Simulating Continuous Fuzzy Systems for Fuzzy Solution Surfaces

ABSTRACT

In our book to appear in print from Springer-Verlag GmbH, Simulating Continuous Fuzzy Systems, Buckley and Jowers, we use crisp continuous simulation under MatlabT/SimulinkT to estimate fuzzy solution trajectories for several different systems of ordinary differential equations (ODEs). Our approach to simulation of a fuzzy system is to evaluate the system with triangular fuzzy parameters, evaluating at the left/vertex/right supports and values in between. Solutions are presented as a graph(s) of the variable(s) of interest, with respect to time. We now investigate two capabilities we identified as interesting for future research. With multiple fuzzy parameters, a solution graph is likely to become overloaded with superfluous information (trajectories which do not contribute to a surface solution). Because of the cost of plotting the superfluous information, certain support values may be skipped. We implement a reduction algorithm for determining solution boundaries, as trajectories are computed. Three dimensional fuzzy solution trajectories are the second improvement we investigate. By including membership computation in our simulations, we produce fuzzy solution surface boundaries. These surfaces enclose the possible solution space. To make this process manageable, we combine this with boundary determination. Our continued research employs MatlabT/SimulinkT, but the algorithm used is applicable to other tools. We use SimulinkT diagrams to model a system. Next we use MatlabT command files to prepare the model for execution (insertion of fuzzy parameter values), execute the model repeatedly while reducing the collected data, and finally preparing the 3-D plot of the solution. In this paper we describe our method and present 3-D solutions of two classical systems of ODE.

Key Words:

Fuzzy systems, Fuzzy differential equations, Simulation, Uncertainty