H ∞ {\displaystyle H\infty } H ∞ {\displaystyle H_{\infty }} H ∞ {\displaystyle H_{\infty }}
考虑具有状态空间实现的连续时间广义系统 P {\displaystyle P}
x ˙ = A x + B 1 w , y = C 2 x + D 21 w {\displaystyle {\begin{aligned}{\dot {x}}&=Ax+B_{1}w,\\y&=C_{2}x+D_{21}w\\\end{aligned}}} 作为输入所需的矩阵是 A , B 1 , B 2 , C 2 , D 21 , D 11 {\displaystyle A,B_{1},B_{2},C_{2},D_{21},D_{11}}
观测器增益 L {\displaystyle L} H ∞ {\displaystyle H\infty }
T ( s ) = C 1 ( s 1 − ( A − L C 2 ) ) − 1 ( B 1 − L D 21 ) + D 11 {\displaystyle {\begin{aligned}T(s)=C_{1}(s1-(A-LC_{2}))^{-1}(B_{1}-LD_{21})+D_{11}\\\end{aligned}}} 被最小化。观测器的形式将是
x ^ ˙ = A x ^ + L ( y − y ^ ) , y ^ = C 2 x ^ {\displaystyle {\begin{aligned}{\dot {\hat {x}}}=A{\hat {x}}+L(y-{\hat {y}}),\\{\hat {y}}=C_{2}{\hat {x}}\\\end{aligned}}} LMI： H ∞ {\displaystyle H_{\infty }} [ 编辑 | 编辑源代码 ] 该 H ∞ {\displaystyle H\infty } P ∈ S n x , G ∈ R n x × n y {\displaystyle P\in \mathbb {S} ^{n_{x}},G\in \mathbb {R} ^{n_{x}\times n_{y}}} γ ∈ R > 0 {\displaystyle \gamma \in \mathbb {R} _{>0}} ζ ( γ ) = γ {\displaystyle \zeta (\gamma )=\gamma } P > 0 {\displaystyle P>0}
[ P A + A T P − G C 2 − C 2 T G T P B 1 − G D 21 C 1 ⋆ − γ 1 D 11 T ⋆ ⋆ − γ 1 ] < 0 {\displaystyle {\begin{aligned}{\begin{bmatrix}PA+A^{T}P-GC_{2}-{C_{2}}^{T}G^{T}&&PB_{1}-GD_{21}&&C_{1}\\\star &&-\gamma 1&&{D_{11}}^{T}\\\star &&\star &&-\gamma 1\end{bmatrix}}<0\\\end{aligned}}}
最佳观测器增益通过 L = P − 1 G {\displaystyle L=P^{-1}G} H ∞ {\displaystyle H_{\infty }} γ {\displaystyle \gamma }
设计 H ∞ {\displaystyle H\infty }
https://github.com/Ricky-10/coding107/blob/master/HinfinityOptimalobserver
