问题描述
设连续系统状态方程和性能指标
X ˙ = f ( t , X , U ) X ( t 0 ) = X 0 J = ϕ [ X ( t f ) , t f ] + ∫ t 0 t f F ( X , U , t ) d t \begin{aligned} \dot{X} & =f(t, X, U) \quad X\left(t_{0}\right)=X_{0} \\ J & =\phi\left[X\left(t_{f}\right), t_{f}\right]+\int_{t_{0}}^{t_{f}} F(X, U, t) d t \end{aligned} X˙J=f(t,X,U)X(t0)=X0=ϕ[X(tf),tf]+∫t0tfF(X,U,t)dt
HJI与HJB方程
HJB方程
设定如下哈密顿函数定义：
H ( X , U , λ , t ) = F ( X , U , t ) + λ T f ( X , U , t ) λ = ∂ V ∂ X \begin{aligned} H(X, U, \lambda, t)=&F(X, U, t)+\lambda^{T} f(X, U, t) \\ \lambda=&\frac{\partial V}{\partial X} \end{aligned} H(X,U,λ,t)=λ=F(X,U,t)+λTf(X,U,t)∂X∂V
HJB方程
− ∂ V ∂ t = min u ∈ Ω H ( X , U , ∂ V ∂ X , t ) = H ∗ ( X , U , ∂ V ∂ X , t ) -\frac{\partial V}{\partial t}=\min _{u \in \Omega} H\left(X, U, \frac{\partial V}{\partial X}, t\right)=H^{*}\left(X, U, \frac{\partial V}{\partial X}, t\right) −∂t∂V=u∈ΩminH(X,U,∂X∂V,t)=H∗(X,U,∂X∂V,t)
此外，参数 λ \lambda λ满足协态方程
λ ˙ = − ∂ H ∂ X \dot{\lambda}=-\frac{\partial H}{\partial X} λ˙=−∂X∂H
横截条件
λ ( t f ) = ∂ ϕ ∂ X ( t f ) \lambda(t_f)=\frac{\partial \phi}{\partial X(t_f)} λ(tf)=∂X(tf)∂ϕ
HJI方程
考虑一个博弈问题其价值函数为
V ( x ) = min u p max u e J V(\boldsymbol{x})=\min _{\boldsymbol{u}_{p}} \max _{\boldsymbol{u}_{e}} J V(x)=upminuemaxJ
HJI方程
− ∂ V ∂ t = ∂ V ∂ x f ( x ) + F ( x ) -\frac{\partial V}{\partial t}=\frac{\partial V}{\partial \boldsymbol{x}}f(x)+F(x) −∂t∂V=∂x∂Vf(x)+F(x)
 构建项目，QT可视化操作，添加自定义测试内容)
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