Abstract
We propose necessary and sufficient Observability conditions for linear time-varying systems with coefficients being time polynomials. These conditions are deduced from the Gabrielov–Khovansky theorem on multiplicity of a zero of a Noetherian function and the Wei–Norman formula for the representation of a solution of a linear time-varying system as a product of matrix exponentials. We define a Noetherian chain consisted of some finite number of usual exponentials corresponding to this system. Our results are formulated in terms of a Noetherian chain generated by these exponential functions and an upper bound of multiplicity of zero of one locally analytic function which is defined with help of the Wei–Norman formula. Relations with Observability conditions of bilinear systems are discussed. The case of two-dimensional systems is examined.