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

and Applied Analysis 3 Definition 1 (see [6]). The system ?̇?(t) = Lx(t) +Mx(t − τ) + Nu(t) with t > t 0 and any τ > 0 is controllable to the origin from time t 0 if for each Φ ∈ Β̂ a , there exists a finite time t 1 > t 0 and an admissible input u(t) defined on [t 0 , t 1 ] such that x(t 1 , t 0 , Φ, u) = 0, where x(t 1 , t 0 , Φ, u) denotes a solution to ?̇?(t) = Lx(t) + Mx(t − τ) + Nu(t) at time t 1 corresponding to initial time t 0 , initial function Φ ∈ Β̂ a , and input u(t), in which L,M, andN are, respectively, the n×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 ?̇?(t) = (L+M)x(t)+Nu(t) with t > t 0 is controllable, then the linear time delay system ?̇?(t) = Lx(t) +Mx(t − τ) +Nu(t) with t > t 0 is controllable in sense of Weiss [6] for any τ > 0. Lemma 4. For any τ > 0, the linear time delay system ?̇?(t) = Lx(t) + Mx(t − τ) + Nu(t) with t > t 0 is controllable in sense of Weiss [6] if the following n × 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 rank n, where L,M ∈ R,N ∈ R, and I n denotes the n × n identity matrix. Proof. Following the same proof procedure as that given by Chen andChou [20], in the abovematrixE 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


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][6][7][8][9][10][11][12][13][14][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][6][7][8][9][10][11][12][13][14][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][18][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.
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][34][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.
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 2 (see [36]).The measure of an  ×  complex matrix  is defined as where ‖ ⋅ ‖ is the induced matrix norm on the  ×  complex matrix.
Proof.Following the same proof procedure as that given by Chen and Chou [20], in the above matrix  of (5), add (+) times the first (block) row to the second, then add ( + ) times the second row to the third, and so on.The result is a matrix The controllability matrix [ (+) ⋅ ⋅ ⋅ (+) −1 ] is of rank  if and only if the matrix in (5) has rank  2 (i.e., the matrix in (5) has rank  2 ).So, from Lemma 3, we can conclude that if the matrix in (5) has rank  2 , then, for any  > 0, the linear time delay system ẋ () = ()+(−)+() with  >  0 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).
While the induced matrix norms are 1-norm, 2-norm, and ∞-norm, the corresponding matrix measures   (⋅), where  = 1, 2, ∞, can be easily calculated as in which   is the th element of the matrix  and   (⋅) denotes the th eigenvalue.
Proof.From the property (iv) in Lemma 8, this lemma can be immediately obtained.
Proof.If the matrix  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  0 in (8) has a full row rank due to that the given combination ( 0 ,  0 ,  0 ) is controllable, and since we know that rank thus, instead of rank Since a matrix has at least rank  2 if it has at least one nonsingular  2 ×  2 submatrix, a sufficient condition for the matrix in (19) to have rank  2 is the nonsingularity of where Thus, from Lemma 8, we have Hence, the matrix  in ( 20) is nonsingular.That is, the matrix . .,  and  = 1, 2, . . ., , has full row rank  2 .So, from the results mentioned previously and Lemma 4, it is ensured that, for any  > 0, the linear interval MIMO system with both state delay and structured parametric uncertainties of (1) is robustly controllable in sense of Weiss [6].
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., Ã = 0, B = 0, and C = 0), the sufficient condition in ( 16) or ( 26) can be, respectively, simplified as or 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).

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 with which is slightly modified from the example given by Chen et al. [23]. Letting then the interval matrices  and  can be represented as  =  0 +  11  11 +  22  22 +  33  33 and  =  0 +  11  11 with  11 = [ and First, following the approach of Cheng and Zhang [21], we have min ( 0  0  ) = 0.055728,  max (     ) = 17.944,  = 0.027486, | 0 ij | ̸ < , and | 0  | ̸ <  for all  and , where ,  0  , and  0  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.
with ‖ Ã‖ ≤ , ‖ B‖ ≤ , ‖ C‖ ≤ , and  = 0.01. Letting Hence, from the results obtained above, we can conclude that, for any  > 0, the linear interval system with both state delay and unstructured parametric uncertainties is robustly controllable in sense of Weiss [6].

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].