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

Authors: Shinn-Horng Chen; Jyh-Horng Chou
Journal: Abstract and Applied Analysis
 (2013)

---

## 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.


---

## Body

## 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 theuncertain/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

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:
(1)x˙(t)=Ax(t)+A~x(t)+Bx(t-τ)+B~x(t-τ)+Cu(t)+C~u(t),
where x(t)∈Rn is the system state vector, u(t)∈Rm 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,
(2)∥A~∥≤β1,∥B~∥≤β2,∥C~∥≤β3,
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∥=maxt∈[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 Rn, and the Rm-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-d0ij≤εij≤d-ij-d0ij, the α×β matrix D=D0+∑i=1α∑j=1βεijDij 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, Dij is an α×β constant matrix with 1 in the ijth entry and 0 elsewhere, and D0=[d0ij]∈[D_,D-] is any given constant matrix. Then, the interval matrices [A_,A¯], [B_,B¯], and [C_,C-] can be written as
(3)[A_,A¯]=A0+∑i=1n∑j=1nεaijAij,[B_,B-]=B0+∑i=1n∑j=1nεbijBij,[C_,C-]=C0+∑j=1n∑k=1mεcjkCjk,
where A_=[a_ij], A¯=[a¯ij], B_=[b_ij], B¯=[b¯ij], C_=[c_ij], C¯=[c¯ij], 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 (for i,j=1,2,…,n and k=1,2,…,m), a_ij-a0ij≤εaij≤a-ij-a0ij, b_ij-b0ij≤εbij≤b-ij-b0ij, and c_jk-c0jk≤εcjk≤c-jk-c0jk, and A0=[a0ij]∈[A_,A-], B0=[b0ij]∈[B_,B-], and C0=[c0jk]∈[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 systemx˙(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 x˙(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 ann×n complex matrix W- is defined as
(4)μ(W-)≡limθ→0(∥I+θW-∥-1)θ,
where ∥·∥ is the induced matrix norm on the n×n complex matrix.Lemma 3 (see [7, 8]).
If the systemx˙(t)=(L+M)x(t)+Nu(t) with t>t0 is controllable, then the linear time delay system x˙(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 anyτ>0, the linear time delay system x˙(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
(5)E=[In0···00···0N-(L+M)In···00···N0····································00···In0···0000···-(L+M)N···00]
has rankn2, where L,M∈Rn×n, N∈Rn×m, and In 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
(6)[In0···00···0N0In···00···N(L+M)N····································00···In0····(L+M)n-2N00···0N(L+M)N···(L+M)n-1N].
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 x˙(t)=Lx(t)+Mx(t-τ)+Nu(t) with t>t0 is controllable in sense of Weiss [6].Remark 5.
From Lemma3, 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 matricesW- 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-∈Cn×n and any nonnegative real number γ;
whereλ(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 μk(·), where k=1,2,∞, can be easily calculated as(i)
μ1(W-)=maxj[Re(w-jj)+∑i=1,i≠jn|w-ij|];(ii)
μ2(W-)=maxi[λi(W-+W-*)/2];(iii)
μ∞(W-)=maxi[Re(w-ii)+∑j=1,j≠in|w-ij|];in which w-ij is the ijth element of the matrix W- and λi(·) denotes the ith eigenvalue.Lemma 7.
For anyγ<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. If μ(-N-)<1, then det(I+N-)≠0.From Lemma4, 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
(7)E=E0+∑i=1n∑j=1nεaijFij+∑i=1n∑j=1nεbijGij+∑j=1n∑k=1mεcjkHjk+F~+G~+H~,
for a_ij-a0ij≤εaij≤a-ij-a0ij, b_ij-b0ij≤εbij≤b-ij-b0ij, c_jk-c0jk≤εcjk≤c-jk-c0jk, i,j=1,2,…,n, and k=1,2,…,m, in which
(8)E0=[In0···00···0C0-(A0+B0)In···00···C00····································00···In0···0000···-(A0+B0)C0···00],(9)Fij=[00···00···00-Aij0···00···00····································00···00···0000···-Aij0···00],(10)Gij=[00···00···00-Bij0···00···00····································00···00···0000···-Bij0···00],(11)Hjk=[00···00···0Cjk00···00···Cjk0····································00···00···0000···0Cjk···00],(12)F~=[00···00···00-A~0···00···00····································00···00···0000···-A~0···00],(13)G~=[00···00···00-B~0···00···00····································00···00···0000···-B~0···00],(14)H~=[00···00···0C~00···00···C~0····································00···00···0000···0C~···00].Let the singular value decomposition ofE0, which has rankn2 due to that the given combination (A0,B0,C0) is controllable, be
(15)E0=U[S0n2×n(m-1)]VH,
where U∈Rn2×n2 and V∈Rn(n+m-1)×n(n+m-1) are the unitary matrices, VH denotes the complex-conjugate transpose of matrix V,  S=diag[σ1,…,σn2], and σ1≥σ2≥⋯≥σn2>0 are the singular values of E0.In what follows, we present a sufficient criterion for ensuring that, for anyτ>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 anyτ>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:
(16)∑i=1n∑j=1nεaijϕij+∑i=1n∑j=1nεbijφij+∑j=1n∑k=1mεcjkθjk+β-1+β-2+β-3<1,
where(17a)ϕij={μ(-S-1UHFijV[In2,0n2×n(m-1)]T),forεaij≥0,-μ(S-1UHFijV[In2,0n2×n(m-1)]T),forεaij<0,(17b)φij={μ(-S-1UHGijV[In2,0n2×n(m-1)]T),forεbij≥0,-μ(S-1UHGijV[In2,0n2×n(m-1)]T),forεbij<0,(17c)θjk={μ(-S-1UHHjkV[In2,0n2×n(m-1)]T),forεcjk≥0,-μ(S-1UHHjkV[In2,0n2×n(m-1)]T),forεcjk<0,(17d)β-1=β1∥S-1UH∥∥V[In2,0n2×n(m-1)]T∥,(17e)β-2=β2∥S-1UH∥∥V[In2,0n2×n(m-1)]T∥,(17f)β-3=β3∥S-1UH∥∥V[In2,0n2×n(m-1)]T∥,i,j=1,2,…,n and k=1,2,…,m, in which the matrices Fij, Gij, Hjk, S, U,  and V are, respectively, defined in (9)–(11) and (15) and In2 denotes the n2×n2 identity matrix.Proof.
If the matrixE 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
(18)rank(E0)=rank(S-1UHE0V),
thus, instead of rank(E0+∑i=1n∑j=1nεaijFij+∑i=1n∑j=1nεbijGij+∑j=1n∑k=1mεcjkHjk+F~+G~+H~), we can discuss the rank of
(19)[In20n2×n(m-1)]+∑i=1n∑j=1nεaijS-1UHFijV+∑i=1n∑j=1nεbijS-1UHGijV+∑j=1n∑k=1mεcjkS-1UHHjkV+S-1UHF~V+S-1UHG~V+S-1UHH~V.
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
(20)L¯=In2+∑i=1n∑j=1nεaijF-ij+∑i=1n∑j=1nεbijG-ij+∑j=1n∑k=1mεcjkH-jk+F-+G-+H-,
where F-ij=S-1UHFijV[In2,0n2×n(m-1)]T, G-ij=S-1UHGijV[In2,0n2×n(m-1)]T, H-jk=S-1UHHjkV[In2,0n2×n(m-1)]T, F-=S-1UHF~V[In2,0n2×n(m-1)]T, G-=S-1UHG~V[In2,0n2×n(m-1)]T, and H-=S-1UHH~V[In2,0n2×n(m-1)]T, for i,j=1,2,…,n and k=1,2,…,m.
Using the properties in Lemmas6 and 7, and from (2) and (16), we have
(21)μ(-∑i=1n∑j=1nεaijF-ij-∑i=1n∑j=1nεbijG-ij-∑j=1n∑k=1mεcjkH-jk-F--G--H-)≤μ(-∑i=1n∑j=1nεaijF-ij)+μ(-∑i=1n∑j=1nεbijG-ij)+μ(-∑j=1n∑k=1mεcjkH-jk)+μ(-F-)+μ(-G-)+μ(-H-)≤μ(-∑i=1n∑j=1nεaijF-ij)+μ(-∑i=1n∑j=1nεbijG-ij)+μ(-∑j=1n∑k=1mεcjkH-jk)+∥F-∥+∥G-∥+∥H-∥≤∑i=1n∑j=1nμ(-εaijF-ij)+∑i=1n∑j=1nμ(-εbijG-ij)+∑j=1n∑k=1mμ(-εcjkH-jk)+∥F-∥+∥G-∥+∥H-∥≤∑i=1n∑j=1nεaijϕij+∑i=1n∑j=1nεbijφij+∑j=1n∑k=1mεcjkθjk+β-1+β-2+β-3<1.
Thus, from Lemma 8, we have
(22)det(L¯)=det(In2+∑i=1n∑j=1nεaijF-ij+∑i=1n∑j=1nεbijG-ij+∑j=1n∑k=1mεcjkH-jk+F-+G-+H-∑j=1nεaijF-ij)≠0.
Hence, the matrix L¯ in (20) is nonsingular. That is, the matrix E=E0+∑i=1n∑j=1nεaijFij+∑i=1n∑j=1nεbijGij+∑j=1n∑k=1mεcjkHik+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(23)x˙(t)=Ax(t)+A~x(t)+Cu(t)+C~u(t).
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 matrixM¯ has full row rankn2, where M¯=M0+∑i=1n∑j=1nεaijFij+∑j=1n∑k=1mεcjkHjk+F~+H~,  and
(24)M0=[In0···00···0C0-A0In···00···C00····································00···In0···0000···-A0C0···00],
in which a_ij-a0ij≤εaij≤a-ij-a0ij, c_jk-c0jk≤εcjk≤c-jk-c0jk, 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 ofM0, which has rankn2 due to that the given pair (A0,C0) is controllable, be
(25)M0=U~[S~0n2×n(m-1)]V~H,
where U~∈Rn2×n2 and V~∈Rn(n+m-1)×n(n+m-1) are the unitary matrices, V~H denotes the complex-conjugate transpose of matrix V~, S~=diag[σ~1,…,σ~n2], and σ~1≥σ~2≥⋯≥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:
(26)∑i=1n∑j=1nεaijϕ~ij+∑j=1n∑k=1mεcjkθ~jk+β~1+β~3<1,
where(27a)ϕ~ij={μ(-S~-1U~HFijV~[In2,0n2×n(m-1)]T),forεaij≥0,-μ(S~-1U~HFijV~[In2,0n2×n(m-1)]T),forεaij<0,(27b)θ~jk={μ(-S~-1U~HHjkV~[In2,0n2×n(m-1)]T),forεcjk≥0,-μ(S~-1U~HHjkV~[In2,0n2×n(m-1)]T),forεcjk<0,(27c)β~1=β1∥S~-1U~H∥∥V~[In2,0n2×n(m-1)]T∥,(27d)β~3=β3∥S~-1U~H∥∥V~[In2,0n2×n(m-1)]T∥,i,j=1,2,…,n and k=1,2,…,m, in which the matrices Fij,  Hjk,  S~,  U~, and V~ are, respectively, defined in (9), (11), and (25) and In2 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
(28)∑i=1n∑j=1nεaijϕij+∑i=1n∑j=1nεbijφij+∑j=1n∑k=1mεcjkθjk<1,
or
(29)∑i=1n∑j=1nεaijϕ~ij+∑j=1n∑k=1mεcjkθ~jk<1.
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(30)x˙(t)=Ax(t)+Cu(t),
with
(31)A=[1-0.2+0.40001-0.1+0.110-24-0.4+0.6],C=[1-0.3+0.600001],
which is slightly modified from the example given by Chen et al. [23].Letting(32)A0=[1000110-24],C0=[100001],
then the interval matrices A and C can be represented as A=A0+εa11A11+εa22A22+εa33A33 and C=C0+εc11C11 with A11=[100000000], A22=[000010000], A33=[000000001], C11=[100000],  εa11∈[-0.2,0.4], εa22∈[-0.1,0.1], εa33∈[-0.4,0.6], and εc11∈[-0.3,0.6].First, following the approach of Cheng and Zhang [21], we have
(33)Q0=[C0A0C0]=[101000010104],Qδ=[|C0||A0||C0|]=[101000010104],λmin(Q0Q0T)=0.055728,λmax(QδQδT)=17.944, δ=0.027486, |δij0|≮δ, and |σij0|≮δ for all i and j, where δ, δij0, and σij0 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
(34)D=[1-0.3+0.601-0.44+1.24000010104-0.3+0.6].
So, we have four subsquare matrices:(35)S1=[1-0.3+0.601-0.44+1.24000010],S2=[1-0.3+0.600001014-0.3+0.6],S3=[1-0.3+0.61-0.44+1.240001004-0.3+0.6],
and S4=[01-0.44+1.240001104-0.3+0.6]. From S2 and S4, it can be obtained that S02 and S04 are nonsingular and the spectral radiuses ρ(∥(S02)-1∥ΔS2)=2.5416≮1 and ρ(∥(S04)-1∥ΔS4)=2.5416≮1; 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)
εa11ϕ~11+εa22ϕ~22+εa33ϕ~33+εc11θ~11≤0.568385<1, for
(36a)εa11∈[0,0.4],εa22∈[0,0.1],εa33∈[0,0.6],εc11∈[0,0.6];(ii)
εa11ϕ~11+εa22ϕ~22+εa33ϕ~33+εc11θ~11≤0.86838<1, for
(36b)εa11∈[0,0.4],εa22∈[0,0.1],εa33∈[0,0.6],εc11∈[-0.3,0];(iii)
εa11ϕ~11+εa22ϕ~22+εa33ϕ~33+εc11θ~11≤0.76838<1, for(36c)εa11∈[0,0.4],εa22∈[0,0.1],εa33∈[-0.4,0],εc11∈[-0.3,0];(iv)
εa11ϕ~11+εa22ϕ~22+εa33ϕ~33+εc11θ~11≤0.46838<1, for
(36d)εa11∈[0,0.4],εa22∈[0,0.1],εa33∈[-0.4,0],εc11∈[0,0.6];(v)
εa11ϕ~11+εa22ϕ~22+εa33ϕ~33+εc11θ~11≤0.56588<1, for
(36e)εa11∈[0,0.4],εa22∈[-0.1,0],εa33∈[0,0.6],εc11∈[0,0.6];(vi)
εa11ϕ~11+εa22ϕ~22+εa33ϕ~33+εc11θ~11≤0.86588<1, for
(36f)εa11∈[0,0.4],εa22∈[-0.1,0],εa33∈[0,0.6],εc11∈[-0.3,0];(vii)
εa11ϕ~11+εa22ϕ~22+εa33ϕ~33+εc11θ~11≤0.76588<1, for
(36g)εa11∈[0,0.4],εa22∈[-0.1,0],εa33∈[-0.4,0],εc11∈[-0.3,0];(viii)
εa11ϕ~11+εa22ϕ~22+εa33ϕ~33+εc11θ~11≤0.46588<1, for
(36h)εa11∈[0,0.4],εa22∈[-0.1,0],εa33∈[-0.4,0],εc11∈[0,0.6];(ix)
εa11ϕ~11+εa22ϕ~22+εa33ϕ~33+εc11θ~11≤0.59031<1, for
(36i)εa11∈[-0.2,0],εa22∈[0,0.1],εa33∈[0,0.6],εc11∈[0,0.6];(x)
εa11ϕ~11+εa22ϕ~22+εa33ϕ~33+εc11θ~11≤0.89031<1, for
(36j)εa11∈[0,0.4],εa22∈[0,0.1],εa33∈[0,0.6],εc11∈[-0.3,0];(xi)
εa11ϕ~11+εa22ϕ~22+εa33ϕ~33+εc11θ~11≤0.790315<1, for
(36k)εa11∈[0,0.4],εa22∈[0,0.1],εa33∈[-0.4,0],εc11∈[-0.3,0];(xii)
εa11ϕ~11+εa22ϕ~22+εa33ϕ~33+εc11θ~11≤0.49031<1, for
(36l)εa11∈[0,0.4],εa22∈[0,0.1],εa33∈[-0.4,0],εc11∈[0,0.6];(xiii)
εa11ϕ~11+εa22ϕ~22+εa33ϕ~33+εc11θ~11≤0.587806<1, for
(36m)εa11∈[0,0.4],εa22∈[-0.1,0],εa33∈[0,0.6],εc11∈[0,0.6];(xiv)
εa11ϕ~11+εa22ϕ~22+εa33ϕ~33+εc11θ~11≤0.88780<1, for
(36n)εa11∈[0,0.4],εa22∈[-0.1,0],εa33∈[0,0.6],εc11∈[-0.3,0];(xv)
εa11ϕ~11+εa22ϕ~22+εa33ϕ~33+εc11θ~11≤0.78780<1, for
(36o)εa11∈[0,0.4],εa22∈[-0.1,0],εa33∈[-0.4,0],εc11∈[-0.3,0];(xvi)
εa11ϕ~11+εa22ϕ~22+εa33ϕ~33+εc11θ~11≤0.48780<1, for
(36p)εa11∈[0,0.4],εa22∈[-0.1,0],εa33∈[-0.4,0],εc11∈[0,0.6].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
(37)A=[1-1-12-0.6+0.80-3-4-3-0.4+0.40],B=[-3-10-2-235-0.5+1.0-10],C=[0001-0.01+1.610],
with ∥A~∥≤β, ∥B~∥≤β, ∥C~∥≤β, and β=0.01.Letting(38)A0=[1-1-120-3-4-30],B0=[-3-10-2-235-10],C0=[000110],
then the interval matrices A, B, and C can be represented as A=A0+εa21A21+εa32A32, B=B0+εb31B31, and C=C0+εc22C22 with A21=[000100000], A32=[000000010], B31=[000000100], C22=[000100], εa21∈[-0.6,0.8], εa32∈[-0.4,0.4], εb31∈[-0.5,1], and εc22∈[-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)
εa21ϕ21+εa32ϕ32+εb31φ31+εc22θ22+β-1+β-2+β-3≤0.92798<1, for
(39a)εa21∈[0,0.8],εa32∈[0,0.4],εb31∈[0,1],εc22∈[0,1.6];(ii)
εa21ϕ21+εa32ϕ32+εb31φ31+εc22θ22+β-1+β-2+β-3≤0.94798<1, for
(39b)εa21∈[0,0.8],εa32∈[0,0.4],εb31∈[0,1],εc22∈[-0.01,0];(iii)
εa21ϕ21+εa32ϕ32+εb31φ31+εc22θ22+β-1+β-2+β-3≤0.84164<1, for
(39c)εa21∈[0,0.8],εa32∈[0,0.4],εb31∈[-0.5,0],εc22∈[-0.01,0];(iv)
εa21ϕ21+εa32ϕ32+εb31φ31+εc22θ22+β-1+β-2+β-3≤0.82164<1, for
(39d)εa21∈[0,0.8],εa32∈[0,0.4],εb31∈[-0.5,0],εc22∈[0,1.6];(v)
εa21ϕ21+εa32ϕ32+εb31φ31+εc22θ22+β-1+β-2+β-3≤0.85587<1, for
(39e)εa21∈[0,0.8],εa32∈[-0.4,0],εb31∈[0,1],εc22∈[0,1.6];(vi)
εa21ϕ21+εa32ϕ32+εb31φ31+εc22θ22+β-1+β-2+β-3≤0.87587<1, for
(39f)εa21∈[0,0.8],εa32∈[-0.4,0],εb31∈[0,1],εc22∈[-0.01,0];(vii)
εa21ϕ21+εa32ϕ32+εb31φ31+εc22θ22+β-1+β-2+β-3≤0.76952<1, for
(39g)εa21∈[0,0.8],εa32∈[-0.4,0],εb31∈[-0.5,0],εc22∈[-0.01,0];(viii)
εa21ϕ21+εa32ϕ32+εb31φ31+εc22θ22+β-1+β-2+β-3≤0.74952<1, for
(39h)εa21∈[0,0.8],εa32∈[-0.4,0],εb31∈[-0.5,0],εc22∈[0,1.6];(ix)
εa21ϕ21+εa32ϕ32+εb31φ31+εc22θ22+β-1+β-2+β-3≤0.84400<1, for
(39i)εa21∈[-0.6,0],εa32∈[0,0.4],εb31∈[0,1],εc22∈[0,1.6];(x)
εa21ϕ21+εa32ϕ32+εb31φ31+εc22θ22+β-1+β-2+β-3≤0.86400<1, for
(39j)εa21∈[-0.6,0],εa32∈[0,0.4],εb31∈[0,1],εc22∈[-0.01,0];(xi)
εa21ϕ21+εa32ϕ32+εb31φ31+εc22θ22+β-1+β-2+β-3≤0.75766<1, for
(39k)εa21∈[-0.6,0],εa32∈[0,0.4],εb31∈[-0.5,0],εc22∈[-0.01,0];(xii)
εa21ϕ21+εa32ϕ32+εb31φ31+εc22θ22+β-1+β-2+β-3≤0.73766<1, for
(39l)εa21∈[-0.6,0],εa32∈[0,0.4],εb31∈[-0.5,0],εc22∈[0,1.6];(xiii)
εa21ϕ21+εa32ϕ32+εb31φ31+εc22θ22+β-1+β-2+β-3≤0.77189<1, for
(39m)εa21∈[-0.6,0],εa32∈[-0.4,0],εb31∈[0,1],εc22∈[0,1.6];(xiv)
εa21ϕ21+εa32ϕ32+εb31φ31+εc22θ22+β-1+β-2+β-3≤0.79189<1, for
(39n)εa21∈[-0.6,0],εa32∈[-0.4,0],εb31∈[0,1],εc22∈[-0.01,0];(xv)
εa21ϕ21+εa32ϕ32+εb31φ31+εc22θ22+β-1+β-2+β-3≤0.68554<1, for
(39o)εa21∈[-0.6,0],εa32∈[-0.4,0],εb31∈[-0.5,0],εc22∈[-0.01,0];(xvi)
εa21ϕ21+εa32ϕ32+εb31φ31+εc22θ22+β-1+β-2+β-3≤0.66554<1, for
(39p)εa21∈[-0.6,0],εa32∈[-0.4,0],εb31∈[-0.5,0],εc22∈[0,1.6].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 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].



---

*Source: 346103-2013-07-29.xml*