Ee291e Lecture 24: Optimal Control 1 Controller Synthesis Problem for Continuous Systems

A continuous system can be though of as a hybrid automaton with: Assumption 1 u and d are piecewise continuous functions of time. Remarks: The trivial dependence on q will be dropped from now on. Strictly speaking execution takes place over a hybrid time trajectory of the form 0 ; 0 0 ]] 1 ; 0 1 ] : : : , where u and d are continuous over each interval (refer to lecture 8). Notice that x() is continuous and piecewise diierentiable. Modulo this technicality, this is a garden variety continuous system with: x(0) 2 Init; _ x(t) = f ? x(t); u(t); d(t) : Since u and d are piecewise continuous execution can be extended over arbitrarily long time horizons. Notice that the continuous system is by deenition non-blocking (since I(q 0) = X). The piecewise continuity assumption eliminates the only other possibility for a controller to \cheat" in a controller synthesis problem with a reachability speciication; under this assumption the system can not be Zeno either.