User:Egm6341.s10.team3.heejun/Meeting 34 transcript

 '''MTG 34: Thu. 25 Mar 10'''


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Note:  Typeset of Transparencies, not lecture transcript  Heejun Chung 20:58, 11 August 2010 (UTC)

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Optimal Control continued, [[media: Egm6341.s10.mtg33.djvu | Page 33-3]]

Initial Condition : $$ \displaystyle \underline{Z}(t_{0}) = \underbrace{\underline{Z}_{0}}_{\color{blue}{given}} $$

Optimal Control Problem : Find $$ \displaystyle \underline{u} $$ such that $$ \displaystyle min\,\, J(\underline{Z},\,\, \underline{u}) $$

Subjected to the following conditions such that $$ \displaystyle \underline{\dot{Z}} = f(\underline{Z},\,\, \underline{u},\,\, t) $$   Dynamics (evolution equation)

Initial Condition:  $$ \displaystyle \underline{Z}(t_{0}) = \underline{Z}_{0} $$

Inequality Constrain:  $$ \displaystyle \underline{g}(\underline{Z},\,\, t) \leq \underline{0} $$

Equality Constrain:  $$ \displaystyle \underline{h}(\underline{Z},\,\, t) = \underline{0} $$


 * Initial Condition can be thought of the particular case of the    Equality Constrain

$$ \displaystyle J(\underline{Z},\,\, t) $$ = objective funtion or performance index

Example 1: Supersonic intercepter

$$ \displaystyle J:= \int^{t_{f}}_{t_{0}} dt = t_{f} - t_{0} $$
 * $$ \displaystyle t_{f}: $$   unknown to be determined. $$ \displaystyle \color{blue}{t_{f} = t_{f}(\underline{Z},\,\, \underline{u})} $$

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Example 2: Bunt maneuver

$$ \displaystyle J = \int^{t_{f}}_{t_{0}} \underbrace{h(t)}_{\color{blue}{altitude}} dt\,\,\,\,\,\,\,\,\,\, $$   Fig. [[media: Egm6341.s10.mtg31.djvu | Page 31-2]]     (Area under curve)

Inequality Constrain $$ \displaystyle \underline{g}(\underline{Z},\,\, \underline{u},\,\, t) \leq \underline{0} $$

e.g., $$ \displaystyle \underbrace{T_{min}}_{\color{blue}{given}} \leq T(t) \leq \underbrace{T_{max}}_{\color{blue}{given}} $$


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& - T(t) \leq T_{min} \forall t \\\end{align}\ $$
 * $$ \displaystyle \begin{align} \Leftrightarrow & \,\,\,\, T(t) \leq T_{max} \forall t \\
 * $$ \displaystyle \begin{align} \Leftrightarrow & \,\,\,\, T(t) \leq T_{max} \forall t \\
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& \color{blue}{g_{2}:=} - T(t) - T_{min} \leq 0 \\\end{align}\ $$
 * $$ \displaystyle \begin{align} \Leftrightarrow \,\, & \color{blue}{g_{1}:=} T(t) - T_{max} \leq 0 \\
 * $$ \displaystyle \begin{align} \Leftrightarrow \,\, & \color{blue}{g_{1}:=} T(t) - T_{max} \leq 0 \\
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 * $$ \displaystyle \Leftrightarrow \underbrace{(g_{1}, g_{2})}_{\color{blue}{\underline{g}}} \underbrace{\leq}_{\color{blue}{\leq}} \underbrace{(0, 0)}_{\color{blue}{\underline{0}}} $$
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Another one(bunt maneuver): $$ \displaystyle h(t) \leq h_{max} $$   (S+Z, 2007)

Solution formulation of the optimal control problem: Direct Transcription : Convert optimal control problem into discrete nonlinear optimization (programming) problem.

$$ \displaystyle \Rightarrow \,\, $$ Discretize problem in time  $$ \displaystyle \color{blue}{\dot{Z} = f(Z,\,\,t)} $$



$$ \displaystyle f_{n} := f(Z_{n},\,\, t_{n}) $$

Hermitian interpolation: $$ \displaystyle Z(t) \cong P_{3}(t) = C_{0} + C_{1}t + C_{2}t^{2} + C_{3}t^{3} $$

[[media: Egm6341.s10.mtg34.djvu | Page 34-4]] 

Def = degree of freedom

$$ \displaystyle d_{1} := Z_{n} $$ $$ \displaystyle d_{2} := \dot{Z}_{n} $$ $$ \displaystyle d_{3} := Z_{n+1} $$ $$ \displaystyle d_{4} := \dot{Z}_{n+1} $$ $$ \displaystyle P_{3}(t) = \sum^{4}_{i=1} \underbrace{N_{i}(t)}_{\color{blue}{basis\,\, function}} d_{i} $$