User:EGM6341.s11.team1.Chiu/Mtg36

Mtg 34: Wed, 23 Mar 11


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page36-1

Abstract formulation cont'd p.35-4

$${\underline{Init. \ cond.}} \ {\underline{Z}}(t_{0})={\underline{Z}}_{0} \ {\color{blue}given}$$

$${\color{blue}\underline{Opt. \ control \ pb.:}} \ Find \ {\underline{u}} \ st. \ min \ J({\underline{z}}, \ {\underline{u}})$$

$$st. \ {\underline{ \overset{ \bullet}{X}}}={\underline{f}}({underline{z}}, \ {underline{u}}, \ t)$$

$${\color{blue}Init. \ cond.:} \ {\underline{z}}(t_{0})={\underline{z}}_{0}$$

$${\color{blue}Ineq. \ constr.:} \ {\underline{g}}({\underline{z}}, \ t) \leqslant {\underline{0}}$$

$${\color{blue}Equal. \ constr.:} \ {\underline{h}}({\underline{z}}, \ t)={\underline{0}}$$

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

Ex1: supersonic interceptor

$$J:= \ \int_{t_{0}}^{t_{f}}dt= {\color{blue}\underset{unkown \ to \ be \ determinded}{ \underbrace}}-t_{0} \, \ t_{f}=t_{f}({underline{z}}, \ {underline{u}})$$

page36-2

Ex2: Bunt manuuver

$$J=\int_{t_{0}}^{t_{f}}{\color{blue} \underset{altitude}{ \underbrace}}(t)dt \ {\color{blue}fig.p.34-1:}Area \ under \ curve$$

$${\color{blue}Ineq. \ constr.:} \ {\underline{g}}({\underline{z}}, \ t) \leqslant {\underline{0}}$$

e.g., \ T_{min} \leqslant T(t) \leqslant T_{max}

$$ \Leftrightarrow \ T(t) \leqslant T_{max} \ \forall t$$

$$ \ \ {\color{red}-}\leqslant \ {\color{red}-}T_ \ \forall t$$

$${\color{blue}g_{1}:=} \ T(t)-T_{max}\leqslant 0$$

$${\color{blue}g_{2}:=} \ {\color{red}-}T(t){\color{red}+}T{min}\leqslant 0$$

$$\Leftrightarrow \ {\color{blue} \underset{ \underbrace}} \ {\color{blue} \underset{ \leqslant}{ {\color{black}\leqslant}}} \ {\color{blue} \underset} $$

page36-3

$${\color{blue}Another \ ineq. \ constr. \ :}h(t)\leqslant h_{max} \ {\color{blue}S \ and \ Z \ 2007}$$

$${\color{blue}Solution \ form \ of \ opt. \ contr. \ : \ Direct \ transcription}$$

$$convert \ \ into$$

$${\color{red}\underline}$$

$${\color{blue}\Rightarrow \ Discretize \ abs. \ form. \ (OESs) \ in \ time} \ {\color{blue} \underset{scalar}{ \underbrace}}=f(z,t)$$


 * nm1.s11.Mtg36.pg3.fig1.svg

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

$${\color{blue}Hermitian \ interp.:} \ z(t) \cong \ P_(t)= \sum_{i=0}^ c_{i}t^{i}$$

page36-4

$${\color{blue}dof \ =}degree. \ of \ freedom$$

$${\color{red}(1) \begin{cases} & \ {\color{black}d_{1}=z_{n}, \ d_{3}=z_{n+1}}\\ & \ {\color{black}d_{2}= \overset{ \bullet}{z}_{n}, \ d_{4}= \overset{ \bullet}{z}_{n+1}} \end{cases}}$$ $$P_(t)= \sum_{i=0}^c_{i}t^{i}=\sum_{i={\color{red}1}}^{\color{blue} \underset{basis \ function}{ \underbrace}} \ {\color{red}(2)} $$

Hermite-Simpson algo:

$$[t_{i}, \ t_{i+1}] \ st \ t(s)=(1-s)t_{i}+st_{i+1} \ {\color{red}(3)}$$

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

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

$$z(s) \overset \overset{=} \sum_{i=0}^c_{i}s^{i}{\color{blue} \begin{cases} & at \ t_{i} \ and t_{i+1}, \ enforce\\ & compliance \ with \ ODE \end{cases}}$$ $${\color{blue} \begin{cases} & {\color{black} \overset{ \bullet}{z}_{i} \overset{=}f_{i} := \ f(z_{i}, \ t_{i})} \\ & {\color{black} \overset{ \bullet}{z}_{i+1} \overset{=}f_{i+1} := \ f(z_{i+1}, \ t_{i+1})} \end{cases}}$$ page36-5

$${\color{blue}In \ general,} \ \forall \ t\in \ ]t_{i}, t_{i+1}[, \ i.e., \ t \neq t_{i} \ and \ t \neq t_{i+1}, \ \overset{ \bullet}{z}_{t} \neq f_{t} \cdot$$