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Lower bounds for stability margin of n-dimensional discrete systems
Computers and Electrical Engineering 29 (2003) 861–871 www.elsevier.com/locate/compeleceng
Lower bounds for stability margin of n-dimensional discrete systems T. Fernando a
a,*
, H. Trinh b, S. Nahavandi
b
Department of Electrical and Electronic Engineering, The University of Western Australia, 35 Stirling Hwy, Crawley, WA 6009, Australia b School of Engineering and Technology, Deakin University, Geelong 3217, Australia Received 28 February 2001; received in revised form 22 August 2001; accepted 6 July 2002
Abstract This paper derives lower bounds for the stability margin of n-dimensional discrete systems in the RoesserÕs state space setting. The lower bounds for stability margin are derived based on the MacLaurine series expansion. Numerical examples are given to illustrate the results. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Multi-dimensional systems; Robust stability; n-dimensional MacLaurine series
1. Introduction The stability analysis of two-dimensional (2-D) discrete systems [1,2] has received considerable attention in the literature. In the early work, Ahmed [3] derived a sufficient condition for checking stability of 2-D systems. The condition is computationally simple. However, it is rather conservative as it is based on matrix norms approach. Based on a 2-D Lyapunov equation approach, [4– 6] derived sufficient conditions for asymptotic stability of 2-D discrete systems. Later, Lu [7] presented a new class of generalized 2-D Lyapunov equations and showed that they can be used to confirm stability of a broader class of 2-D discrete systems. The stability results of 2-D systems have also been extended to n-D systems based on solving an n-D Lyapunov equation [8,9]. On the other hand, the subject of stability margin, i.e. ‘‘how stable’’ or ‘‘how far’’ a system from being unstable has also been widely studied over the years [10–12]. Early studies on stability *
Corresponding author. Tel.: +61-89-380-3954; fax: +61-89-380-1065. E-mail address: [email protected] (T. Fernando).
0045-7906/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0045-7906(03)00038-7
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Nomenclature Rn real vector space of dimension n identity matrix of order n In detðAÞ determinant of matrix A qðAÞ spectral radius of matrix A kmax ðAÞ, kmin ðAÞ maximum, minimum eigenvalue of matrix A, respectively jAj modulus matrix of A transpose of matrix A AT kAk matrix norm of A, kAk ¼ rmax ðAÞ ½kmax ðAT AÞ1=2 W1 0 W ¼ W1 W2 direct sum, i.e. W ¼ 0 W2 A > ð P ÞB matrix ðA BÞ is positive definite (semi-definite)
margin [10,11] have been carried out in the frequency domain. The development of the timedomain representation [1,2] of multi-dimensional systems and of the Lyapunov based methods [4– 7] have made it possible to investigate stability margins in the state-space domain. Agathoklis [12] derived expressions for lower bounds for stability margin based on solving a 2-D Lyapunov equation. These reported lower bounds in [12] are for 2-D discrete systems in the RoesserÕs state space setting. So far, a literature search indicates that lower bounds for stability margin in the time-domain has not been reported for n-D systems. This paper investigates the margin of stability from a state-space approach and considers a general n-D systems in RoesserÕs state space setting. Lower bounds for stability margin are derived based on the MacLaurine series expansion. The algorithm for computing the bounds involves the computation of a power series for nominally stable n-D systems. The bounds are computationally easy to derive. It will be shown, by a numerical example, that for the 2-D case, the results of this paper can provide less conservative margins than the results of Agathoklis [12], and it does not require the solution to a multi-dimensional Lyapunov equation. This paper is organized as follows: In Section 2, the n-D discrete state-space model is introduced and some previous stability results using state-space methods are discussed. In Section 3, robust stability conditions for n-D discrete systems under structured perturbations are derived based on the MacLaurine series expansion. These stability conditions are then extended to derive lower bounds for stability margin. In Section 4, two numerical examples are given to illustrate the results. Finally, Section 5 concludes the paper.
2. Preliminaries Linear shift invariant n-D discrete systems can be represented by the following state space model [8,9]:
T. Fernando et al. / Computers and Electrical Engineering 29 (2003) 861–871
2
3
2
x1 ði1 þ 1; i2 ; . . . ; in Þ A11 6 x2 ði1 ; i2 þ 1; . . . ; in Þ 7 6 A21 6 7 6 6 7 ¼ 6 .. .. 4 5 4 . . An1 xn ði1 ; i2 ; . . . ; in þ 1Þ
32
3
2
863
3
A12 A22 .. .
.. .
x1 ði1 ; i2 ; . . . ; in Þ A1n B1 6 7 7 6 A2n 76 x2 ði1 ; i2 ; . . . ; in Þ 7 6 B2 7 7 7 þ 6 .. 7uði1 ; i2 ; . . . ; in Þ; .. 76 .. 4 5 5 4 . 5 . .
An2
Ann
xn ði1 ; i2 ; . . . ; in Þ
Bn ð1aÞ
2
yði1 ; i2 ; . . . ; in Þ ¼ ½ C1
C2
3 x1 ði1 ; . . . ; in Þ 6 7 .. Cn 4 5; .
ð1bÞ
xn ði1 ; . . . ; in Þ where xi 2 Rmi , i ¼ 1; . . . ; n represent the states, u is the input and y is the output. The stability of system (1) depends on the location of zeros of the following characteristic polynomial: 3 2 Im1 z1 A11
z1 A12
z1 A1n 6 z2 A21 Im2 z2 A22
z2 A2n 7 7: ð2Þ Cðz1 ; z2 ; . . . ; zn Þ ¼ det6 5 4
zn An2 Imn zn Ann
zn An1 The n-D stability condition requires that the characteristic polynomial has no zeros inside the n closed n-disk U , i.e., n
Cðz1 ; z2 ; . . . ; zn Þ 6¼ 0 in U ;
ð3Þ
where n
U ¼ fðz1 ; . . . ; zn Þjjz1 j 6 1; . . . ; jzn j 6 1g: It was reported in [8,9] that a sufficient condition for the stability of the system described in (1) is the existence of matrices W > 0 and Q > 0 satisfying the following n-D Lyapunov equation: W AT WA ¼ Q;
ð4aÞ
where 2
A11 6 A21 6 A ¼ 6 .. 4 .
A12 A22 .. .
.. .
3 A1n A2n 7 7 .. 7; . 5
An1
An2
Ann
ð4bÞ
W is the direct sum of symmetric matrices Wi ði ¼ 1; 2; . . . ; nÞ of dimensions ðmi mi Þ, i.e., W ¼ W1 W2 Wn and Q is a symmetric matrix of dimension ðm1 þ þ mn Þ2 . For such stable systems, the stability margin is referred to as the distance between the boundary of the unit n-disc to the closest stable pole of the system. If the system characterized by (1) is stable then the stability margins are ri ði ¼ 1; . . . ; nÞ and r. They are defined as the largest n-disc where the polynomial Cðz1 ; z2 ; . . . ; zn Þ has no zeros [10], i.e.,
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Cðz1 ; z2 ; . . . ; zn Þ 6¼ 0 in Uri2 ¼ fðz1 ; z2 ; . . . ; zn Þjjzi j < 1 þ ri ; jzj j < 1 8j 6¼ ig;
ð5Þ
Cðz1 ; z2 ; . . . ; zn Þ 6¼ 0 in Ur2 ¼ fðz1 ; z2 ; . . . ; zn Þjjzi j < 1 þ r 8i ¼ 1; . . . ; ng:
ð6Þ
Agathoklis [12] derived lower bounds for stability margins based on the 2-D Lyapunov equation approach: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Q1 Q2
1
1
1 T r1 P kmax ½M1
1; where M1 ¼ In W1 Q1 þ W1 Q2 Q3 Q2 ; Q ¼ ; ð7Þ QT2 Q3 r2 P rP
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 kmax ½M2
1; where M2 ¼ Im W2 1 Q3 þ W2 1 QT2 Q 1 1 Q2 ;
ð8Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 kmax ½M
1; where M ¼ IðnþmÞ W 1 Q:
ð9Þ
To calculate the above lower bounds for stability margin (7)–(9), it is required to compute W and Q that satisfy Eq. (4). Furthermore, the above result applies to 2-D systems only. In this paper, lower bounds for stability margin are derived for n-D systems. The approach proposed in this paper is based on the MacLaurine series expansion and it does not require the solution to a n-D Lyapunov equation.
3. Proposed algorithm In this section, to proceed with the development of the algorithm we first present new robust stability conditions for n-D uncertain discrete systems of the form 3 2 3 32 2 x1 ði1 ; . . . ; in Þ x1 ði1 þ 1; i2 ; . . . ; in Þ A11 þ DA11 A12 þ DA12 A1n þ DA1n 6 x2 ði1 ; i2 þ 1; . . . ; in Þ 7 6 A21 þ DA21 A22 þ DA22 A2n þ DA2n 76 x2 ði1 ; . . . ; in Þ 7 7 6 7 76 6 7¼6 7 76 6 .. .. .. .. .. .. 5 4 5 54 4 . . . . . . An1 þ DAn1 An2 þ DAn2 2 3 B1 6 B2 7 6 7 þ 6 .. 7uði1 ; . . . ; in Þ; 4 . 5 Bn
xn ði1 ; i2 ; . . . ; in þ 1Þ
Ann þ DAnn
xn ði1 ; . . . ; in Þ
ð10aÞ 2
yði1 ; i2 ; . . . ; in Þ ¼ ½ C1
C2
3 x1 ði1 ; . . . ; in Þ 6 7 .. Cn 4 5; .
ð10bÞ
xn ði1 ; . . . ; in Þ where perturbation matrices DAkl ðk; l ¼ 1; 2; . . . ; nÞ are assumed to be independently structured with jDAkl j 6 qEkl ðk; l ¼ 1; 2; . . . ; nÞ, q is a positive constant number, and Ekl are non-negative
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matrices representing highly structured information for perturbations on the entries of Akl . To proceed with the main results, let us first introduce the following results: Lemma 1 [13]. For any ðn nÞ matrix R, the following statement is true fqðRÞ < 1g implies fdetðIn RÞ 6¼ 0g:
ð11Þ
Theorem 1. Let Xi 2 Rðm1 þ þmn Þðm1 þ þmn Þ 8i ¼ 1; 2; . . . ; n be defined as 2 3 f0g Xi ¼ 4 Ai1 Ain 5 8i ¼ 1; 2; . . . ; n f0g 1 Pn and the MacLaurine series for I j¼1 zj Xj be expressed as ! 1 !i n 1 n X X X zj Xj ¼ zj Xj : Gðz1 ; z2 ; . . . ; zn Þ ¼ I
j¼1
i¼0
ð12Þ
ð13Þ
j¼1
Then jGðz1 ; z2 ; . . . ; zn Þj 6 I þ
n X
n
S j S 8ðz1 ; z2 ; . . . ; zn Þ 2 U ;
ð14Þ
j¼1
where 0 2 3 0 S1 B 6 T2 6 6 S2 7 B 6 . 6 7 B 6 . 7 ¼ B I 6 .. 6 4 .. 5 B @ 4 Tn 1 Sn Tn 2
T1 0 .. .
T1 T2 .. .
.. .
Tn
Tn 1
0 Tn
T1 T2 .. .
31 1
7C 7C 7C 7C 7C Tn 1 5A 0
2
3 T1 6 T2 7 6 7 6 .. 7 4 . 5
ð15Þ
Tn
and Tj ¼
1 X ðXj Þi
8j ¼ 1; 2; . . . ; n:
ð16Þ
i¼1
Proof of Theorem 1. Let us introduce the set W ¼ fX1 ; X2 ; . . . ; Xn g. Let W1 ¼ W2 ¼ ¼ Wk ¼ W Sk and U ¼ i¼1 Wi . Now let vk be the set of all k element subsets of the S set U. Also let e 2 vk , and the set He be the set of all permutations of e. Let us introduce Nk ¼ e2vk He . Now we can define the following sum X Pk ¼ C: ð17Þ C2Nk
From (13) and its MacLaurine expansion, Gðz1 ; z2 ; . . . ; zn Þ can be expanded as follows n X Gðz1 ; z2 ; . . . ; zn Þ ¼ I þ Sj ; j¼1
ð18Þ
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T. Fernando et al. / Computers and Electrical Engineering 29 (2003) 861–871
where
"
Sj ¼ Xj zj I þ
1 X
# Pk
8j ¼ 1; 2; . . . ; n:
ð19Þ
k¼1
Eq. (19) can be rewritten as follows 1 0 ! 1 n X X C B Sj ¼ I þ Xkj zjk B Sk C A 8j ¼ 1; 2; . . . ; n: @ k¼1
ð20Þ
k¼1 k6¼j
From Eq. (20), it is clear that for jzj j 6 1 8j ¼ 1; 2; . . . ; n the following bounds for Sj 8j ¼ 1; 2; . . . ; n can be obtained 1 1 0 0 n X n X C B B jSj jC ð21Þ jSi j 6 Ti @I þ Sj A 6 Ti @I þ A 8i ¼ 1; 2; . . . ; n: j¼1 j¼1 j6¼i j6¼i Solving inequalities in (21), gives: 0 2 2 3 0 T1 T1 S1 B 6 T2 0 T2 6 6 S2 7 B 6 .. 6 7 B .. .. 6 .. 7 6 B I 6 . . . 6 4 . 5 B @ 4 Tn 1 Tn 1 Sn Tn Tn Accordingly, this completes the proof
.. .
T1 T2 .. .
31 1
7C 7C 7C 7C 7C 0 Tn 1 5A Tn 0 of Theorem 1.
3 2 3 T1 S1 6 T2 7 6 S 2 7 6 7 6 7 6 .. 7 6 .. 7: 4 . 5 4 . 5 2
Tn
ð22Þ
Sn
Remark 1. The power series (16) will always converge provided that the spectral radius of matrix Xj is less than 1, i.e. ð23aÞ qðXj Þ < 1 8j ¼ 1; 2; . . . n; where Xj is as defined in Eq. (12). Accordingly, provided that condition (23a) is satisfied, the power series can be computed in a finite number of terms. It is also clear from (22) that matrix S defined in (14) exists provided that condition (23a) holds and 3 2 T1 T1 0 T1 6 T2 0 T2 T2 7 7 6 6 . .. 7 .. .. .. < 1: ð23bÞ q6 .. . 7 . . . 7 6 4 Tn 1 Tn 1 0 Tn 1 5 Tn Tn Tn 0 Remark 2. For the case where jXi j ¼ Xi 8i ¼ 1; 2; . . . ; n and matrix ðX1 þ þ Xn Þ is asymptotically stable, then it can be shown in a similar way to that in [14], S ¼ ðI X1 Xn Þ 1 . When jXi j 6¼ Xi and if matrix ðjX1 j þ þ jXn jÞ is asymptotically stable, then S satisfies the following upper bound: S 6 ðI jX1 j jXn jÞ 1 .
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In the following, we present new robust stability conditions for system (10) based on the result derived in Theorem 1. We assume that the nominal system is stable and that matrix S exists. The results are given in the following theorem. Theorem 2. System (10) is robustly stable provided that the following inequality is satisfied 1 ; q< 2 3 q½SE E11 E1n 6 . .. 7 .. where matrix S is defined in Eqs. (14), and E ¼ 4 .. . 5. . En1 Enn
ð24Þ
Proof of Theorem 2. The stability condition of system (10) can also be expressed as det½I z1 X1 zn Xn ðz1 DX1 þ þ zn DXn Þ ¼ det½I z1 X1 zn Xn det½I ðI z1 X1 zn Xn Þ 1 ðz1 DX1 þ þ zn DXn Þ 6¼ 0; n
for ðz1 ; . . . ; zn Þ 2 U ; ð25Þ where
2
3 f0g DXi ¼ 4 DAi1 DAin 5 8i ¼ 1; 2; . . . ; n: ð26Þ f0g Since the nominal system is assumed to be asymptotically stable, det½I z1 X1 zn Xn 6¼ 0 n for ðz1 ; . . . ; zn Þ 2 U . Therefore condition (25) reduces to det½I ðI z1 X1 zn Xn Þ 1 ðz1 DX1 þ þ zn DXn Þ 6¼ 0
n
for ðz1 ; . . . ; zn Þ 2 U :
ð27Þ
Substituting (14) into (27) and using Lemma 1 gives qðGðz1 ; . . . ; zn Þðz1 DX1 þ þ zn DXn ÞÞ < 1
n
for ðz1 ; . . . ; zn Þ 2 U :
ð28Þ
Again, by using the properties of spectral radius and by considering the structure of the uncertainty as defined in (26), the following inequality can be obtained qðGðz1 ; . . . ; zn Þðz1 DX1 þ þ zn DXn ÞÞ 6 qðjðGðz1 ; . . . ; zn Þjðjz1 jDX1 j þ þ jzn jDXn jÞÞ 6 qðjGðz1 ; . . . ; zn ÞjqEÞ
n
for ðz1 ; . . . ; zn Þ 2 U :
ð29Þ
Substituting the upper bound for jGðz1 ; . . . ; zn Þj (i.e. (14) from Theorem 1) into the right hand side of (29), gives qðjGðz1 ; . . . ; zn ÞjqEÞ 6 qðqSEÞ
n
for ðz1 ; . . . ; zn Þ 2 U :
ð30Þ
Accordingly, provided that the stability condition in Theorem 2 is satisfied, the system (10) is robustly stable. This completes the Proof of Theorem 2. Now the lower bounds for stability margin can be derived as a special case of Theorem 2, i.e. by choosing the structure of the uncertainty we can obtain the required lower bounds. Let us now introduce the following Lemma and Corollary.
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Lemma 2. For qj P 0 8j ¼ 1; . . . ; n, if C1 ðz1 ; z2 ; . . . ; zn Þ 2 Im1 z1 ð1 þ q1 ÞA11
z1 ð1 þ q1 ÞA12 6 z2 ð1 þ q2 ÞA21 Im2 z2 ð1 þ q2 ÞA22 6 ¼ det6 . 6 .. 4
zn ð1 þ qn ÞAn1 6¼ 0
z1 ð1 þ q1 ÞA1n
3
z2 ð1 þ q2 ÞA2n .. .
7 7 7 7 5
zn ð1 þ qn ÞAnn 1
Imn zn ð1 þ qn ÞAnn
n
8ðz1 ; . . . ; zn Þ 2 U ;
then
ð31Þ
2
Im1 z01 A11 6 z02 A21 6 0 0 Cðz1 ; . . . ; zn Þ ¼ det 6 .. 4 .
z0n An1
z01 A12 Im2 z02 A22
..
. 0
zn Ann 1
z01 A1n
z02 A2n .. . Imn z0n Ann
3 7 7 7 6¼ 0 in U n ; 5
ð32Þ
where
U n ¼ ðz01 ; . . . ; z0n Þjz01 j < 1 þ q1 ; . . . ; jz0n j < 1 þ qn :
ð33Þ
Proof of Lemma 2. The result in (32) follows immediately if the variable transformation z0i ¼ ð1 þ qi Þzi 8i ¼ 1; 2; . . . ; n is applied to (31). Corollary 1. Let z0i ¼ ð1 þ qi Þzi and z0j ¼ zj 8j ¼ 1; . . . ; i 1; i þ 1; . . . ; n (i.e. choose qi 6¼ 0 and qj ¼ 0 8j ¼ 1; . . . ; i 1; i þ 1; . . . ; nÞ, then 3 2
z01 A12
z01 A1n Im1 z01 A11 6 z02 A21 Im2 z02 A22
z02 A2n 7 7 6 0 0 ð34Þ Cðz1 ; . . . ; zn Þ ¼ det6 7 6¼ 0 in Uqin ; .. .. .. 5 4 . . .
z0n An1
z0n Ann 1
Imn z0n Ann
where
Uqin ¼ ðz01 ; . . . ; z0n Þjz0i j < 1 þ qi ; jz01 j < 1; . . . ; jz0i 1 j < 1; jz0iþ1 j < 1; . . . ; jz0n j < 1 :
ð35Þ
The lower bounds on the margin of stability of stable systems characterized by (1) can then be easily deduced from the results of Theorem 2, Lemma 2 and Corollary 1. They are now expressed in the following theorem. Theorem 3. The lower bounds for stability margin ri 8i ¼ 1; 2; . . . ; n and r are provided by the upper bound of qi 8i ¼ 1; 2; . . . ; n and q respectively, and satisfying the following conditions 1 f0g q@S 4 jAi1 j f0g
qi < 0 2
31 jAin j 5A
8i ¼ 1; 2; . . . ; n
1 jA11 j B 6 .. .. q@S 4 . . jAn1 j
and q < 0 2
31 : jA1n j .. 7C . 5A jAnn j ð36Þ
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869
3
f0g Proof of Theorem 3. From Theorem 2, by letting E ¼ 4 jAi1 j f0g
jAin j 5 then it is clear that
provided the condition (36) is satisfied, the characteristic polynomial Cðz1 ; . . . ; zi 1 ; zi ð1 þ qi Þ; ziþ1 ; . . . ; zn Þ 6¼ 0
n
ð37Þ
in U :
From Corollary 1 it is clear that (37) implies Cðz1 ; . . . ; zn Þ 6¼ 0
in Uqin ;
ð38Þ
where Uqin ¼ fðz1 ; . . . ; zn Þjjzi j < 1 þ qi ; jz1 j < 1; . . . ; jzi 1 j < 1; jziþ1 j < 1; . . . ; jzn j < 1g:
ð39Þ
Because condition (24) only provides a sufficient condition for stability, the value of qi satisfying (36) is only a conservative estimate in the above equation. Therefore the n-disc Uqin is not the largest n-disc where the characteristic equation Cðz1 ; . . . ; zn Þ 6¼ 0. From the definition of stability margin (i.e. (5) and (6)), it follows that qi is a lower bound for ri . Similarly, by choosing 2 3 jA11 j jA1n j 6 .. .. 7 ð40Þ E ¼ 4 ... . . 5 jAn1 j
jAnn j
and by following the above same lines of proof as for qi , q is a lower bound for r. This completes the Proof of Theorem 3.
4. Numerical Examples Let us look at two numerical examples to illustrate the results of this paper. Example 1. Consider the following 2-D Rosser model with 0:4 0:3 1 0 0:1 0:03 A11 ¼ ; A12 ¼ ; A21 ¼
0:3 0 0 0 0:1 0:2 Using (15) and (16), matrix 2 3:0655 1:7422 6 0:9196 1:5227 S¼6 4 1:0557 0:8683 0:7016 0:6524
and A22 ¼
0:1 0:2
0:9 : 0
S for this system can be computed as 3 4:2576 3:8318 1:2773 1:1495 7 7: 2:8552 2:5696 5 1:2522 2:1270
Using Eq. (36) the lower bounds can be obtained and are shown in Table 1. Table 1 also shows the lower bounds derived by the method proposed by Agathoklis [12] with
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Table 1 Comparison of lower bounds for stability margin This paper
Agathoklis [12]
r1 ¼ 0:3961 r2 ¼ 0:3823 r ¼ 0:1984
r1 ¼ 0:2929 r2 ¼ 0:1123 r ¼ 0:0869
Table 2 Lower bounds for stability margin for the 3-D Rosser model r1
r2
r3
r
0.2576
0.4263
0.6814
0.1254
2
4 61 W ¼6 40 0
1 5 0 0
3 2 0 0 2:9700 6 0:4340 0 0 7 7 and Q ¼ 6 4 1:5800 8 0 5 0 10
0:7200
0:4340 4:2328
1:5760 0:2160
1:5800
1:5760 3:5200
0:7200
3
0:7200 0:2160 7 7
0:7200 5 3:5200
being the positive definite solution of the 2-D Lyapunov equation (4). As can be seen in Table 1, the lower bounds computed using the method proposed in this paper are less conservative than those calculated by solving a 2-D Lyapunov equation. Example 2. Now consider 0:4 0:3 A11 ¼
0:3 0 A21 ¼ ½ 0:1 0:03 A31 ¼ ½ 0:1 0:2
the following 3-D Rosser model with 1 0 A12 ¼ A13 ¼ 0 0 : A22 ¼ 0:1 A23 ¼ 1 A32 ¼ 0:2 A33 ¼ 0
Using (15) and (16), matrix 2 3:4926 2:1954 6 1:0478 1:6586 S¼6 4 1:3422 1:1722 0:8273 0:7857
S for this system can be found as 3 4:9895 4:9895 1:4968 1:4968 7 7: 3:3459 3:3459 5 1:4675 2:4675
Using (36) the lower bounds can be obtained and are shown in Table 2.
5. Conclusion In this paper, a method based on the MacLaurine series expansion for computing lower bounds for stability margin has been presented. The method does not require to solve a n-D Lyapunov equation. It was shown in numerical Example 1 that the general result for n-D systems when applied to a 2-D system could compute lower bounds which are less conservative than bounds computed by solving a 2-D Lyapunov equation.
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References [1] Roesser RP. A discrete state space model for linear image processing. IEEE Trans Automat Contr 1975;20:1–10. [2] Fornasini E, Marchesini G. Doubly-indexed dynamical systems: state-space models and structural properties. Math Syst Theory 1978;12:59–72. [3] Ahmed ARE. On the stability of two-dimensional discrete systems. IEEE Trans Automat Contr 1980;25:551–2. [4] Lodge JH, Fahmy NM. Stability and overflow oscillations in 2-D state-space digital filters. IEEE Trans Acoust Speech Signal Process 1981;29:1161–71. [5] Anderson BDO, Agathoklis P, Jury EI, Mansour M. Stability and the matrix Lyapunov equation for discrete 2dimensional systems. IEEE Trans Circ Syst 1986;33:261–7. [6] Hinamoto T. 2-D Lyapunov equation and filter design based on the Fornasini–Marchesini second model. IEEE Trans Circ Syst 1993;40:102–10. [7] Lu WS. On a Lyapunov approach to stability analysis of 2-D digital filters. IEEE Trans Circ Syst 1994;41:665–9. [8] Xiao C, Hill DJ, Agathoklis P. Stability and the Lyapunov equation for n-dimensional digital systems. IEEE Trans Circ Syst 1997;44:614–21. [9] Agathoklis P. The Lyapunov equation for n-dimensional discrete systems. IEEE Trans Circ Syst 1988;35:448–51. [10] Walach E, Zeheb E. N -dimensional stability margins computation and a variable transformation. IEEE Trans Acoust Speech Signal Process 1982;30:887–94. [11] Agathoklis P, Jury EI, Mansour M. The margin of stability of 2-D linear discrete systems. IEEE Trans Acoust Speech Signal Process 1982;30:869–74. [12] Agathoklis P. Lower bounds for the stability margin of discrete two-dimensional systems based on the twodimensional Lyapunov equation. IEEE Trans Circ Syst 1988;35:745–9. [13] Chou JH. Pole-assignment robustness in a specified disk. Syst Contr Lett 1991;16:41–4. [14] Shafai B, Chen J, Kothandaraman M. Explicit formulas for stability radii of nonnegative and Metzlerian matrices. IEEE Trans Automat Contr 1997;42:265–70. Tyrone Fernando obtained his B.E. (Hons) and PhD degrees in electrical engineering from the University of Melbourne, Australia in 1990 and 1996 respectively. In 1995 he was a visiting lecturer in the Department of Electrical and Computer Engineering at Monash University, Melbourne, Australia. He has been with the Department of Electrical and Electronic Engineering, University of Western Australia since 1996 where he is currently a Senior Lecturer. He has taught a wide range of courses in control, signal processing, systems and network theory. His main research interests are in the fields of biomedical engineering, multi-dimensional systems theory and robust control. Hieu Trinh obtained his BEng (Hons), MEngSc and PhD degrees in electrical engineering, all from The University of Melbourne in 1990, 1992 and 1996. Between March 1995 and December 1996, he was a postdoctoral research fellow in the Department of Electrical and Electronic Engineering, The University of Melbourne. Between January 1997 and December 2000, he was a lecturer at James Cook University, Townsville, Australia. He joined Deakin University in January 2001 where he is currently a Senior Lecturer. He has taught a wide range of courses in control, power systems, electric machines, electronics and network theory. His current research activities are in the areas of systems and control theory, multi-dimensional (n-D) systems theory, intelligent control, optimisation, fault diagnosis and fault-tolerant control, robotics, control and analysis of power systems, and application of control theory to industrial systems. So far, he has published over 75 research papers on a wide range of topics in Systems and Control Theory, and their application to real-life systems. Saeid Nahavandi received BSc (Hons), MSc and a PhD in Automation from Durham University (UK). In 1991 he joined Massey University (NZ) where he taught and led research in robotics. In 1998 he became an Associate Professor at Deakin University (AU) and the leader for the Intelligent Systems research group and also Manager for the Cooperative Research Center for CAST Metals Manufacturing. In 2002 Professor Nahavandi took the position of Chair in Engineering in the same university. Dr. Nahavandi has published over 100 reviewed papers and delivered several invited plenary lectures at international conferences. He is the recipient of four international awards, best paper award at the World Automation Congress (USA) and the Young Engineer of the Year award. Professor Nahavandi is the founder of the World Manufacturing Congress series and the Autonomous Intelligent Systems Congress series. He is a Fellow of IEAust and IEE. His current research interests include modeling and control and the application of soft computing to industrial processes.
Lower bounds for stability margin of n-dimensional discrete systems T. Fernando a
a,*
, H. Trinh b, S. Nahavandi
b
Department of Electrical and Electronic Engineering, The University of Western Australia, 35 Stirling Hwy, Crawley, WA 6009, Australia b School of Engineering and Technology, Deakin University, Geelong 3217, Australia Received 28 February 2001; received in revised form 22 August 2001; accepted 6 July 2002
Abstract This paper derives lower bounds for the stability margin of n-dimensional discrete systems in the RoesserÕs state space setting. The lower bounds for stability margin are derived based on the MacLaurine series expansion. Numerical examples are given to illustrate the results. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Multi-dimensional systems; Robust stability; n-dimensional MacLaurine series
1. Introduction The stability analysis of two-dimensional (2-D) discrete systems [1,2] has received considerable attention in the literature. In the early work, Ahmed [3] derived a sufficient condition for checking stability of 2-D systems. The condition is computationally simple. However, it is rather conservative as it is based on matrix norms approach. Based on a 2-D Lyapunov equation approach, [4– 6] derived sufficient conditions for asymptotic stability of 2-D discrete systems. Later, Lu [7] presented a new class of generalized 2-D Lyapunov equations and showed that they can be used to confirm stability of a broader class of 2-D discrete systems. The stability results of 2-D systems have also been extended to n-D systems based on solving an n-D Lyapunov equation [8,9]. On the other hand, the subject of stability margin, i.e. ‘‘how stable’’ or ‘‘how far’’ a system from being unstable has also been widely studied over the years [10–12]. Early studies on stability *
Corresponding author. Tel.: +61-89-380-3954; fax: +61-89-380-1065. E-mail address: [email protected] (T. Fernando).
0045-7906/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0045-7906(03)00038-7
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Nomenclature Rn real vector space of dimension n identity matrix of order n In detðAÞ determinant of matrix A qðAÞ spectral radius of matrix A kmax ðAÞ, kmin ðAÞ maximum, minimum eigenvalue of matrix A, respectively jAj modulus matrix of A transpose of matrix A AT kAk matrix norm of A, kAk ¼ rmax ðAÞ ½kmax ðAT AÞ1=2 W1 0 W ¼ W1 W2 direct sum, i.e. W ¼ 0 W2 A > ð P ÞB matrix ðA BÞ is positive definite (semi-definite)
margin [10,11] have been carried out in the frequency domain. The development of the timedomain representation [1,2] of multi-dimensional systems and of the Lyapunov based methods [4– 7] have made it possible to investigate stability margins in the state-space domain. Agathoklis [12] derived expressions for lower bounds for stability margin based on solving a 2-D Lyapunov equation. These reported lower bounds in [12] are for 2-D discrete systems in the RoesserÕs state space setting. So far, a literature search indicates that lower bounds for stability margin in the time-domain has not been reported for n-D systems. This paper investigates the margin of stability from a state-space approach and considers a general n-D systems in RoesserÕs state space setting. Lower bounds for stability margin are derived based on the MacLaurine series expansion. The algorithm for computing the bounds involves the computation of a power series for nominally stable n-D systems. The bounds are computationally easy to derive. It will be shown, by a numerical example, that for the 2-D case, the results of this paper can provide less conservative margins than the results of Agathoklis [12], and it does not require the solution to a multi-dimensional Lyapunov equation. This paper is organized as follows: In Section 2, the n-D discrete state-space model is introduced and some previous stability results using state-space methods are discussed. In Section 3, robust stability conditions for n-D discrete systems under structured perturbations are derived based on the MacLaurine series expansion. These stability conditions are then extended to derive lower bounds for stability margin. In Section 4, two numerical examples are given to illustrate the results. Finally, Section 5 concludes the paper.
2. Preliminaries Linear shift invariant n-D discrete systems can be represented by the following state space model [8,9]:
T. Fernando et al. / Computers and Electrical Engineering 29 (2003) 861–871
2
3
2
x1 ði1 þ 1; i2 ; . . . ; in Þ A11 6 x2 ði1 ; i2 þ 1; . . . ; in Þ 7 6 A21 6 7 6 6 7 ¼ 6 .. .. 4 5 4 . . An1 xn ði1 ; i2 ; . . . ; in þ 1Þ
32
3
2
863
3
A12 A22 .. .
.. .
x1 ði1 ; i2 ; . . . ; in Þ A1n B1 6 7 7 6 A2n 76 x2 ði1 ; i2 ; . . . ; in Þ 7 6 B2 7 7 7 þ 6 .. 7uði1 ; i2 ; . . . ; in Þ; .. 76 .. 4 5 5 4 . 5 . .
An2
Ann
xn ði1 ; i2 ; . . . ; in Þ
Bn ð1aÞ
2
yði1 ; i2 ; . . . ; in Þ ¼ ½ C1
C2
3 x1 ði1 ; . . . ; in Þ 6 7 .. Cn 4 5; .
ð1bÞ
xn ði1 ; . . . ; in Þ where xi 2 Rmi , i ¼ 1; . . . ; n represent the states, u is the input and y is the output. The stability of system (1) depends on the location of zeros of the following characteristic polynomial: 3 2 Im1 z1 A11
z1 A12
z1 A1n 6 z2 A21 Im2 z2 A22
z2 A2n 7 7: ð2Þ Cðz1 ; z2 ; . . . ; zn Þ ¼ det6 5 4
zn An2 Imn zn Ann
zn An1 The n-D stability condition requires that the characteristic polynomial has no zeros inside the n closed n-disk U , i.e., n
Cðz1 ; z2 ; . . . ; zn Þ 6¼ 0 in U ;
ð3Þ
where n
U ¼ fðz1 ; . . . ; zn Þjjz1 j 6 1; . . . ; jzn j 6 1g: It was reported in [8,9] that a sufficient condition for the stability of the system described in (1) is the existence of matrices W > 0 and Q > 0 satisfying the following n-D Lyapunov equation: W AT WA ¼ Q;
ð4aÞ
where 2
A11 6 A21 6 A ¼ 6 .. 4 .
A12 A22 .. .
.. .
3 A1n A2n 7 7 .. 7; . 5
An1
An2
Ann
ð4bÞ
W is the direct sum of symmetric matrices Wi ði ¼ 1; 2; . . . ; nÞ of dimensions ðmi mi Þ, i.e., W ¼ W1 W2 Wn and Q is a symmetric matrix of dimension ðm1 þ þ mn Þ2 . For such stable systems, the stability margin is referred to as the distance between the boundary of the unit n-disc to the closest stable pole of the system. If the system characterized by (1) is stable then the stability margins are ri ði ¼ 1; . . . ; nÞ and r. They are defined as the largest n-disc where the polynomial Cðz1 ; z2 ; . . . ; zn Þ has no zeros [10], i.e.,
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Cðz1 ; z2 ; . . . ; zn Þ 6¼ 0 in Uri2 ¼ fðz1 ; z2 ; . . . ; zn Þjjzi j < 1 þ ri ; jzj j < 1 8j 6¼ ig;
ð5Þ
Cðz1 ; z2 ; . . . ; zn Þ 6¼ 0 in Ur2 ¼ fðz1 ; z2 ; . . . ; zn Þjjzi j < 1 þ r 8i ¼ 1; . . . ; ng:
ð6Þ
Agathoklis [12] derived lower bounds for stability margins based on the 2-D Lyapunov equation approach: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Q1 Q2
1
1
1 T r1 P kmax ½M1
1; where M1 ¼ In W1 Q1 þ W1 Q2 Q3 Q2 ; Q ¼ ; ð7Þ QT2 Q3 r2 P rP
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 kmax ½M2
1; where M2 ¼ Im W2 1 Q3 þ W2 1 QT2 Q 1 1 Q2 ;
ð8Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 kmax ½M
1; where M ¼ IðnþmÞ W 1 Q:
ð9Þ
To calculate the above lower bounds for stability margin (7)–(9), it is required to compute W and Q that satisfy Eq. (4). Furthermore, the above result applies to 2-D systems only. In this paper, lower bounds for stability margin are derived for n-D systems. The approach proposed in this paper is based on the MacLaurine series expansion and it does not require the solution to a n-D Lyapunov equation.
3. Proposed algorithm In this section, to proceed with the development of the algorithm we first present new robust stability conditions for n-D uncertain discrete systems of the form 3 2 3 32 2 x1 ði1 ; . . . ; in Þ x1 ði1 þ 1; i2 ; . . . ; in Þ A11 þ DA11 A12 þ DA12 A1n þ DA1n 6 x2 ði1 ; i2 þ 1; . . . ; in Þ 7 6 A21 þ DA21 A22 þ DA22 A2n þ DA2n 76 x2 ði1 ; . . . ; in Þ 7 7 6 7 76 6 7¼6 7 76 6 .. .. .. .. .. .. 5 4 5 54 4 . . . . . . An1 þ DAn1 An2 þ DAn2 2 3 B1 6 B2 7 6 7 þ 6 .. 7uði1 ; . . . ; in Þ; 4 . 5 Bn
xn ði1 ; i2 ; . . . ; in þ 1Þ
Ann þ DAnn
xn ði1 ; . . . ; in Þ
ð10aÞ 2
yði1 ; i2 ; . . . ; in Þ ¼ ½ C1
C2
3 x1 ði1 ; . . . ; in Þ 6 7 .. Cn 4 5; .
ð10bÞ
xn ði1 ; . . . ; in Þ where perturbation matrices DAkl ðk; l ¼ 1; 2; . . . ; nÞ are assumed to be independently structured with jDAkl j 6 qEkl ðk; l ¼ 1; 2; . . . ; nÞ, q is a positive constant number, and Ekl are non-negative
T. Fernando et al. / Computers and Electrical Engineering 29 (2003) 861–871
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matrices representing highly structured information for perturbations on the entries of Akl . To proceed with the main results, let us first introduce the following results: Lemma 1 [13]. For any ðn nÞ matrix R, the following statement is true fqðRÞ < 1g implies fdetðIn RÞ 6¼ 0g:
ð11Þ
Theorem 1. Let Xi 2 Rðm1 þ þmn Þðm1 þ þmn Þ 8i ¼ 1; 2; . . . ; n be defined as 2 3 f0g Xi ¼ 4 Ai1 Ain 5 8i ¼ 1; 2; . . . ; n f0g 1 Pn and the MacLaurine series for I j¼1 zj Xj be expressed as ! 1 !i n 1 n X X X zj Xj ¼ zj Xj : Gðz1 ; z2 ; . . . ; zn Þ ¼ I
j¼1
i¼0
ð12Þ
ð13Þ
j¼1
Then jGðz1 ; z2 ; . . . ; zn Þj 6 I þ
n X
n
S j S 8ðz1 ; z2 ; . . . ; zn Þ 2 U ;
ð14Þ
j¼1
where 0 2 3 0 S1 B 6 T2 6 6 S2 7 B 6 . 6 7 B 6 . 7 ¼ B I 6 .. 6 4 .. 5 B @ 4 Tn 1 Sn Tn 2
T1 0 .. .
T1 T2 .. .
.. .
Tn
Tn 1
0 Tn
T1 T2 .. .
31 1
7C 7C 7C 7C 7C Tn 1 5A 0
2
3 T1 6 T2 7 6 7 6 .. 7 4 . 5
ð15Þ
Tn
and Tj ¼
1 X ðXj Þi
8j ¼ 1; 2; . . . ; n:
ð16Þ
i¼1
Proof of Theorem 1. Let us introduce the set W ¼ fX1 ; X2 ; . . . ; Xn g. Let W1 ¼ W2 ¼ ¼ Wk ¼ W Sk and U ¼ i¼1 Wi . Now let vk be the set of all k element subsets of the S set U. Also let e 2 vk , and the set He be the set of all permutations of e. Let us introduce Nk ¼ e2vk He . Now we can define the following sum X Pk ¼ C: ð17Þ C2Nk
From (13) and its MacLaurine expansion, Gðz1 ; z2 ; . . . ; zn Þ can be expanded as follows n X Gðz1 ; z2 ; . . . ; zn Þ ¼ I þ Sj ; j¼1
ð18Þ
866
T. Fernando et al. / Computers and Electrical Engineering 29 (2003) 861–871
where
"
Sj ¼ Xj zj I þ
1 X
# Pk
8j ¼ 1; 2; . . . ; n:
ð19Þ
k¼1
Eq. (19) can be rewritten as follows 1 0 ! 1 n X X C B Sj ¼ I þ Xkj zjk B Sk C A 8j ¼ 1; 2; . . . ; n: @ k¼1
ð20Þ
k¼1 k6¼j
From Eq. (20), it is clear that for jzj j 6 1 8j ¼ 1; 2; . . . ; n the following bounds for Sj 8j ¼ 1; 2; . . . ; n can be obtained 1 1 0 0 n X n X C B B jSj jC ð21Þ jSi j 6 Ti @I þ Sj A 6 Ti @I þ A 8i ¼ 1; 2; . . . ; n: j¼1 j¼1 j6¼i j6¼i Solving inequalities in (21), gives: 0 2 2 3 0 T1 T1 S1 B 6 T2 0 T2 6 6 S2 7 B 6 .. 6 7 B .. .. 6 .. 7 6 B I 6 . . . 6 4 . 5 B @ 4 Tn 1 Tn 1 Sn Tn Tn Accordingly, this completes the proof
.. .
T1 T2 .. .
31 1
7C 7C 7C 7C 7C 0 Tn 1 5A Tn 0 of Theorem 1.
3 2 3 T1 S1 6 T2 7 6 S 2 7 6 7 6 7 6 .. 7 6 .. 7: 4 . 5 4 . 5 2
Tn
ð22Þ
Sn
Remark 1. The power series (16) will always converge provided that the spectral radius of matrix Xj is less than 1, i.e. ð23aÞ qðXj Þ < 1 8j ¼ 1; 2; . . . n; where Xj is as defined in Eq. (12). Accordingly, provided that condition (23a) is satisfied, the power series can be computed in a finite number of terms. It is also clear from (22) that matrix S defined in (14) exists provided that condition (23a) holds and 3 2 T1 T1 0 T1 6 T2 0 T2 T2 7 7 6 6 . .. 7 .. .. .. < 1: ð23bÞ q6 .. . 7 . . . 7 6 4 Tn 1 Tn 1 0 Tn 1 5 Tn Tn Tn 0 Remark 2. For the case where jXi j ¼ Xi 8i ¼ 1; 2; . . . ; n and matrix ðX1 þ þ Xn Þ is asymptotically stable, then it can be shown in a similar way to that in [14], S ¼ ðI X1 Xn Þ 1 . When jXi j 6¼ Xi and if matrix ðjX1 j þ þ jXn jÞ is asymptotically stable, then S satisfies the following upper bound: S 6 ðI jX1 j jXn jÞ 1 .
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In the following, we present new robust stability conditions for system (10) based on the result derived in Theorem 1. We assume that the nominal system is stable and that matrix S exists. The results are given in the following theorem. Theorem 2. System (10) is robustly stable provided that the following inequality is satisfied 1 ; q< 2 3 q½SE E11 E1n 6 . .. 7 .. where matrix S is defined in Eqs. (14), and E ¼ 4 .. . 5. . En1 Enn
ð24Þ
Proof of Theorem 2. The stability condition of system (10) can also be expressed as det½I z1 X1 zn Xn ðz1 DX1 þ þ zn DXn Þ ¼ det½I z1 X1 zn Xn det½I ðI z1 X1 zn Xn Þ 1 ðz1 DX1 þ þ zn DXn Þ 6¼ 0; n
for ðz1 ; . . . ; zn Þ 2 U ; ð25Þ where
2
3 f0g DXi ¼ 4 DAi1 DAin 5 8i ¼ 1; 2; . . . ; n: ð26Þ f0g Since the nominal system is assumed to be asymptotically stable, det½I z1 X1 zn Xn 6¼ 0 n for ðz1 ; . . . ; zn Þ 2 U . Therefore condition (25) reduces to det½I ðI z1 X1 zn Xn Þ 1 ðz1 DX1 þ þ zn DXn Þ 6¼ 0
n
for ðz1 ; . . . ; zn Þ 2 U :
ð27Þ
Substituting (14) into (27) and using Lemma 1 gives qðGðz1 ; . . . ; zn Þðz1 DX1 þ þ zn DXn ÞÞ < 1
n
for ðz1 ; . . . ; zn Þ 2 U :
ð28Þ
Again, by using the properties of spectral radius and by considering the structure of the uncertainty as defined in (26), the following inequality can be obtained qðGðz1 ; . . . ; zn Þðz1 DX1 þ þ zn DXn ÞÞ 6 qðjðGðz1 ; . . . ; zn Þjðjz1 jDX1 j þ þ jzn jDXn jÞÞ 6 qðjGðz1 ; . . . ; zn ÞjqEÞ
n
for ðz1 ; . . . ; zn Þ 2 U :
ð29Þ
Substituting the upper bound for jGðz1 ; . . . ; zn Þj (i.e. (14) from Theorem 1) into the right hand side of (29), gives qðjGðz1 ; . . . ; zn ÞjqEÞ 6 qðqSEÞ
n
for ðz1 ; . . . ; zn Þ 2 U :
ð30Þ
Accordingly, provided that the stability condition in Theorem 2 is satisfied, the system (10) is robustly stable. This completes the Proof of Theorem 2. Now the lower bounds for stability margin can be derived as a special case of Theorem 2, i.e. by choosing the structure of the uncertainty we can obtain the required lower bounds. Let us now introduce the following Lemma and Corollary.
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T. Fernando et al. / Computers and Electrical Engineering 29 (2003) 861–871
Lemma 2. For qj P 0 8j ¼ 1; . . . ; n, if C1 ðz1 ; z2 ; . . . ; zn Þ 2 Im1 z1 ð1 þ q1 ÞA11
z1 ð1 þ q1 ÞA12 6 z2 ð1 þ q2 ÞA21 Im2 z2 ð1 þ q2 ÞA22 6 ¼ det6 . 6 .. 4
zn ð1 þ qn ÞAn1 6¼ 0
z1 ð1 þ q1 ÞA1n
3
z2 ð1 þ q2 ÞA2n .. .
7 7 7 7 5
zn ð1 þ qn ÞAnn 1
Imn zn ð1 þ qn ÞAnn
n
8ðz1 ; . . . ; zn Þ 2 U ;
then
ð31Þ
2
Im1 z01 A11 6 z02 A21 6 0 0 Cðz1 ; . . . ; zn Þ ¼ det 6 .. 4 .
z0n An1
z01 A12 Im2 z02 A22
..
. 0
zn Ann 1
z01 A1n
z02 A2n .. . Imn z0n Ann
3 7 7 7 6¼ 0 in U n ; 5
ð32Þ
where
U n ¼ ðz01 ; . . . ; z0n Þjz01 j < 1 þ q1 ; . . . ; jz0n j < 1 þ qn :
ð33Þ
Proof of Lemma 2. The result in (32) follows immediately if the variable transformation z0i ¼ ð1 þ qi Þzi 8i ¼ 1; 2; . . . ; n is applied to (31). Corollary 1. Let z0i ¼ ð1 þ qi Þzi and z0j ¼ zj 8j ¼ 1; . . . ; i 1; i þ 1; . . . ; n (i.e. choose qi 6¼ 0 and qj ¼ 0 8j ¼ 1; . . . ; i 1; i þ 1; . . . ; nÞ, then 3 2
z01 A12
z01 A1n Im1 z01 A11 6 z02 A21 Im2 z02 A22
z02 A2n 7 7 6 0 0 ð34Þ Cðz1 ; . . . ; zn Þ ¼ det6 7 6¼ 0 in Uqin ; .. .. .. 5 4 . . .
z0n An1
z0n Ann 1
Imn z0n Ann
where
Uqin ¼ ðz01 ; . . . ; z0n Þjz0i j < 1 þ qi ; jz01 j < 1; . . . ; jz0i 1 j < 1; jz0iþ1 j < 1; . . . ; jz0n j < 1 :
ð35Þ
The lower bounds on the margin of stability of stable systems characterized by (1) can then be easily deduced from the results of Theorem 2, Lemma 2 and Corollary 1. They are now expressed in the following theorem. Theorem 3. The lower bounds for stability margin ri 8i ¼ 1; 2; . . . ; n and r are provided by the upper bound of qi 8i ¼ 1; 2; . . . ; n and q respectively, and satisfying the following conditions 1 f0g q@S 4 jAi1 j f0g
qi < 0 2
31 jAin j 5A
8i ¼ 1; 2; . . . ; n
1 jA11 j B 6 .. .. q@S 4 . . jAn1 j
and q < 0 2
31 : jA1n j .. 7C . 5A jAnn j ð36Þ
T. Fernando et al. / Computers and Electrical Engineering 29 (2003) 861–871
2
869
3
f0g Proof of Theorem 3. From Theorem 2, by letting E ¼ 4 jAi1 j f0g
jAin j 5 then it is clear that
provided the condition (36) is satisfied, the characteristic polynomial Cðz1 ; . . . ; zi 1 ; zi ð1 þ qi Þ; ziþ1 ; . . . ; zn Þ 6¼ 0
n
ð37Þ
in U :
From Corollary 1 it is clear that (37) implies Cðz1 ; . . . ; zn Þ 6¼ 0
in Uqin ;
ð38Þ
where Uqin ¼ fðz1 ; . . . ; zn Þjjzi j < 1 þ qi ; jz1 j < 1; . . . ; jzi 1 j < 1; jziþ1 j < 1; . . . ; jzn j < 1g:
ð39Þ
Because condition (24) only provides a sufficient condition for stability, the value of qi satisfying (36) is only a conservative estimate in the above equation. Therefore the n-disc Uqin is not the largest n-disc where the characteristic equation Cðz1 ; . . . ; zn Þ 6¼ 0. From the definition of stability margin (i.e. (5) and (6)), it follows that qi is a lower bound for ri . Similarly, by choosing 2 3 jA11 j jA1n j 6 .. .. 7 ð40Þ E ¼ 4 ... . . 5 jAn1 j
jAnn j
and by following the above same lines of proof as for qi , q is a lower bound for r. This completes the Proof of Theorem 3.
4. Numerical Examples Let us look at two numerical examples to illustrate the results of this paper. Example 1. Consider the following 2-D Rosser model with 0:4 0:3 1 0 0:1 0:03 A11 ¼ ; A12 ¼ ; A21 ¼
0:3 0 0 0 0:1 0:2 Using (15) and (16), matrix 2 3:0655 1:7422 6 0:9196 1:5227 S¼6 4 1:0557 0:8683 0:7016 0:6524
and A22 ¼
0:1 0:2
0:9 : 0
S for this system can be computed as 3 4:2576 3:8318 1:2773 1:1495 7 7: 2:8552 2:5696 5 1:2522 2:1270
Using Eq. (36) the lower bounds can be obtained and are shown in Table 1. Table 1 also shows the lower bounds derived by the method proposed by Agathoklis [12] with
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T. Fernando et al. / Computers and Electrical Engineering 29 (2003) 861–871
Table 1 Comparison of lower bounds for stability margin This paper
Agathoklis [12]
r1 ¼ 0:3961 r2 ¼ 0:3823 r ¼ 0:1984
r1 ¼ 0:2929 r2 ¼ 0:1123 r ¼ 0:0869
Table 2 Lower bounds for stability margin for the 3-D Rosser model r1
r2
r3
r
0.2576
0.4263
0.6814
0.1254
2
4 61 W ¼6 40 0
1 5 0 0
3 2 0 0 2:9700 6 0:4340 0 0 7 7 and Q ¼ 6 4 1:5800 8 0 5 0 10
0:7200
0:4340 4:2328
1:5760 0:2160
1:5800
1:5760 3:5200
0:7200
3
0:7200 0:2160 7 7
0:7200 5 3:5200
being the positive definite solution of the 2-D Lyapunov equation (4). As can be seen in Table 1, the lower bounds computed using the method proposed in this paper are less conservative than those calculated by solving a 2-D Lyapunov equation. Example 2. Now consider 0:4 0:3 A11 ¼
0:3 0 A21 ¼ ½ 0:1 0:03 A31 ¼ ½ 0:1 0:2
the following 3-D Rosser model with 1 0 A12 ¼ A13 ¼ 0 0 : A22 ¼ 0:1 A23 ¼ 1 A32 ¼ 0:2 A33 ¼ 0
Using (15) and (16), matrix 2 3:4926 2:1954 6 1:0478 1:6586 S¼6 4 1:3422 1:1722 0:8273 0:7857
S for this system can be found as 3 4:9895 4:9895 1:4968 1:4968 7 7: 3:3459 3:3459 5 1:4675 2:4675
Using (36) the lower bounds can be obtained and are shown in Table 2.
5. Conclusion In this paper, a method based on the MacLaurine series expansion for computing lower bounds for stability margin has been presented. The method does not require to solve a n-D Lyapunov equation. It was shown in numerical Example 1 that the general result for n-D systems when applied to a 2-D system could compute lower bounds which are less conservative than bounds computed by solving a 2-D Lyapunov equation.
T. Fernando et al. / Computers and Electrical Engineering 29 (2003) 861–871
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References [1] Roesser RP. A discrete state space model for linear image processing. IEEE Trans Automat Contr 1975;20:1–10. [2] Fornasini E, Marchesini G. Doubly-indexed dynamical systems: state-space models and structural properties. Math Syst Theory 1978;12:59–72. [3] Ahmed ARE. On the stability of two-dimensional discrete systems. IEEE Trans Automat Contr 1980;25:551–2. [4] Lodge JH, Fahmy NM. Stability and overflow oscillations in 2-D state-space digital filters. IEEE Trans Acoust Speech Signal Process 1981;29:1161–71. [5] Anderson BDO, Agathoklis P, Jury EI, Mansour M. Stability and the matrix Lyapunov equation for discrete 2dimensional systems. IEEE Trans Circ Syst 1986;33:261–7. [6] Hinamoto T. 2-D Lyapunov equation and filter design based on the Fornasini–Marchesini second model. IEEE Trans Circ Syst 1993;40:102–10. [7] Lu WS. On a Lyapunov approach to stability analysis of 2-D digital filters. IEEE Trans Circ Syst 1994;41:665–9. [8] Xiao C, Hill DJ, Agathoklis P. Stability and the Lyapunov equation for n-dimensional digital systems. IEEE Trans Circ Syst 1997;44:614–21. [9] Agathoklis P. The Lyapunov equation for n-dimensional discrete systems. IEEE Trans Circ Syst 1988;35:448–51. [10] Walach E, Zeheb E. N -dimensional stability margins computation and a variable transformation. IEEE Trans Acoust Speech Signal Process 1982;30:887–94. [11] Agathoklis P, Jury EI, Mansour M. The margin of stability of 2-D linear discrete systems. IEEE Trans Acoust Speech Signal Process 1982;30:869–74. [12] Agathoklis P. Lower bounds for the stability margin of discrete two-dimensional systems based on the twodimensional Lyapunov equation. IEEE Trans Circ Syst 1988;35:745–9. [13] Chou JH. Pole-assignment robustness in a specified disk. Syst Contr Lett 1991;16:41–4. [14] Shafai B, Chen J, Kothandaraman M. Explicit formulas for stability radii of nonnegative and Metzlerian matrices. IEEE Trans Automat Contr 1997;42:265–70. Tyrone Fernando obtained his B.E. (Hons) and PhD degrees in electrical engineering from the University of Melbourne, Australia in 1990 and 1996 respectively. In 1995 he was a visiting lecturer in the Department of Electrical and Computer Engineering at Monash University, Melbourne, Australia. He has been with the Department of Electrical and Electronic Engineering, University of Western Australia since 1996 where he is currently a Senior Lecturer. He has taught a wide range of courses in control, signal processing, systems and network theory. His main research interests are in the fields of biomedical engineering, multi-dimensional systems theory and robust control. Hieu Trinh obtained his BEng (Hons), MEngSc and PhD degrees in electrical engineering, all from The University of Melbourne in 1990, 1992 and 1996. Between March 1995 and December 1996, he was a postdoctoral research fellow in the Department of Electrical and Electronic Engineering, The University of Melbourne. Between January 1997 and December 2000, he was a lecturer at James Cook University, Townsville, Australia. He joined Deakin University in January 2001 where he is currently a Senior Lecturer. He has taught a wide range of courses in control, power systems, electric machines, electronics and network theory. His current research activities are in the areas of systems and control theory, multi-dimensional (n-D) systems theory, intelligent control, optimisation, fault diagnosis and fault-tolerant control, robotics, control and analysis of power systems, and application of control theory to industrial systems. So far, he has published over 75 research papers on a wide range of topics in Systems and Control Theory, and their application to real-life systems. Saeid Nahavandi received BSc (Hons), MSc and a PhD in Automation from Durham University (UK). In 1991 he joined Massey University (NZ) where he taught and led research in robotics. In 1998 he became an Associate Professor at Deakin University (AU) and the leader for the Intelligent Systems research group and also Manager for the Cooperative Research Center for CAST Metals Manufacturing. In 2002 Professor Nahavandi took the position of Chair in Engineering in the same university. Dr. Nahavandi has published over 100 reviewed papers and delivered several invited plenary lectures at international conferences. He is the recipient of four international awards, best paper award at the World Automation Congress (USA) and the Young Engineer of the Year award. Professor Nahavandi is the founder of the World Manufacturing Congress series and the Autonomous Intelligent Systems Congress series. He is a Fellow of IEAust and IEE. His current research interests include modeling and control and the application of soft computing to industrial processes.