LMIs in Control/Click here to continue/LMIs in system and stability Theory/L2-Gain of Systems with Multiplicative noise

The System


\begin{align} x(k+1)&=Ax(k)+B_ww(k)+\sum_{i=1}^{L}(A_ix(k)+B_{w,i}w(k))p_i(k),\quad x(0)=0,\\ z(k)&=C_zx(k)+D_{zw}w(k)+\sum_{i=1}^{L}(C_{z,i}x(k)+D_{zw,i}w(k))p_i(k), \end{align}$$ where $$ p(0), p(1),\dots $$, are independent, identically distributed random variables with $$ Ep(k) = 0, Ep(k)p^\top(k) = \Sigma = diag(\sigma_1,\dots,\sigma_L)$$ and $$x(0)$$ is independent of the process $$p$$.

The Data
The matrices $$ A,B_w,\{A_i.B_{w,i}\}_{i=1}^L,C_z,D_{zw},\{C_{z,i},D_{zw,i}\}_{i=1}^L,\{\sigma_i\}_{i=1}^L $$.

The LMI:


\begin{align} &\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^2_i\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{align}$$

Implementation
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/a34713575cd8ae9831cb5b7eb759d0fd45a8c37f

Conclusion
The optimal $$ \gamma $$ returns an upper bound on the $$ \mathcal{L}_2 $$ gain of the system. .

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