# 控制李亚普诺夫函数

${\displaystyle {\dot {x}}=f(x,u)}$

${\displaystyle x\in \mathbf {R} ^{n}}$为状态向量，
${\displaystyle u\in \mathbf {R} ^{m}}$为控制向量

${\displaystyle \forall x\neq 0,\exists u\qquad {\dot {V}}(x,u)=\nabla V(x)\cdot f(x,u)<0.}$

Artstein定理：动态系统有可微分控制李亚普诺夫函数的充份必要条件是存在一个可以稳定系统的回授u(x)。

${\displaystyle u^{*}(x)=\operatorname {*} {argmin}_{u}\nabla V(x)\cdot f(x,u)}$

## 例子

${\displaystyle m(1+q^{2}){\ddot {q}}+b{\dot {q}}+K_{0}q+K_{1}q^{3}=u}$

${\displaystyle r={\dot {e}}+\alpha e}$

${\displaystyle V={\frac {1}{2}}r^{2}}$

${\displaystyle q\neq 0}$, ${\displaystyle {\dot {q}}\neq 0}$，上述函数皆为正定。

${\displaystyle {\dot {V}}=r{\dot {r}}}$
${\displaystyle {\dot {V}}=({\dot {e}}+\alpha e)({\ddot {e}}+\alpha {\dot {e}})}$

${\displaystyle {\dot {V}}=-\kappa V}$

${\displaystyle V}$是全域的正定，上式则为全域的指数稳定。

${\displaystyle ({\ddot {e}}+\alpha {\dot {e}})=({\ddot {q}}_{d}-{\ddot {q}}+\alpha {\dot {e}})}$

${\displaystyle ({\ddot {q}}_{d}-{\ddot {q}}+\alpha {\dot {e}})=-{\frac {\kappa }{2}}({\dot {e}}+\alpha e)}$

${\displaystyle ({\ddot {q}}_{d}-{\frac {u-K_{0}q-K_{1}q^{3}-b{\dot {q}}}{m(1+q^{2})}}+\alpha {\dot {e}})=-{\frac {\kappa }{2}}({\dot {e}}+\alpha e)}$

${\displaystyle u=m(1+q^{2})({\ddot {q}}_{d}+\alpha {\dot {e}}+{\frac {\kappa }{2}}r)+K_{0}q+K_{1}q^{3}+b{\dot {q}}}$

${\displaystyle {\dot {V}}=-\kappa V}$

${\displaystyle V=V(0)e^{-\kappa t}}$

${\displaystyle r{\dot {r}}=-{\frac {\kappa }{2}}r^{2}}$
${\displaystyle {\dot {r}}=-{\frac {\kappa }{2}}r}$
${\displaystyle r=r(0)e^{-{\frac {\kappa }{2}}t}}$
${\displaystyle {\dot {e}}+\alpha e=({\dot {e}}(0)+\alpha e(0))e^{-{\frac {\kappa }{2}}t}}$

## 脚注

1. ^ Freeman (46)