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

Mtg 34: Wed, 23 Mar 11

page36-1 Abstract formulation cont'd p.35-4 ${\displaystyle {\underline {Init.\ cond.}}\ {\underline {Z}}(t_{0})={\underline {Z}}_{0}\ {\color {blue}given}}$ ${\displaystyle {\color {blue}{\underline {Opt.\ control\ pb.:}}}\ Find\ {\underline {u}}\ st.\ min\ J({\underline {z}},\ {\underline {u}})}$ ${\displaystyle st.\ {\underline {\overset {\bullet }{X}}}={\underline {f}}({underline{z}},\ {underline{u}},\ t)}$ ${\displaystyle {\color {blue}Init.\ cond.:}\ {\underline {z}}(t_{0})={\underline {z}}_{0}}$ ${\displaystyle {\color {blue}Ineq.\ constr.:}\ {\underline {g}}({\underline {z}},\ t)\leqslant {\underline {0}}}$ ${\displaystyle {\color {blue}Equal.\ constr.:}\ {\underline {h}}({\underline {z}},\ t)={\underline {0}}}$ J(z,u)=Obj. funcion or Performance Index Ex1: supersonic interceptor ${\displaystyle J:=\ \int _{t_{0}}^{t_{f}}dt={\color {blue}{\underset {unkown\ to\ be\ determinded}{\underbrace {\color {black}t_{f}} }}}-t_{0}\ ,\ t_{f}=t_{f}({underline{z}},\ {underline{u}})}$ page36-2 Ex2: Bunt manuuver ${\displaystyle J=\int _{t_{0}}^{t_{f}}{\color {blue}{\underset {altitude}{\underbrace {\color {black}h} }}}(t)dt\ {\color {blue}fig.p.34-1:}Area\ under\ curve}$ ${\displaystyle {\color {blue}Ineq.\ constr.:}\ {\underline {g}}({\underline {z}},\ t)\leqslant {\underline {0}}}$ <maht>e.g., \ T_{min} \leqslant T(t) \leqslant T_{max}</math> ${\displaystyle \Leftrightarrow \ T(t)\leqslant T_{max}\ \forall t}$ ${\displaystyle \ \ {\color {red}-}\leqslant \ {\color {red}-}T_{\color {red}min}\ \forall t}$ ${\displaystyle {\color {blue}g_{1}:=}\ T(t)-T_{max}\leqslant 0}$ ${\displaystyle {\color {blue}g_{2}:=}\ {\color {red}-}T(t){\color {red}+}T{min}\leqslant 0}$ ${\displaystyle \Leftrightarrow \ {\color {blue}{\underset {\underline {g}}{\underbrace {\color {black}{\begin{Bmatrix}\ {\color {blue}g_{1}}\\{\color {blue}g_{2}}\end{Bmatrix}}} }}}\ {\color {blue}{\underset {\leqslant }{\color {black}\leqslant }}}\ {\color {blue}{\underset {\underline {0}}{\color {black}{\begin{Bmatrix}\ 0\\0\end{Bmatrix}}}}}}$ page36-3 ${\displaystyle {\color {blue}Another\ ineq.\ constr.\$ :}h(t)\leqslant h_{max}\ {\color {blue}S\ and\ Z\ 2007}} ${\displaystyle {\color {blue}Solution\ form\ of\ opt.\ contr.\$ :\ Direct\ transcription}} ${\displaystyle convert\ {\color {red}{\underline {\color {black}continous\ opt.\ contr.\ pb.}}}\ into}$ ${\displaystyle {\color {red}{\underline {\color {black}discrete\ nonlinear\ programming\ (opt.)\ pb.}}}}$ ${\displaystyle {\color {blue}\Rightarrow \ Discretize\ abs.\ form.\ (OESs)\ in\ time}\ {\color {blue}{\underset {scalar}{\underbrace {\color {black}{\overset {\bullet }{z}}} }}}=f(z,t)}$ ${\displaystyle f_{n}:=f(z_{n},t_{n})}$ ${\displaystyle {\color {blue}Hermitian\ interp.:}\ z(t)\cong \ P_{\color {red}3}(t)=\sum _{i=0}^{\color {red}3}c_{i}t^{i}}$ page36-4 ${\displaystyle {\color {blue}dof\ =}degree.\ of\ freedom}$ ${\displaystyle {\color {red}(1){\begin{cases}&\ {\color {black}d_{1}=z_{n},\ d_{3}=z_{n+1}}\\&\ {\color {black}d_{2}={\overset {\bullet }{z}}_{n},\ d_{4}={\overset {\bullet }{z}}_{n+1}}\end{cases}}}}$ ${\displaystyle P_{\color {red}(3)}(t)=\sum _{i=0}^{\color {red}3}c_{i}t^{i}=\sum _{i={\color {red}1}}^{\color {red}4}{\color {blue}{\underset {basis\ function}{\underbrace {\color {black}N_{i}(t)d_{i}} }}}\ {\color {red}(2)}}$ Hermite-Simpson algo: ${\displaystyle [t_{i},\ t_{i+1}]\ st\ t(s)=(1-s)t_{i}+st_{i+1}\ {\color {red}(3)}}$ ${\displaystyle s=0\ \Rightarrow \ t(0)=t_{i}}$ ${\displaystyle s=1\ \Rightarrow \ t(1)=t_{i+1}}$ ${\displaystyle z(s){\overset {\color {red}(4)}{\overset {\color {red}(4)}{=}}}\sum _{i=0}^{\color {red}3}c_{i}s^{i}{\color {blue}{\begin{cases}&at\ t_{i}\ andt_{i+1},\ enforce\\&compliance\ with\ ODE\end{cases}}}}$ ${\displaystyle {\color {blue}{\begin{cases}&{\color {black}{\overset {\bullet }{z}}_{i}{\overset {\color {red}(5)}{=}}f_{i}:=\ f(z_{i},\ t_{i})}\\&{\color {black}{\overset {\bullet }{z}}_{i+1}{\overset {\color {red}(6)}{=}}f_{i+1}:=\ f(z_{i+1},\ t_{i+1})}\end{cases}}}}$ page36-5 ${\displaystyle {\color {blue}In\ general,}\ \forall \ t\in \ ]t_{i},t_{i+1}[,\ i.e.,\ t\neq t_{i}\ and\ t\neq t_{i+1},\ {\overset {\bullet }{z}}_{t}\neq f_{t}\cdot }$