Using Swarm Intelligence for Optimization of Parameters in Approximations of Fractional-Order Operators

This chapter applies Particle Swarm Optimization (PSO) to rational approximation of fractional order differential or integral operators. These operators are the building blocks of Fractional Order Controllers, that often can improve performance and robustness of control loops. However, the implementation of fractional order operators requires a rational approximation specified by a transfer function, i.e. by a set of zeros and poles. Since the quality of the approximation in the frequency domain can be measured by the linearity of the Bode magnitude plot and by the “flatness” of the Bode phase plot in a given frequency range, the zeros and poles must be properly set. Namely, they must guarantee stability and minimum-phase properties, while enforcing zero-pole interlacing. Hence, the PSO must satisfy these requirements in optimizing the zero-pole location. Finally, to enlighten the crucial role of the zero-pole distribution, the outputs of the PSO optimization are compared with the results of classical schemes. The comparison shows that the PSO algorithm improves the quality of the approximation, especially in the Bode phase plot.