Stability analysis of generalized cellular neural networks

IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 1998

This paper gives a new sufficient condition for complete stability of a nonsymmetric cellular neural network (CNN). The convergence theorem of the Gauss-Seidel method, which is an iterative technique for solving a linear algebraic equation, plays an important role in our proof. It is also shown that the existence of a stable equilibrium point does not imply complete stability of a nonsymmetric CNN.

2008 7th World Congress on Intelligent Control and Automation, 2008

A novel criterion for the global asymptotic stability of two-dimensional (2-D) discrete systems described by the Roesser model employing saturation arithmetic is presented. The criterion is compared with previously reported criteria. Numerical examples showing the effectiveness of the present criterion are given.

IEEE Transactions on Circuits and Systems I: Regular Papers, 2004

Cellular neural networks are dynamical systems, described by a large set of coupled nonlinear differential equations. The equilibrium point analysis is an important step for understanding the global dynamics and for providing design rules. We yield a set of sufficient conditions (and a simple algorithm for checking them) ensuring the existence of at least one stable equilibrium point. Such conditions give rise to simple constraints, that extend the class of CNN, for which the existence of a stable equilibrium point is rigorously proved. In addition, they are suitable for design and easy to check, because they are directly expressed in term of the template elements.

Proceedings of the 2003 International Symposium on Circuits and Systems, 2003. ISCAS '03.

This paper presents new criteria for the existence of stable equilibrium points in the total saturation region for cellular neural networks (CNNs). It is shown that the results obtained can be used to derive some complete stability conditions for some special classes of CNNs such as positive cell-linking CNNs, opposite-sign CNNs and dominant-template CNNs. Our results are also compared with the previous results derived in the literature for the existence of stable equilibrium points for CNNs.

Cellular neural networks (CNNs) are recurrent artificial neural networks. Due to their cyclic connections and to the neurons' nonlinear activation functions, recurrent neural networks are nonlinear dynamic systems, which display stable and unstable fixed points, limit cycles and chaotic behaviour. Since the field of neural networks is still a young one, improving the stability conditions for such systems is an obvious and quasi-permanent task. This paper focuses on CNNs affected by time delays. We are interested to obtain sufficient conditions for the asymptotical stability of a cellular neural network with time delay feedback and zero control templates. For this purpose we shall use a method suggested by Malkin [8], where the "exact" Liapunov-Krasovskii functional will be constructed according the procedure proposed by Kharitonov [6] for stability analysis of uncertain linear time delay systems.

IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 2003

This brief gives new sufficient conditions for the complete stability and global asymptotic stability of a delayed cellular neural network. The results are obtained through the construction of a nontrivial Lyapunov functional. Illustrative examples provided show that those conditions are significantly weaker than those in existing literature. Index Terms-Cellular neural network (CNN), complete stability, delay, global asymptotic stability, image processing. matrix D, respectively. The above work was based on the construction Manuscript

2001

It is shown that first-order autonomous space-invariant cellular neural networks (CNNs) may exhibit a complex dynamic behavior (i.e. equilibrium point and limit cycle bifurcation, strange and chaotic attractors). The most significant limit cycle bifurcation processes, leading to chaos, are investigated through the computation of the corresponding Floquet's multipliers and Lyapunov exponents. It is worth noting that most practical CNN implementations exploit first order cells and spaceinvariant templates: so far no example of complex dynamics has been shown in first-order autonomous space-invariant CNNs.

IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 1993

Abstruct-Complete stabsty of cellular neural networks and their associated dynamical system precludes the possibility of any periodic or chaotic behavior and is an important property to establish. In this paper complete stability of the opposite-sign cellular neural network is established provided that the template values fall within the range ( p -1)/2 < s < p -1. The results in this paper extend the parameter range from previously known results.

Proceedings of ISCAS'95 - International Symposium on Circuits and Systems, 1995

The aim of this paper is to show that discrete time Generalized Cellular Neural Networks, with feedforward, feedback or cascade interconnections between CNNs can be represented as NL q s, a concept introduced in 12]. NL q s are nonlinear systems in state space form with the typical feature of having a number of q layers with alternating linear and nonlinear operators that satisfy a sector condition. It can be shown that many systems and problems arising in neural networks, systems and control are special cases of NL q s. Su cient conditions for global asymptotic stability and dissipativity with nite L 2 -gain are available. For q = 1 the criteria are closely related to known results in H 1 and control theory.

In this study linear stability of a class of three neuron cellular network with transmission delay had been studied. Approach: The model for the problem was first presented. The problem is then formulated analytically and numerical simulations pertaining to the model are carried out. Results: A necessary and sufficient condition for asymptotic stability of trivial steady state in the absence of delay is derived. Then a delay dependent sufficient condition for local asymptotic stability of trivial, steady state and sufficient condition for no stability switching of trivial steady for such a network are derived. Numerical simulation results of the model were presented. Conclusion/Recommendations: From numerical simulation, it appears that there may be a possibility of multiple steady states of the model. It may be possible to investigate the condition for the existence of periodic solutions of the non-linear model analytically.

Computational Intelligence and Bioinspired Systems, Lectures Notes in Computer Science, 3512/2005, pp. 366-373, 2005

In this paper we give a sufficient condition to imply global asymptotic stability of a delayed cellular neural network of the forṁ

International Journal of Circuit Theory and Applications, 2005

This paper presents a cellular neural network (CNN) scheme employing a new non-linear activation function, called trapezoidal activation function (TAF). The new CNN structure can classify linearly non-separable data points and realize Boolean operations (including eXclusive OR) by using only a single-layer CNN. In order to simplify the stability analysis, a feedback matrix W is deÿned as a function of the feedback template A and 2D equations are converted to 1D equations. The stability conditions of CNN with TAF are investigated and a su cient condition for the existence of a unique equilibrium and global asymptotic stability is derived. By processing several examples of synthetic images, the analytically derived stability condition is also conÿrmed. 394 E. BILGILI,İ. C. G OKNAR AND O. N. UCAN CNN stability is analysed for the standard activation function as in References , (v) global exponential stability conditions of CNN via a new Lyapunov function are stated in Reference [9]. It is well known that the standard uncoupled CNN single-layer structures, extremely useful for realizing Boolean functions, are not capable of classifying linearly nonseparable data. The parity is a binary function of the inputs, which returns a high output if the number of inputs set to 1 is odd and a low output if that number is even. Therefore, for n inputs, the parity problem consists of being able to divide the n-dimensional input space into disjoint decision regions such that all input patterns in the same region yield the same output and, thus is linearly non-separable. Uncoupled CNN can only classify linearly separable data, that is can only separate the input space with hyper-planes [10]. Recently, a single perceptron-like cell with: (i) double threshold, (ii) implemented using only ÿve MOS transistors, (iii) capable of classifying data which are not linearly separable has been reported in References .

Proceedings of the 2003 International Symposium on Circuits and Systems, 2003. ISCAS '03., 2000

The paper introduces a general class of neural networks where the neuron activations are modeled by discontinuous functions. The neural networks have an additive interconnecting structure and they include as particular cases the Hopfield neural networks (HNNs), and the standard Cellular Neural Networks (CNNs), in the limiting situation where the HNNs and CNNs possess neurons with infinite gain. Conditions are obtained which ensure global convergence toward the unique equilibrium point in finite time, where the convergence time can be easily estimated on the basis of the relevant neural network parameters. These conditions are based on the concept of Lyapunov Diagonally Stable (LDS) neuron interconnection matrices, and are applicable to general non-symmetric neural networks.

IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 2003

This paper presents a novel class of cellular neural networks (CNNs), where output of a cell in the CNN is given by the piecewise-linear (PWL) function having multiple constant regions or a quantization function. CNNs with one of these output functions allow us to extend CNNs to image processing with multiple gray levels. Since each cell of the original CNN has the PWL output function with two saturation regions, the image-processing tasks are mainly developed for black and white output images. Hence, the proposed architecture will extend the promising nature of the CNN further. Moreover, the hysteresis characteristics are introduced for these functions, which make tolerance to a noise robust. It is demonstrated mathematically that under a mild assumption, the stability of the CNN which has an output function with hysteresis characteristics is guaranteed, and the impressive simulation results are also presented.

IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 2002

It is shown that first-order autonomous space-invariant cellular neural networks (CNNs) may exhibit a complex dynamic behavior (i.e., equilibrium point and limit cycle bifurcation, strange and chaotic attractors). The most significant limit cycle bifurcation processes, leading to chaos, are investigated through the computation of the corresponding Floquet's multipliers and Lyapunov exponents. It is worth noting that most practical CNN implementations exploit first-order cells and space-invariant templates: so far no example of complex dynamics has been shown in first-order autonomous space-invariant CNNs.