Journal of the Franklin Institute 350 (2013) 2277–2290
Algebraic criterion for robust controllability of
continuous linear time-delay systems with
parametric uncertainties
Shinn-Horng Chen
a, Jyh-Horng Chou
a,b,c,naDepartment of Mechanical/Electrical Engineering, National Kaohsiung University of Applied Sciences, 415 Chien-Kung
Road, Kaohsiung 807, Taiwan, Republic of China
bInstitute of Electrical Engineering as well as Department of Mechanical and Automation Engineering, National
Kaohsiung First University of Science and Technology, 1 University Road, Yenchao, Kaohsiung 824, Taiwan, Republic of China
cDepartment of Healthcare Administration and Medical Informatics, Kaohsiung Medical University, 100 Shi-Chuan
1st Road, Kaohsiung 807, Taiwan, Republic of China
Received 19 September 2011; received in revised form 16 February 2013; accepted 12 June 2013 Available online 22 June 2013
Abstract
The robust controllability problem for the continuous linear time-delay systems with structured parametric uncertainties is studied in this paper. A new sufficient algebraic criterion for the robust controllability of uncertain linear time-delay systems is established. The proposed sufficient condition can provide the explicit relationship of the bounds on system uncertainties for guaranteeing the controllability property. Three numerical examples are given to illustrate the application of the proposed sufficient algebraic criterion and to compare the results with those obtained from the approaches in the literature. & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction
It is well recognized that time-delay phenomenon is ubiquitous in nature and engineering, including mechanical engineering, aeronautics and astronautics, ecology, biology, information
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nCorresponding author at: Institute of Electrical Engineering, National Kaohsiung First University of Science and
Technology, 1 University Road, Yenchao, Kaohsiung 824, Taiwan, Republic of China. Tel.:+886 7 6011000; fax:+886 7 6011066.
technology, economics, and so on. Time-delay effect may occur naturally because of the inherent
characteristics of some system components, or part of the control process [9]. Besides,
controllability plays a central role throughout the history of modern control theory and engineering because it has close connection to eigenvalue assignment, optimal control, and controller design [14]. Therefore, the controllability problem of continuous linear time-delay systems has been studied by some researchers (see, for example,[20,21,10,12,13,22,18,9,19]). On the other hand, in fact, in many cases it is very difficult, if not impossible, to obtain the accurate values of some system parameters. This is due to the inaccurate measurement, unaccessibility to the system parameters, or variation of the parameters. These system uncertainties may destroy the controllability property of the linear time-delay systems. By using the result presented by Hewer[10]as well as Levsen and Nazaroff[12], the robust controllability problem of uncertain continuous linear time-delay systems can be transferred into that of uncertain continuous linear delay-free systems. Some methods for studying the robust controllability problem of uncertain continuous linear delay-free systems have been proposed by some researchers (see, for example,[15,8,6,4,17,2]; and therein references). Therefore, these proposed methods can be used to solve the robust controllability problem of uncertain continuous linear time-delay systems. Most notably, the approaches proposed by Elizondo and
Collado [8], Cheng and Zhang [4] as well as Chen and Chou [2] give algebraically elegant
derivations. However, the approach of Elizondo and Collado [8] is only suitable for the
unidirectional uncertainty case, the parametric uncertainties considered by Cheng and Zhang[4]
must satisfy the sign-invariant condition, and all the uncertain elements of the interval matrices
considered by Chen and Chou[2]must have the same variations.
The purpose of this paper is to present an alternative new approach for investigating the robust controllability problem of the continuous linear time-delay systems with system uncertainties.
The presented new approach provides an algebraically elegant derivation. A new sufficient
algebraic criterion is proposed to guarantee the robust controllability of uncertain linear time-delay systems. The robust controllability problem for uncertain continuous linear time-delay-free systems studied in this paper is more general than those considered by Elizondo and Collado[8]
as well as Cheng and Zhang[4]. The mathematical means used in this paper is also very different
from those used by Elizondo and Collado[8] as well as Cheng and Zhang[4]. The proposed
sufficient condition can provide the explicit relationship of the bounds on system uncertainties for having the robust controllability property. Three numerical examples are also given in this paper to illustrate the application of the proposed sufficient algebraic criterion and to compare the
results with those obtained from the approaches of Elizondo and Collado [8] and Cheng and
Zhang[4].
2. Robust controllability analysis
Consider the following continuous linear time-delay system with system uncertainties:
_xðtÞ ¼ ðA þ ΔAÞxðtÞ þ ðB þ ΔBÞxðt−τÞ þ ðC þ ΔCÞuðtÞ; t4t0; ð1Þ
where xðtÞ∈Rn is the system state vector, uðtÞ∈Rm is the input vector, τ40 denotes the time delay, A; B and C are, respectively, the n n; n n and n m constant matrices, as well as ΔA; ΔB and ΔC are, respectively, the uncertain matrices existing in the system matrices A and B, and in the input matrix C due to the inaccurate measurement, unaccessibility to the system parameters, or variation of the parameters. LetΒ be the Banach space of real n-vector-valued continuous functions defined on the interval ½t0−τ; t0 with the uniform norm, i.e., if Φ∈Β, we
have ∥Φ∥ ¼ maxt∈½t0−τ;t0jΦðtÞj. The initia1 function space is assumed to be Β, the space of
continuous functions mapping ½t0−τ; t0 into Rn, and the Rm-valued control function uðtÞ is
measurable and bounded on everyfinite time interval[21].
In many interesting problems, we have only a small number of uncertainties, but these uncertainties may enter into many entries of the system and input matrices [23,1,16]. For
example, consider a two-mass system with an uncertain stiffness given by Sinha [16]. The
system matrix A is A¼ 0 0 1 0 0 0 0 1 −~k ~k 0 0 ~k −~k 0 0 2 6 6 6 4 3 7 7 7 5¼ 0 0 1 0 0 0 0 1 −k0 k0 0 0 k0 −k0 0 0 2 6 6 6 4 3 7 7 7 5þ ε 0 0 0 0 0 0 0 0 −1 1 0 0 1 −1 0 0 2 6 6 6 4 3 7 7 7 5;
where the parametric uncertaintyε enters into four entries of the system matrix. So, the forms are the elemental parametric uncertain forms for the general and practical cases[16]. Therefore, in
this paper, we suppose that the uncertain matricesΔA; ΔB and ΔC take the forms
ΔA ¼ ∑m k¼ 1 εkAk; ΔB ¼ ∑ m k¼ 1 εkBk; and ΔC ¼ ∑ m k¼ 1 εkCk; ð2Þ
whereεkðk ¼ 1; 2; …; mÞ are the elemental parametric uncertainties, as well as Ak; Bkand Ckare,
respectively, the given n n; n n and n m constant matrices which are prescribed a prior to denote the linearly dependent information on the elemental parametric uncertaintiesεk; in which
k¼ 1; 2; …; m:
Before we investigate the robust property of controllability for the uncertain linear time-delay system in Eqs.(1)and(2), the following definitions and lemmas need to be introduced first.
Definition 1. Weiss [21]
The system_xðtÞ ¼ AxðtÞ þ Bxðt−τÞ þ CuðtÞ with t4t0is controllable to the origin from time t0
if, for eachΦ∈Β, there exists a finite time t14t0 and an admissible input uðtÞ defined on ½t0:t1
such that xðt1; t0; Φ; uÞ ¼ 0, where xðt1; t0; Φ; uÞ denotes a solution to _xðtÞ ¼ AxðtÞ þ Bxðt−τÞ þ
CuðtÞ at time t1 corresponding to initial time t0, initial function Φ∈Β and input uðtÞ.
Definition 2. Desoer and Vidyasagar [7]
The measure of a matrix W∈Cnn is defined as
μðWÞ≡lim
θ-0
ð∥I þ θW∥−1Þ
θ ;
where∥∥ is the induced matrix norm on Cnn.
Lemma 1. Hewer [10],and Levsen and Nazaroff [12]
If the system _xðtÞ ¼ ðA þ BÞxðtÞ þ CuðtÞ with t4t0 is controllable, then the linear time-delay
system _xðtÞ ¼ AxðtÞ þ Bxðt−τÞ þ CuðtÞ with t4t0 is controllable in sense of Weiss[21]for any
Lemma 2. For anyτ40; the linear time-delay system _xðtÞ ¼ AxðtÞ þ Bxðt−τÞ þ CuðtÞ with t4t0
is controllable in sense of Weiss[21], if the following n2 nðn þ m−1Þ controllability matrix
Q¼ In 0 0 0 0 C −ðA þ BÞ In 0 0 C 0 0 0 In 0 0 0 0 0 −ðA þ BÞ C 0 0 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 ð3Þ
has rank n2; where A; B∈Rnn; C∈Rnm and I
n denotes the n n identity matrix.
Proof. Following the similar proof procedure as that in the work of Chen and Chou[3], in the
above matrix Q of Eq. (3), add ðA þ BÞ times the first (block) row to the second, then add
ðA þ BÞ times the second row to the third, and so on. The result is a matrix
ð4Þ
The controllability matrix C ðA þ BÞC ðA þ BÞn−1C
h i
is of rank n if and only if the matrix in Eq.(4) has rank n2 (i.e., the matrix in Eq.(3) has rank n2). And, the system
_xðtÞ ¼ ðA þ BÞxðtÞ þ CuðtÞ with t4t0 is controllable, if and only if
rankð C ðA þ BÞC ðA þ BÞn−1C
h i
Þ ¼ n[11]. So, fromLemma 1, we can conclude that, if the matrix in Eq. (3) has rank n2, then, for any τ40, the linear time-delay system
_xðtÞ ¼ AxðtÞ þ Bxðt−τÞ þ CuðtÞ with t4t0 is controllable in sense of Weiss[21]. Q.E.D.
Lemma 3. Desoer and Vidyasagar [7]
The matrix measures of the matrices W and V; namely μðWÞ and μðVÞ respectively, are well
defined for any norm and have the following properties: (i)μð7IÞ ¼ 71; for the identity matrix I;
(ii)−∥W∥≤−μð−WÞ≤ReðλðWÞÞ≤μðWÞ≤∥W∥; for any norm ∥∥ and any matrix W∈Cnn;
(iii)μðW þ VÞ≤μðWÞ þ μðVÞ; for any two matrices W; V∈Cnn;
(iv) μðγWÞ ¼ γμðWÞ; for any matrix W∈Cnn and any non-negative real numberγ;
whereλðWÞ denotes any eigenvalue of W; and ReðλðWÞÞ denotes the real part of λðWÞ:
Lemma 4. For anyγo0 and any matrix W∈Cnn; μðγWÞ ¼ −γμð−WÞ:
Lemma 5. Chen and Chou [3]
Let N∈Cnn: If μð−NÞo1; then detðI þ NÞ≠0:
Proof. From the property (ii) in Lemma 3 and since μð−NÞo1; we can get that
ReðλðNÞÞ≥−μð−NÞ4−1: This implies that λðNÞ≠−1: So, we have the stated result. Q.E.D.
FromLemma 2, it is known that, for anyτ40, the uncertain linear time-delay system in Eqs.
(1) and (2) is robustly controllable in sense of Weiss [21], if the following n2 nðn þ m−1Þ matrix
~Q ¼ Q þ ∑m
k¼ 1
εkEk ð5Þ
has full row rank n2; where Q is given in Eq.(3), and
Ek¼ 0 0 0 0 0 Ck −ðAkþ BkÞ 0 0 0 Ck 0 0 0 0 0 0 0 0 0 −ðAkþ BkÞ Ck 0 0 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 : ð6Þ
Let the singular value decomposition of Q; which has rank n2; be
Q¼ U S 0n2nðm−1Þ
h i
VH; ð7Þ
where U∈Rn2n2
and V∈Rnðnþm−1Þnðnþm−1Þ are the unitary matrices, VH denotes the
complex-conjugate transpose of matrix V; S ¼ diag½s1; …; sn2; and s1≥s2≥⋯≥sn240 are the singular
values of Q.
In what follows, we present a sufficient criterion for ensuring that, for any τ40, the uncertain linear time-delay system in Eqs.(1)and(2)is robustly controllable in sense of Weiss[21].
Theorem. For anyτ40, the uncertain linear time-delay system in Eqs.(1)and(2)is robustly
controllable in sense of Weiss [21], if the matrix Q in Eq. (3) has a full row rank, and if the following condition holds
∑m k¼ 1 εkfko1; ð8Þ where fk¼ μð−S−1UHE kV½In2; 0n2nðm−1ÞTÞ; −μðS−1UHE kV½In2; 0n2nðm−1ÞTÞ; for for εk≥0; εko0; (
the matrices Ek; S; U and V are, respectively, defined in Eqs.(6)and(7), and In2denotes the
Proof. Since the matrix Q in Eq.(3)has a full row rank and we know that
rankðQÞ ¼ rankðS−1UHQVÞ; ð9Þ
thus, instead of rankð ~QÞ; we can discuss the rank of In2 0n2nðm−1Þ
h i
þ ∑m
k¼ 1
εkRk; ð10Þ
where Rk¼ S−1UHEkV; for k ¼ 1; 2; …; m: Since a matrix has at least rank n2if it has at least one
nonsingular n2 n2submatrix, a sufficient condition for the matrix in Eq.(10)to have rank n2is
the nonsingularity of G¼ In2þ ∑ m k¼ 1 εkRk; ð11Þ where Rk¼ S−1UHEkV½In2; 0n2nðm−1ÞT:
Using the properties in Lemmas 3 and 4, and from Eq.(8), we have μð− ∑m k¼ 1 εkRkÞ ¼ μ − ∑ m k¼ 1 εk S−1UHEkV½In2; 0n2nðm−1ÞT ≤ ∑m k¼ 1 μ −εk S−1UHEkV½In2; 0n2nðm−1ÞT ¼ ∑m k¼ 1 εkϕko1: ð12Þ
Thus, from Lemma 5, we have
detðGÞ ¼ detðIn2þ ∑
m k¼ 1
εkRkÞ≠0: ð13Þ
Hence, the matrix G in Eq.(11)is nonsingular. That is, the matrix ~Q in Eq.(5)has full row rank n2: Therefore, from the results mentioned-above andLemma 2, it is ensured that, for any τ40, the uncertain linear time-delay system in Eqs.(1)and(2)is robustly controllable in sense of Weiss[21]. Q.E.D.
Remark 1. The proposed sufficient condition in Eq.(8)can give the explicit relationship of the
bounds on εk for guaranteeing the robust controllability. In addition, the bounds, that are
obtained by using the proposed sufficient condition, on εk are not necessarily symmetric with
respect to the origin of the parameter space regardingεk, in which k¼ 1; 2; …; m: On the other
hand, if the parametric uncertaintiesεk(k¼ 1; 2; ⋯; m) are unknown, then the inequality in Eq.
(8) can be rewritten as εoð1=∑m
k¼ 1ϕkÞ, where jεkj≤ε, to estimate the maximal bound on
parametric uncertainties for guaranteeing the robust controllability.
3. Illustrative examples
In this section, three numerical examples are given to illustrate the applications of the
proposed sufficient algebraic criterion. We will also compare the results of the proposed
sufficient condition for linear system having no state delay with those obtained from the
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