Using steady-state response for predicting stability boundaries in switched systems under PWM with linear and bilinear plants

Switching systems under Pulse Width Modulation (PWM) are commonly utilized in many industrial applications. Due to their associated nonlinearities, such systems are prone to exhibit a large variety of complex dynamics and undesired behaviors. In general, slow dynamics in these systems can be predicted and analyzed by conventional averaging procedures. However, fast dynamics instabilities such as period doubling (PD) and saddle-node (SN) bifurcations cannot be detected by average models and analyzing them requires the use of additional sophisticated tools. In this chapter, closed-form conditions for predicting the boundary of these bifurcations in a class of PWM systems with linear and bilinear plants are obtained using a time-domain asymptotic approach. Previous studies have obtained similar boundaries by either solving the eigenvalue problem of the monodromy matrix of the Poincaré map or performing a Fourier series expansion of the feedback signal. While the former approach is general and can be applied to linear as well as bilinear plants, the later approach is applicable only to PWM systems with linear plants. The conditions for fast scale instability boundaries presented in this chapter are obtained from the steady-state analysis of the Poincaré map using an asymptotic approach without resorting to frequency-domain Fourier analysis and without using the monodromy matrix of the Poincaré map. The obtained expressions are simpler than the previously reported ones and allow to understand the effect of different system’s parameters on its stability. In PWM systems with linear plants, under certain practical conditions concerning these parameters, the matrix form expression can be approximated by standard polynomial functions expressed in terms of the operating duty cycle weighted by the Markov parameters of the linear part of the system.