LMIs in Control/pages/Stability 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 LMI: The Lure's System's Stability

The following feasibility problem should be solved as sufficient condition for the stability of the above Lur'e system.

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

Implementation

https://codeocean.com/capsule/0232754/tree

Conclusion

If the feasibility problem with LMI constraints has solution, then the Lure's system is stable.

Remark

The LMI is only a sufficient condition for the existence of a Lur’e Lyapunov function that proves stability of Lur'e system . It is also necessary when there is only one nonlinearity, i.e., when $n_p=1$.

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.