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The Identification of Coupled Map Lattice Models for Autonomous Cellular Neural Network Patterns This is a repository copy of The identification of coupled map lattice models for autonomous cellular neural network patterns. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/74608/ Monograph: Pan, Y., Billings, S.A. and Zhao, Y. (2007) The identification of coupled map lattice models for autonomous cellular neural network patterns. Research Report. ACSE Research Report no. 949 . Automatic Control and Systems Engineering, University of Sheffield Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ ｔｨ･＠ｉ､･ｮｴｩｦｩ｣｡ｴｩｯｮ＠ｯｦ＠ｃｯｵｰｬ･､＠ｍ｡ｰ＠ｌ｡ｴｴｩ｣･＠ｭｯ､･ｬｳ＠ｦｯｲ ａｵｴｯｮｯｭｯｵｳ＠ｃ･ｬｬｵｬ｡ｲ＠ｎ･ｵｲ｡ｬ＠ｎ･ｴｷｯｲｫ＠ｐ｡ｴｴ･ｲｮｳ ｐ｡ｮＬ＠ｙＮＬ＠ｂｩｬｬｩｮｧｳＬ＠ｓＮ＠ａＮ＠｡ｮ､＠ｚｨ｡ｯＬ＠ｙＮ ｄ･ｰ｡ｲｴｭ･ｮｴ＠ｯｦ＠ａｵｴｯｭ｡ｴｩ｣＠ｃｯｮｴｲｯｬ＠｡ｮ､＠ｓｹｳｴ･ｭｳ＠ｅｮｧｩｮ･･ｲｩｮｧ ｕｮｩｶ･ｲｳｩｴｹ＠ｯｦ＠ｓｨ･ｦｦｩ･ｬ､ ｓｨ･ｦｦｩ･ｬ､Ｌ＠ｓＱ＠Ｓｊｄ ｕｋ ｒ･ｳ･｡ｲ｣ｨ＠ｒ･ｰｯｲｴ＠ｎｯＮ＠ＹＴＹ ａｰｲｩｬ＠ＲＰＰＷ The Identification of Coupled Map Lattice models for Autonomous Cellular Neural Network Patterns Pan, Y., Billings, S. A. and Zhao, Y. Department of Automatic Control & Systems Engineering Sheffield University, Sheffield, S1 3JD, UK March, 2007 Abstract The identification problem for spatiotemporal patterns which are generated by autonomous Cellular Neural Networks (CNN) is inves- tigated in this paper. The application of traditional identification algorithms to these special spatiotemporal systems can produce poor models due to the inherent piecewise nonlinear structure of CNN. To solve this problem, a new type of Coupled Map Lattice model with output constraints and corresponding identification algorithms are proposed in the present study. Numerical examples show that the identified CML models have good prediction capabilities even over the long term and the main dynamics of the original patterns appears to be well represented. 1 Introduction Spatiotemporal pattern formation is common in many disciplines of science and engineering including chemistry, physics and biology. For example, pe- riodic, chaotic, oscillatory and Turing patterns have been observed in con- centrations of chemically reacting and diffusing systems [17] [21] [10]. The reaction-diffusion dynamics of spatial pattern formation have also been used to model the development of patterns in fish, seashells and animal coats [3] [19] [20]. Similar structures such as travelling waves and Turing patterns 1 have been observed in biological systems [18]. As a consequence of the com- mon occurrence of spatial patterns, the discovery of new pattern formation phenomenon is potentially of great research interest. Several authors have therefore studied the analysis and simulation of a variety of spatiotempo- ral models which produce various interesting spatial patterns. The Partial Differential Equation (PDE) models where the variables evolve continuously over the continuous space and time domain have been used to describe a wide class of spatiotemporal behaviours, including the Gray-Scott model [11], Fitz-Hugh-Nagumo model and Belousov-Zhabotinsky model. Cellular Neural Networks (CNN) [7] which are defined by the coupling of cells continuously evolving over a discrete spatial lattice have also been widely applied to model complex spatiotemporal patterns. Because of the simplicity and easy implementation in hardware, CNN’s have found numerous applica- tions in image and video signal processing, and in pattern recognition. Apart from providing an alternative paradigm for simulating nonlinear Partial dif- ferential Equations, CNN models have been shown to generate propagating waves, patches, checkerboard patterns, stripes and reaction-diffusion type patterns [2] [23] [13]. Further research on the emergence and complexity of spatiotemporal systems has revealed that CNN can give rise to many inter- esting patterns by tuning the parameters in the CNN templates. Hence, the problem of deriving the corresponding CNN templates for specific spatiotem- poral behaviours has also been recently investigated [13]. The Coupled Map Lattice model which was initially introduced in the 1980s by Kaneko [14][15] has been widely used to model spatiotemporal sys- tems. The CML model is discrete in time and space and has a continuous state value. A CML is a d-dimension lattice where each site evolves in time through a discrete map which describes the influence of the past state and neighboring sites. It has been shown that CML models can exhibit complex spatiotemporal behaviors, including chaos, intermittency, travelling waves and Turing patterns [16]. Compared with PDE models, CML models are computationally more efficient and have been used to study spatiotemporal systems in a wide class of scientific subjects. Given the various patterns produced by CNN models, the inverse problem of how to model these spatiotemporal behaviours from observed data will be considered in this paper. As a subset of CNN patterns, the identification problem of autonomous CNN patterns using CML models is the main focus of the present study. Due to the special nonlinear structure of CNN, it is not easy to obtain CML models for these patterns using general identification algorithms. In the present study, an identification algorithm based on a CML model is introduced based on an amended Orthogonal Forward Regression (OFR) algorithms for these special classes of spatiotemporal behaviours. The 2 coefficients of the identified model are adjusted by using an amended error sequence from the model prediction to enhance the prediction performance of the final model. The paper is arranged as follows. Section 2 gives a general description of autonomous Cellular Neural Networks and Coupled Map Lattice models for spatiotemporal systems. Section 3 introduces a correlation based method to select the sampling interval for the identification data. Then, the new iden- tification method which is based on a modified OFR algorithm is proposed. Some numerical examples of CNN patterns are included in Section 4 to illus- trate the application of the new identification methods and to demonstrate the performance of the identified CML models to identify these systems. 2 Autonomous Cellular Neural Networks and Coupled Map Lattice Models 2.1 Coupled Map Lattice models Consider the input-output CML model for time and spatially invariant spa- tiotemporal systems[8] [9] ny nu ny my nu mu yi(k)= f(q yi(k), q ui(k), q s yi(k), q s ui(k)) + εi(k) (1) where i Id is the spatial index of a d-dimensional space and k = 1, 2,..., ∈ is the temporal index; yi(k), ui(k) and εi(k) are the output, input, model residual sequences respectively, and qn(k) is a temporal backward shift oper- ator qn =(q−1, q−2, ..., q−n) (2) so that ny q yi(k)=(yi(k 1), yi(k 2), ..., yi(k ny)) nu − − − (3) q ui(k)=(ui(k 1), ui(k 2), ..., ui(k nu)) − − − In (1), sm is a multi valued spatial shift operator 1 2 m sm =(sp , sp , ..., sp ) (4) where pj Id is the spatial translation multi index, such that ∈ my s yi =(yi−p1 , yi−p2 , ..., yi−pmy ) mu (5) s ui =(ui−p1 , ui−p2 , ..., ui−pmu ) The parameters my, mu denote the maximum spatial radius associated with the output y and input u. 3 2.2 Autonomous Cellular Neural Networks A CNN is said to be autonomous if the cells do not have external inputs. The standard form of an autonomous CNN defined over a two dimensional N N array can be expressed by the following equation, × x˙ i,j = xi,j + zi,j + an,lyn,l (6a) − n,l∈S Xi,j yi,j = g(xi,j) (6b) where (i, j) N N denotes the spatial site and xi,j, yi,j are state space variables and∈ output× variables respectively. The output function g( ) here is defined as the three segment piecewise linear saturation function. · 1 g(xi,j)= ( xi,j +1 xi,j 1 ) (7) 2 | |−| − | The CNN dynamics are usually restricted to be inside the boundary of the array N N. Consequently, additional boundary conditions need to be specified× for model (6). Three kinds of boundary conditions are most com- monly used, Dirichlet (fixed), Neumann (zero flux) and Toroidal (periodic) boundary conditions. With specific boundary conditions, the dynamics of the standard CNN (6) is then only determined by the CNN template which is composed of the threshold zi,j and the neighbourhood coupling matrix A = an,l . If the neighbourhood radius of every cell is set to be r = 1, the feedback{ template} A of a two dimensional CNN can be described as follows. a−1,−1 a−1,0 a−1,1 A = a ,− a , a , (8) 0 1 0 0 0 1 a1,−1 a1,0 a1,1 The behaviours of different pattern formations of standard autonomous CNN models have been extensively studied with various A-templates using state space stability theory. Perturbing the unstable equilibrium state with small and random disturbances, the symmetry of the unstable equilibrium is broken and complex patterns will be formed. As can be seen from the piecewise func- tion g( ), the output trajectories of all cells can be divided into three regions: a linear· region and two saturation regions.
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