State deadbeat response and observability in multi-modal systems

This paper deals with two aspects of multimodal systems. First we show that any completely controllable multi-modal system, with state dimension n not exceeding 3, is capable through feedback of state dead-beat response. We conjecture that the result holds for all n, as is the case for the classical single-mode system. Certain properties of multi-modal systems indicate that they vary significantly from the single-mode systems. For example, the controllable set is not in general a subspace, and furthermore, the number of steps necessary to reach all states in the controllable set is not bounded by the state dimension. In this paper, we obtain bounds for this number in the case of a completely controllable system with n≤3, and use them to establish state deadbeat response. The second portion of this paper refines the controllability canonical form for a multi-modal system. This is accomplished through the introduction of a notion of observability, dual to controllability for these systems. An amplified version of this paper, including the proofs omitted here, will appear under the present title in a forthcoming issue of the SIAM Journal on Control and Optimization.