![{\displaystyle {\begin{aligned}x(k+1)&=Ax(k)+B_{w}w(k)+\sum _{i=1}^{L}(A_{i}x(k)+B_{w,i}w(k))p_{i}(k),\quad x(0)=0,\\z(k)&=C_{z}x(k)+D_{zw}w(k)+\sum _{i=1}^{L}(C_{z,i}x(k)+D_{zw,i}w(k))p_{i}(k),\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc5cc5b56f382be9c9451967e8cadadc23deb4c2)
where
, are independent, identically distributed random variables with
and
is independent of the process
.
The matrices
.
![{\displaystyle {\begin{aligned}&\min _{\{P\succ 0,\gamma ^{2}\}}\gamma ^{2}\\&\quad s.t.{\begin{bmatrix}A&B_{w}\\C_{z}&D_{zw}\end{bmatrix}}^{\top }{\begin{bmatrix}P&0\\0&I\end{bmatrix}}{\begin{bmatrix}A&B_{w}\\C_{z}&D_{zw}\end{bmatrix}}-{\begin{bmatrix}P&0\\0&\gamma ^{2}I\end{bmatrix}}+\sum _{i=1}^{L}\sigma _{i}^{2}{\begin{bmatrix}A_{i}&B_{w,i}\\C_{z,i}&D_{zw,i}\end{bmatrix}}^{\top }{\begin{bmatrix}P&0\\0&I\end{bmatrix}}{\begin{bmatrix}A_{i}&B_{w,i}\\C_{z,i}&D_{zw,i}\end{bmatrix}}^{\top }\preceq 0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b72f604e21834aa1da42a5e28df6758b9e289be)
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/a34713575cd8ae9831cb5b7eb759d0fd45a8c37f
The optimal
returns an upper bound on the
gain of the system. .
It is straightforward to apply scaling method [Boyd, sec.6.3.4] to obtain component-wise results.