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A Necessary Algebraic Condition for Controllability and Observability of Linear Time-Varying Systems

Abstract:

In this note, we give an algebraic condition which is necessary for the system x′(t) = A(t)x(t) + B(t)u(t), y(t) = C(t)x(t), either to be totally controllable or to be totally observable, where x ∈ Rd, u ∈ Rp, y ∈ Rq, and the matrix functions A, B and C are (d - 2), (d - 1) and (d - 1) times continuously differentiable, respectively. All conditions presented here are in terms of known quantities and therefore easily verified. Our conditions can be used to rule out large classes of time-varying systems which cannot be controlled and/or observed no matter what the nonzero time-varying coefficients are. This work is motivated by the deep result of Silverman and Meadows.