最优控制理论与系统

2020-05-28 217 阅读1分钟

一、末端时刻固定时的最优解

1.末端受约束情况

{\underset {u(t)}{\operatorname {minJ}}}=\varphi [x(t_f)]+\int_{t_0}^{t_f}L(x,u,t)dt,\quad t_f固定

s.t.\quad \dot{x}(t)=f(x,u,t),\quad x(t_0)=x_0

\psi[x(t_f)]=0

$最优解的必要条件是：$

$(1)\ x(t)和\lambda(t)满足下列正则方程：$

\dot{x}(t)=\frac{\partial{H}}{\partial{\lambda}}

\dot{\lambda}(t)=-\frac{\partial{H}}{\partial{x}}

$式中$

H(x,u,\lambda,t)=L(x,u,t)+\lambda^T(t)f(x,u,t)

$(2)\ 边界条件$

x(t_0)=x_0

\lambda(t_f)=\frac{\partial{\varphi}}{\partial{x(t_f)}}+\frac{\partial{\psi^T}}{\partial{x(t_f)}}\gamma(t_f)

\psi[x(t_f)]=0

$(3)\ 极值条件$

\frac{\partial{H}}{\partial{u}}=0