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

The System

$\begin{array}{cc}x(k+1)& =Ax(k)+B_{w}w(k)+\sum _{i=1}^L(A_{i}x(k)+B_{w,i}w(k))p_i(k),x(0)=0,\\ z(k)& =C_{z}x(k)+D_{zw}w(k)+\sum _{i=1}^L(C_{z,i}x(k)+D_{zw,i}w(k))p_i(k),\end{array}$

where $p(0),p(1),\dots$, are independent, identically distributed random variables with $Ep(k)=0,Ep(k)p^{\mathrm{\top }}(k)=\mathrm{\Sigma }=diag({\sigma }_1,\dots ,{\sigma }_L)$ 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:

$\begin{array}{cc}& \underset{\{P\succ 0,{\gamma }^2\}}{\min }{\gamma }^2\\ & s.t.{\left[\begin{array}{cc}A& B_w\\ C_z& D_{zw}\end{array}\right]}^{\mathrm{\top }}\left[\begin{array}{cc}P& 0\\ 0& I\end{array}\right]\left[\begin{array}{cc}A& B_w\\ C_z& D_{zw}\end{array}\right]-\left[\begin{array}{cc}P& 0\\ 0& {\gamma }^{2}I\end{array}\right]+\sum _{i=1}^L{\sigma }_i^2{\left[\begin{array}{cc}{A}_i& B_{w,i}\\ C_{z,i}& D_{zw,i}\end{array}\right]}^{\mathrm{\top }}\left[\begin{array}{cc}P& 0\\ 0& I\end{array}\right]{\left[\begin{array}{cc}{A}_i& B_{w,i}\\ C_{z,i}& D_{zw,i}\end{array}\right]}^{\mathrm{\top }}⪯0\end{array}$

Implementation

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

Conclusion

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

Remark

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

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.