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

Abstract

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 components 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 controlling 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 controllability 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 parametric 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 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 considered 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 controllability 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

Let 𝐷={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: 𝑥̇(𝑡)(𝑡)+(𝑡)()+()(𝑡)+(𝑡),=𝐴𝑥𝐴𝑥+𝐵𝑥𝑡−𝜏𝐵𝑥𝑡−𝜏+𝐶𝑢𝐶𝑢 (1) where 𝑥(𝑡)∈𝑅^𝑛 is the system state vector, 𝑢(𝑡)∈𝑅^𝑚 is 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, and 𝑛×𝑚 interval matrices, and the unstructured parametric matrices 𝐴, 𝐵, and 𝐶 are assumed to be bounded, that is, 𝐴≤𝛽₁,𝐵≤𝛽₂,𝐶≤𝛽₃, (2) where 𝛽₁, 𝛽₂, and 𝛽₃ are nonnegative real constant numbers, and ‖⋅‖ denotes any matrix norm. Let Β_𝑎 be the Banach space of real 𝑛-vector-valued continuous functions defined on the interval [𝑡₀−𝜏,𝑡₀] with the uniform norm; that is, if Φ∈Β_𝑎, we have ‖Φ‖=max_{𝑡∈[𝑡0−𝜏,𝑡0]}|Φ(𝑡)|. The initial function space is assumed to be Β_𝑎, the space of continuous functions mapping [𝑡₀−𝜏,𝑡₀] into 𝑅^𝑛, and the 𝑅^𝑚-valued control function 𝑢(𝑡) is measurable and bounded on every finite time interval [6]. The system in (1), called the linear interval 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_𝑖𝑗 denotes that it varies between 𝐷 and 𝐷, in which 𝐷=[𝑑_𝑖𝑗] and 𝐷=[𝑑_𝑖𝑗] are, respectively, the lower bound and upper bound matrices of interval matrix, 𝐷_𝑖𝑗 is an 𝛼×𝛽 constant matrix with 1 in the 𝑖𝑗th entry and 0 elsewhere, and 𝐷₀=[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+^𝑛𝑛_𝑖=1𝑗=1𝜀_𝑎𝑖𝑗𝐴_𝑖𝑗,𝐵,𝐵=𝐵₀+^𝑛𝑛_𝑖=1𝑗=1𝜀_𝑏𝑖𝑗𝐵_𝑖𝑗,𝐶,𝐶=𝐶₀+^𝑛𝑚_𝑗=1𝑘=1𝜀_𝑐𝑗𝑘𝐶_𝑗𝑘, (3) where 𝐴=[𝑎_𝑖𝑗], 𝐴=[𝑎_𝑖𝑗], 𝐵=[𝑏_𝑖𝑗], 𝐵=[𝑏_𝑖𝑗], 𝐶=[𝑐_𝑖𝑗], 𝐶=[𝑐_𝑖𝑗], 𝐴_𝑖𝑗, 𝐵_𝑖𝑗, and 𝐶_𝑗𝑘 are, respectively, 𝑛×𝑛, 𝑛×𝑛, and 𝑛×𝑚 constant matrices with 1 in the 𝑖𝑗th or 𝑗𝑘th entry and 0 elsewhere (for 𝑖,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, and 𝑐_𝑗𝑘−𝑐_0𝑗𝑘≤𝜀_𝑐𝑗𝑘≤𝑐_𝑗𝑘−𝑐_0𝑗𝑘, and 𝐴₀=[𝑎_0𝑖𝑗]∈[𝐴,𝐴], 𝐵₀=[𝑏_0𝑖𝑗]∈[𝐵,𝐵], and 𝐶₀=[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 controllability for the linear interval system with both state delay and unstructured parametric uncertainties of (1), the following definitions and lemmas need to be introduced first.

Definition 1 (see [6]). The system 𝑥̇(𝑡)=𝐿𝑥(𝑡)+𝑀𝑥(𝑡−𝜏)+𝑁𝑢(𝑡) with 𝑡>𝑡₀ and any 𝜏>0 is controllable to the origin from time 𝑡₀ if for each Φ∈Β_𝑎, there exists a finite time 𝑡₁>𝑡₀ and an admissible input 𝑢(𝑡) defined on [𝑡₀,𝑡₁] such that 𝑥(𝑡₁,𝑡₀,Φ,𝑢)=0, where 𝑥(𝑡₁,𝑡₀,Φ,𝑢) denotes a solution to 𝑥̇(𝑡)=𝐿𝑥(𝑡)+𝑀𝑥(𝑡−𝜏)+𝑁𝑢(𝑡) at time 𝑡₁ corresponding to initial time 𝑡₀, initial function Φ∈Β_𝑎, and input 𝑢(𝑡), in which 𝐿, 𝑀, and 𝑁 are, respectively, the 𝑛×𝑛, 𝑛×𝑛, and 𝑛×𝑚 matrices.

Definition 2 (see [36]). The measure of an 𝑛×𝑛 complex matrix 𝑊 is defined as 𝜇𝑊≡lim_𝜃→0𝐼+𝜃𝑊−1𝜃, (4) where ‖⋅‖ is the induced matrix norm on the 𝑛×𝑛 complex matrix.

Lemma 3 (see [7, 8]). If the system 𝑥̇(𝑡)=(𝐿+𝑀)𝑥(𝑡)+𝑁𝑢(𝑡) with 𝑡>𝑡₀ is controllable, then the linear time delay system 𝑥̇(𝑡)=𝐿𝑥(𝑡)+𝑀𝑥(𝑡−𝜏)+𝑁𝑢(𝑡) with 𝑡>𝑡₀ is controllable in sense of Weiss [6] for any 𝜏>0.

Lemma 4. For any 𝜏>0, the linear time delay system 𝑥̇(𝑡)=𝐿𝑥(𝑡)+𝑀𝑥(𝑡−𝜏)+𝑁𝑢(𝑡) with 𝑡>𝑡₀ is controllable in sense of Weiss [6] if the following 𝑛²×𝑛(𝑛+𝑚−1) controllability matrix 𝐼𝐸=^𝑛−()𝐼0⋅⋅⋅00⋅⋅⋅0𝑁𝐿+𝑀_𝑛⋅⋅⋅00⋅⋅⋅𝑁0⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅00⋅⋅⋅𝐼_𝑛()0⋅⋅⋅0000⋅⋅⋅−𝐿+𝑀𝑁⋅⋅⋅00 (5) has rank𝑛², where 𝐿,𝑀∈𝑅^𝑛×𝑛, 𝑁∈𝑅^𝑛×𝑚, and 𝐼_𝑛 denotes the 𝑛×𝑛 identity matrix.

Proof. Following the same proof procedure as that given by Chen and Chou [20], in the above matrix 𝐸 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 𝐼_𝑛0⋅⋅⋅00⋅⋅⋅0𝑁0𝐼_𝑛⋅⋅⋅00⋅⋅⋅𝑁(𝐿+𝑀)𝑁⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅00⋅⋅⋅𝐼_𝑛0⋅⋅⋅⋅(𝐿+𝑀)^𝑛−2𝑁00⋅⋅⋅0𝑁(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 any 𝜏>0, the linear time delay system 𝑥̇(𝑡)=𝐿𝑥(𝑡)+𝑀𝑥(𝑡−𝜏)+𝑁𝑢(𝑡) with 𝑡>𝑡₀ is controllable in sense of Weiss [6].

Remark 5. From Lemma 3, we know that the robust controllability 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 calculate 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 Lemma 4 to check the rank of controllability matrix in (5).

Lemma 6 (see [36]). The matrix measures of the matrices 𝑊 and 𝑉, 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 matrices 𝑊 and 𝑉;(iv)𝜇(𝛾𝑊)=𝛾𝜇(𝑊), for any matrix 𝑊∈𝐶^𝑛×𝑛 and any nonnegative real number 𝛾;
where 𝜆(𝑊) denotes any eigenvalue of 𝑊 and Re(𝜆(𝑊)) denotes the real part of 𝜆(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 which 𝑤_𝑖𝑗 is the 𝑖𝑗th element of the matrix 𝑊 and 𝜆_𝑖(⋅) denotes the 𝑖th eigenvalue.

Lemma 7. For any 𝛾<0 and any 𝑛×𝑛 complex matrix 𝑊, 𝜇(𝛾𝑊)=−𝛾𝜇(−𝑊).

Proof. From the property (iv) in Lemma 8, this lemma can be immediately obtained.

Lemma 8 (see [37]). Let 𝑁∈𝐶^𝑛×𝑛. If 𝜇(−𝑁)<1, then det(𝐼+𝑁)≠0.

From Lemma 4, it is known that, for any 𝜏>0, the interval system with both state delay and unstructured parametric uncertainties in (1) is robustly controllable on [0,𝑇] in sense of Weiss [6] if the 𝑛²×𝑛(𝑛+𝑚−1) matrix 𝐸 has full row rank𝑛², where 𝐸=𝐸₀+^𝑛𝑛_𝑖=1𝑗=1𝜀_𝑎𝑖𝑗𝐹_𝑖𝑗+^𝑛𝑛_𝑖=1𝑗=1𝜀_𝑏𝑖𝑗𝐺_𝑖𝑗+^𝑛𝑚_𝑗=1𝑘=1𝜀_𝑐𝑗𝑘𝐻_𝑗𝑘+𝐹+𝐺+𝐻, (7) for 𝑎_𝑖𝑗−𝑎_0𝑖𝑗≤𝜀_𝑎𝑖𝑗≤𝑎_𝑖𝑗−𝑎_0𝑖𝑗, 𝑏_𝑖𝑗−𝑏_0𝑖𝑗≤𝜀_𝑏𝑖𝑗≤𝑏_𝑖𝑗−𝑏_0𝑖𝑗, 𝑐_𝑗𝑘−𝑐_0𝑗𝑘≤𝜀_𝑐𝑗𝑘≤𝑐_𝑗𝑘−𝑐_0𝑗𝑘, 𝑖,𝑗=1,2,…,𝑛, and 𝑘=1,2,…,𝑚, in which 𝐸₀=𝐼_𝑛0⋅⋅⋅00⋅⋅⋅0𝐶₀−𝐴₀+𝐵₀𝐼_𝑛⋅⋅⋅00⋅⋅⋅𝐶₀0⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅00⋅⋅⋅𝐼_𝑛𝐴0⋅⋅⋅0000⋅⋅⋅−₀+𝐵₀𝐶₀𝐹⋅⋅⋅00_𝑖𝑗,=00⋅⋅⋅00⋅⋅⋅00−𝐴_𝑖𝑗0⋅⋅⋅00⋅⋅⋅00⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅00⋅⋅⋅00⋅⋅⋅0000⋅⋅⋅−𝐴_𝑖𝑗𝐺0⋅⋅⋅00_𝑖𝑗,=00⋅⋅⋅00⋅⋅⋅00−𝐵_𝑖𝑗0⋅⋅⋅00⋅⋅⋅00⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅00⋅⋅⋅00⋅⋅⋅0000⋅⋅⋅−𝐵_𝑖𝑗𝐻0⋅⋅⋅00_𝑗𝑘,=00⋅⋅⋅00⋅⋅⋅0𝐶_𝑗𝑘00⋅⋅⋅00⋅⋅⋅𝐶_𝑗𝑘0⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅00⋅⋅⋅00⋅⋅⋅0000⋅⋅⋅0𝐶_𝑗𝑘−−𝐶⋅⋅⋅00,𝐹=00⋅⋅⋅00⋅⋅⋅00𝐴0⋅⋅⋅00⋅⋅⋅00⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅00⋅⋅⋅00⋅⋅⋅0000⋅⋅⋅−𝐴0⋅⋅⋅00,𝐺=00⋅⋅⋅00⋅⋅⋅00𝐵0⋅⋅⋅00⋅⋅⋅00⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅00⋅⋅⋅00⋅⋅⋅0000⋅⋅⋅−𝐵0⋅⋅⋅00,𝐻=00⋅⋅⋅00⋅⋅⋅000⋅⋅⋅00⋅⋅⋅𝐶0⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅00⋅⋅⋅00⋅⋅⋅0000⋅⋅⋅0𝐶⋅⋅⋅00. (8)(9)(10)(11)(12)(13)(14)

Let the singular value decomposition of 𝐸₀, which has rank𝑛² due to that the given combination (𝐴₀,𝐵₀,𝐶₀) is controllable, be 𝐸₀=𝑈𝑆0_{𝑛2×𝑛(𝑚−1)}𝑉^𝐻, (15) where 𝑈∈𝑅^𝑛2×𝑛2 and 𝑉∈𝑅^{𝑛(𝑛+𝑚−1)×𝑛(𝑛+𝑚−1)} are the unitary matrices, 𝑉^𝐻 denotes the complex-conjugate transpose of matrix 𝑉,  𝑆=diag[𝜎₁,…,𝜎_𝑛2], and 𝜎₁≥𝜎₂≥⋯≥𝜎_𝑛2>0 are the singular values of 𝐸₀.

In what follows, we present a sufficient criterion for ensuring that, for any 𝜏>0, the linear interval system with both state delay and unstructured parametric uncertainties of (1) is robustly controllable on [0,𝑇] in sense of Weiss [6].

Theorem 9. For any 𝜏>0, the linear interval MIMO system with both state delay and unstructured parametric uncertainties of (1) is robustly controllable in sense of Weiss [6] if the matrix 𝐸₀ in (8) has a full row rank, and if the following conditions simultaneously hold: ^𝑛𝑛_𝑖=1𝑗=1𝜀_𝑎𝑖𝑗𝜙_𝑖𝑗+^𝑛𝑛_𝑖=1𝑗=1𝜀_𝑏𝑖𝑗𝜑_𝑖𝑗+^𝑛𝑚_𝑗=1𝑘=1𝜀_𝑐𝑗𝑘𝜃_𝑗𝑘+𝛽₁+𝛽₂+𝛽₃<1, (16) where𝜙_𝑖𝑗=𝜇−𝑆⁻¹𝑈^𝐻𝐹_𝑖𝑗𝑉[𝐼_𝑛2,0_{𝑛2×𝑛(𝑚−1)}]^𝑇,𝑓𝑜𝑟𝜀_𝑎𝑖𝑗𝑆≥0,−𝜇⁻¹𝑈^𝐻𝐹_𝑖𝑗𝑉𝐼_𝑛2,0_{𝑛2×𝑛(𝑚−1)}^𝑇,𝑓𝑜𝑟𝜀_𝑎𝑖𝑗𝜑<0,_𝑖𝑗=𝜇−𝑆⁻¹𝑈^𝐻𝐺_𝑖𝑗𝑉𝐼_𝑛2,0_{𝑛2×𝑛(𝑚−1)}^𝑇,𝑓𝑜𝑟𝜀_𝑏𝑖𝑗𝑆≥0,−𝜇⁻¹𝑈^𝐻𝐺_𝑖𝑗𝑉𝐼_𝑛2,0_{𝑛2×𝑛(𝑚−1)}^𝑇,𝑓𝑜𝑟𝜀_𝑏𝑖𝑗𝜃<0,_𝑗𝑘=𝜇−𝑆⁻¹𝑈^𝐻𝐻_𝑗𝑘𝑉𝐼_𝑛2,0_{𝑛2×𝑛(𝑚−1)}^𝑇,𝑓𝑜𝑟𝜀_𝑐𝑗𝑘𝑆≥0,−𝜇⁻¹𝑈^𝐻𝐻_𝑗𝑘𝑉𝐼_𝑛2,0_{𝑛2×𝑛(𝑚−1)}^𝑇,𝑓𝑜𝑟𝜀_𝑐𝑗𝑘<0,𝛽₁=𝛽₁𝑆⁻¹𝑈^𝐻𝑉𝐼_𝑛2,0_{𝑛2×𝑛(𝑚−1)}^𝑇,𝛽₂=𝛽₂𝑆⁻¹𝑈^𝐻𝑉𝐼_𝑛2,0_{𝑛2×𝑛(𝑚−1)}^𝑇,𝛽₃=𝛽₃𝑆⁻¹𝑈^𝐻𝑉𝐼_𝑛2,0_{𝑛2×𝑛(𝑚−1)}^𝑇, (17a)(17b)(17c)(17d)(17e)(17f)𝑖,𝑗=1,2,…,𝑛 and 𝑘=1,2,…,𝑚, in which the matrices 𝐹_𝑖𝑗, 𝐺_𝑖𝑗, 𝐻_𝑗𝑘, 𝑆, 𝑈, and 𝑉 are, respectively, defined in (9)–(11) and (15) and 𝐼_𝑛2 denotes the 𝑛²×𝑛² identity matrix.

Proof. If the matrix 𝐸 in (7) has full row rank, then the linear interval system with both state delay and unstructured parametric uncertainties of (1) is robustly controllable. Since the matrix 𝐸₀ in (8) has a full row rank due to that the given combination (𝐴₀,𝐵₀,𝐶₀) is controllable, and since we know that rank𝐸₀=rank𝑆⁻¹𝑈^𝐻𝐸₀𝑉, (18) thus, instead of rank(𝐸₀+∑^𝑛_𝑖=1∑^𝑛_𝑗=1𝜀_𝑎𝑖𝑗𝐹_𝑖𝑗+∑^𝑛_𝑖=1∑^𝑛_𝑗=1𝜀_𝑏𝑖𝑗𝐺_𝑖𝑗+∑^𝑛_𝑗=1∑^𝑚_𝑘=1𝜀_𝑐𝑗𝑘𝐻_𝑗𝑘+𝐹+𝐺+𝐻), we can discuss the rank of 𝐼_𝑛20_{𝑛2×𝑛(𝑚−1)}+^𝑛𝑛_𝑖=1𝑗=1𝜀_𝑎𝑖𝑗𝑆⁻¹𝑈^𝐻𝐹_𝑖𝑗𝑉+^𝑛𝑛_𝑖=1𝑗=1𝜀_𝑏𝑖𝑗𝑆⁻¹𝑈^𝐻𝐺_𝑖𝑗𝑉+^𝑛𝑚_𝑗=1𝑘=1𝜀_𝑐𝑗𝑘𝑆⁻¹𝑈^𝐻𝐻_𝑗𝑘𝑉+𝑆⁻¹𝑈^𝐻𝐹𝑉+𝑆⁻¹𝑈^𝐻𝐺𝑉+𝑆⁻¹𝑈^𝐻𝐻𝑉. (19) Since a matrix has at least rank𝑛² if it has at least one nonsingular 𝑛²×𝑛² submatrix, a sufficient condition for the matrix in (19) to have rank𝑛² is the nonsingularity of 𝐿=𝐼_𝑛2+^𝑛𝑛_𝑖=1𝑗=1𝜀_𝑎𝑖𝑗𝐹_𝑖𝑗+^𝑛𝑛_𝑖=1𝑗=1𝜀_𝑏𝑖𝑗𝐺_𝑖𝑗+^𝑛𝑚_𝑗=1𝑘=1𝜀_𝑐𝑗𝑘𝐻_𝑗𝑘+𝐹+𝐺+𝐻, (20) where 𝐹_𝑖𝑗=𝑆⁻¹𝑈^𝐻𝐹_𝑖𝑗𝑉[𝐼_𝑛2,0_{𝑛2×𝑛(𝑚−1)}]^𝑇, 𝐺_𝑖𝑗=𝑆⁻¹𝑈^𝐻𝐺_𝑖𝑗𝑉[𝐼_𝑛2,0_{𝑛2×𝑛(𝑚−1)}]^𝑇, 𝐻_𝑗𝑘=𝑆⁻¹𝑈^𝐻𝐻_𝑗𝑘𝑉[𝐼_𝑛2,0_{𝑛2×𝑛(𝑚−1)}]^𝑇, 𝐹=𝑆⁻¹𝑈^𝐻𝐹𝑉[𝐼_𝑛2,0_{𝑛2×𝑛(𝑚−1)}]^𝑇, 𝐺=𝑆⁻¹𝑈^𝐻𝐺𝑉[𝐼_𝑛2,0_{𝑛2×𝑛(𝑚−1)}]^𝑇, and 𝐻=𝑆⁻¹𝑈^𝐻𝐻𝑉[𝐼_𝑛2,0_{𝑛2×𝑛(𝑚−1)}]^𝑇, for 𝑖,𝑗=1,2,…,𝑛 and 𝑘=1,2,…,𝑚.
Using the properties in Lemmas 6 and 7, and from (2) and (16), we have 𝜇−^𝑛𝑛_𝑖=1𝑗=1𝜀_𝑎𝑖𝑗𝐹_𝑖𝑗−^𝑛𝑛_𝑖=1𝑗=1𝜀_𝑏𝑖𝑗𝐺_𝑖𝑗−^𝑛𝑚_𝑗=1𝑘=1𝜀_𝑐𝑗𝑘𝐻_𝑗𝑘−𝐹−𝐺−𝐻≤𝜇−^𝑛𝑛_𝑖=1𝑗=1𝜀_𝑎𝑖𝑗𝐹_𝑖𝑗+𝜇−^𝑛𝑛_𝑖=1𝑗=1𝜀_𝑏𝑖𝑗𝐺_𝑖𝑗+𝜇−^𝑛𝑚_𝑗=1𝑘=1𝜀_𝑐𝑗𝑘𝐻_𝑗𝑘+𝜇−𝐹+𝜇−𝐺+𝜇−𝐻≤𝜇−^𝑛𝑛_𝑖=1𝑗=1𝜀_𝑎𝑖𝑗𝐹_𝑖𝑗+𝜇−^𝑛𝑛_𝑖=1𝑗=1𝜀_𝑏𝑖𝑗𝐺_𝑖𝑗+𝜇−^𝑛𝑚_𝑗=1𝑘=1𝜀_𝑐𝑗𝑘𝐻_𝑗𝑘+𝐹+𝐺+𝐻≤^𝑛𝑛_𝑖=1𝑗=1𝜇−𝜀_𝑎𝑖𝑗𝐹_𝑖𝑗+^𝑛𝑛_𝑖=1𝑗=1𝜇−𝜀_𝑏𝑖𝑗𝐺_𝑖𝑗+^𝑛𝑚_𝑗=1𝑘=1𝜇−𝜀_𝑐𝑗𝑘𝐻_𝑗𝑘+𝐹+𝐺+𝐻≤^𝑛𝑛_𝑖=1𝑗=1𝜀_𝑎𝑖𝑗𝜙_𝑖𝑗+^𝑛𝑛_𝑖=1𝑗=1𝜀_𝑏𝑖𝑗𝜑_𝑖𝑗+^𝑛𝑚_𝑗=1𝑘=1𝜀_𝑐𝑗𝑘𝜃_𝑗𝑘+𝛽₁+𝛽₂+𝛽₃<1. (21) Thus, from Lemma 8, we have det𝐿=det𝐼_𝑛2+^𝑛𝑛_𝑖=1𝑗=1𝜀_𝑎𝑖𝑗𝐹_𝑖𝑗+^𝑛𝑛_𝑖=1𝑗=1𝜀_𝑏𝑖𝑗𝐺_𝑖𝑗+^𝑛𝑚_𝑗=1𝑘=1𝜀_𝑐𝑗𝑘𝐻_𝑗𝑘+𝐹+𝐺+𝐻≠0. (22) Hence, the matrix 𝐿 in (20) is nonsingular. That is, the matrix 𝐸=𝐸₀+∑^𝑛_𝑖=1∑^𝑛_𝑗=1𝜀_𝑎𝑖𝑗𝐹_𝑖𝑗+∑^𝑛_𝑖=1∑^𝑛_𝑗=1𝜀_𝑏𝑖𝑗𝐺_𝑖𝑗+∑^𝑛_𝑗=1∑^𝑚_𝑘=1𝜀_𝑐𝑗𝑘𝐻_𝑖𝑘+𝐹+𝐺+𝐻, for 𝑖,𝑗=1,2,…,𝑛 and 𝑘=1,2,…,𝑚, has full row rank𝑛². So, from the results mentioned previously and Lemma 4, it is ensured that, for any 𝜏>0, the linear interval MIMO system with both state delay and structured parametric uncertainties of (1) is robustly controllable in sense of Weiss [6].

3. Linear Interval Systems without State Delay

In this section, we consider a linear interval MIMO system with unstructured parametric uncertainties having the form of 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐴𝑥(𝑡)+𝐶𝑢(𝑡)+𝐶𝑢(𝑡). (23) The system in (23), called the linear interval MIMO system with unstructured parametric uncertainties, is said to be controllable if each pair (𝐴,𝐶) is controllable, where 𝐴=𝐴+𝐴, 𝐶=𝐶+𝐶, 𝐴∈[𝐴,𝐴], and C∈[C,C].

Following the same proof procedure given in the aforementioned theorem, we can get the following corollary to ensure that the linear interval system with unstructured parametric uncertainties is robustly controllable if the matrix 𝑀 has full row rank𝑛², where 𝑀=𝑀₀+∑^𝑛_𝑖=1∑^𝑛_𝑗=1𝜀_𝑎𝑖𝑗𝐹_𝑖𝑗+∑^𝑛_𝑗=1∑^𝑚_𝑘=1𝜀_𝑐𝑗𝑘𝐻_𝑗𝑘+𝐹+𝐻, and 𝑀₀=𝐼_𝑛0⋅⋅⋅00⋅⋅⋅0𝐶₀−𝐴₀𝐼_𝑛⋅⋅⋅00⋅⋅⋅𝐶₀0⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅00⋅⋅⋅𝐼_𝑛0⋅⋅⋅0000⋅⋅⋅−𝐴₀𝐶₀⋅⋅⋅00, (24) in which 𝑎_𝑖𝑗−𝑎_0𝑖𝑗≤𝜀_𝑎𝑖𝑗≤𝑎_𝑖𝑗−𝑎_0𝑖𝑗, 𝑐_𝑗𝑘−𝑐_0𝑗𝑘≤𝜀_𝑐𝑗𝑘≤𝑐_𝑗𝑘−𝑐_0𝑗𝑘, for 𝑖,𝑗=1,2,…,𝑛 and 𝑘=1,2,…,𝑚.𝐴₀∈[𝐴,𝐴] and 𝐶₀∈[𝐶,𝐶] are given constant matrices, and the pair (𝐴₀,𝐶₀) is controllable.

Let the singular value decomposition of 𝑀₀, which has rank𝑛² due to that the given pair (𝐴₀,𝐶₀) is controllable, be 𝑀₀=𝑈𝑆0_{𝑛2×𝑛(𝑚−1)}𝑉^𝐻, (25) where 𝑈∈𝑅^𝑛2×𝑛2 and 𝑉∈𝑅^{𝑛(𝑛+𝑚−1)×𝑛(𝑛+𝑚−1)} are the unitary matrices, 𝑉^𝐻 denotes the complex-conjugate transpose of matrix 𝑉, 𝑆=diag[𝜎₁,…,𝜎_𝑛2], and 𝜎₁≥𝜎₂≥⋯≥𝜎_𝑛2>0 are the singular values of 𝑀₀.

Corollary 10. The linear interval MIMO system with unstructured parametric uncertainties in (23) is robustly controllable, if the given pair (𝐴₀,𝐶₀) is controllable, and if the following conditions simultaneously hold: ^𝑛𝑛_𝑖=1𝑗=1𝜀_𝑎𝑖𝑗𝜙_𝑖𝑗+^𝑛𝑚_𝑗=1𝑘=1𝜀_𝑐𝑗𝑘𝜃_𝑗𝑘+𝛽₁+𝛽₃<1, (26) where𝜙_𝑖𝑗=𝜇−𝑆⁻¹𝑈^𝐻𝐹_𝑖𝑗𝑉𝐼_𝑛2,0_{𝑛2×𝑛(𝑚−1)}^𝑇,𝑓𝑜𝑟𝜀_𝑎𝑖𝑗𝑆≥0,−𝜇⁻¹𝑈^𝐻𝐹_𝑖𝑗𝑉𝐼_𝑛2,0_{𝑛2×𝑛(𝑚−1)}^𝑇,𝑓𝑜𝑟𝜀_𝑎𝑖𝑗𝜃<0,_𝑗𝑘=𝜇−𝑆⁻¹𝑈^𝐻𝐻_𝑗𝑘𝑉𝐼_𝑛2,0_{𝑛2×𝑛(𝑚−1)}^𝑇,𝑓𝑜𝑟𝜀_𝑐𝑗𝑘𝑆≥0,−𝜇⁻¹𝑈^𝐻𝐻_𝑗𝑘𝑉𝐼_𝑛2,0_{𝑛2×𝑛(𝑚−1)}^𝑇,𝑓𝑜𝑟𝜀_𝑐𝑗𝑘𝛽<0,₁=𝛽₁𝑆⁻¹𝑈^𝐻𝑉𝐼_𝑛2,0_{𝑛2×𝑛(𝑚−1)}^𝑇,𝛽₃=𝛽₃𝑆⁻¹𝑈^𝐻𝑉𝐼_𝑛2,0_{𝑛2×𝑛(𝑚−1)}^𝑇, (27a)(27b)(27c)(27d)𝑖,𝑗=1,2,…,𝑛 and 𝑘=1,2,…,𝑚, in which the matrices 𝐹_𝑖𝑗,  𝐻_𝑗𝑘,  𝑆,  𝑈, and 𝑉 are, respectively, defined in (9), (11), and (25) and 𝐼_𝑛2 denotes the 𝑛²×𝑛² identity matrix.

Proof. The proof procedure is the same as that of the aforementioned Theorem, hence omitted here.

Remark 11. If we only consider the robust controllability problem of linear interval systems with/without state delay (i.e., 𝐴=0, 𝐵=0, and 𝐶=0), the sufficient condition in (16) or (26) can be, respectively, simplified as ^𝑛𝑛_𝑖=1𝑗=1𝜀_𝑎𝑖𝑗𝜙_𝑖𝑗+^𝑛𝑛_𝑖=1𝑗=1𝜀_𝑏𝑖𝑗𝜑_𝑖𝑗+^𝑛𝑚_𝑗=1𝑘=1𝜀_𝑐𝑗𝑘𝜃_𝑗𝑘<1, (28) or ^𝑛𝑛_𝑖=1𝑗=1𝜀_𝑎𝑖𝑗𝜙_𝑖𝑗+^𝑛𝑚_𝑗=1𝑘=1𝜀_𝑐𝑗𝑘𝜃_𝑗𝑘<1. (29) Furthermore, it can be found that the result given by Chen and Chou [20] can be viewed as a special case of (16) and (28). On the other hand, by applying the evolutionary optimization methods [38] to choose the best matrices 𝐴₀, 𝐵₀, and 𝐶₀, the conservatism of (16), (26), (28), and (29) can be reduced.

4. Illustrative Examples

In this section, two numerical examples are given to illustrate the applications of the proposed sufficient algebraic criteria. We will also compare the conservatism of the proposed sufficient condition for linear interval system having no state delay and no unstructured parametric uncertainties with those results reported recently in the literature.

Example 1. Consider a linear interval system having no state delay and no unstructured parametric uncertainties as 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐶𝑢(𝑡), (30) with 1𝐴=^+0.4−0.20001^+0.1_−0.110−24^+0.6_−0.41,𝐶=^+0.6−0.300001, (31) which is slightly modified from the example given by Chen et al. [23].

Letting 𝐴₀=1000110−24,𝐶₀=100001, (32) then the interval matrices 𝐴 and 𝐶 can be represented as 𝐴=𝐴₀+𝜀_𝑎11𝐴₁₁+𝜀_𝑎22𝐴₂₂+𝜀_𝑎33𝐴₃₃ and 𝐶=𝐶₀+𝜀_𝑐11𝐶₁₁ with 𝐴₁₁=¹⁰⁰⁰⁰⁰₀₀₀, 𝐴₂₂=⁰⁰⁰⁰¹⁰₀₀₀, 𝐴₃₃=⁰⁰⁰⁰⁰⁰₀₀₁, 𝐶₁₁=¹⁰⁰⁰₀₀,  𝜀_𝑎11∈−0.2,0.4, 𝜀_𝑎22∈−0.1,0.1, 𝜀_𝑎33∈−0.4,0.6, and 𝜀_𝑐11∈−0.3,0.6.

First, following the approach of Cheng and Zhang [21], we have 𝑄₀=𝐶₀𝐴₀𝐶₀=,𝑄101000010104_𝛿=𝐶₀𝐴₀𝐶₀=,101000010104 (33)𝜆_min(𝑄₀𝑄₀^𝑇)=0.055728,𝜆_max(𝑄_𝛿𝑄_𝛿^𝑇)=17.944, 𝛿=0.027486, |𝛿⁰_ij|<𝛿̸, and |𝜎⁰_𝑖𝑗|<𝛿̸ for all 𝑖 and 𝑗, where 𝛿, 𝛿⁰_𝑖𝑗, and 𝜎⁰_𝑖𝑗 are detailedly defined in the work of Cheng and Zhang [21]. Hence, the condition of Cheng and Zhang [21] is not satisfied. Then, no conclusion can be made. That is, the sufficient condition of Cheng and Zhang [21] cannot be applied in this example.

Next, we use the method proposed by Ahn et al. [22] and Chen et al. [23] to test the controllability. The controllability matrix 𝐷 is calculated as 1𝐷=^+0.6−0.301^+1.24_−0.44000010104^+0.6_−0.3. (34) So, we have four subsquare matrices:𝑆¹=1^+0.6_−0.301^+1.24_−0.44,𝑆000010₂=1^+0.6_−0.300001014^+0.6_−0.3,𝑆³=1^+0.6_−0.31^+1.24_−0.440001004^+0.6_−0.3, (35) and 𝑆⁴=^{01+1.24−0.440001}_{104+0.6−0.3}. From 𝑆² and S⁴, it can be obtained that 𝑆²₀ and 𝑆⁴₀ are nonsingular and the spectral radiuses 𝜌(‖(𝑆²₀)⁻¹‖Δ𝑆²)=2.5416<1̸ and 𝜌(‖(𝑆⁴₀)⁻¹‖Δ𝑆⁴)=2.5416<1̸; thus, the condition of Ahn et al. [22] and Chen et al. [23] is not satisfied either. Then, no conclusion can be made. That is, the sufficient condition of Ahn et al. [22] and Chen et al. [23] cannot be applied in this example. On the other hand, all the elements in the interval matrices of (30) can independently vary with different direction and magnitude. That is, the interval matrices in (30) do not satisfy the assumption that all the elements in the interval matrices must have the same variations. So, the result of Chen and Chou [19, 20, 31] cannot be used in this example.

Now, applying the sufficient criterion in the proposed condition of (29) with 2-norm-based matrix measure for the robust controllability, we have rank(𝑀₀)=9, and we get(i)𝜀_𝑎11𝜙₁₁+𝜀_𝑎22𝜙₂₂+𝜀_𝑎33𝜙₃₃+𝜀_𝑐11𝜃₁₁≤0.568385<1, for 𝜀_𝑎11∈0,0.4,𝜀_𝑎22∈,𝜀0,0.1_𝑎33∈0,0.6,𝜀_𝑐11∈0,0.6; (36a)(ii)𝜀_𝑎11𝜙₁₁+𝜀_𝑎22𝜙₂₂+𝜀_𝑎33𝜙₃₃+𝜀_𝑐11𝜃₁₁≤0.86838<1, for 𝜀_𝑎11∈0,0.4,𝜀_𝑎22∈,𝜀0,0.1_𝑎33∈0,0.6,𝜀_𝑐11∈−0.3,0; (36b)(iii)𝜀_𝑎11𝜙₁₁+𝜀_𝑎22𝜙₂₂+𝜀_𝑎33𝜙₃₃+𝜀_𝑐11𝜃₁₁≤0.76838<1, for𝜀_𝑎11∈0,0.4,𝜀_𝑎22∈,𝜀0,0.1_𝑎33∈−0.4,0,𝜀_𝑐11∈−0.3,0; (36c)(iv)𝜀_𝑎11𝜙₁₁+𝜀_𝑎22𝜙₂₂+𝜀_𝑎33𝜙₃₃+𝜀_𝑐11𝜃₁₁≤0.46838<1, for 𝜀_𝑎11∈0,0.4,𝜀_𝑎22∈,𝜀0,0.1_𝑎33∈−0.4,0,𝜀_𝑐11∈0,0.6; (36d)(v)𝜀_𝑎11𝜙₁₁+𝜀_𝑎22𝜙₂₂+𝜀_𝑎33𝜙₃₃+𝜀_𝑐11𝜃₁₁≤0.56588<1, for 𝜀_𝑎11∈0,0.4,𝜀_𝑎22∈,𝜀−0.1,0_𝑎33∈0,0.6,𝜀_𝑐11∈0,0.6; (36e)(vi)𝜀_𝑎11𝜙₁₁+𝜀_𝑎22𝜙₂₂+𝜀_𝑎33𝜙₃₃+𝜀_𝑐11𝜃₁₁≤0.86588<1, for 𝜀_𝑎11∈0,0.4,𝜀_𝑎22∈,𝜀−0.1,0_𝑎33∈0,0.6,𝜀_𝑐11∈−0.3,0; (36f)(vii)𝜀_𝑎11𝜙₁₁+𝜀_𝑎22𝜙₂₂+𝜀_𝑎33𝜙₃₃+𝜀_𝑐11𝜃₁₁≤0.76588<1, for 𝜀_𝑎11∈0,0.4,𝜀_𝑎22∈,𝜀−0.1,0_𝑎33∈−0.4,0,𝜀_𝑐11∈−0.3,0; (36g)(viii)𝜀_𝑎11𝜙₁₁+𝜀_𝑎22𝜙₂₂+𝜀_𝑎33𝜙₃₃+𝜀_𝑐11𝜃₁₁≤0.46588<1, for 𝜀_𝑎11∈0,0.4,𝜀_𝑎22∈,𝜀−0.1,0_𝑎33∈−0.4,0,𝜀_𝑐11∈0,0.6; (36h)(ix)𝜀_𝑎11𝜙₁₁+𝜀_𝑎22𝜙₂₂+𝜀_𝑎33𝜙₃₃+𝜀_𝑐11𝜃₁₁≤0.59031<1, for 𝜀_𝑎11∈−0.2,0,𝜀_𝑎22∈,𝜀0,0.1_𝑎33∈0,0.6,𝜀_𝑐11∈0,0.6; (36i)(x)𝜀_𝑎11𝜙₁₁+𝜀_𝑎22𝜙₂₂+𝜀_𝑎33𝜙₃₃+𝜀_𝑐11𝜃₁₁≤0.89031<1, for 𝜀_𝑎11∈0,0.4,𝜀_𝑎22∈,𝜀0,0.1_𝑎33∈0,0.6,𝜀_𝑐11∈−0.3,0; (36j)(xi)𝜀_𝑎11𝜙₁₁+𝜀_𝑎22𝜙₂₂+𝜀_𝑎33𝜙₃₃+𝜀_𝑐11𝜃₁₁≤0.790315<1, for 𝜀_𝑎11∈0,0.4,𝜀_𝑎22∈,𝜀0,0.1_𝑎33∈−0.4,0,𝜀_𝑐11∈−0.3,0; (36k)(xii)𝜀_𝑎11𝜙₁₁+𝜀_𝑎22𝜙₂₂+𝜀_𝑎33𝜙₃₃+𝜀_𝑐11𝜃₁₁≤0.49031<1, for 𝜀_𝑎11∈0,0.4,𝜀_𝑎22∈,𝜀0,0.1_𝑎33∈−0.4,0,𝜀_𝑐11∈0,0.6; (36l)(xiii)𝜀_𝑎11𝜙₁₁+𝜀_𝑎22𝜙₂₂+𝜀_𝑎33𝜙₃₃+𝜀_𝑐11𝜃₁₁≤0.587806<1, for 𝜀_𝑎11∈0,0.4,𝜀_𝑎22∈,𝜀−0.1,0_𝑎33∈0,0.6,𝜀_𝑐11∈0,0.6; (36m)(xiv)𝜀_𝑎11𝜙₁₁+𝜀_𝑎22𝜙₂₂+𝜀_𝑎33𝜙₃₃+𝜀_𝑐11𝜃₁₁≤0.88780<1, for 𝜀_𝑎11∈0,0.4,𝜀_𝑎22∈,𝜀−0.1,0_𝑎33∈0,0.6,𝜀_𝑐11∈−0.3,0; (36n)(xv)𝜀_𝑎11𝜙₁₁+𝜀_𝑎22𝜙₂₂+𝜀_𝑎33𝜙₃₃+𝜀_𝑐11𝜃₁₁≤0.78780<1, for 𝜀_𝑎11∈0,0.4,𝜀_𝑎22∈,𝜀−0.1,0_𝑎33∈−0.4,0,𝜀_𝑐11∈−0.3,0; (36o)(xvi)𝜀_𝑎11𝜙₁₁+𝜀_𝑎22𝜙₂₂+𝜀_𝑎33𝜙₃₃+𝜀_𝑐11𝜃₁₁≤0.48780<1, for 𝜀_𝑎11∈0,0.4,𝜀_𝑎22∈,𝜀−0.1,0_𝑎33∈−0.4,0,𝜀_𝑐11∈0,0.6. (36p)

Then, from the results obtained above in (36a), (36b), (36c), (36d), (36e), (36f), (36g), (36h), (36i), (36j), (36k), (36l), (36m), (36n), (36o), and (36p), we can conclude that the linear interval system in (30) is robustly controllable. From this example, it is clear that our proposed approach gives less conservative results than the existing methods of Cheng and Zhang [21], Ahn et al. [22], Chen et al. [23], and Chen and Chou [19, 20, 31].

Example 2. Consider a linear interval system with both state delay and unstructured parametric uncertainties of (1) having 𝐴=21−1−1_+0.8−0.60−3−4−3^+0.4_−0.40,5𝐵=−3−10−2−23_+1.0−0.5,−10𝐶=0001^+1.6_−0.01,10 (37) with ‖𝐴‖≤𝛽, ‖𝐵‖≤𝛽, ‖𝐶‖≤𝛽, and 𝛽=0.01.

Letting 𝐴₀=,𝐵1−1−120−3−4−30₀=−3−10−2−235−10,𝐶₀=,000110 (38) then the interval matrices 𝐴, 𝐵, and 𝐶 can be represented as 𝐴=𝐴₀+𝜀_𝑎21𝐴₂₁+𝜀_𝑎32𝐴₃₂, 𝐵=𝐵₀+𝜀_𝑏31𝐵₃₁, and 𝐶=𝐶₀+𝜀_𝑐22𝐶₂₂ with 𝐴₂₁=⁰⁰⁰¹⁰⁰₀₀₀, 𝐴₃₂=⁰⁰⁰⁰⁰⁰₀₁₀, 𝐵₃₁=⁰⁰⁰⁰⁰⁰₁₀₀, 𝐶₂₂=⁰⁰⁰¹₀₀, 𝜀_𝑎21∈−0.6,0.8, 𝜀_𝑎32∈−0.4,0.4, 𝜀_𝑏31∈−0.5,1, and 𝜀_𝑐22∈−0.01,1.6. Now, applying the sufficient criterion in the proposed theorem with 2-norm-based matrix measure for the robust controllability, we get rank(𝐸₀)=9, and we have(i)𝜀_𝑎21𝜙₂₁+𝜀_𝑎32𝜙₃₂+𝜀_𝑏31𝜑₃₁+𝜀_𝑐22𝜃₂₂+𝛽₁+𝛽₂+𝛽₃≤0.92798<1, for 𝜀_𝑎21∈0,0.8,𝜀_𝑎32∈,𝜀0,0.4_𝑏31∈0,1,𝜀_𝑐22∈0,1.6; (39a)(ii)𝜀_𝑎21𝜙₂₁+𝜀_𝑎32𝜙₃₂+𝜀_𝑏31𝜑₃₁+𝜀_𝑐22𝜃₂₂+𝛽₁+𝛽₂+𝛽₃≤0.94798<1, for 𝜀_𝑎21∈0,0.8,𝜀_𝑎32∈,𝜀0,0.4_𝑏31∈0,1,𝜀_𝑐22∈−0.01,0; (39b)(iii)𝜀_𝑎21𝜙₂₁+𝜀_𝑎32𝜙₃₂+𝜀_𝑏31𝜑₃₁+𝜀_𝑐22𝜃₂₂+𝛽₁+𝛽₂+𝛽₃≤0.84164<1, for 𝜀_𝑎21∈0,0.8,𝜀_𝑎32∈,𝜀0,0.4_𝑏31∈−0.5,0,𝜀_𝑐22∈−0.01,0; (39c)(iv)𝜀_𝑎21𝜙₂₁+𝜀_𝑎32𝜙₃₂+𝜀_𝑏31𝜑₃₁+𝜀_𝑐22𝜃₂₂+𝛽₁+𝛽₂+𝛽₃≤0.82164<1, for 𝜀_𝑎21∈0,0.8,𝜀_𝑎32∈,𝜀0,0.4_𝑏31∈−0.5,0,𝜀_𝑐22∈0,1.6; (39d)(v)𝜀_𝑎21𝜙₂₁+𝜀_𝑎32𝜙₃₂+𝜀_𝑏31𝜑₃₁+𝜀_𝑐22𝜃₂₂+𝛽₁+𝛽₂+𝛽₃≤0.85587<1, for 𝜀_𝑎21∈0,0.8,𝜀_𝑎32∈,𝜀−0.4,0_𝑏31∈0,1,𝜀_𝑐22∈0,1.6; (39e)(vi)𝜀_𝑎21𝜙₂₁+𝜀_𝑎32𝜙₃₂+𝜀_𝑏31𝜑₃₁+𝜀_𝑐22𝜃₂₂+𝛽₁+𝛽₂+𝛽₃≤0.87587<1, for 𝜀_𝑎21∈0,0.8,𝜀_𝑎32∈,𝜀−0.4,0_𝑏31∈0,1,𝜀_𝑐22∈−0.01,0; (39f)(vii)𝜀_𝑎21𝜙₂₁+𝜀_𝑎32𝜙₃₂+𝜀_𝑏31𝜑₃₁+𝜀_𝑐22𝜃₂₂+𝛽₁+𝛽₂+𝛽₃≤0.76952<1, for 𝜀_𝑎21∈0,0.8,𝜀_𝑎32∈,𝜀−0.4,0_𝑏31∈−0.5,0,𝜀_𝑐22∈−0.01,0; (39g)(viii)𝜀_𝑎21𝜙₂₁+𝜀_𝑎32𝜙₃₂+𝜀_𝑏31𝜑₃₁+𝜀_𝑐22𝜃₂₂+𝛽₁+𝛽₂+𝛽₃≤0.74952<1, for 𝜀_𝑎21∈0,0.8,𝜀_𝑎32∈,𝜀−0.4,0_𝑏31∈−0.5,0,𝜀_𝑐22∈0,1.6; (39h)(ix)𝜀_𝑎21𝜙₂₁+𝜀_𝑎32𝜙₃₂+𝜀_𝑏31𝜑₃₁+𝜀_𝑐22𝜃₂₂+𝛽₁+𝛽₂+𝛽₃≤0.84400<1, for 𝜀_𝑎21∈−0.6,0,𝜀_𝑎32∈,𝜀0,0.4_𝑏31∈0,1,𝜀_𝑐22∈0,1.6; (39i)(x)𝜀_𝑎21𝜙₂₁+𝜀_𝑎32𝜙₃₂+𝜀_𝑏31𝜑₃₁+𝜀_𝑐22𝜃₂₂+𝛽₁+𝛽₂+𝛽₃≤0.86400<1, for 𝜀_𝑎21∈−0.6,0,𝜀_𝑎32∈,𝜀0,0.4_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∈,𝜀0,0.4_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∈,𝜀0,0.4_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∈,𝜀−0.4,0_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∈,𝜀−0.4,0_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∈,𝜀−0.4,0_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∈,𝜀−0.4,0_𝑏31∈−0.5,0,𝜀_𝑐22∈0,1.6. (39p)

Hence, from the results obtained above, we can conclude that, for any 𝜏>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 unstructured 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 algebraically elegant criteria for the robust controllability of linear interval MIMO systems with/without state delay and with unstructured 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 unstructured 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.