Fourier state-space analysis of linear discrete-time periodic systems

By the Floquet-Lyapunov transformation, a linear periodically time-varying discrete-time system has a state-space representation (A,B(n),C(n)) having time-invariant autonomous dynamics, while the time periodicity is contained in the input and output matrices. In this paper it is shown that such a system also has state-space realizations of the form (A,B(n),C) and (A,B,C(n)), where the time periodicity is contained at the input, or alternatively the output, matrix only, and whose state dimensions are bounded by the product of the system order and the period length. The system representations are applied to construct a state-space realization of the optimal linear time-invariant approximation of a time-periodic system.