Simulation of Continuous Fuzzy Systems

From the Introduction (slightly edited) ---

In Simulating Continuous Fuzzy Systems [ref.] we use crisp continuous simulation to estimate fuzzy solution trajectories for several different systems of ordinary differential equations (ODEs). Our approach to simulation of fuzzy ODE systems is to evaluate a system with triangular fuzzy parameters, evaluating at left/vertex/right supports and values in between. Solutions are presented as a graph(s) of 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 an understanding of a solution). Because of cost of processing and plotting superfluous information, critical support values may be skipped. We implement a reduction algorithm for determining solution boundaries, as trajectories are computed. Keeping aware of membership values, quantization considerations, and solution neighborhoods, we collect a fraction of the data collected by brute force methods, and yet provide better coverage of fuzzy parameters. 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, it is combined with boundary determination. We are concerned with improvements to methods of using crisp simulation to compute, and render for visualization, estimated solution surfaces of continuous fuzzy systems. Our previous work in this area of simulating continuous fuzzy systems demonstrated using crisp simulation to estimate fuzzy solution trajectories [3 refs.] A wide variety of examples are considered in those works; e.g., queuing networks, infectious disease models, chemical reactions, etc. As continuous systems are usually described by a system of ordinary differential equations (ODEs), our examples are all governed by fuzzy differential equations. Many parameters in a system of ODEs may be imprecisely known; they may be estimated by experts, or estimated from data. Uncertainty in these parameter values make them fuzzy number estimators. Fuzzy number parameter values result in a system of fuzzy ODEs [ref]; i.e., a continuous fuzzy system. Before, we used crisp continuous simulation to compute multiple solution trajectories which we combined to estimate fuzzy trajectories. We continue this theme. For completeness we repeat parts of the prior work, but with interesting improvements. Our continued research employs MatlabTM/SimulinkTM , but the algorithm used is applicable to other tools. We use SimulinkTM diagrams to model a system. Next we use MatlabTM command files to prepare the model for execution (insertion of fuzzy parameter values), execute the model repeatedly while reducing collected data, and finally preparing plots of the solution. We render fuzzy solution surfaces using C++ and OpenGL, a (free) graphics library package. Here we describe our method and present 3-D solutions of two classical systems of ODE.

Omitted: Text on (some of the) method details !
The next section discusses briefy an example of our prior simulations. Then we discuss how we capture boundaries of the fuzzy trajectories and a lattice of points to define a fuzzy solution surface. Next we discuss how to render that surface for visualization. We present two examples: (1) the Belousov-Zhabotinskii (BZ) chemical reaction model [2 refs.] and (2) an infectious disease model [ref.]. The last section contains a brief summary, conclusions and plans for future research.