Stability Analysis of Delayed Cellular Neural ... - Semantic Scholar
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 51, NO. 11, NOVEMBER 2004
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Stability Analysis of Delayed Cellular Neural Networks Described Using Cloning Templates Zhigang Zeng, Jun Wang, Senior Member, IEEE, and Xiaoxin Liao
Abstract—In this paper, we show that any -dimensional delayed cellular neural network described using cloning templates can have no more than 3 isolated equilibrium points and 2 of these equilibrium points located in saturation regions are locally exponentially stable. In addition, we give the conditions for the equilibrium points to be locally exponentially stable when the equilibrium points locate the designated saturation region. These conditions improve and extend the existing stability results in the literature. The conditions are also very easy to be verified and can be checked by direct examination of the templates, regardless of the number of cells. Finally, the validity and performance of the results are illustrated by use of two numerical examples. Index Terms—Cellular neural network (CNN), cloning template, isolated equilibria, stability.
I. INTRODUCTION
C
ELLULAR neural networks (CNNs) are arrays of dynamical cells that are suitable for the formulation and solution of many complex computational problems. In recent years, CNNs [1] have attracted great attention due to their significant potential in applications. It is well known that the stability of CNNs and delayed CNNs (DCNNs) are critical for their applications, especially in signal processing, image processing, solving nonlinear algebraic and transcendental equations, and solving optimization problems [1]–[4]. A preliminary step for investigating the dynamics of CNNs and DCNNs is the stability analysis. As CNNs and DCNNs are large-scale nonlinear dynamical systems, their stability analysis is a nontrivial task. In the past decade, the stability of CNNs and DCNNs has been widely investigated; e.g., [5]–[28]. However, most of these results apply to general networks and do not really exploit the two main characteristics of CNNs: the local connectivity and the space-invariance structure. CNN design is based on the cloning templates suitable for given applications. Therefore, it is important for the stability conditions to be directly expressed in terms of their templates.
Manuscript received March 1, 2004; revised May 9, 2004. This works was supported by the Hong Kong Research Grants Council under Grant CUHK4165/03E and the Young Foundation of the Education Department of Hubei Province, China. This paper was recommended by Associate Editor C.-T. Lin. Z. Zeng was with Department of Automation, University of Science and Technology of China, Hefei, China. He is now with the Department of Automation and Computer-Aided Engineering, The Chinese University of Hong Kong, Hong Kong, and also with the Department of Mathematics, Hubei Normal University, Huangshi, China (e-mail: [email protected]). J. Wang is with the Department of Automation and Computer-Aided Engineering, The Chinese University of Hong Kong, Hong Kong (e-mail: [email protected]). X. Liao is with the Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, China. Digital Object Identifier 10.1109/TCSI.2004.836855
In this paper, we study stability of DCNNs described using templates. The stability conditions can be checked by direct examination of the templates, regardless of the number of cells. This seems to be an advantage in comparison with the results in [25]–[27] in dealing with large-scale DCNNs. Specifically, the results in this paper consist of four theorems and five corol-dimensional CNN laries. Theorem 1 shows that any or DCNN can have no more than isolated equilibrium -dimensional CNN and DCNN can points, shows that the isolated equilibrium points which are locally exhave ponentially stable, and gives the estimates of attractive domain locally exponentially stable equilibrium points. of such Theorem 2 gives the conditions for an equilibrium to be locally exponentially stable when the equilibrium locates in the saturation region. Theorem 3 gives the conditions for an equilibrium to be locally exponentially stable when the equilibrium locates in . Theorem 4 gives the conditions for an equilibrium to be locally exponentially stable when the equilibrium locates in the other regions. The remaining part of this paper consists of six sections. In Section II, relevant background information is given. In Section III, the number of equilibria of DCNNs is obtained. In Section IV, the number of equilibria of DCNNs without external input is obtained. In Section V, two illustrative examples are provided with simulation results. Finally, concluding remarks are given in Section VI. II. PRELIMINARIES Consider a two-dimensional (2-D) DCNN described by two space-invariant templates where the cells are arranged on a rectrows and columns. The dyangular array composed of namics of such a DCNN is governed by the following normalized equations:
(1) where denotes the state of the cell located at the crossing between the th row and th column of the network, and are positive integers is denoting neighborhood radiuses, the feedback cloning template defined by a
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real matrix, cloning template defined by a
is the delay feedback real matrix is the activation function
defined by (2) is the time delay upper bounded by a constant , and is an external input. An alternative expression for the state equation of a DCNN can be obtained by ordering the cells in some way (e.g., by rows or by columns) and by cascading the state variables into a state . vector The following compact form is then obtained: (3) where the coefficient matrices and are obtained and . The input vector is through the templates vector-valued activation function obtained through
. Denote
, so can be divided into
subspaces
or
or (4)
and
can be divided into three subspaces
Fig. 1.
Decompose 2-D space.
is continuous and satisfies (1) and , for . Also, simply denote as the solution of (1). Definition 1: The set is said to be a globally of (1) with any attractive set of (1), if for the solution , there exists initial condition depending on such that for . Definition 2: The set is said to be a globally exponentially of (1) with any attractive set of (1), if for the solution , there exist coninitial condition and (depending on ) such that for stants
Definition 3: The point is said to be an isolated equilibis an equilibrium point of (1) and there rium point of (1), if such that exists is not an equilibrium point of (1). Definition 4: The equilibrium point of (1) is said to be locally exponentially stable in region , if there exist constants such that for
or
Hence, gion up of
is made up of one part, the saturation reis made up of parts, is made parts. For example, when , and all of the parts of are depicted in Fig. 1. be the space of continuous functions Let mapping into with norm defined , by where . Denote as the vector norm of the vector . The initial condition of DCNN (1) is assumed to be (5) . Denote as the sowhere lution of (1) with initial condition (5), it means that
where is the solution of the neural network (1) with , and is said to any initial condition be a locally exponentially attractive set of the equilibrium point . When is said to be globally exponentially stable. be a bounded and close set in Lemma 1 [29]: Let be a mapping on complete matric space , is where measurement in . If and there exists a constant such that , then such that . there exists is a nonsingular -matrix, then Lemma 2 [8]: If the DCNN (3) is globally exponentially stable, where is the identity matrix
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Consider the following coupled system: (6)
locally exponenIn addition, if (9) holds, then (1) has tially stable equilibrium points located in the saturation region, where the saturation region
(7) or where
and
are
matrices, is the into . space of continuous functions mapping Lemma 3: If (7) is globally exponentially stable, then (6) is also globally exponentially stable. Proof: By applying the variant format of constants, soluof (6) can be expressed as tion
Proof: If
, then from (2)
(10)
(11)
Since (7) is globally exponentially stable, there exist constants such that . Hence, , where . Then, when , for
Hence, from (2), for DCNN (1), there exists an equilibrium point if and only if for
when ; i.e., (6) is also globally exponentially stable.
(12)
III. NUMBER OF EQUILIBRIA In the following, we always assume that , and
are empty. Denote
(8) Then,
, where
is defined in (4).
A. Equilibria in Theorem 1: The DCNN (1) has exactly librium points if and only if
isolated equi-
(9)
-dimensional linear algebraic equaEquation (12) is an tion, its solution is an isolated point, a line segment, or a higher dimensional region. Proof of necessary condition: Since solution of (12) is one of the following cases: an isolated point, a line segment, or a higher dimensional region ( -dimensional region), if (1) has neither more nor less than isolated equilibrium points, then (1) has neither more nor less than one isolated equilibrium point . Specially, choose to be empty, let located in be an equilibrium point of (1). For can equal , and can also equal to . From (12), if , then to
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If choose
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 51, NO. 11, NOVEMBER 2004
, choose , then
if
,
(18) , for
From (15) and (17), when
(13) Similarly, if
, then
Similarly, when
, for
(14) Since for , (13) and (14) hold, it implies that (9) holds. Proof of sufficiency: According to (9), the coefficient matrix of (12) is nonsingular, ; i.e., the solution of hence (12) has only a solution in (12) is an isolated point. Since is divided into subspaces , the DCNN (1) has no more than isolated equilibrium points. For , let (15) (16)
(17)
Hence, from (16) and (18), addition, for
. In
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From (9) and Lemma 1, there exists such that . Choose (19) , when From (17) and . From (19) and (9), when
Then, i.e., is of (1). an isolated equilibrium point located in is divided into subspaces , Since the DCNN (1) has neither more nor less than isolated equilibrium points. to be empty, then Specially, choose is a saturation region. According to the above proof, is an equilibrium point located in a saturation region. Obviously, the number of such saturation in . Hence,(1) has equilibrium region is , points located in saturation regions. Let then when and locate in the same saturation region, . So when , and locate in the same saturation region (20) Obviously, (20) is exponentially stable. Hence, is locally exponentially stable equilibrium point of (1). According to the definition of in (1), when . Example 1: Consider
(21)
Similarly, when Proposition: If (21) has neither more nor less than isolated equilibrium points, then (22) Proof: and only if
is an equilibrium point of (21) if
(23) Consider the region depicted in Fig. 1
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In above region, (for example, only in ), it is impossible that (21) has two isolated equilibrium points. In fact, in , (23) is a 2-D algebraic equation (24)
In
, (23) is a 2-D algebraic equation (32)
If the solution of (32) is an isolated point in
, then (33)
The solution of (24) is an isolated point; i.e., , or is empty. Hence, in , (21) has an isolated equilibrium point or not isolated equilibrium point (when ); i.e., it is impossible that (21) has two isolated equilibrium points located . in In , (23) is a 2-D algebraic equation
Equations (27), (29), (31), and (33) imply that (22) holds. Remark: Since , if (9) holds, then when
(25) The solution of (25) is an isolated point or a beeline or a plane; i.e., it is impossible that (21) has two isolated equilibrium points . located in In any one of , it is similar to prove that (21) has no more than one isolated equilibrium point. isoIf (21) has neither more nor less than lated equilibrium points, then there exists an isolated equilibrespectively. rium point located In , (23) is a 2-D algebraic equation (26) If the solution of (26) is an isolated point in
when
Hence, (22) holds. Similarly, if (22) holds then (9) holds in Example 1. B. Equilibria in the Saturation Region Let or . Theorem 2: If there exists
, then
such that
(27) In
, (23) is a 2-D algebraic equation (28)
If the solution of (28) is an isolated point in
, then (29)
In
, (23) is a 2-D algebraic equation (30)
If the solution of (30) is an isolated point in
, then (31)
(34) then (1) has exactly one locally exponentially stable equilibrium point located in the saturation region , where saturation region
ZENG et al.: STABILITY ANALYSIS OF DELAYED CNNs
Proof: For , then from (1)
,
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, let
C. Equilibria in Theorem 3: If
(35) . Choose , then , and is an equifrom (34), librium point of (35). Similar to the proof of Theorem 1, is locally exponentially stable equilibrium point located in saturaof (35); i.e., (1) has neither more nor tion region less than one isolated equilibrium points located in saturation regions , and it is locally exponentially stable. Corollary 1: If Obviously, If
(36)
, then
then (1) has one locally exponentially stable equilibrium point . located in the saturation region , according to Proof: Choose Theorem 2, (1) has a locally exponentially stable equilibrium . point located in the saturation region Corollary 2: If
then (1) has neither more nor less than one globally exponen. tially stable isolated equilibrium point located in , let Proof: For
Since , according to Lemma 1, similar to the proof of Theorem 1, there exists such that ; is an isolated equilibrium located in of (1). i.e., From (3) and (36), is a nonsingular -matrix. According to Lemma 2, the equilibrium point of (3) is globally exponentially stable; i.e., (1) is globally exponentially stable. D. Equilibria in Other Regions Theorem 4: If for
then (1) has a locally exponentially stable equilibrium point located in saturation region . , according Proof: Choose to Theorem 2, (1) has a locally exponentially stable equilibrium . point located in saturation region
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for
, similar to the proof of Theorem 1, Since such from (38), (39), and (40), there exists is an isolated equilibrium point located in that of (1). From (37), for
(37) for If
, consider subsystem of (1)
(41) then (1) has neither more nor less than one isolated equilib, and it is locally exponentially rium located in is defined by (4). stable, where , let Proof: For
and the subsystem of (1)
(38) (42) From (41)
(39) (40)
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from (42)
, its cloning template is . The matrix , defined by (3), composed of the template has the circulant form
..
..
.
..
.
(43)
.
Also, for example, for a 2-D space-invariant CNN with a neighborhood radius , its cloning template is a real matrix; i.e.,
where
.. .
..
.
.. .
..
.
.. .
.. .
..
.
.. .
..
.
.. .
, and , and .
When
For simplicity, as required in most applications, in the following, . The matrix , defined by (3), we always assume that composed of the template has the form
or or
; when .
Hence, (41) and (42) can be rewritten as .. .
.. .
.. .
.. .
..
.. .
.
.. .
where
respectively, where is a constant. According to Lemma 2, (37) implies that the equilibrium of (41) is globally exponentially stable. Since the equilibrium of (41) is globally exponentially stable, according to Lemma 3, the equilibrium of (42) is also globally of (1) is exponentially stable. Hence the equilibrium point locally exponentially stable in .
.. .
.. .
.. .
..
.
.. .
.. .
.. .
.. .
.. .
..
.
.. .
.. .
IV. NUMBER OF EQUILIBRIA WHEN EXTERNAL INPUT VECTOR IS NULL Throughout this section, we assume that . The dynamic rule of a DCNN can be completely specified by its in the DCNN (3) depend on the cloning templates. and established order among the cells and on the cloning templates and the delay cloning template. For example, for a one-dimensional (1-D) space-invariant CNN with a neighborhood radius
.. .
and
.. .
.. .
..
.
.. .
can be similarly defined.
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A. One-Dimensional Space-Invariant DCNNs
B. Two-Dimensional Space-Invariant DCNNs , for
Corollary 5: If
, for
Corollary 3: If
, and
, and
(44)
then (1) has two locally exponentially stable equilibrium points and , located in saturation region respectively. or Proof: Choose , (44) implies that (34) holds. According to Theorem 2, Corollary 3 holds. Corollary 4: If , for , and
(45)
then (1) has two locally exponentially stable equilibrium points located in saturation region and , respectively. or Proof: Choose , (45) implies that (34) holds. According to Theorem 2, Corollary 4 holds. To show the usefulness of Corollaries 3 and 4, we discuss a simple example. We consider the DCNN described by the 1D . According to template Corollary 3, if (46)
where , then (1) has two locally exponentially stable equilibrium points located in saturation region and , respectively. Proof: Choose or , conditions of Corollary 5 imply that (34) holds. According to Theorem 2, Corollary 5 holds. V. ILLUSTRATIVE EXAMPLES In this section, we give two examples to illustrate the new results. Example 2: Consider a 2-D space-invariant CNN with a . Its cloning template is a neighborhood radius real matrix; i.e.,
Assume the delay cloning template then from (1) and (3)
then (1) has two locally exponentially stable equilibrium points located in saturation region and , respectively. According to Corollary 4, if (47) then (1) has two locally exponentially stable equilibrium points located in saturation region and , respectively. From (46) and (47), , then (1) has two locally exponenif tially stable equilibrium points located in saturation regions; if , then (1) has four locally exponentially stable equilibrium points located in saturation regions. From , then (1) has [25], it is known that if , such a at least one stable equilibrium point. When criterion coincides with that in [26]. For a DCNN composed of cells, the condition of the theorem in [27] is always stronger than those of Corollaries 3 and 4, regardless of the number of cells.
and
,
(48)
Hence, from (48)
(49)
According to Theorem 1, the CNN (49) has isolated equilibrium points and of them are locally exponentially stable. The are depicted in Fig. 2. transient behaviors of
ZENG et al.: STABILITY ANALYSIS OF DELAYED CNNs
Fig. 2.
Transient behavior of x
;x
;x
;x
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in Example 2.
Fig. 3. Transient behavior of x
Example 3: Consider a 2-D space-invariant CNN with a . Its cloning template is a 3 3 neighborhood radius real matrix; i.e.,
;x
;x
;x
in Example 3.
of cells. The analytical results are the improvement and extension of the existing stability results in the literature.
REFERENCES
Assume that the delay cloning template , then from (48)
and
(50)
According to Theorem 4, (50) has an isolated equilibrium point
transient behaviors of
, and it is locally exponentially stable. The are depicted in Fig. 3.
VI. CONCLUDING REMARKS In this paper, we show that any -dimensional DCNNs described by a template can have no more than isolated of them are locally exponentially equilibrium points and stable located in saturation regions. In addition, we also give the conditions for an equilibrium to be locally exponentially stable when the equilibrium point locates the designated saturation region. These easily testable conditions can be checked by direct examination of the cloning templates, regardless of the number
[1] L. O. Chua and T. Roska, “Stability of a class of nonreciprocal cellular neural networks,” IEEE Trans. Circuits Syst., vol. 37, pp. 1520–1527, Dec. 1990. [2] , “Cellular neural networks with nonlinear and delay-type template elements and nonunuform grids,” Int. J. Circuit Theory Applicat., vol. 20, pp. 449–451, 1992. [3] T. Roska, C. W. Wu, M. Balsi, and L. O. Chua, “Stability and dynamics of delay-type general and cellular neural networks,” IEEE Trans. Circuits Syst. I, vol. 39, pp. 487–490, June 1992. [4] T. Roska, C. W. Wu, and L. O. Chua, “Stability of cellular neural networks with dominant nonlinear and delay-type templates,” IEEE Trans. Circuits Syst. I, vol. 40, pp. 270–272, Apr. 1993. [5] P. P. Civalleri, M. Gilli, and L. Pandolfi, “On stability of cellular neural networks with delay,” IEEE Trans. Circuits Syst. I, vol. 40, pp. 157–164, Mar. 1993. [6] X. X. Liao and J. Wang, “Algebraic criteria for global exponential stability of cellular neural networks with multiple time delays,” IEEE Trans. Circuits Syst. I, vol. 50, pp. 268–274, Feb. 2003. [7] N. Takahashi, “A new sufficient condition for complete stability of cellular neural networks with delay,” IEEE Trans. Circuits Syst. I, vol. 47, pp. 793–799, June 2000. [8] Z. Zeng, J. Wang, and X. X. Liao, “Global exponential stability of a general class of recurrent neural networks with time-varying delays,” IEEE Trans. Circuits Syst. I, vol. 50, pp. 1353–1358, Oct. 2003. [9] Z. Yi, A. Heng, and K. S. Leung, “Convergence analysis of cellular neural networks with unbounded delay,” IEEE Trans. Circuits Syst. I, vol. 48, pp. 680–687, June 2001. [10] T. L. Liao and F. C. Wang, “Global stability for cellular neural networks with time delay,” IEEE Trans. Neural Networks, vol. 11, pp. 1481–1484, Nov. 2000. [11] J. Cao, “Global stability conditions for delayed CNNs,” IEEE Trans. Circuits Syst. I, vol. 48, pp. 1330–1333, Nov. 2001. [12] M. Dong, “Global exponential stability and existence of periodic solutions of CNNs with delays,” Phys. Lett. A, vol. 300, pp. 49–57, 2002. [13] J. Cao, “New results concerning exponential stability and periodic solutions of delayed cellular neural networks,” Phys. Lett. A, vol. 307, pp. 136–147, 2003. [14] X. M. Li, L. H. Huang, and H. Zhu, “Global stability of cellular neural networks with constant and variable delays,” Nonlin. Anal., vol. 53, pp. 319–333, 2003. [15] H. Jiang, Z. Li, and Z. Teng, “Boundedness and stability for nonautonomous cellular neural networks with delay,” Phys. Lett. A, vol. 306, pp. 313–325, 2003.
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[16] G.-J. Yu, C.-Y. Lu, J. S.-H. Tsai, T.-J. Su, and B.-D. Liu, “Stability of cellular neural networks with time-varying delay,” IEEE Trans. Circuits Syst. I, vol. 50, pp. 677–678, May 2003. [17] X. X. Liao and J. Wang, “Global dissipativity of continuous-time recurrent neural networks with time delay,” Phys. Rev. E, vol. 68, pp. 1–7, 2003. [18] X. X. Liao, “Mathematical theory of cellular neural networks (I),” Sci. China (A), vol. 24, pp. 902–910, 1994. [19] , “Mathematical theory of cellular neural networks (II),” Sci. China (A), vol. 38, pp. 542–551, 1995. [20] S. Arik and V. Tavsanoglu, “Equilibrium analysis of delayed CNNs,” IEEE Trans. Circuits Syst. I, vol. 45, pp. 168–171, Feb. 2000. [21] S. Arik, “On the global asymptotic stability of delayed cellular neural networks,” IEEE Trans. Circuits Syst. I, vol. 47, pp. 571–574, Apr. 2000. [22] S. Mohamad and K. Gopalsamy, “Exponential stability of continuous-time and discrete-time cellular neural networks with delays,” Appl. Math. Comput., vol. 135, pp. 17–38, 2003. [23] S. Arik, “An improved global stability result for delayed cellular neural networks,” IEEE Trans. Circuits Syst. I, vol. 49, pp. 1211–1214, Aug. 2002. , “An analysis of global asymptotic stability of delayed cel[24] lular neural networks,” IEEE Trans. Neural Networks, vol. 13, pp. 1239–1242, Sept. 2002. [25] P. P. Civalleri and M. Gilli, “Practical stability criteria for cellular neural networks,” Electron. Lett., vol. 33, pp. 970–971, 1997. [26] M. P. Joy and V. Tavsanoglu, “A new parameter range for the stability of opposite sign cellular neural networks,” IEEE Trans. Circuits Syst.I, vol. 40, pp. 204–207, Mar. 1993. [27] S. Arik and V. Tavsanoglu, “Equilibrium analysis of nonsymmetric CNNs,” Int. J. Circuit Theory Applicat., vol. 24, pp. 269–274, 1996. [28] T.-L. Liao and F.-C. Wang, “Global stability condition for cellular neural networks with delay,” Electron. Lett., vol. 35, pp. 1348–1349, 1999. [29] K. Yosida, Functional Analysis. Berlin, Germany: Springer-Verlag, 1978.
Zhigang Zeng received the B.S. degree in mathematics from Hubei Normal University, Huangshi, China, and the M.S. degree in ecological mathematics from Hubei University, Hubei, China, in 1993 and 1996, respectively, and the Ph.D. degree in systems analysis and integration from Huazhong University of Science and Technology, Wuhan, China, in 2003. He was a Postdoctoral Research Fellow in the Department of Automation, University of Science and Technology, Hefei, China. He is currently a Research Associate in the Department of Automation and Computer-Aided Engineering, Chinese University of Hong Kong, Hong Kong. He is also an Associate Professor at Hubei Normal University. His current research interests include neural networks and stability analysis of dynamic systems.
Jun Wang (S’89–M’90–SM’93) received the B.S. degree in electrical engineering and the M.S. degree in systems engineering from Dalian University of Technology, China, and the Ph.D. degree in systems engineering from Case Western Reserve University, Cleveland, OH. He is a Professor in the Department of Automation and Computer-Aided Engineering, Chinese University of Hong Kong, Hong Kong. He was an Associate Professor at the University of North Dakota, Grand Forks, until 1995. His current research interests include neural networks and their engineering applications. He is an Associate Editor of the IEEE TRANSACTIONS ON NEURAL NETWORKS and IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: Parts B and C.
Xiaoxin Liao received the B.S. degree in mathematics from Wuhan University, Wuhan, China, in 1963. He is a Professor and Ph.D. Advisor in the Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, China. He is also a Chief Advisory Professor in the School of Automation, Wuhan University of Technology, Wuhan, China. His current research interests include neural networks, chaotic synchronization, and automatic control. Mr. Liao is on the editoria board of the Journal of Mathematics and Applied Mathematics in China.
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Stability Analysis of Delayed Cellular Neural Networks Described Using Cloning Templates Zhigang Zeng, Jun Wang, Senior Member, IEEE, and Xiaoxin Liao
Abstract—In this paper, we show that any -dimensional delayed cellular neural network described using cloning templates can have no more than 3 isolated equilibrium points and 2 of these equilibrium points located in saturation regions are locally exponentially stable. In addition, we give the conditions for the equilibrium points to be locally exponentially stable when the equilibrium points locate the designated saturation region. These conditions improve and extend the existing stability results in the literature. The conditions are also very easy to be verified and can be checked by direct examination of the templates, regardless of the number of cells. Finally, the validity and performance of the results are illustrated by use of two numerical examples. Index Terms—Cellular neural network (CNN), cloning template, isolated equilibria, stability.
I. INTRODUCTION
C
ELLULAR neural networks (CNNs) are arrays of dynamical cells that are suitable for the formulation and solution of many complex computational problems. In recent years, CNNs [1] have attracted great attention due to their significant potential in applications. It is well known that the stability of CNNs and delayed CNNs (DCNNs) are critical for their applications, especially in signal processing, image processing, solving nonlinear algebraic and transcendental equations, and solving optimization problems [1]–[4]. A preliminary step for investigating the dynamics of CNNs and DCNNs is the stability analysis. As CNNs and DCNNs are large-scale nonlinear dynamical systems, their stability analysis is a nontrivial task. In the past decade, the stability of CNNs and DCNNs has been widely investigated; e.g., [5]–[28]. However, most of these results apply to general networks and do not really exploit the two main characteristics of CNNs: the local connectivity and the space-invariance structure. CNN design is based on the cloning templates suitable for given applications. Therefore, it is important for the stability conditions to be directly expressed in terms of their templates.
Manuscript received March 1, 2004; revised May 9, 2004. This works was supported by the Hong Kong Research Grants Council under Grant CUHK4165/03E and the Young Foundation of the Education Department of Hubei Province, China. This paper was recommended by Associate Editor C.-T. Lin. Z. Zeng was with Department of Automation, University of Science and Technology of China, Hefei, China. He is now with the Department of Automation and Computer-Aided Engineering, The Chinese University of Hong Kong, Hong Kong, and also with the Department of Mathematics, Hubei Normal University, Huangshi, China (e-mail: [email protected]). J. Wang is with the Department of Automation and Computer-Aided Engineering, The Chinese University of Hong Kong, Hong Kong (e-mail: [email protected]). X. Liao is with the Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, China. Digital Object Identifier 10.1109/TCSI.2004.836855
In this paper, we study stability of DCNNs described using templates. The stability conditions can be checked by direct examination of the templates, regardless of the number of cells. This seems to be an advantage in comparison with the results in [25]–[27] in dealing with large-scale DCNNs. Specifically, the results in this paper consist of four theorems and five corol-dimensional CNN laries. Theorem 1 shows that any or DCNN can have no more than isolated equilibrium -dimensional CNN and DCNN can points, shows that the isolated equilibrium points which are locally exhave ponentially stable, and gives the estimates of attractive domain locally exponentially stable equilibrium points. of such Theorem 2 gives the conditions for an equilibrium to be locally exponentially stable when the equilibrium locates in the saturation region. Theorem 3 gives the conditions for an equilibrium to be locally exponentially stable when the equilibrium locates in . Theorem 4 gives the conditions for an equilibrium to be locally exponentially stable when the equilibrium locates in the other regions. The remaining part of this paper consists of six sections. In Section II, relevant background information is given. In Section III, the number of equilibria of DCNNs is obtained. In Section IV, the number of equilibria of DCNNs without external input is obtained. In Section V, two illustrative examples are provided with simulation results. Finally, concluding remarks are given in Section VI. II. PRELIMINARIES Consider a two-dimensional (2-D) DCNN described by two space-invariant templates where the cells are arranged on a rectrows and columns. The dyangular array composed of namics of such a DCNN is governed by the following normalized equations:
(1) where denotes the state of the cell located at the crossing between the th row and th column of the network, and are positive integers is denoting neighborhood radiuses, the feedback cloning template defined by a
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real matrix, cloning template defined by a
is the delay feedback real matrix is the activation function
defined by (2) is the time delay upper bounded by a constant , and is an external input. An alternative expression for the state equation of a DCNN can be obtained by ordering the cells in some way (e.g., by rows or by columns) and by cascading the state variables into a state . vector The following compact form is then obtained: (3) where the coefficient matrices and are obtained and . The input vector is through the templates vector-valued activation function obtained through
. Denote
, so can be divided into
subspaces
or
or (4)
and
can be divided into three subspaces
Fig. 1.
Decompose 2-D space.
is continuous and satisfies (1) and , for . Also, simply denote as the solution of (1). Definition 1: The set is said to be a globally of (1) with any attractive set of (1), if for the solution , there exists initial condition depending on such that for . Definition 2: The set is said to be a globally exponentially of (1) with any attractive set of (1), if for the solution , there exist coninitial condition and (depending on ) such that for stants
Definition 3: The point is said to be an isolated equilibis an equilibrium point of (1) and there rium point of (1), if such that exists is not an equilibrium point of (1). Definition 4: The equilibrium point of (1) is said to be locally exponentially stable in region , if there exist constants such that for
or
Hence, gion up of
is made up of one part, the saturation reis made up of parts, is made parts. For example, when , and all of the parts of are depicted in Fig. 1. be the space of continuous functions Let mapping into with norm defined , by where . Denote as the vector norm of the vector . The initial condition of DCNN (1) is assumed to be (5) . Denote as the sowhere lution of (1) with initial condition (5), it means that
where is the solution of the neural network (1) with , and is said to any initial condition be a locally exponentially attractive set of the equilibrium point . When is said to be globally exponentially stable. be a bounded and close set in Lemma 1 [29]: Let be a mapping on complete matric space , is where measurement in . If and there exists a constant such that , then such that . there exists is a nonsingular -matrix, then Lemma 2 [8]: If the DCNN (3) is globally exponentially stable, where is the identity matrix
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Consider the following coupled system: (6)
locally exponenIn addition, if (9) holds, then (1) has tially stable equilibrium points located in the saturation region, where the saturation region
(7) or where
and
are
matrices, is the into . space of continuous functions mapping Lemma 3: If (7) is globally exponentially stable, then (6) is also globally exponentially stable. Proof: By applying the variant format of constants, soluof (6) can be expressed as tion
Proof: If
, then from (2)
(10)
(11)
Since (7) is globally exponentially stable, there exist constants such that . Hence, , where . Then, when , for
Hence, from (2), for DCNN (1), there exists an equilibrium point if and only if for
when ; i.e., (6) is also globally exponentially stable.
(12)
III. NUMBER OF EQUILIBRIA In the following, we always assume that , and
are empty. Denote
(8) Then,
, where
is defined in (4).
A. Equilibria in Theorem 1: The DCNN (1) has exactly librium points if and only if
isolated equi-
(9)
-dimensional linear algebraic equaEquation (12) is an tion, its solution is an isolated point, a line segment, or a higher dimensional region. Proof of necessary condition: Since solution of (12) is one of the following cases: an isolated point, a line segment, or a higher dimensional region ( -dimensional region), if (1) has neither more nor less than isolated equilibrium points, then (1) has neither more nor less than one isolated equilibrium point . Specially, choose to be empty, let located in be an equilibrium point of (1). For can equal , and can also equal to . From (12), if , then to
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If choose
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 51, NO. 11, NOVEMBER 2004
, choose , then
if
,
(18) , for
From (15) and (17), when
(13) Similarly, if
, then
Similarly, when
, for
(14) Since for , (13) and (14) hold, it implies that (9) holds. Proof of sufficiency: According to (9), the coefficient matrix of (12) is nonsingular, ; i.e., the solution of hence (12) has only a solution in (12) is an isolated point. Since is divided into subspaces , the DCNN (1) has no more than isolated equilibrium points. For , let (15) (16)
(17)
Hence, from (16) and (18), addition, for
. In
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From (9) and Lemma 1, there exists such that . Choose (19) , when From (17) and . From (19) and (9), when
Then, i.e., is of (1). an isolated equilibrium point located in is divided into subspaces , Since the DCNN (1) has neither more nor less than isolated equilibrium points. to be empty, then Specially, choose is a saturation region. According to the above proof, is an equilibrium point located in a saturation region. Obviously, the number of such saturation in . Hence,(1) has equilibrium region is , points located in saturation regions. Let then when and locate in the same saturation region, . So when , and locate in the same saturation region (20) Obviously, (20) is exponentially stable. Hence, is locally exponentially stable equilibrium point of (1). According to the definition of in (1), when . Example 1: Consider
(21)
Similarly, when Proposition: If (21) has neither more nor less than isolated equilibrium points, then (22) Proof: and only if
is an equilibrium point of (21) if
(23) Consider the region depicted in Fig. 1
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In above region, (for example, only in ), it is impossible that (21) has two isolated equilibrium points. In fact, in , (23) is a 2-D algebraic equation (24)
In
, (23) is a 2-D algebraic equation (32)
If the solution of (32) is an isolated point in
, then (33)
The solution of (24) is an isolated point; i.e., , or is empty. Hence, in , (21) has an isolated equilibrium point or not isolated equilibrium point (when ); i.e., it is impossible that (21) has two isolated equilibrium points located . in In , (23) is a 2-D algebraic equation
Equations (27), (29), (31), and (33) imply that (22) holds. Remark: Since , if (9) holds, then when
(25) The solution of (25) is an isolated point or a beeline or a plane; i.e., it is impossible that (21) has two isolated equilibrium points . located in In any one of , it is similar to prove that (21) has no more than one isolated equilibrium point. isoIf (21) has neither more nor less than lated equilibrium points, then there exists an isolated equilibrespectively. rium point located In , (23) is a 2-D algebraic equation (26) If the solution of (26) is an isolated point in
when
Hence, (22) holds. Similarly, if (22) holds then (9) holds in Example 1. B. Equilibria in the Saturation Region Let or . Theorem 2: If there exists
, then
such that
(27) In
, (23) is a 2-D algebraic equation (28)
If the solution of (28) is an isolated point in
, then (29)
In
, (23) is a 2-D algebraic equation (30)
If the solution of (30) is an isolated point in
, then (31)
(34) then (1) has exactly one locally exponentially stable equilibrium point located in the saturation region , where saturation region
ZENG et al.: STABILITY ANALYSIS OF DELAYED CNNs
Proof: For , then from (1)
,
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, let
C. Equilibria in Theorem 3: If
(35) . Choose , then , and is an equifrom (34), librium point of (35). Similar to the proof of Theorem 1, is locally exponentially stable equilibrium point located in saturaof (35); i.e., (1) has neither more nor tion region less than one isolated equilibrium points located in saturation regions , and it is locally exponentially stable. Corollary 1: If Obviously, If
(36)
, then
then (1) has one locally exponentially stable equilibrium point . located in the saturation region , according to Proof: Choose Theorem 2, (1) has a locally exponentially stable equilibrium . point located in the saturation region Corollary 2: If
then (1) has neither more nor less than one globally exponen. tially stable isolated equilibrium point located in , let Proof: For
Since , according to Lemma 1, similar to the proof of Theorem 1, there exists such that ; is an isolated equilibrium located in of (1). i.e., From (3) and (36), is a nonsingular -matrix. According to Lemma 2, the equilibrium point of (3) is globally exponentially stable; i.e., (1) is globally exponentially stable. D. Equilibria in Other Regions Theorem 4: If for
then (1) has a locally exponentially stable equilibrium point located in saturation region . , according Proof: Choose to Theorem 2, (1) has a locally exponentially stable equilibrium . point located in saturation region
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for
, similar to the proof of Theorem 1, Since such from (38), (39), and (40), there exists is an isolated equilibrium point located in that of (1). From (37), for
(37) for If
, consider subsystem of (1)
(41) then (1) has neither more nor less than one isolated equilib, and it is locally exponentially rium located in is defined by (4). stable, where , let Proof: For
and the subsystem of (1)
(38) (42) From (41)
(39) (40)
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from (42)
, its cloning template is . The matrix , defined by (3), composed of the template has the circulant form
..
..
.
..
.
(43)
.
Also, for example, for a 2-D space-invariant CNN with a neighborhood radius , its cloning template is a real matrix; i.e.,
where
.. .
..
.
.. .
..
.
.. .
.. .
..
.
.. .
..
.
.. .
, and , and .
When
For simplicity, as required in most applications, in the following, . The matrix , defined by (3), we always assume that composed of the template has the form
or or
; when .
Hence, (41) and (42) can be rewritten as .. .
.. .
.. .
.. .
..
.. .
.
.. .
where
respectively, where is a constant. According to Lemma 2, (37) implies that the equilibrium of (41) is globally exponentially stable. Since the equilibrium of (41) is globally exponentially stable, according to Lemma 3, the equilibrium of (42) is also globally of (1) is exponentially stable. Hence the equilibrium point locally exponentially stable in .
.. .
.. .
.. .
..
.
.. .
.. .
.. .
.. .
.. .
..
.
.. .
.. .
IV. NUMBER OF EQUILIBRIA WHEN EXTERNAL INPUT VECTOR IS NULL Throughout this section, we assume that . The dynamic rule of a DCNN can be completely specified by its in the DCNN (3) depend on the cloning templates. and established order among the cells and on the cloning templates and the delay cloning template. For example, for a one-dimensional (1-D) space-invariant CNN with a neighborhood radius
.. .
and
.. .
.. .
..
.
.. .
can be similarly defined.
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A. One-Dimensional Space-Invariant DCNNs
B. Two-Dimensional Space-Invariant DCNNs , for
Corollary 5: If
, for
Corollary 3: If
, and
, and
(44)
then (1) has two locally exponentially stable equilibrium points and , located in saturation region respectively. or Proof: Choose , (44) implies that (34) holds. According to Theorem 2, Corollary 3 holds. Corollary 4: If , for , and
(45)
then (1) has two locally exponentially stable equilibrium points located in saturation region and , respectively. or Proof: Choose , (45) implies that (34) holds. According to Theorem 2, Corollary 4 holds. To show the usefulness of Corollaries 3 and 4, we discuss a simple example. We consider the DCNN described by the 1D . According to template Corollary 3, if (46)
where , then (1) has two locally exponentially stable equilibrium points located in saturation region and , respectively. Proof: Choose or , conditions of Corollary 5 imply that (34) holds. According to Theorem 2, Corollary 5 holds. V. ILLUSTRATIVE EXAMPLES In this section, we give two examples to illustrate the new results. Example 2: Consider a 2-D space-invariant CNN with a . Its cloning template is a neighborhood radius real matrix; i.e.,
Assume the delay cloning template then from (1) and (3)
then (1) has two locally exponentially stable equilibrium points located in saturation region and , respectively. According to Corollary 4, if (47) then (1) has two locally exponentially stable equilibrium points located in saturation region and , respectively. From (46) and (47), , then (1) has two locally exponenif tially stable equilibrium points located in saturation regions; if , then (1) has four locally exponentially stable equilibrium points located in saturation regions. From , then (1) has [25], it is known that if , such a at least one stable equilibrium point. When criterion coincides with that in [26]. For a DCNN composed of cells, the condition of the theorem in [27] is always stronger than those of Corollaries 3 and 4, regardless of the number of cells.
and
,
(48)
Hence, from (48)
(49)
According to Theorem 1, the CNN (49) has isolated equilibrium points and of them are locally exponentially stable. The are depicted in Fig. 2. transient behaviors of
ZENG et al.: STABILITY ANALYSIS OF DELAYED CNNs
Fig. 2.
Transient behavior of x
;x
;x
;x
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in Example 2.
Fig. 3. Transient behavior of x
Example 3: Consider a 2-D space-invariant CNN with a . Its cloning template is a 3 3 neighborhood radius real matrix; i.e.,
;x
;x
;x
in Example 3.
of cells. The analytical results are the improvement and extension of the existing stability results in the literature.
REFERENCES
Assume that the delay cloning template , then from (48)
and
(50)
According to Theorem 4, (50) has an isolated equilibrium point
transient behaviors of
, and it is locally exponentially stable. The are depicted in Fig. 3.
VI. CONCLUDING REMARKS In this paper, we show that any -dimensional DCNNs described by a template can have no more than isolated of them are locally exponentially equilibrium points and stable located in saturation regions. In addition, we also give the conditions for an equilibrium to be locally exponentially stable when the equilibrium point locates the designated saturation region. These easily testable conditions can be checked by direct examination of the cloning templates, regardless of the number
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[16] G.-J. Yu, C.-Y. Lu, J. S.-H. Tsai, T.-J. Su, and B.-D. Liu, “Stability of cellular neural networks with time-varying delay,” IEEE Trans. Circuits Syst. I, vol. 50, pp. 677–678, May 2003. [17] X. X. Liao and J. Wang, “Global dissipativity of continuous-time recurrent neural networks with time delay,” Phys. Rev. E, vol. 68, pp. 1–7, 2003. [18] X. X. Liao, “Mathematical theory of cellular neural networks (I),” Sci. China (A), vol. 24, pp. 902–910, 1994. [19] , “Mathematical theory of cellular neural networks (II),” Sci. China (A), vol. 38, pp. 542–551, 1995. [20] S. Arik and V. Tavsanoglu, “Equilibrium analysis of delayed CNNs,” IEEE Trans. Circuits Syst. I, vol. 45, pp. 168–171, Feb. 2000. [21] S. Arik, “On the global asymptotic stability of delayed cellular neural networks,” IEEE Trans. Circuits Syst. I, vol. 47, pp. 571–574, Apr. 2000. [22] S. Mohamad and K. Gopalsamy, “Exponential stability of continuous-time and discrete-time cellular neural networks with delays,” Appl. Math. Comput., vol. 135, pp. 17–38, 2003. [23] S. Arik, “An improved global stability result for delayed cellular neural networks,” IEEE Trans. Circuits Syst. I, vol. 49, pp. 1211–1214, Aug. 2002. , “An analysis of global asymptotic stability of delayed cel[24] lular neural networks,” IEEE Trans. Neural Networks, vol. 13, pp. 1239–1242, Sept. 2002. [25] P. P. Civalleri and M. Gilli, “Practical stability criteria for cellular neural networks,” Electron. Lett., vol. 33, pp. 970–971, 1997. [26] M. P. Joy and V. Tavsanoglu, “A new parameter range for the stability of opposite sign cellular neural networks,” IEEE Trans. Circuits Syst.I, vol. 40, pp. 204–207, Mar. 1993. [27] S. Arik and V. Tavsanoglu, “Equilibrium analysis of nonsymmetric CNNs,” Int. J. Circuit Theory Applicat., vol. 24, pp. 269–274, 1996. [28] T.-L. Liao and F.-C. Wang, “Global stability condition for cellular neural networks with delay,” Electron. Lett., vol. 35, pp. 1348–1349, 1999. [29] K. Yosida, Functional Analysis. Berlin, Germany: Springer-Verlag, 1978.
Zhigang Zeng received the B.S. degree in mathematics from Hubei Normal University, Huangshi, China, and the M.S. degree in ecological mathematics from Hubei University, Hubei, China, in 1993 and 1996, respectively, and the Ph.D. degree in systems analysis and integration from Huazhong University of Science and Technology, Wuhan, China, in 2003. He was a Postdoctoral Research Fellow in the Department of Automation, University of Science and Technology, Hefei, China. He is currently a Research Associate in the Department of Automation and Computer-Aided Engineering, Chinese University of Hong Kong, Hong Kong. He is also an Associate Professor at Hubei Normal University. His current research interests include neural networks and stability analysis of dynamic systems.
Jun Wang (S’89–M’90–SM’93) received the B.S. degree in electrical engineering and the M.S. degree in systems engineering from Dalian University of Technology, China, and the Ph.D. degree in systems engineering from Case Western Reserve University, Cleveland, OH. He is a Professor in the Department of Automation and Computer-Aided Engineering, Chinese University of Hong Kong, Hong Kong. He was an Associate Professor at the University of North Dakota, Grand Forks, until 1995. His current research interests include neural networks and their engineering applications. He is an Associate Editor of the IEEE TRANSACTIONS ON NEURAL NETWORKS and IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: Parts B and C.
Xiaoxin Liao received the B.S. degree in mathematics from Wuhan University, Wuhan, China, in 1963. He is a Professor and Ph.D. Advisor in the Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, China. He is also a Chief Advisory Professor in the School of Automation, Wuhan University of Technology, Wuhan, China. His current research interests include neural networks, chaotic synchronization, and automatic control. Mr. Liao is on the editoria board of the Journal of Mathematics and Applied Mathematics in China.
