Admissibility of Trajectories for Control Systems Related by Smooth Mappings

Abstract

We examine the problem of relating a pair of nonlinear control systems by a smooth mapping between their state spaces that sends trajectories of one system onto trajectories of the other. This problem is fundamental to certain notions of hierarchical structure or state aggregation for control systems in which one wishes to relate a low-level, complex system, to a high-level, simpler system. Pappas, Lafferriere, and Sastry have recently introduced the concept of Φ-related systems (where Φ refers to the mapping between the systems’ state spaces) and have shown that this concept is equivalent to the property that Φ sends trajectories of one system onto trajectories of the other. However, this equivalence does not address any regularity properties of the controls (such as measurability or piecewise smoothness). Thus, in principle, one is not allowed to work with specified “admissible” classes of trajectories generated by corresponding classes of “admissible” controls. In this paper we identify several situations in which one can be assured that Φ-related systems do indeed send appropriately defined admissible trajectories of one system onto admissible trajectories of the other.