Rigorous discrete time linearization of periodically switched circuits with respect to duty cycle perturbations

Abstract

ABSTRACTThis paper considers the well-known problem of deriving a linear model of dynamics of periodically switched circuits w.r.t. small perturbations in duty cycle (or switching instances) as external control inputs. A rigorous approach to this problem is developed and is shown that the linearized model is shift invariant and discrete time in nature. This is at variance with the well-known model, which is linear time invariant continuous time referred as state space averaging (SSA) model. SSA model ignores commutativity conditions in matrices of state space model due to varying parameters over intervals as well as the discrete nature of control input. The proposed method of linearization considers the problem of linearization in a neighbourhood of a periodic solution. The monodromy matrix for state transition over all phases of switching is considered to account for non-commuting matrices of parameters. Similarly discrete nature of the input changing once in every period of switching leads to the discrete model. This methodology is applicable for multiple independently switched circuits and takes into account orders of switching once the nominal periodic solution over which linearization is sought is fixed. This paper gives the detailed theory as well as illustrative examples to prove the usefulness of the proposed methodology.