Cellular Fuzzy Networks on complex dynamics modeling

M. Branciforte,G. Di Bernardo, M. Lavorgna, L. Occhipinti

Soft Computing Group, STMicroelectronics, Stradale Primosole 50, 95121 Catania, ItalyTel. +39 095 7407939 – Fax +39 095 7407717

[marco.branciforte, giovanni.di-bernardo, mario.lavorgna, luigi.occhipinti]@st.com

Abstract-

The CNNs paradigm, based on Chua's circuit [1] is extended to cellular dynamic systems, named Cellular Fuzzy Network (CFN), whose functionality is based on fuzzy logic. A functional description of the basic CFN cell is reported showing the feasibility of the system and an approach based on both CNNs and CFNs, for complex phenomena modeling, is given.

Key-words: Cellular Fuzzy Processor, Reaction-Diffusion, Complexity.

I. Introduction

Classical formalization of natural phenomena by means of macro-models are characterized by a fundamental simplicity because of their analytical description and cannot be extended to those phenomena that are based on the microscopic level interactions among simple units. On the contrary, all those cases that generate, under particular conditions, self-organization phenomena manifest a Complex behavior. In few words, complexity concerns with those evolutions in which final configuration, among these, is a spatial pattern or a periodic active waves propagation. All of these phenomena derive from Reaction-Diffusion (RD) mechanism described by:

(1)

A large number of experiments have already evidenced as CNNs is an useful support for reproducing RD dynamics. This is due to peculiarities of CNNs architecture. Each isolated cell, behaving such as an oscillator, realizes the reactive part of the RD process. Moreover, local interconnections make interacting cells with their neighborhood, producing the diffusive effect [2].

The whole mechanism is governed by values assumed by parameters appearing into the state equations system, constituting those matrixes named cloning templates. The right choice of these templates affects completely the evolution of the CNNs, and it represents the difficult task of CNNs programming.

II. Cellular Fuzzy Processor

It was outlined that, for modeling a particular phenomenon with the aid of a CNNs it is needed a detailed knowledge of those physical laws that govern the evolution, and then to translate them into a state-equations system, namely to program the CNNs by means of cloning templates. Unfortunately, there is not yet an

established methodology for determining the cloning templates that impose the desired evolution. Basically, the choice of templates is demanded to the ability of the user, but for solving complex tasks, some automatic learning based on numerical research method have been applied, such as Gradient method and Genetic Algorithm. They succeeded in many cases and failed in many others, especially when the determination of a combination of more templates, to be applied sequentially, was requested by the task. With the aim to offer an easier way of programming the CNNs, we proposed the Cellular Fuzzy Network (CFN) [3], an architecture that maintains the characteristics to be cellular and distributed, but programmable by means linguistic statements. The linguistic expressions allow us to describe any given system behavior, at the simple cell level, even when we have only a rough knowledge on the system. The original problem of determining parameters values of cloning templates, now becomes that to define a set of fuzzy rules, yielding a complete linguistic description of the system dynamic, and to tune the membership functions of state (output) variables involved into the rules.

III. CFN description and hw realization

Cellular Fuzzy Network is architecture composed by identical processing elements, named fuzzy cells. As well as CNNs, local interconnections allow each single cell to interact only with its nearby cells. Instead of equation system, the CFN is governed by means two set of fuzzy rules. The first one, R1, determines the internal cell dynamic, while R2, the second one, implements the logical interconnections among the cells and their neighborhood. It is following an example of the two kinds of fuzzy rules applied to CFN at the kth -processing step.

R1: If Xij(k) Is low And Nrij(k) Is high Then Xij (k+1) Is small_positive

R2: If X(i-1)j(k) Is low And Xi(j+1)(k) Is high And ... Then Nrij(k) Is high

An experimental digital architecture of a dedicated Cellular Fuzzy Processor (CFP), implementing a CFN, has been designed in order to make microprocessor available for both fast complex phenomena simulation and advanced image processing applications [3]. CFP is a hybrid fuzzy-algebraic architecture, constituted by a fuzzy core and an ALU, which can process both linguistic and mathematical computations. The ALU can be also programmed to support feedback and control templates allowing both fuzzy and mathematical processing within a single integration step. It is also utilized when numerical integration must be solved. It is easy programmable for computing any kind of numerical integration method (Runge-Kutta, Eulero, ecc.).

IV. Complex phenomena simulation via cellular systems

The authors developed a useful and powerful software tool able to reproduce any kind of Reaction-Diffusion-based phenomena by simulating a RD via cellular architecture evolution [4].

As we know, the shape and dimensions of the pattern and the way to propagate of active waves depend, among other parameters, on the initial conditions, the boundary conditions, the geometry and dimensions of the spatial domains. The software developed allows the user to set up all these conditions and the dimension of the network. Moreover it has been implemented a Cellular Fuzzy Processing software simulation, where fuzzy processing is implemented through a default set of MFs and crisps, easily re-tunable by the user. The software was designed to simulate the following nonlinear system with both methodology, mathematical and fuzzy processing [5]:

(1)

where:

(2)

s is the coupling parameter between the two layers, is feedback parameter and ii are the bias currents.

The cellular system is considered to be discrete in space and continuos in time. Therefore, the differential equation system is integrated by using 4 th order Runge-Kutta method and the Laplacian operator computed by its discrete approximation:

(3)

As discussed in the previous section, CFP is hybrid architecture allowing fuzzy-algebraic computation. Therefore, in software simulation, we exploited this characteristic setting up two fuzzy sets of rules that replace some terms in equations system (1) and going on using 4th order Runge-Kutta method for the numerical integration. In a real application of CFP, this should involve the ALU for both product computation and numerical integration. The first set of fuzzy rules replaces the piece-wise linear function realizing a fuzzy non-linear function, while the second is implemented instead of the Laplacian operator. Following are reported the fuzzy rules and the membership functions (fig. 1) used in the computation of the laplacian operator, where y is the output of the cell to be processed and Nr is the average value of output of the neighborhood cells:

If (y is low) and (Nr is low) Then LO is ZeroIf (y is low) and (Nr is medium) Then LO is SmallPosIf (y is low) and (Nr is high) Then LO is PositiveIf (y is medium) and (Nr is low) Then LO is SmallNegIf (y is medium) and (Nr is medium) Then LO is ZeroIf (y is medium) and (Nr is high) Then LO is SmallPosIf (y is high) and (Nr is low) Then LO is NegativeIf (y is high) and (Nr is medium) Then LO is SmallNegIf (y is high) and (Nr is high) Then LO is Zero

As a generalization of reaction-diffusion phenomena, these complex dynamic behaviors derive from continuous interaction between the "activator" and "inhibitor" morphogenesis (using the terminology introduced by Turing) in biological cell arrays [6]. To allow propagation through such a medium of more than one wave is necessary that active medium itself were re-excitable. Then, a second variable is required, and the resulting minimal system will have two equations. This is the reason why these phenomena are described by a RD equations system of at least second order. The resulting wave traveling along the active media are named “auto-waves” [7] because they are self sustaining waves, that is they exploit the energy of the active media on which they propagate.

As an example of the proposed approach, are reported in figures. 2-3 the final results of two simulation concerning respectively with a Turing Pattern formation and a four-armed vortex.

Fig. 4 The Hexapod

1

0

-1 0 1

Low Medium High1

0

-1 0 1

Low Medium HighMBF on 'x' MBF on 'Nr'

Fig. 1. Fuzzy rules and membership functions used for pattern formation.

V. Application of Auto-waves in Robotics

A typical example of auto wave is a wave front traveling on a plane. If the plane is represented by a CFN on which Periodic Ring Border Conditions are imposed, the wave front never extinguishes, until the CFN is power supplied. This permanent wave has been used as driving signal for controlling the movement of a simple walking hexapod. Moreover, the speed of the traveling wave is strictly related to the gait of the robot, so by controlling the phase shift between two nearby cells it is possible to switch from a fast gait to a slow gait, and viceversa.

In figure 4 it is depicted the hexapod built in ST laboratories used to prove this theory.

References

[1] L.O.Chua, L.Yang “Cellular Neural Networks: Theory/Applications.” Trans. on Circuits and Systems Vol.35, pp. 1257-1272, 1988.

[2] L.O. Chua, M. Hasler, G.S. Moschytz, J. Neirynck, "Autonomous Cellular Neural Networks: A Unified Paradigm for pattern Formation and Active Wave Propagation", IEEE Trans. On CAS-I, Vol.42, NO.10, pp. 559-577, Oct. 1995.

[3] R. Caponetto, M. Lavorgna, L. Occhipinti, "Fuzzy Cellular Systems: Characteristics and Architecture", in Fuzzy Hardware, by Kluwer Ac. Ed., Chapt. 14th, 1997.

[4] P. Arena, S. Baglio, L. Fortuna, G. Manganaro, “State Controlled CNN: A New Strategy for Generating High Complex Dynamics”, IEEE. Trans on Fundamentals Vol. E-79-A, N. 10 Oct 1996.

[5] P. Arena, R. Caponetto, L. Fortuna, G. Manganaro, “Cellular neural networks to explore complexity”, Soft Computing Journal, Vol.1 N.3, September 1997

[6] A.M. Turing, "The chemical basis of morphogenesis", Phil. Trans. Roy. Soc. Lond., Vol. 237 (B.641), pp. 37-72, 1952.

[7] V.I.Krinsky, "Autowaves: Results, Problems, Outlooks", in Self-Organization: Autowaves and Structures Far from Equilibrium, Springer-Verlag, Berlin, 1984, pp.9-18.

Fig. 2 Turing Pattern in a 60 x 60 CNN: a) Initial Conditions; b) Steady State.

a) b)

b)a)

Fig. 3 Complex Phenomenon: Multi-armed vortex; a) Initial Conditions; b) Periodic State