑛×𝑚, and𝐼_𝑛denotes the

𝑛 × 𝑛 identity matrix.

Proof. Following the same proof procedure as that given by

Chen and Chou [20], in the above matrix𝐸 of (5), add(𝐿+𝑀) times the first (block) row to the second, then add(𝐿 + 𝑀) times the second row to the third, and so on. The result is a matrix [ [ [ [ [ [ 𝐼𝑛 0 ⋅ ⋅ ⋅ 0 0 ⋅ ⋅ ⋅ 0 𝑁 0 𝐼𝑛 ⋅ ⋅ ⋅ 0 0 ⋅ ⋅ ⋅ 𝑁 (𝐿 + 𝑀) 𝑁 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 0 0 ⋅ ⋅ ⋅ 𝐼𝑛 0 ⋅ ⋅ ⋅ ⋅ (𝐿 + 𝑀)𝑛−2𝑁 0 0 ⋅ ⋅ ⋅ 0 𝑁 (𝐿 + 𝑀) 𝑁 ⋅ ⋅ ⋅ (𝐿 + 𝑀)𝑛−1𝑁 ] ] ] ] ] ] . (6)

The controllability matrix[𝑁 (𝐿+𝑀)𝑁 ⋅ ⋅ ⋅ (𝐿+𝑀)𝑛−1𝑁] is of rank𝑛 if and only if the matrix in (5) has rank𝑛2(i.e., the matrix in (5) has rank𝑛2). So, from Lemma 3, we can conclude that if the matrix in (5) has rank𝑛2, then, for any𝜏 > 0, the linear time delay system ̇𝑥(𝑡) = 𝐿𝑥(𝑡)+𝑀𝑥(𝑡−𝜏)+𝑁𝑢(𝑡) with𝑡 > 𝑡₀is controllable in sense of Weiss [6].

Remark 5. From Lemma3, we know that the robust con-trollability problem of linear system with state delay can be converted to the rank preservation problem of controllability matrix. Due to the parametric uncertainties being interval matrices and unstructured uncertainties, it is difficult to cal-culate their matrix exponentiation and product operations for checking the rank of controllability matrix for linear time delay systems. To solve this difficulty, we can apply Lemma4

to check the rank of controllability matrix in (5).

Lemma 6 (see [36]). The matrix measures of the matrices𝑊

and𝑉, namely, 𝜇(𝑊) and 𝜇(𝑉), respectively, are well defined for any norm and have the following properties:

(i)𝜇(±𝐼) = ±1, for the identity matrix 𝐼;

(ii)−‖𝑊‖ ≤ −𝜇(−𝑊) ≤ Re(𝜆(𝑊)) ≤ 𝜇(𝑊) ≤ ‖𝑊‖, for

any norm‖ ⋅ ‖ and any 𝑛 × 𝑛 complex matrix 𝑊;

(iii)𝜇(𝑊 + 𝑉) ≤ 𝜇(𝑊) + 𝜇(𝑉), for any two 𝑛 × 𝑛 complex

matrices𝑊 and 𝑉;

(iv)𝜇(𝛾𝑊) = 𝛾𝜇(𝑊), for any matrix 𝑊 ∈ 𝐶𝑛×𝑛and any nonnegative real number𝛾;

where𝜆(𝑊) denotes any eigenvalue of 𝑊 and Re(𝜆(𝑊)) de-notes the real part of𝜆(𝑊).

While the induced matrix norms are 1-norm, 2-norm, and ∞-norm, the corresponding matrix measures 𝜇_𝑘(⋅), where 𝑘 = 1, 2, ∞, can be easily calculated as

(i)𝜇₁(𝑊) = max_𝑗[Re(𝑤_𝑗𝑗) + ∑𝑛_{𝑖=1,𝑖 ̸= 𝑗}|𝑤_𝑖𝑗|]; (ii)𝜇₂(𝑊) = max_𝑖[𝜆_𝑖(𝑊 + 𝑊∗)/2];

(iii)𝜇_∞(𝑊) = max_𝑖[Re(𝑤_𝑖𝑖) + ∑𝑛_{𝑗=1,𝑗 ̸= 𝑖}|𝑤_𝑖𝑗|];

in which𝑤_𝑖𝑗is the𝑖𝑗th element of the matrix 𝑊 and 𝜆_𝑖(⋅) denotes the𝑖th eigenvalue.

Lemma 7. For any 𝛾 < 0 and any 𝑛 × 𝑛 complex matrix 𝑊,

𝜇(𝛾𝑊) = −𝛾𝜇(−𝑊).

Proof. From the property (iv) in Lemma8, this lemma can be immediately obtained.

Lemma 8 (see [37]). Let𝑁 ∈ 𝐶𝑛×𝑛. If𝜇(−𝑁) < 1, then det(𝐼+

𝑁) ̸= 0.

From Lemma 4, it is known that, for any𝜏 > 0, the interval system with both state delay and unstructured par-ametric uncertainties in (1) is robustly controllable on[0, 𝑇] in sense of Weiss [6] if the𝑛2× 𝑛(𝑛 + 𝑚 − 1) matrix 𝐸 has full row rank𝑛2, where

𝐸 = 𝐸₀+∑𝑛 𝑖=1 𝑛 ∑ 𝑗=1 𝜀_𝑎𝑖𝑗𝐹_𝑖𝑗+∑𝑛 𝑖=1 𝑛 ∑ 𝑗=1 𝜀_𝑏𝑖𝑗𝐺_𝑖𝑗 +∑𝑛 𝑗=1 𝑚 ∑ 𝑘=1 𝜀𝑐𝑗𝑘𝐻𝑗𝑘+ ̃𝐹 + ̃𝐺 + ̃𝐻, (7)

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10 Abstract and Applied Analysis

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