# Pseudo-continuous multi-dimensional multi-mode systems

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## Abstract

The dynamics of multi-mode multi-dimensional (*M* ^{3} *D*) hybrid systems is described. Such systems have modes of different dimensions, for which the state space is defined as a fibre bundle. The implications of the behavior at the mode transitions is investigated in detail, for which pseudo-continuity is introduced. An *M* ^{3} *D* system is pseudo-continuous if instantaneous switching via higher dimensional modes does not have any effect. Canonical forms and parameterizations are derived for pseudo-continuous *M* ^{3} *D* systems. The system may be actively controlled (exo-*M* ^{3} *D*), or passively switched via a fixed switching surface (auto-*M* ^{3} *D*). *M* ^{3} *D* systems are of interest in the approximate and reduced order modeling for nonlinear systems and the remote control over one-way communication channels. The optimal timing (switching) control for such *M* ^{3} *D* systems is solved in the general case. Necessary conditions for a stationary solution are derived and shown to extend those of the equal dimension case (Egerstedt et al. 2003). We also give a specific solution for the linear quadratic problem, involving a generalization of the Riccati equation. This problem is of interest in deriving neighboring extremal solutions for the control under small perturbations of s nominal solution. Some suggestions towards determining the optimal mode sequence are given. We illustrate the problem with the optimal control for a spring assisted high jump (aka man on a trampoline).

## Keywords

Hybrid systems Switched systems Optimal control Structure Canonical form Parameterization## References

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