Advanced PMP Solution Analysis
Explore the analysis of a basic PMP solution to find the optimal control that minimizes a given function, incorporating necessary conditions and constraints. Understand the dimensions of the PMP solution, including unknowns, state, co-state, and control variables. Learn how to handle additional constraints like final state and time constraints using Lagrange multipliers.
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Presentation Transcript
Analysis of Basic PMP Solution Find control u(t) which minimizes the function: ? x(t=0) = x0, T fixed, ? = ?(?,?) ? ?,? = ? ?,? ?? + ? ? ? 0 To find a necessary condition to minimize J(x,u), we augment the cost function with the dynamics constraints ???(? ?,? ?)?? = 0 ?? ?,? ?? ? ?? + ?(? ? ) ? ?,?,? = ? ?,? + 0 Extremize the augmented cost w.r.t. variations x(t); u(t); (t); ? ?? ??+ ?? ? ?? +?? ?? ?? ???? dt ? ? = ???? + 0 ?? ?? ??? + ? ?,? ? ?,? + ???(?,?) ?? ? + ??0?? 0
Dimension of PMP Solution Unknowns n state variables x(t) n co-state variables ?(t) m control variables u(t) Optimal control satisfies: ? ?? ? n o.d.e.s with I.C.s ? = = ?(?,?) ? 0 = ?0 2 point B.V. problem ? ?? ?? ? =?? n o.d.e.s with T.C.s ??? ? = ?? ??= 0 ; m algebraic equations
Additional Constraints? Example: what if final state constraints i(x(T))=0, i=1, ,p are desired: Add constraints to cost using additional Lagrange multipliers Multipliers ? ???,? = ? ?,? ?? + ? ? ? + ?1?1? ? + + ????? ? 0 Cost augmentation Extremizing the constrained, augmented the cost function yields ? ?? ? n o.d.e.s with I.C.s ? = = ?(?,?) ? 0 = ?0 ? ?? ?? ? =?? ?? ??(? ? ) ? n o.d.e.s with T.C.s = ??(? ? ) + ?? ??= 0 ; m algebraic equations ?1? ? = 0; ; ??? ? = 0 p algebraic equations
Additional Constraints? Example: what if final time is undetermined? Final time T is an additional variable H(T) = 0 gives one additional constraint equation!
