The poles and zeros of a linear time-varying system

AbstractFor linear time-varying discrete-time and continuous-time systems, a notion of poles and zeros is developed in terms of factorizations of operator polynomials with time-varying coefficients. In the discrete-time case, it is shown that the poles can be computed by solving a nonlinear recursion with time-varying coefficients. In the continuous-time case, the poles can be calculated by solving a nonlinear differential equation with time-varying coefficients. The theory is applied to the study of the zero-input response and asymptotic stability. It is shown that if a time-varying analogue of the Vandermonde matrix is invertible, the zero-input response can be decomposed into a sum of modes associated with the poles. Stability is then studied in terms of the components of the modal decomposition