Testing Stability of Digital Filters Using Multimodal Particle Swarm Optimization with Phase Analysis

In this paper, a novel meta-heuristic method for evaluation of digital filter stability is presented. The proposed method is very general because it allows one to evaluate stability of systems whose characteristic equations are not based on polynomials. The method combines an efficient evolutionary algorithm represented by the particle swarm optimization and the phase analysis of a complex function in the characteristic equation. The method generates randomly distributed particles (i.e., a swarm) within the unit circle on the complex plane and extracts the phase quadrant of function value in position of each particle. By determining the function phase quadrants, regions of immediate vicinity of unstable zeros, called candidate regions, are detected. In these regions, both real and imaginary parts of the complex function change signs. Then, the candidate regions are explored by subsequently generated swarms. When sizes of the candidate regions are reduced to a value of assumed accuracy, then the occurrence of unstable zero is verified with the use of discrete Cauchy's argument principle. The algorithm is evaluated in four benchmarks for integer- and fractional-order digital filters and systems. The numerical results show that the algorithm is able to evaluate the stability of digital filters very fast even with a small number of particles in subsequent swarms. However, the multimodal particle swarm optimization with phase analysis may not be computationally efficient in stability tests of systems with complicated phase portraits.