LMIs in Control/pages/L2 gain of systems with multiplicative noise

The System

{\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}}}

where ${\displaystyle p(0),p(1),\dots }$, are independent, identically distributed random variables with ${\displaystyle Ep(k)=0,Ep(k)p^{\top }(k)=\Sigma =diag(\sigma _{1},\dots ,\sigma _{L})}$ and ${\displaystyle x(0)}$ is independent of the process ${\displaystyle p}$.

The Data

The matrices ${\displaystyle A,B_{w},\{A_{i}.B_{w,i}\}_{i=1}^{L},C_{z},D_{zw},\{C_{z,i},D_{zw,i}\}_{i=1}^{L},\{\sigma _{i}\}_{i=1}^{L}}$.

The LMI:

{\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}}}

Conclusion

The optimal ${\displaystyle \gamma }$ returns an upper bound on the ${\displaystyle {\mathcal {L}}_{2}}$ gain of the system. .

Remark

It is straightforward to apply scaling method [Boyd, sec.6.3.4] to obtain component-wise results.