LMIs in Control/pages/L2 gain of Lure systems

The System

$\begin{array}{cc}\dot{x}(t)& =Ax(t)+B_{p}p(t)+B_{w}w(t),\\ z(t)& =C_{z}x(t)\\ p_i(t)& ={\varphi }_i(q_i(t)),i=1,\dots ,n_p\\ q& =C_{q}x,\\ 0& \le \sigma {\varphi }_i(\sigma )\le {\sigma }^2\mathrm{\forall }\sigma \in ℝ\end{array}$

The Data

The matrices $A,B_p,B_w,C_q,C_z$.

The Optimization Problem:

The following semi-definite problem should be solved.

$\begin{array}{cc}& \underset{\{P\succ 0,\mathrm{\Lambda }=diag({\lambda }_1,\dots ,{\lambda }_{n_p})⪰0,T=diag({\tau }_1,\dots ,{\tau }_{n_p})⪰0\}}{\min }{\gamma }^2\\ & s.t.\left[\begin{array}{ccc}{A}^{\mathrm{\top }}P+PA+C_z^{\mathrm{\top }}{C}_z& P{B}_p+A^{\mathrm{\top }}{C}_q^{\mathrm{\top }}\mathrm{\Lambda }+C_q^{\mathrm{\top }}T& P{B}_w\\ B_p^{\mathrm{\top }}P+\mathrm{\Lambda }{C}_{q}A+T{C}_q& \mathrm{\Lambda }{C}_q{B}_p+B_p^{\mathrm{\top }}{C}_q^{\mathrm{\top }}\mathrm{\Lambda }-2T& \mathrm{\Lambda }{C}_q{B}_w\\ B_w^{\mathrm{\top }}P& B_w^{\mathrm{\top }}{C}_q^{\mathrm{\top }}\mathrm{\Lambda }& -{\gamma }^{2}I\end{array}\right]⪯0\\ \end{array}$

Implementation

https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/12a7039f9e3d966e24b43fd58a3cce3725282ed2

Conclusion

The value function returns the square of the smallest provable upper bound on the ${ℒ}_2$ gain of the Lure's system.

Remark

The Lyapunov function which is used to proof is similar to the one for the systems with unknown parameters.

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.