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Stability analysis of linear 2-D systems
ARTICLE IN PRESS
Signal Processing 88 (2008) 2078–2084 www.elsevier.com/locate/sigpro
Stability analysis of linear 2-D systems Tao Liu Department of Communication Engineering, Information Engineering School, University of Science and Technology Beijing, Beijing 100083, China Received 9 July 2007; received in revised form 2 February 2008; accepted 13 February 2008 Available online 20 February 2008
Abstract The present paper is concerned with stability analysis of linear two-dimensional systems described by Fornasini– Marchesini state-space model. Necessary and sufficient conditions for asymptotic stability of the systems are obtained first. Several simple stability criteria are derived via the nonnegative matrix theory, which are sharper than those in literature. When all the parameter matrices are nonnegative, the criteria are necessary and sufficient for stability of the system. Illustrative examples are provided. r 2008 Elsevier B.V. All rights reserved. Keywords: Stability; Two-dimensional systems; Nonnegative matrix
1. Introduction We are concerned with the stability of twodimensional (2-D) systems described by Fornasini– Marchesini state-space model xði þ 1; j þ 1Þ ¼ A1 xði þ 1; jÞ þ A2 xði; j þ 1Þ þ A3 xði; jÞ, n
nn
(1)
where xði; jÞ 2 R , Ai 2 R for i ¼ 1, 2 and 3 are constant matrices. The model is presented by Fornasini and Marchesini [1] for 2-D state-space digital filters. The main goal of the present paper is to find computable stability criteria for system (1) based on nonnegative matrices theory. Along the line of [2], necessary and sufficient conditions for asymptotic stability of the systems are obtained. Then several computable stability criteria are derived via the nonnegative matrix theory. When all the parameter matrices are nonnegative, the criteria are necessary E-mail address: [email protected]
and sufficient for stability of the system. It is shown that our criteria are sharper than those in [2–4]. Now Fornasini–Marchesini state-space model has been applied in many practical problems, for example, the control of sheet forming processes [5], circuits, signal processing and discretization of some partial differential equations with initial-boundary conditions [6]. The investigation for the stability of (1) has attracted much attention. Several methods have been proposed, for example, using Lyapunov function [7,8], using LMI technique [9] and using the nonnegative matrices theory [2–4]. For linear 2-D model in general case, stability has been discussed in [10–12]. Recently an inviable approach is proposed by [12] which is related to the Schur 1-D stability test. Notation: Let W 2 Cnn with elements wjk , jW j denotes the nonnegative matrix in Rnn with elements jwjk j. The rðW Þ stands for the spectral radius of W . Let S ¼ fsjk g and V ¼ fvjk g be matrices in Rnn , we write SXV if and only if sjk Xvjk for each ðj; kÞ.
0165-1684/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2008.02.007
ARTICLE IN PRESS T. Liu / Signal Processing 88 (2008) 2078–2084
Which implies that
2. Necessary and sufficient conditions
z1 lj ðA1 Þa1
Define hðz1 ; z2 Þ ¼ det½I z1 A1 z2 A2 z1 z2 A3 ,
(2)
where I denotes the n n identity matrix. The following lemma gives a necessary and sufficient condition for stability of system (1). Lemma 2.1 (Huang [10] and Sˇiljak [11]). System (1) is asymptotically stable if and only if ¯ hðz1 ; z2 Þa0 for all z1 ; z2 2 U, ¯ ¼ fðz1 ; z2 Þ : jz1 jp1; jz2 jp1g. where U
for jz1 jp1 and j ¼ 1; . . . ; n,
(9)
lj ðF Þ stands for the jth eigenvalue of matrix F . Now we will show that rðA1 Þo1. Assume that rðA1 ÞX1, then there is some k such that jlk ðA1 ÞjX1. We can take z1 ¼ 1=lk ðA1 Þ, and have 1 p1 jz1 j ¼ lk ðA1 Þ and z1 lk ðA1 Þ ¼ 1
An equivalent version of the above lemma is as follows. Lemma 2.2 (Huang [10] and Sˇiljak [11]). System (1) is asymptotically stable if and only if
which contracts (9). From (8), the matrix ðI z1 A1 Þ1 exists for jz1 jp1. From (2) and (7), we get hðz1 ; z2 Þ ¼ det½I z1 A1 det½I z2 V ðz1 Þ
(10)
for jz1 j ¼ 1 and jz2 jp1, which implies that det½I z1 A1 a0 and det½I z2 V ðz1 Þa0.
hðz1 ; 0Þa0 for jz1 jp1
Since det½I z1 A1 a0, (10) implies
and hðz1 ; z2 Þa0
for jz1 j ¼ 1 and jz2 jp1.
(3)
Using the above lemma, we can obtain the following theorem. Theorem 2.1. System (1) is asymptotically stable if and only if one of the following two conditions holds: rðA1 Þo1
ðiÞ
2079
and
rðV ðzÞÞo1
for jzj ¼ 1,
and
rðW ðzÞÞo1
(4) for jzj ¼ 1,
where the matrix
(11) (12)
for jz1 j ¼ 1. Since the matrix V ðz1 Þ is unitarily similar to an upper triangular matrix O [13], from rðV ðz1 ÞÞo1 for jz1 j ¼ 1, we have det½I z2 V ðz1 Þ ¼ det½I z2 Oa0
1
W ðzÞ ¼ ðI zA2 Þ ðA1 þ zA3 Þ.
(5)
Here, rðF Þ stands for the spectral radius of a complex-valued matrix F . Proof. We will prove part (i) of the theorem first. Necessity: If system (1) is asymptotically stable, from Lemma 2.2, we have hðz1 ; 0Þa0 for jz1 jp1
(6)
and for jz1 j ¼ 1 and jz2 jp1.
(7)
Since any matrix is unitarily similar to an upper triangular matrix D [13], from (2) and (6), we have hðz1 ; 0Þ ¼ det½I z1 A1 ¼ det½I z1 Da0 for jz1 jp1.
hðz1 ; 0Þ ¼ det½I z1 A1 a0. hðz1 ; z2 Þ ¼ det½I z1 A1 det½I z2 V ðz1 Þ
V ðzÞ ¼ ðI zA1 Þ1 ðA2 þ zA3 Þ.
hðz1 ; z2 Þa0
for jz1 j ¼ 1 and jz2 jp1. In the same way for the proof of rðA1 Þo1, we have rðV ðzÞÞo1 for jzj ¼ 1. Sufficiency: If condition (i) is satisfied, from rðA1 Þo1, det½I z1 A1 a0 for jz1 jp1. We get Using (2) we get
where the matrix
ðiiÞ rðA2 Þo1
det½I z2 V ðz1 Þa0
(8)
for jz1 j ¼ 1 and jz2 jp1.
(13)
Using (13) we get hðz1 ; z2 Þ ¼ det½I z1 A1 det½I z2 V ðz1 Þa0 for jz1 j ¼ 1 and jz2 jp1.
(14)
From (11) and (14), system (1) is asymptotically stable by means of Lemma 2.2. Similar to the proof process of (i), we can show that condition (ii) is necessary and sufficient for the asymptotic stability of system (1). The proof is completed. & Remark 2.1. Since the conditions of Theorem 2.1 only involve one complex variable z with jzj ¼ 1, it is easier to check Theorem 2.1 than Lemmas 2.1 and 2.2.
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3. Simple stability criteria
and
In the section, several stability criteria will be presented via the nonnegative matrix theory. It is easier to check these criteria than those of the above section. We need the following definitions and lemmas. Lemma 3.1 (Lancaster [13]). Let W 2 Cnn and V 2 Rnn . If jW jpV , then rðW ÞprðV Þ.
W qþ1 pW q .
Proof. We will prove part (i) of the theorem first. By means of Lemma 3.2, for jzj ¼ 1 we have ðI zA1 Þ1 ¼ I þ zA1 þ ðzA1 Þ2 þ ¼ I þ zðI zA1 Þ1 A1
Lemma 3.2 (Lancaster [13]). If rðW Þo1, then ðI W Þ1 exists and
and
ðI W Þ1 ¼ I þ W þ W 2 þ .
jðI zA1 Þ1 j ¼ jI þ zA1 þ ðzA1 Þ2 þ j ¼ ðI jA1 jÞ1 .
(i) Assume rðjA1 jÞo1. Define the matrix þ ðI jA1 jÞ1 jA1 ðA1 A2 þ A3 Þj,
(15)
and for an integer qX1, define the matrix q X
jAj1 ðA1 A2 þ A3 Þj
jA1qþ1 ðA1 A2
þ A3 Þj.
V ðzÞ ¼ ðI zA1 Þ1 ðA2 þ zA3 Þ ¼ ðI zA1 Þ1 A2 þ ½I þ zðI zA1 Þ1 A1 zA3
j¼1
þ ðI jA1 jÞ
(24)
From (23), for jzj ¼ 1, we have
V 0 ¼ jA2 j þ jA1 A2 þ A3 j
1
(23)
pI þ jA1 j þ jA1 j2 þ
Definition 3.1. Consider system (1).
V q ¼ jA2 j þ jA1 A2 þ A3 j þ
(22)
(16)
¼ ½I þ zA1 þ ðzA1 Þ2 þ A2 þ zA3 þ z2 ðI zA1 Þ1 A1 A3 ¼ A2 þ zðA1 A2 þ A3 Þ þ ½ðzA1 Þ2 þ ðzA1 Þ3 þ A2 þ z2 ðI zA1 Þ1 A1 A3
(ii) Assume rðjA2 jÞo1. Define the matrix
¼ A2 þ zðA1 A2 þ A3 Þ þ ½I þ zA1 þ ðzA1 Þ2
W 0 ¼ jA1 j þ jA2 A1 þ A3 j
þ ðzA1 Þ2 A2 þ z2 ðI zA1 Þ1 A1 A3
þ ðI jA2 jÞ1 jA2 ðA2 A1 þ A3 Þj,
(17)
and for an integer qX1, define the matrix W q ¼ jA1 j þ jA2 A1 þ A3 j þ
q X
þ z2 ðI zA1 Þ1 A1 A3
jAj2 ðA2 A1 þ A3 Þj
j¼1
þ ðI jA2 jÞ1 jAqþ1 2 ðA2 A1 þ A3 Þj.
¼ A2 þ zðA1 A2 þ A3 Þ þ z2 ðI zA1 Þ1 ðA1 Þ2 A2 ¼ A2 þ zðA1 A2 þ A3 Þ þ z2 ðI zA1 Þ1 A1 ðA1 A2 þ A3 Þ.
(18) From (24), for jzj ¼ 1, jV ðzÞj ¼ jA2 þ zðA1 A2 þ A3 Þ
Theorem 3.1. Consider system (1).
þ z2 ðI zA1 Þ1 A1 ðA1 A2 þ A3 Þj
(i) Assume rðjA1 jÞo1, then, for qX1 and jzj ¼ 1,
pjA2 j þ jA1 A2 þ A3 j þ jðI zA1 Þ1 A1 ðA1 A2 þ A3 Þj
jV ðzÞj ¼ jðI zA1 Þ ðA2 þ zA3 ÞjpV q pV 0 1
(19)
¼ jA2 j þ jA1 A2 þ A3 j þ j½I þ zA1 þ ðzA1 Þ2
(20)
pjA2 j þ jA1 A2 þ A3 j q X jðA1 Þj ðA1 A2 þ A3 Þj þ
þ A1 ðA1 A2 þ A3 Þj
and V qþ1 pV q .
(ii) Assume rðjA2 jÞo1, then, for qX1 and jzj ¼ 1, jW ðzÞj ¼ jðI zA2 Þ ðA1 þ zA3 ÞjpW q pW 0 1
(21)
j¼1
þ ðI jA1 jÞ1 jðA1 Þqþ1 ðA1 A2 þ A3 Þj ¼ V q,
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where the matrix V 0 is given by (15) in Definition 3.1.
i.e., jV ðzÞjpV q .
(25)
V q ¼ jA2 j þ jA1 A2 þ A3 j þ
q X
jðA1 Þj ðA1 A2 þ A3 Þj
j¼1 1
þ ðI jA1 jÞ jðA1 Þ ¼ jA2 j þ jA1 A2 þ A3 j
qþ1
ðA1 A2 þ A3 Þj
rðW 0 Þo1,
where the matrix W 0 is given by (17) in Definition 3.1. Further criteria for stability of system (1) are as follows. Theorem 3.3. System (1) is asymptotically stable if one of the following two conditions holds:
2
þ ½jA1 ðA1 A2 þ A3 Þj þ jðA1 Þ ðA1 A2 þ A3 Þj þ þ jðA1 Þq ðA1 A2 þ A3 Þj
ðiÞ
þ ðI þ jA1 j þ jA1 j2 þ ÞjðA1 Þqþ1 ðA1 A2 þ A3 Þj
pjA2 j þ jA1 A2 þ A3 j
rðjA1 jÞo1 and
þ ðI þ jA1 j þ jA1 j þ
rðV q Þo1
for qX1,
where the matrix V q is given by (16) in Definition 3.1. and
ðiiÞ rðjA2 jÞo1
2
rðW q Þo1
for qX1,
where the matrix W q is given by (18) in Definition 3.1.
þ jA1 jq1 ÞjA1 ðA1 A2 þ A3 Þj
Remark 3.1. Theorem 3.3 is sharper than Theorem 3.2 since rðV q ÞprðV 0 Þ and rðW q ÞprðW 0 Þ.
þ ðjA1 jq þ jA1 jqþ1 þ ÞjA1 ðA1 A2 þ A3 Þj ¼ jA2 j þ jA1 A2 þ A3 j þ ðI þ jA1 j þ jA1 j2
When all the elements of matrices A1 ; A2 and A3 are nonnegative, we have the following corollary.
þ ÞjA1 ðA1 A2 þ A3 Þj ¼ jA2 j þ jA1 A2 þ A3 j
Corollary 3.1. When all the elements of matrices A1 ; A2 and A3 are nonnegative. System (1) is asymptotically stable if and only if one of the following two conditions holds:
þ ðI jA1 jÞ1 jA1 ðA1 A2 þ A3 Þj ¼ V 0, i.e., V q pV 0 .
and
ðiiÞ rðjA2 jÞo1
ðiÞ (26)
From (25) and (26), we have jV ðzÞjpV q pV 0 , i.e., inequality (19) holds. Similarly, we can prove the inequality V qþ1 pV q for qX1. Similar to the proof process of (i), we can show that condition (ii) holds. The proof is completed. & Since jV ðzÞj ¼ jV ðzÞj and jW ðzÞj ¼ jW ðzÞj, we have rðV ðzÞÞprðjV ðzÞjÞ and rðW ðzÞÞprðjW ðzÞjÞ by Lemma 3.1. According to Theorem 3.1, for jzj ¼ 1 we have
rðA1 Þo1 and
rðV 0 Þo1,
where the matrix V 0 ¼ ðI A1 Þ1 ðA2 þ A3 Þ. ðiiÞ rðA2 Þo1
and
rðW 0 Þo1,
where the matrix W 0 ¼ ðI A2 Þ1 ðA1 þ A3 Þ. Proof. Sufficiency: According to Theorem 3.2, we have that V 0 ¼ jA2 j þ jA1 A2 þ A3 j þ ðI jA1 jÞ1 jA1 ðA1 A2 þ A3 Þj
rðV ðzÞÞprðjV ðzÞjÞprðV q ÞprðV 0 Þ
¼ A2 þ A1 A2 þ A3 þ ðI A1 Þ1 A1 ðA1 A2 þ A3 Þ
and
¼ ðI A1 Þ1 ½ðI A1 ÞðA2 þ A1 A2 þ A3 Þ
rðW ðzÞÞprðjW ðzÞjÞprðW q ÞprðW 0 Þ.
þ A1 ðA1 A2 þ A3 Þ ¼ ðI A1 Þ1 ½A2 þ A1 A2 þ A3 A1 A2
By means of Theorem 2.1, we can directly derive the following criteria, Theorems 3.2 and 3.3 for the asymptotic stability of system (1). Theorem 3.2. System (1) is asymptotically stable if one of the following two conditions holds: ðiÞ
rðjA1 jÞo1
and
rðV 0 Þo1,
A1 A1 A2 A1 A3 þ A1 A1 A2 þ A1 A3 ¼ ðI A1 Þ1 ðA2 þ A3 Þ, i.e., V 0 ¼ ðI A1 Þ1 ðA2 þ A3 Þ.
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matrix ðI jA1 jÞ1 ðjA2 j þ jA3 jÞ.
Similar to the above, we have that
V 0 ¼ jA2 j þ jA1 A2 þ A3 j
W 0 ¼ ðI A2 Þ1 ðA1 þ A3 Þ.
þ ðI jA1 jÞ1 jA1 ðA1 A2 þ A3 Þj, ¼ ðI jA1 jÞ1 ½ðI jA1 jÞðjA2 j þ jA1 A2 þ A3 jÞ
In the case, rðjA1 jÞ ¼ rðA1 Þ and rðjA2 jÞ ¼ rðA2 Þ. The sufficiency is proved. Necessity: By Theorem 2.1, let z ¼ 1 in (4) and (5), the necessity is proved. The proof is completed. &
þ jA1 ðA1 A2 þ A3 Þj ¼ ðI jA1 jÞ1 ½jA2 j þ jA1 A2 þ A3 j jA1 jjA2 j jA1 jjA1 A2 þ A3 j þ jA1 ðA1 A2 þ A3 Þj pðI jA1 jÞ1 ½jA2 j þ jA1 jjA2 j þ jA3 j jA1 jjA2 j
Remark 3.2. The approach proposed can be applied to Roesser’s state-space model since Roesser’s model can be transferred into Fornasini–Marchesini model [6]. Similar stability criteria may be obtained.
jA1 jjA1 A2 þ A3 j þ jA1 jjA1 A2 þ A3 j ¼ ðI jA1 jÞ1 ðjA2 j þ jA3 jÞ. According to Lemma 3.1, we obtain
Now we compare our criteria with those of [3,4]. The stability criterion in [3,4] for system (1) is as follows. Lemma 3.3. System (1) is asymptotically stable if a ¼ rðjA1 j þ jA2 j þ jA3 jÞo1.
(27)
The following lemma is the main result of [3,4]. Lemma 3.4 (Bose and Trautman [3] and Su and Bhaya [4]). The following conditions on matrices A1 ; A2 ; A3 2 Rnn are equivalent: ðaÞ rðjA1 jÞo1
and
1
r½ðI jA1 jÞ ðjA2 j þ jA3 jÞo1. ðbÞ rðjA2 jÞo1 and r½ðI jA2 jÞ1 ðjA1 j þ jA3 jÞo1. ðcÞ rðjA3 jÞo1
and
1
r½ðI jA3 jÞ ðjA1 j þ jA2 jÞo1.
ðiÞ
rðjA1 jÞo1 and
rðV 0 Þo1
from ðaÞ rðjA1 jÞo1
and
1
r½ðI jA1 jÞ ðjA2 j þ jA3 jÞo1. Hence (d) in Lemma 3.4 ) (i) in Theorem 3.2. Similarly we can prove (b) in Lemma 3.4 ) (ii) in Theorem 3.2, then (d) in Lemma 3.4 ) (ii) in Theorem 3.2. The proof is completed. & Remark 3.3. Theorem 3.3 is also sharper than Lemma 3.3 according to Theorem 3.1. It is easier to check these stability criteria of Theorems 3.2 and 3.3 than those of Theorem 2.1. Now we will compare our criteria with those of [2]. We need the following definition. Definition 3.2 (Hu and Liu [2]). If rðjA1 jÞo1 let b1 ¼ r½jA2 j þ jA3 j þ ðI jA1 jÞ1 ðjA1 A2 j þ jA1 A3 jÞ. (28)
ðdÞ a ¼ rðjA1 j þ jA2 j þ jA3 jÞo1. If rðjA2 jÞo1 let We will show that our conditions are sharper than those of [3,4].
b2 ¼ r½jA1 j þ jA3 j þ ðI jA2 jÞ1 ðjA2 A1 j þ jA2 A3 jÞ.
(29) If rðjA3 jÞo1 let
Theorem 3.4. If a ¼ rðjA1 j þ jA2 j þ jA3 jÞo1, then both conditions (i) and (ii) in Theorem 3.2 hold. i.e., Theorem 3.2 is sharper than Lemma 3.3. Proof. We will show that (d) in Lemma 3.4 ) (i) in Theorem 3.2. By means of Lemma 3.4, (d) ) (a), it is sufficient to show (a) of Lemma 3.4 ) (i) of Theorem 3.2. We compare matrix V 0 with
b3 ¼ r½jA1 j þ jA2 j þ ðI jA3 jÞ1 ðjA3 A1 j þ jA3 A2 jÞ.
(30) The stability criteria in [2] for system (1) are as follows. Lemma 3.5 (Hu and Liu [2]). The state-space model (1) is asymptotically stable if one of the
ARTICLE IN PRESS T. Liu / Signal Processing 88 (2008) 2078–2084
following three conditions holds: ðaÞ rðjA1 jÞo1
and
b1 o1.
ðbÞ rðjA2 jÞo1
and
b2 o1.
ðcÞ rðjA3 jÞo1
and
b3 o1.
2083
Remark 3.4. Theorem 3.5 shows that condition (i) (condition (ii)) in Theorem 3.2 is sharper than condition (a) (condition (b)) in Lemma 3.5 when A1 ; A2 ; A3 2 R1 . But in the general case, we do not know whether the relation is true. 4. Numerical examples
Comparing Theorem 3.2 with that of [2], we have the following result. Theorem 3.5. Suppose all A1 ; A2 and A3 are scalar, i.e., A1 ; A2 ; A3 2 R1 : (I) If rðjA1 jÞo1 and b1 o1, then rðjA1 jÞo1 and rðV 0 Þo1. (II) If rðjA2 jÞo1 and b2 o1, then rðjA2 jÞo1 and rðW 0 Þo1. Here the matrices V 0 and W 0 are defined by (15) and (17), respectively. Proof. By the definitions of V 0 and b1 , we have V 0 ¼ jA2 j þ jA1 A2 þ A3 j þ ðI jA1 jÞ1 jA1 ðA1 A2 þ A3 Þj, b1 ¼ r½jA2 j þ jA3 j þ ðI jA1 jÞ1 ðjA1 A2 j þ jA1 A3 jÞ. Since A1 ; A2 ; A3 2 R1 ,
We will illustrate our stability criteria (Theorem 3.2) for linear 2-D system (1) through several examples. All computations in this section are carried out by Matlab 7.0. Example 4.1. In the state-space model (1), let A1 ¼ 0:8, A2 ¼ 0:7 and A3 ¼ 0:6. We have rðjA1 jÞ ¼ 0:8, rðjA2 jÞ ¼ 0:7 and a ¼ 2:1. Since a ¼ 2:141, we cannot determine whether system (1) is stable by Lemma 3.3, the result of [3,4]. We have that b1 ¼ 6:5; b2 ¼ 4:6667 and b3 ¼ 3:75. Since they are all greater that 1, we cannot determine whether system (1) is stable by Lemma 3.5, the result of [2]. Since rðV 0 Þ ¼ 0:9o1 (or rðW 0 Þ ¼ 0:9333o1), condition (i) (or (ii)) of Theorem 3.2 is satisfied. System (1) is asymptotically stable by Theorem 3.2. Example 4.2. In system (1), let 0:72 0 0:7 A1 ¼ ; A2 ¼ 0 0:72 0:2
rðV 0 Þ b1 ¼ jA2 j þ jA1 A2 þ A3 j þ ð1 jA1 jÞ1 jA1 ðA1 A2 þ A3 Þj
and
1
½jA2 j þ jA3 j þ ð1 jA1 jÞ ðjA1 A2 j þ jA1 A3 jÞ ¼ jA1 A2 þ A3 j jA3 j þ ð1 jA1 jÞ1 ½jA1 ðA1 A2 þ A3 Þj ðjA1 A2 j þ jA1 A3 jÞ
A3 ¼
0:5
0:1
0:15
0:5
.
We have rðjA1 jÞ ¼ 0:72, rðjA2 jÞ ¼ 0:8414 and a ¼ 2:1846. Since a41, we cannot determine whether system (1) is stable by Lemma 3.3, the result of [3,4]. We have that b1 ¼ 5:2306; b2 ¼ 8:4729 and b3 ¼ 4:139. Since they are all greater that 1, we cannot determine whether system (1) is stable by Lemma 3.5, the result of [2]. On the other hand, we have that rðV 0 Þ ¼ 0:9247o1 and rðW 0 Þ ¼ 0:8259o1. Since rðV 0 Þo1 (or rðW 0 o1), condition (i) (or (ii)) of Theorem 3.2 is satisfied. System (1) is asymptotically stable by Theorem 3.2.
pjA1 A2 j þ ð1 jA1 jÞ ½jA1 A1 A2 j 1
þ jA1 A3 j jA1 A2 j jA1 A3 j ¼ jA1 A2 j þ ð1 jA1 jÞ1 ðjA1 A1 A2 j jA1 A2 jÞ ¼ jA1 A2 j þ ð1 jA1 jÞ1 ½ðjA1 j 1ÞjA1 A2 j ¼ jA1 A2 j jA1 A2 j ¼ 0, i.e., rðV 0 Þpb1 .
0:1 0:7
(31)
According to (31) and b1 o1, we have rðV 0 Þpb1 o1. Part (I) is proved. Similar to the above, we can prove part (II). The proof is completed. &
Example 4.3 (See Example 2.1 in Hu and Liu [2]). In the state-space model (1), let 0:2 0:1 0:3 0:1 A1 ¼ ; A2 ¼ 0:02 0:1 0:2 0:1
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and A3 ¼
0:4 0:33 0:1
0:3
.
Again, we have rðjA1 jÞ ¼ 0:2171, rðjA2 jÞ ¼ 0:3732 and a ¼ 1:1578. Since a41, we cannot determine whether system (1) is stable by Lemma 3.3, the result of [3,4]. We have that b1 ¼ 1:11; b2 ¼ 1:251 and b3 ¼ 0:935. Since b3 ¼ 0:935o1, condition (c) of Lemma 3.5 [2] is satisfied. System (1) is asymptotically stable by Lemma 3.5, the result of [2]. To apply Theorem 3.2, we compute rðV 0 Þ ¼ 1:0264 and rðW 0 Þ ¼ 0:9666. Since rðW 0 Þo1, condition (ii) of Theorem 3.2 is satisfied. It also shows that system (1) is asymptotically stable by Theorem 3.2. Examples 4.1 and 4.2 show that Theorem 3.2 is sharper than Lemmas 3.3 and 3.5 which are main results in the literature. When the conditions in Lemmas 3.3 and 3.5 are not satisfied, the conditions in Theorem 3.2 are satisfied. Example 4.3 shows that the two conditions in Theorem 3.2 are not equivalent since rðV 0 Þ41 and rðW 0 Þo1, although the two conditions in Theorem 2.1 are equivalent by which the two conditions in Theorem 3.2 are derived. The two conditions in Theorem 3.2 are only sufficient for stability. 5. Conclusions Necessary and sufficient conditions (Theorem 2.1) for stability of the Fornasini–Marchesini model are presented which are easier than those (Lemmas 2.1 and 2.2) in literature. Then new sufficient conditions (Theorems 3.2 and 3.3) for stability of the system, which are sharper than those in [2–4] and computable, are derived. When all the parameter matrices are nonnegative, the criteria are necessary and sufficient for stability of the system.
The approach proposed can be applied to Roesser’s state-space model [6]. Similar stability criteria may be obtained. Acknowledgements The author is greatly indebted to the associate editor and the three anonymous referees for careful reviews and valuable suggestions. References [1] E. Fornasini, G. Marchesini, State-space realization theory of two-dimensional filters, IEEE Trans. Automat. Control 21 (1976) 484–492. [2] G.D. Hu, M. Liu, Simple criteria for stability of twodimensional linear systems, IEEE Trans. Signal Process. 53 (2005) 4720–4723. [3] T. Bose, D.A. Trautman, Two’s complement quantization in two-dimensional digital state-space filters, IEEE Trans. Signal Process. 40 (1992) 2589–2592. [4] Y. Su, A. Bhaya, On the Bose–Trautman condition for stability of two-dimensional linear systems, IEEE Trans. Signal Process. 46 (1998) 2069–2070. [5] M.B. Zarrop, P.E. Wellstead, Two-dimensional and EM techniques for cross directional estimation and control, IEE Proc. Control Theory Appl. 149 (2002) 457–462. [6] T. Kaczorek, Two-Dimensional Linear Systems, Springer, Berlin, 1985. [7] H. Kar, V. Sigh, Stability of 2-D systems described by the Fornasini–Marchesini first model, IEEE Trans. Signal Process. 51 (2003) 1675–1676. [8] T. Ooba, On stability analysis of 2-D systems based on 2-D Lyapunov matrix inequalities, IEEE Trans. Circuit Syst. I 47 (2000) 1263–1265. [9] C. Du, L. Xie, Stability analysis and stabilization of uncertain two-dimensional discrete systems: an LMI approach, IEEE Trans. Circuit Syst. I 46 (1999) 1371–1374. [10] T.S. Huang, Stability of two-dimensional recursive filters, IEEE Trans. Audio Electroacoustics 20 (1972) 158–163. [11] D.D. Sˇiljak, Stability criteria for two-variable polynomials, IEEE Trans. Circuit Syst. 22 (1975) 185–189. [12] Y. Bistritz, On an inviable approach for derivation of 2-D stability tests, IEEE Trans. Circuit Syst. II 52 (2005) 713–718. [13] P. Lancaster, The Theory of Matrices with Applications, Academic Press, Inc., Orlando, 1985.
Signal Processing 88 (2008) 2078–2084 www.elsevier.com/locate/sigpro
Stability analysis of linear 2-D systems Tao Liu Department of Communication Engineering, Information Engineering School, University of Science and Technology Beijing, Beijing 100083, China Received 9 July 2007; received in revised form 2 February 2008; accepted 13 February 2008 Available online 20 February 2008
Abstract The present paper is concerned with stability analysis of linear two-dimensional systems described by Fornasini– Marchesini state-space model. Necessary and sufficient conditions for asymptotic stability of the systems are obtained first. Several simple stability criteria are derived via the nonnegative matrix theory, which are sharper than those in literature. When all the parameter matrices are nonnegative, the criteria are necessary and sufficient for stability of the system. Illustrative examples are provided. r 2008 Elsevier B.V. All rights reserved. Keywords: Stability; Two-dimensional systems; Nonnegative matrix
1. Introduction We are concerned with the stability of twodimensional (2-D) systems described by Fornasini– Marchesini state-space model xði þ 1; j þ 1Þ ¼ A1 xði þ 1; jÞ þ A2 xði; j þ 1Þ þ A3 xði; jÞ, n
nn
(1)
where xði; jÞ 2 R , Ai 2 R for i ¼ 1, 2 and 3 are constant matrices. The model is presented by Fornasini and Marchesini [1] for 2-D state-space digital filters. The main goal of the present paper is to find computable stability criteria for system (1) based on nonnegative matrices theory. Along the line of [2], necessary and sufficient conditions for asymptotic stability of the systems are obtained. Then several computable stability criteria are derived via the nonnegative matrix theory. When all the parameter matrices are nonnegative, the criteria are necessary E-mail address: [email protected]
and sufficient for stability of the system. It is shown that our criteria are sharper than those in [2–4]. Now Fornasini–Marchesini state-space model has been applied in many practical problems, for example, the control of sheet forming processes [5], circuits, signal processing and discretization of some partial differential equations with initial-boundary conditions [6]. The investigation for the stability of (1) has attracted much attention. Several methods have been proposed, for example, using Lyapunov function [7,8], using LMI technique [9] and using the nonnegative matrices theory [2–4]. For linear 2-D model in general case, stability has been discussed in [10–12]. Recently an inviable approach is proposed by [12] which is related to the Schur 1-D stability test. Notation: Let W 2 Cnn with elements wjk , jW j denotes the nonnegative matrix in Rnn with elements jwjk j. The rðW Þ stands for the spectral radius of W . Let S ¼ fsjk g and V ¼ fvjk g be matrices in Rnn , we write SXV if and only if sjk Xvjk for each ðj; kÞ.
0165-1684/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2008.02.007
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Which implies that
2. Necessary and sufficient conditions
z1 lj ðA1 Þa1
Define hðz1 ; z2 Þ ¼ det½I z1 A1 z2 A2 z1 z2 A3 ,
(2)
where I denotes the n n identity matrix. The following lemma gives a necessary and sufficient condition for stability of system (1). Lemma 2.1 (Huang [10] and Sˇiljak [11]). System (1) is asymptotically stable if and only if ¯ hðz1 ; z2 Þa0 for all z1 ; z2 2 U, ¯ ¼ fðz1 ; z2 Þ : jz1 jp1; jz2 jp1g. where U
for jz1 jp1 and j ¼ 1; . . . ; n,
(9)
lj ðF Þ stands for the jth eigenvalue of matrix F . Now we will show that rðA1 Þo1. Assume that rðA1 ÞX1, then there is some k such that jlk ðA1 ÞjX1. We can take z1 ¼ 1=lk ðA1 Þ, and have 1 p1 jz1 j ¼ lk ðA1 Þ and z1 lk ðA1 Þ ¼ 1
An equivalent version of the above lemma is as follows. Lemma 2.2 (Huang [10] and Sˇiljak [11]). System (1) is asymptotically stable if and only if
which contracts (9). From (8), the matrix ðI z1 A1 Þ1 exists for jz1 jp1. From (2) and (7), we get hðz1 ; z2 Þ ¼ det½I z1 A1 det½I z2 V ðz1 Þ
(10)
for jz1 j ¼ 1 and jz2 jp1, which implies that det½I z1 A1 a0 and det½I z2 V ðz1 Þa0.
hðz1 ; 0Þa0 for jz1 jp1
Since det½I z1 A1 a0, (10) implies
and hðz1 ; z2 Þa0
for jz1 j ¼ 1 and jz2 jp1.
(3)
Using the above lemma, we can obtain the following theorem. Theorem 2.1. System (1) is asymptotically stable if and only if one of the following two conditions holds: rðA1 Þo1
ðiÞ
2079
and
rðV ðzÞÞo1
for jzj ¼ 1,
and
rðW ðzÞÞo1
(4) for jzj ¼ 1,
where the matrix
(11) (12)
for jz1 j ¼ 1. Since the matrix V ðz1 Þ is unitarily similar to an upper triangular matrix O [13], from rðV ðz1 ÞÞo1 for jz1 j ¼ 1, we have det½I z2 V ðz1 Þ ¼ det½I z2 Oa0
1
W ðzÞ ¼ ðI zA2 Þ ðA1 þ zA3 Þ.
(5)
Here, rðF Þ stands for the spectral radius of a complex-valued matrix F . Proof. We will prove part (i) of the theorem first. Necessity: If system (1) is asymptotically stable, from Lemma 2.2, we have hðz1 ; 0Þa0 for jz1 jp1
(6)
and for jz1 j ¼ 1 and jz2 jp1.
(7)
Since any matrix is unitarily similar to an upper triangular matrix D [13], from (2) and (6), we have hðz1 ; 0Þ ¼ det½I z1 A1 ¼ det½I z1 Da0 for jz1 jp1.
hðz1 ; 0Þ ¼ det½I z1 A1 a0. hðz1 ; z2 Þ ¼ det½I z1 A1 det½I z2 V ðz1 Þ
V ðzÞ ¼ ðI zA1 Þ1 ðA2 þ zA3 Þ.
hðz1 ; z2 Þa0
for jz1 j ¼ 1 and jz2 jp1. In the same way for the proof of rðA1 Þo1, we have rðV ðzÞÞo1 for jzj ¼ 1. Sufficiency: If condition (i) is satisfied, from rðA1 Þo1, det½I z1 A1 a0 for jz1 jp1. We get Using (2) we get
where the matrix
ðiiÞ rðA2 Þo1
det½I z2 V ðz1 Þa0
(8)
for jz1 j ¼ 1 and jz2 jp1.
(13)
Using (13) we get hðz1 ; z2 Þ ¼ det½I z1 A1 det½I z2 V ðz1 Þa0 for jz1 j ¼ 1 and jz2 jp1.
(14)
From (11) and (14), system (1) is asymptotically stable by means of Lemma 2.2. Similar to the proof process of (i), we can show that condition (ii) is necessary and sufficient for the asymptotic stability of system (1). The proof is completed. & Remark 2.1. Since the conditions of Theorem 2.1 only involve one complex variable z with jzj ¼ 1, it is easier to check Theorem 2.1 than Lemmas 2.1 and 2.2.
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3. Simple stability criteria
and
In the section, several stability criteria will be presented via the nonnegative matrix theory. It is easier to check these criteria than those of the above section. We need the following definitions and lemmas. Lemma 3.1 (Lancaster [13]). Let W 2 Cnn and V 2 Rnn . If jW jpV , then rðW ÞprðV Þ.
W qþ1 pW q .
Proof. We will prove part (i) of the theorem first. By means of Lemma 3.2, for jzj ¼ 1 we have ðI zA1 Þ1 ¼ I þ zA1 þ ðzA1 Þ2 þ ¼ I þ zðI zA1 Þ1 A1
Lemma 3.2 (Lancaster [13]). If rðW Þo1, then ðI W Þ1 exists and
and
ðI W Þ1 ¼ I þ W þ W 2 þ .
jðI zA1 Þ1 j ¼ jI þ zA1 þ ðzA1 Þ2 þ j ¼ ðI jA1 jÞ1 .
(i) Assume rðjA1 jÞo1. Define the matrix þ ðI jA1 jÞ1 jA1 ðA1 A2 þ A3 Þj,
(15)
and for an integer qX1, define the matrix q X
jAj1 ðA1 A2 þ A3 Þj
jA1qþ1 ðA1 A2
þ A3 Þj.
V ðzÞ ¼ ðI zA1 Þ1 ðA2 þ zA3 Þ ¼ ðI zA1 Þ1 A2 þ ½I þ zðI zA1 Þ1 A1 zA3
j¼1
þ ðI jA1 jÞ
(24)
From (23), for jzj ¼ 1, we have
V 0 ¼ jA2 j þ jA1 A2 þ A3 j
1
(23)
pI þ jA1 j þ jA1 j2 þ
Definition 3.1. Consider system (1).
V q ¼ jA2 j þ jA1 A2 þ A3 j þ
(22)
(16)
¼ ½I þ zA1 þ ðzA1 Þ2 þ A2 þ zA3 þ z2 ðI zA1 Þ1 A1 A3 ¼ A2 þ zðA1 A2 þ A3 Þ þ ½ðzA1 Þ2 þ ðzA1 Þ3 þ A2 þ z2 ðI zA1 Þ1 A1 A3
(ii) Assume rðjA2 jÞo1. Define the matrix
¼ A2 þ zðA1 A2 þ A3 Þ þ ½I þ zA1 þ ðzA1 Þ2
W 0 ¼ jA1 j þ jA2 A1 þ A3 j
þ ðzA1 Þ2 A2 þ z2 ðI zA1 Þ1 A1 A3
þ ðI jA2 jÞ1 jA2 ðA2 A1 þ A3 Þj,
(17)
and for an integer qX1, define the matrix W q ¼ jA1 j þ jA2 A1 þ A3 j þ
q X
þ z2 ðI zA1 Þ1 A1 A3
jAj2 ðA2 A1 þ A3 Þj
j¼1
þ ðI jA2 jÞ1 jAqþ1 2 ðA2 A1 þ A3 Þj.
¼ A2 þ zðA1 A2 þ A3 Þ þ z2 ðI zA1 Þ1 ðA1 Þ2 A2 ¼ A2 þ zðA1 A2 þ A3 Þ þ z2 ðI zA1 Þ1 A1 ðA1 A2 þ A3 Þ.
(18) From (24), for jzj ¼ 1, jV ðzÞj ¼ jA2 þ zðA1 A2 þ A3 Þ
Theorem 3.1. Consider system (1).
þ z2 ðI zA1 Þ1 A1 ðA1 A2 þ A3 Þj
(i) Assume rðjA1 jÞo1, then, for qX1 and jzj ¼ 1,
pjA2 j þ jA1 A2 þ A3 j þ jðI zA1 Þ1 A1 ðA1 A2 þ A3 Þj
jV ðzÞj ¼ jðI zA1 Þ ðA2 þ zA3 ÞjpV q pV 0 1
(19)
¼ jA2 j þ jA1 A2 þ A3 j þ j½I þ zA1 þ ðzA1 Þ2
(20)
pjA2 j þ jA1 A2 þ A3 j q X jðA1 Þj ðA1 A2 þ A3 Þj þ
þ A1 ðA1 A2 þ A3 Þj
and V qþ1 pV q .
(ii) Assume rðjA2 jÞo1, then, for qX1 and jzj ¼ 1, jW ðzÞj ¼ jðI zA2 Þ ðA1 þ zA3 ÞjpW q pW 0 1
(21)
j¼1
þ ðI jA1 jÞ1 jðA1 Þqþ1 ðA1 A2 þ A3 Þj ¼ V q,
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where the matrix V 0 is given by (15) in Definition 3.1.
i.e., jV ðzÞjpV q .
(25)
V q ¼ jA2 j þ jA1 A2 þ A3 j þ
q X
jðA1 Þj ðA1 A2 þ A3 Þj
j¼1 1
þ ðI jA1 jÞ jðA1 Þ ¼ jA2 j þ jA1 A2 þ A3 j
qþ1
ðA1 A2 þ A3 Þj
rðW 0 Þo1,
where the matrix W 0 is given by (17) in Definition 3.1. Further criteria for stability of system (1) are as follows. Theorem 3.3. System (1) is asymptotically stable if one of the following two conditions holds:
2
þ ½jA1 ðA1 A2 þ A3 Þj þ jðA1 Þ ðA1 A2 þ A3 Þj þ þ jðA1 Þq ðA1 A2 þ A3 Þj
ðiÞ
þ ðI þ jA1 j þ jA1 j2 þ ÞjðA1 Þqþ1 ðA1 A2 þ A3 Þj
pjA2 j þ jA1 A2 þ A3 j
rðjA1 jÞo1 and
þ ðI þ jA1 j þ jA1 j þ
rðV q Þo1
for qX1,
where the matrix V q is given by (16) in Definition 3.1. and
ðiiÞ rðjA2 jÞo1
2
rðW q Þo1
for qX1,
where the matrix W q is given by (18) in Definition 3.1.
þ jA1 jq1 ÞjA1 ðA1 A2 þ A3 Þj
Remark 3.1. Theorem 3.3 is sharper than Theorem 3.2 since rðV q ÞprðV 0 Þ and rðW q ÞprðW 0 Þ.
þ ðjA1 jq þ jA1 jqþ1 þ ÞjA1 ðA1 A2 þ A3 Þj ¼ jA2 j þ jA1 A2 þ A3 j þ ðI þ jA1 j þ jA1 j2
When all the elements of matrices A1 ; A2 and A3 are nonnegative, we have the following corollary.
þ ÞjA1 ðA1 A2 þ A3 Þj ¼ jA2 j þ jA1 A2 þ A3 j
Corollary 3.1. When all the elements of matrices A1 ; A2 and A3 are nonnegative. System (1) is asymptotically stable if and only if one of the following two conditions holds:
þ ðI jA1 jÞ1 jA1 ðA1 A2 þ A3 Þj ¼ V 0, i.e., V q pV 0 .
and
ðiiÞ rðjA2 jÞo1
ðiÞ (26)
From (25) and (26), we have jV ðzÞjpV q pV 0 , i.e., inequality (19) holds. Similarly, we can prove the inequality V qþ1 pV q for qX1. Similar to the proof process of (i), we can show that condition (ii) holds. The proof is completed. & Since jV ðzÞj ¼ jV ðzÞj and jW ðzÞj ¼ jW ðzÞj, we have rðV ðzÞÞprðjV ðzÞjÞ and rðW ðzÞÞprðjW ðzÞjÞ by Lemma 3.1. According to Theorem 3.1, for jzj ¼ 1 we have
rðA1 Þo1 and
rðV 0 Þo1,
where the matrix V 0 ¼ ðI A1 Þ1 ðA2 þ A3 Þ. ðiiÞ rðA2 Þo1
and
rðW 0 Þo1,
where the matrix W 0 ¼ ðI A2 Þ1 ðA1 þ A3 Þ. Proof. Sufficiency: According to Theorem 3.2, we have that V 0 ¼ jA2 j þ jA1 A2 þ A3 j þ ðI jA1 jÞ1 jA1 ðA1 A2 þ A3 Þj
rðV ðzÞÞprðjV ðzÞjÞprðV q ÞprðV 0 Þ
¼ A2 þ A1 A2 þ A3 þ ðI A1 Þ1 A1 ðA1 A2 þ A3 Þ
and
¼ ðI A1 Þ1 ½ðI A1 ÞðA2 þ A1 A2 þ A3 Þ
rðW ðzÞÞprðjW ðzÞjÞprðW q ÞprðW 0 Þ.
þ A1 ðA1 A2 þ A3 Þ ¼ ðI A1 Þ1 ½A2 þ A1 A2 þ A3 A1 A2
By means of Theorem 2.1, we can directly derive the following criteria, Theorems 3.2 and 3.3 for the asymptotic stability of system (1). Theorem 3.2. System (1) is asymptotically stable if one of the following two conditions holds: ðiÞ
rðjA1 jÞo1
and
rðV 0 Þo1,
A1 A1 A2 A1 A3 þ A1 A1 A2 þ A1 A3 ¼ ðI A1 Þ1 ðA2 þ A3 Þ, i.e., V 0 ¼ ðI A1 Þ1 ðA2 þ A3 Þ.
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matrix ðI jA1 jÞ1 ðjA2 j þ jA3 jÞ.
Similar to the above, we have that
V 0 ¼ jA2 j þ jA1 A2 þ A3 j
W 0 ¼ ðI A2 Þ1 ðA1 þ A3 Þ.
þ ðI jA1 jÞ1 jA1 ðA1 A2 þ A3 Þj, ¼ ðI jA1 jÞ1 ½ðI jA1 jÞðjA2 j þ jA1 A2 þ A3 jÞ
In the case, rðjA1 jÞ ¼ rðA1 Þ and rðjA2 jÞ ¼ rðA2 Þ. The sufficiency is proved. Necessity: By Theorem 2.1, let z ¼ 1 in (4) and (5), the necessity is proved. The proof is completed. &
þ jA1 ðA1 A2 þ A3 Þj ¼ ðI jA1 jÞ1 ½jA2 j þ jA1 A2 þ A3 j jA1 jjA2 j jA1 jjA1 A2 þ A3 j þ jA1 ðA1 A2 þ A3 Þj pðI jA1 jÞ1 ½jA2 j þ jA1 jjA2 j þ jA3 j jA1 jjA2 j
Remark 3.2. The approach proposed can be applied to Roesser’s state-space model since Roesser’s model can be transferred into Fornasini–Marchesini model [6]. Similar stability criteria may be obtained.
jA1 jjA1 A2 þ A3 j þ jA1 jjA1 A2 þ A3 j ¼ ðI jA1 jÞ1 ðjA2 j þ jA3 jÞ. According to Lemma 3.1, we obtain
Now we compare our criteria with those of [3,4]. The stability criterion in [3,4] for system (1) is as follows. Lemma 3.3. System (1) is asymptotically stable if a ¼ rðjA1 j þ jA2 j þ jA3 jÞo1.
(27)
The following lemma is the main result of [3,4]. Lemma 3.4 (Bose and Trautman [3] and Su and Bhaya [4]). The following conditions on matrices A1 ; A2 ; A3 2 Rnn are equivalent: ðaÞ rðjA1 jÞo1
and
1
r½ðI jA1 jÞ ðjA2 j þ jA3 jÞo1. ðbÞ rðjA2 jÞo1 and r½ðI jA2 jÞ1 ðjA1 j þ jA3 jÞo1. ðcÞ rðjA3 jÞo1
and
1
r½ðI jA3 jÞ ðjA1 j þ jA2 jÞo1.
ðiÞ
rðjA1 jÞo1 and
rðV 0 Þo1
from ðaÞ rðjA1 jÞo1
and
1
r½ðI jA1 jÞ ðjA2 j þ jA3 jÞo1. Hence (d) in Lemma 3.4 ) (i) in Theorem 3.2. Similarly we can prove (b) in Lemma 3.4 ) (ii) in Theorem 3.2, then (d) in Lemma 3.4 ) (ii) in Theorem 3.2. The proof is completed. & Remark 3.3. Theorem 3.3 is also sharper than Lemma 3.3 according to Theorem 3.1. It is easier to check these stability criteria of Theorems 3.2 and 3.3 than those of Theorem 2.1. Now we will compare our criteria with those of [2]. We need the following definition. Definition 3.2 (Hu and Liu [2]). If rðjA1 jÞo1 let b1 ¼ r½jA2 j þ jA3 j þ ðI jA1 jÞ1 ðjA1 A2 j þ jA1 A3 jÞ. (28)
ðdÞ a ¼ rðjA1 j þ jA2 j þ jA3 jÞo1. If rðjA2 jÞo1 let We will show that our conditions are sharper than those of [3,4].
b2 ¼ r½jA1 j þ jA3 j þ ðI jA2 jÞ1 ðjA2 A1 j þ jA2 A3 jÞ.
(29) If rðjA3 jÞo1 let
Theorem 3.4. If a ¼ rðjA1 j þ jA2 j þ jA3 jÞo1, then both conditions (i) and (ii) in Theorem 3.2 hold. i.e., Theorem 3.2 is sharper than Lemma 3.3. Proof. We will show that (d) in Lemma 3.4 ) (i) in Theorem 3.2. By means of Lemma 3.4, (d) ) (a), it is sufficient to show (a) of Lemma 3.4 ) (i) of Theorem 3.2. We compare matrix V 0 with
b3 ¼ r½jA1 j þ jA2 j þ ðI jA3 jÞ1 ðjA3 A1 j þ jA3 A2 jÞ.
(30) The stability criteria in [2] for system (1) are as follows. Lemma 3.5 (Hu and Liu [2]). The state-space model (1) is asymptotically stable if one of the
ARTICLE IN PRESS T. Liu / Signal Processing 88 (2008) 2078–2084
following three conditions holds: ðaÞ rðjA1 jÞo1
and
b1 o1.
ðbÞ rðjA2 jÞo1
and
b2 o1.
ðcÞ rðjA3 jÞo1
and
b3 o1.
2083
Remark 3.4. Theorem 3.5 shows that condition (i) (condition (ii)) in Theorem 3.2 is sharper than condition (a) (condition (b)) in Lemma 3.5 when A1 ; A2 ; A3 2 R1 . But in the general case, we do not know whether the relation is true. 4. Numerical examples
Comparing Theorem 3.2 with that of [2], we have the following result. Theorem 3.5. Suppose all A1 ; A2 and A3 are scalar, i.e., A1 ; A2 ; A3 2 R1 : (I) If rðjA1 jÞo1 and b1 o1, then rðjA1 jÞo1 and rðV 0 Þo1. (II) If rðjA2 jÞo1 and b2 o1, then rðjA2 jÞo1 and rðW 0 Þo1. Here the matrices V 0 and W 0 are defined by (15) and (17), respectively. Proof. By the definitions of V 0 and b1 , we have V 0 ¼ jA2 j þ jA1 A2 þ A3 j þ ðI jA1 jÞ1 jA1 ðA1 A2 þ A3 Þj, b1 ¼ r½jA2 j þ jA3 j þ ðI jA1 jÞ1 ðjA1 A2 j þ jA1 A3 jÞ. Since A1 ; A2 ; A3 2 R1 ,
We will illustrate our stability criteria (Theorem 3.2) for linear 2-D system (1) through several examples. All computations in this section are carried out by Matlab 7.0. Example 4.1. In the state-space model (1), let A1 ¼ 0:8, A2 ¼ 0:7 and A3 ¼ 0:6. We have rðjA1 jÞ ¼ 0:8, rðjA2 jÞ ¼ 0:7 and a ¼ 2:1. Since a ¼ 2:141, we cannot determine whether system (1) is stable by Lemma 3.3, the result of [3,4]. We have that b1 ¼ 6:5; b2 ¼ 4:6667 and b3 ¼ 3:75. Since they are all greater that 1, we cannot determine whether system (1) is stable by Lemma 3.5, the result of [2]. Since rðV 0 Þ ¼ 0:9o1 (or rðW 0 Þ ¼ 0:9333o1), condition (i) (or (ii)) of Theorem 3.2 is satisfied. System (1) is asymptotically stable by Theorem 3.2. Example 4.2. In system (1), let 0:72 0 0:7 A1 ¼ ; A2 ¼ 0 0:72 0:2
rðV 0 Þ b1 ¼ jA2 j þ jA1 A2 þ A3 j þ ð1 jA1 jÞ1 jA1 ðA1 A2 þ A3 Þj
and
1
½jA2 j þ jA3 j þ ð1 jA1 jÞ ðjA1 A2 j þ jA1 A3 jÞ ¼ jA1 A2 þ A3 j jA3 j þ ð1 jA1 jÞ1 ½jA1 ðA1 A2 þ A3 Þj ðjA1 A2 j þ jA1 A3 jÞ
A3 ¼
0:5
0:1
0:15
0:5
.
We have rðjA1 jÞ ¼ 0:72, rðjA2 jÞ ¼ 0:8414 and a ¼ 2:1846. Since a41, we cannot determine whether system (1) is stable by Lemma 3.3, the result of [3,4]. We have that b1 ¼ 5:2306; b2 ¼ 8:4729 and b3 ¼ 4:139. Since they are all greater that 1, we cannot determine whether system (1) is stable by Lemma 3.5, the result of [2]. On the other hand, we have that rðV 0 Þ ¼ 0:9247o1 and rðW 0 Þ ¼ 0:8259o1. Since rðV 0 Þo1 (or rðW 0 o1), condition (i) (or (ii)) of Theorem 3.2 is satisfied. System (1) is asymptotically stable by Theorem 3.2.
pjA1 A2 j þ ð1 jA1 jÞ ½jA1 A1 A2 j 1
þ jA1 A3 j jA1 A2 j jA1 A3 j ¼ jA1 A2 j þ ð1 jA1 jÞ1 ðjA1 A1 A2 j jA1 A2 jÞ ¼ jA1 A2 j þ ð1 jA1 jÞ1 ½ðjA1 j 1ÞjA1 A2 j ¼ jA1 A2 j jA1 A2 j ¼ 0, i.e., rðV 0 Þpb1 .
0:1 0:7
(31)
According to (31) and b1 o1, we have rðV 0 Þpb1 o1. Part (I) is proved. Similar to the above, we can prove part (II). The proof is completed. &
Example 4.3 (See Example 2.1 in Hu and Liu [2]). In the state-space model (1), let 0:2 0:1 0:3 0:1 A1 ¼ ; A2 ¼ 0:02 0:1 0:2 0:1
ARTICLE IN PRESS T. Liu / Signal Processing 88 (2008) 2078–2084
2084
and A3 ¼
0:4 0:33 0:1
0:3
.
Again, we have rðjA1 jÞ ¼ 0:2171, rðjA2 jÞ ¼ 0:3732 and a ¼ 1:1578. Since a41, we cannot determine whether system (1) is stable by Lemma 3.3, the result of [3,4]. We have that b1 ¼ 1:11; b2 ¼ 1:251 and b3 ¼ 0:935. Since b3 ¼ 0:935o1, condition (c) of Lemma 3.5 [2] is satisfied. System (1) is asymptotically stable by Lemma 3.5, the result of [2]. To apply Theorem 3.2, we compute rðV 0 Þ ¼ 1:0264 and rðW 0 Þ ¼ 0:9666. Since rðW 0 Þo1, condition (ii) of Theorem 3.2 is satisfied. It also shows that system (1) is asymptotically stable by Theorem 3.2. Examples 4.1 and 4.2 show that Theorem 3.2 is sharper than Lemmas 3.3 and 3.5 which are main results in the literature. When the conditions in Lemmas 3.3 and 3.5 are not satisfied, the conditions in Theorem 3.2 are satisfied. Example 4.3 shows that the two conditions in Theorem 3.2 are not equivalent since rðV 0 Þ41 and rðW 0 Þo1, although the two conditions in Theorem 2.1 are equivalent by which the two conditions in Theorem 3.2 are derived. The two conditions in Theorem 3.2 are only sufficient for stability. 5. Conclusions Necessary and sufficient conditions (Theorem 2.1) for stability of the Fornasini–Marchesini model are presented which are easier than those (Lemmas 2.1 and 2.2) in literature. Then new sufficient conditions (Theorems 3.2 and 3.3) for stability of the system, which are sharper than those in [2–4] and computable, are derived. When all the parameter matrices are nonnegative, the criteria are necessary and sufficient for stability of the system.
The approach proposed can be applied to Roesser’s state-space model [6]. Similar stability criteria may be obtained. Acknowledgements The author is greatly indebted to the associate editor and the three anonymous referees for careful reviews and valuable suggestions. References [1] E. Fornasini, G. Marchesini, State-space realization theory of two-dimensional filters, IEEE Trans. Automat. Control 21 (1976) 484–492. [2] G.D. Hu, M. Liu, Simple criteria for stability of twodimensional linear systems, IEEE Trans. Signal Process. 53 (2005) 4720–4723. [3] T. Bose, D.A. Trautman, Two’s complement quantization in two-dimensional digital state-space filters, IEEE Trans. Signal Process. 40 (1992) 2589–2592. [4] Y. Su, A. Bhaya, On the Bose–Trautman condition for stability of two-dimensional linear systems, IEEE Trans. Signal Process. 46 (1998) 2069–2070. [5] M.B. Zarrop, P.E. Wellstead, Two-dimensional and EM techniques for cross directional estimation and control, IEE Proc. Control Theory Appl. 149 (2002) 457–462. [6] T. Kaczorek, Two-Dimensional Linear Systems, Springer, Berlin, 1985. [7] H. Kar, V. Sigh, Stability of 2-D systems described by the Fornasini–Marchesini first model, IEEE Trans. Signal Process. 51 (2003) 1675–1676. [8] T. Ooba, On stability analysis of 2-D systems based on 2-D Lyapunov matrix inequalities, IEEE Trans. Circuit Syst. I 47 (2000) 1263–1265. [9] C. Du, L. Xie, Stability analysis and stabilization of uncertain two-dimensional discrete systems: an LMI approach, IEEE Trans. Circuit Syst. I 46 (1999) 1371–1374. [10] T.S. Huang, Stability of two-dimensional recursive filters, IEEE Trans. Audio Electroacoustics 20 (1972) 158–163. [11] D.D. Sˇiljak, Stability criteria for two-variable polynomials, IEEE Trans. Circuit Syst. 22 (1975) 185–189. [12] Y. Bistritz, On an inviable approach for derivation of 2-D stability tests, IEEE Trans. Circuit Syst. II 52 (2005) 713–718. [13] P. Lancaster, The Theory of Matrices with Applications, Academic Press, Inc., Orlando, 1985.