LMIs in Control/pages/Output Energy Bounds for Lure System

The System

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

The Data

The matrices $A,B_p,B_w,C_q,C_z,x(0)$.

The Optimization Problem:

The following optimization problem should be to find the tightest upper bound for the output energy of the above Lur'e system.

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

Implementation

https://github.com/mkhajenejad/Mohammad-Khajenejad/blob/master/LMIs%20for%20Output%20Energy%20Bounds%20of%20Lure's%20Systems

Conclusion

The value function returns the the lowest bound for the energy function of the Lure's systems, i.e., $J=\underset{0}{\overset{\mathrm{\infty }}{\int }}{z}^{\mathrm{\top }}zdt$ with initial conditions $x(0)$.

Remark

The key step in the proof is to satisfy $\frac{d}{dt}V(x)+z^{\mathrm{\top }}z\le 0$, where $V(.)$ is Lyapunov function in a special form.

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.