Research Article | Open Access

Shinn-Horng Chen, Jyh-Horng Chou, "Controllability Robustness of Linear Interval Systems with/without State Delay and with Unstructured Parametric Uncertainties", *Abstract and Applied Analysis*, vol. 2013, Article ID 346103, 10 pages, 2013. https://doi.org/10.1155/2013/346103

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

**Academic Editor:**Elena Braverman

#### 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, 5â€“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, 5â€“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* [17â€“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, 21â€“31] and references therein). But, till now, only a few researchers studied the controllability issue of the interval MIMO systems without state delay [16, 19â€“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 [33â€“35]. To the authorsâ€™ 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 and be real matrices satisfying , that is, , and . The set of matrices is called an interval matrix. Consider a linear interval MIMO system with both state delay and unstructured parametric uncertainties as the following form: where is the system state vector, is the control input vector, denotes the time delay, , , and are, respectively, the , , and interval matrices, and the unstructured parametric matrices , , and are assumed to be bounded, that is, where , , and are nonnegative real constant numbers, and denotes any matrix norm. Let be the Banach space of real -vector-valued continuous functions defined on the interval with the uniform norm; that is, if , we have . The initial function space is assumed to be , the space of continuous functions mapping into , and the -valued control function 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 is controllable, where , , , , , and .

For an interval matrix and for , the matrix denotes that it varies between and , in which and are, respectively, the lower bound and upper bound matrices of interval matrix, is an constant matrix with 1 in the th entry and 0 elsewhere, and is any given constant matrix. Then, the interval matrices , , and can be written as where , , , , , , , , and are, respectively, , , and constant matrices with 1 in the th or th entry and 0 elsewhere (for and ), , , and , and , , and are, respectively, any given , , and constant matrices with that the combination 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 with and any is controllable to the origin from time if for each , there exists a finite time and an admissible input defined on such that , where denotes a solution to at time corresponding to initial time , initial function , and input , in which , , and are, respectively, the , , and matrices.

*Definition 2 (see [36]). * The measure of an complex matrix is defined as
where is the induced matrix norm on the complex matrix.

Lemma 3 (see [7, 8]). *If the system with is controllable, then the linear time delay system with is controllable in sense of Weiss [6] for any .*

Lemma 4. *For any , the linear time delay system with is controllable in sense of Weiss [6] if the following controllability matrix
**
has , where , , and denotes the identity 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 is of â€‰ if and only if the matrix in (5) has (i.e., the matrix in (5) has ). So, from Lemma 3, we can conclude that if the matrix in (5) has , then, for any , the linear time delay system with 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 and , namely, and , respectively, are well defined for any norm and have the following properties:*(i)*, for the identity matrix ;*(ii)*, for any norm and any complex matrix ;*(iii)*, for any two complex matrices and ;*(iv)*, for any matrix and any nonnegative real number ;**
where denotes any eigenvalue of and denotes the real part of .*

While the induced matrix norms are 1-norm, 2-norm, and -norm, the corresponding matrix measures , where , can be easily calculated as(i); (ii);(iii); in which is the th element of the matrix and denotes the th eigenvalue.

Lemma 7. *For any and any complex matrix , .*

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

Lemma 8 (see [37]). *Let . If , then .*

From Lemma 4, it is known that, for any , the interval system with both state delay and unstructured parametric uncertainties in (1) is robustly controllable on in sense of Weiss [6] if the matrix has full row , where for , , , , and , in which

Let the singular value decomposition of , which has due to that the given combination is controllable, be where and are the unitary matrices, denotes the complex-conjugate transpose of matrix ,â€‰â€‰, and are the singular values of .

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

Theorem 9. *For any , 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 in (8) has a full row rank, and if the following conditions simultaneously hold:
**
where** and , in which the matrices , , , , , and are, respectively, defined in (9)â€“(11) and (15) and denotes the identity matrix.*

*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 in (8) has a full row rank due to that the given combination is controllable, and since we know that
thus, instead of , we can discuss the rank of
Since a matrix has at least if it has at least one nonsingular submatrix, a sufficient condition for the matrix in (19) to have is the nonsingularity of
where , , , , , and , for and .

Using the properties in Lemmas 6 and 7, and from (2) and (16), we have
Thus, from Lemma 8, we have
Hence, the matrix in (20) is nonsingular. That is, the matrix , for and , has full row . So, from the results mentioned previously and Lemma 4, it is ensured that, for any , 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 The system in (23), called the linear interval MIMO system with unstructured parametric uncertainties, is said to be controllable if each pair is controllable, where , , , and .

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 has full row , where , and in which , , for and . and are given constant matrices, and the pair is controllable.

Let the singular value decomposition of , which has due to that the given pair is controllable, be where and are the unitary matrices, denotes the complex-conjugate transpose of matrix , , and are the singular values of .

Corollary 10. *The linear interval MIMO system with unstructured parametric uncertainties in (23) is robustly controllable, if the given pair is controllable, and if the following conditions simultaneously hold:
**
where** and , in which the matrices ,â€‰â€‰,â€‰â€‰,â€‰â€‰, and are, respectively, defined in (9), (11), and (25) and denotes the 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., , , and ), 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). On the other hand, by applying the evolutionary optimization methods [38] to choose the best matrices , , and , 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
with
which is slightly modified from the example given by Chen et al. [23].

Letting then the interval matrices and can be represented as and with , , , ,â€‰ , , , and .

First, following the approach of Cheng and Zhang [21], we have ,, , , and for all and , where , , and 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 is calculated as So, we have four subsquare matrices: and . From and , it can be obtained that and are nonsingular and the spectral radiuses and ; 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 , and we get(i), for (ii), for (iii), for(iv), for (v), for (vi), for (vii), for (viii), for (ix), for (x), for (xi), for (xii), for (xiii), for (xiv), for (xv), for (xvi), for

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