Symbolic Stability Boundaries and Root Locus Plots in Time-Periodic Systems

Abstract

Abstract The recent development of a symbolic computational algorithm for approximating the parameter-dependent fundamental solution matrix of linear time-periodic systems is used to symbolically compute stability boundaries as an explicit function of the system parameters and to construct root locus plots. By evaluating this matrix at the end of the principal period, the parameter-dependent Floquet Transition Matrix (FTM), or the linear part of the Poincaré map, is obtained. The sub sequent use of well-known criteria enables one to obtain the equations for the stability boundaries in the parameter space as polynomials of the system parameters. The symbolic nature of the method also allows one to obtain root locus plots in the complex plane as a function of the system parameters. The roots of the FTM (Floquet multipliers) must lie within the unit circle for stability. Further, the technique can successfully be applied to periodic systems whose internal excitation is strong. The symbolic software Mathematica is used here to perform all symbolic calculations. Two illustrative example problems, viz., a parametrically excited simple pendulum and a double inverted pendulum subjected to a periodic follower force, are included.