University of Florida/Egm6321/f09.team1.gzc/Mtg36

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page36-1

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$\underline{I n i t . c o n d .} \underline{Z} (t_{0}) = {\underline{Z}}_{0} g i v e n$

$\underline{O p t . c o n t r o l p b . :} F i n d \underline{u} s t . m i n J (\underline{z}, \underline{u})$

$s t . \underline{\overset{∙}{X}} = \underline{f} (u n d e r l i n e z, u n d e r l i n e u, t)$

$I n i t . c o n d . : \underline{z} (t_{0}) = {\underline{z}}_{0}$

$I n e q . c o n s t r . : \underline{g} (\underline{z}, t) ⩽ \underline{0}$

$E q u a l . c o n s t r . : \underline{h} (\underline{z}, t) = \underline{0}$

J(z,u)=Obj. funcion or Performance Index

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$J : = \int_{t_{0}}^{t_{f}} d t = \underset{u n k o w n t o b e d e t e r m i n d e d}{\underset{⏟}{t_{f}}} - t_{0}, t_{f} = t_{f} (u n d e r l i n e z, u n d e r l i n e u)$

page36-2

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$J = \int_{t_{0}}^{t_{f}} \underset{a l t i t u d e}{\underset{⏟}{h}} (t) d t f i g . p . 34 - 1 : A r e a u n d e r c u r v e$

$I n e q . c o n s t r . : \underline{g} (\underline{z}, t) ⩽ \underline{0}$

<maht>e.g., \ T_{min} \leqslant T(t) \leqslant T_{max}</math>

$\Leftrightarrow T (t) ⩽ T_{m a x} \forall t$

$- ⩽ - T_{m i n} \forall t$

$g_{1} : = T (t) - T_{m a x} ⩽ 0$

$g_{2} : = - T (t) + T m i n ⩽ 0$

$\Leftrightarrow \underset{\underline{g}}{\underset{⏟}{{\begin{matrix} g_{1} \\ g_{2} \end{matrix}}}} \underset{⩽}{⩽} \underset{\underline{0}}{{\begin{matrix} 0 \\ 0 \end{matrix}}}$

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$A n o t h e r i n e q . c o n s t r . : h (t) ⩽ h_{m a x} S a n d Z 2007$

$S o l u t i o n f o r m o f o p t . c o n t r . : D i r e c t t r a n s c r i p t i o n$

$c o n v e r t \underline{c o n t i n o u s o p t . c o n t r . p b .} i n t o$

$\underline{d i s c r e t e n o n l i n e a r p r o g r a m m i n g (o p t .) p b .}$

$\Rightarrow D i s c r e t i z e a b s . f o r m . (O E S s) i n t i m e \underset{s c a l a r}{\underset{⏟}{\overset{∙}{z}}} = f (z, t)$

$f_{n} : = f (z_{n}, t_{n})$

$H e r m i t i a n i n t e r p . : z (t) ≅ P_{3} (t) = \sum_{i = 0}^{3} c_{i} t^{i}$

page36-4

$d o f = d e g r e e . o f f r e e d o m$

$(1) {\begin{matrix} d_{1} = z_{n}, d_{3} = z_{n + 1} \\ d_{2} = {\overset{∙}{z}}_{n}, d_{4} = {\overset{∙}{z}}_{n + 1} \end{matrix}$

 $P_{(3)} (t) = \sum_{i = 0}^{3} c_{i} t^{i} = \sum_{i = 1}^{4} \underset{b a s i s f u n c t i o n}{\underset{⏟}{N_{i} (t) d_{i}}} (2)$

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$[t_{i}, t_{i + 1}] s t t (s) = (1 - s) t_{i} + s t_{i + 1} (3)$

$s = 0 \Rightarrow t (0) = t_{i}$

$s = 1 \Rightarrow t (1) = t_{i + 1}$

$z (s) \overset{(4)}{\overset{(4)}{=}} \sum_{i = 0}^{3} c_{i} s^{i} {\begin{matrix} a t t_{i} a n d t_{i + 1}, e n f o r c e \\ c o m p l i a n c e w i t h O D E \end{matrix}$

${\begin{matrix} {\overset{∙}{z}}_{i} \overset{(5)}{=} f_{i} : = f (z_{i}, t_{i}) \\ {\overset{∙}{z}}_{i + 1} \overset{(6)}{=} f_{i + 1} : = f (z_{i + 1}, t_{i + 1}) \end{matrix}$

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$I n g e n e r a l, \forall t \in] t_{i}, t_{i + 1} [, i . e ., t \neq t_{i} a n d t \neq t_{i + 1}, {\overset{∙}{z}}_{t} \neq f_{t} \cdot$

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University of Florida/Egm6321/f09.team1.gzc/Mtg36

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