LMIs in Control/pages/Dissipativity of Affine Parametric Varying Systems

The System

$\begin{array}{cc}\dot{x}(t)& =Ax(t)+B_{w}w(t),\\ z(t)& =C_z(\theta )x(t)+D_{zw}(\theta )w(t),\end{array}$

where $C_z$ and $D_{zw}$ depend affinely on parameter $\theta \in {ℝ}^p$.

The Data

The matrices $A,B_w,C_z(.),D_{zw}(.)$.

The Optimization Problem:

Solve the following semi-definite program

$\begin{array}{cc}& \underset{\{P\succ 0,\gamma ,\theta \}}{\min }\gamma \\ & s.t.\left[\begin{array}{cc}{A}^{\mathrm{\top }}P+PA& P{B}_w-C_z(\theta {)}^{\mathrm{\top }}\\ B_w^{\mathrm{\top }}P-C_z(\theta )& 2\gamma I-D_{zw}(\theta )-D_{zw}(\theta {)}^{\mathrm{\top }}\end{array}\right]⪯0.\end{array}$

Implementation

https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/b6cd6b81f75be4a2052ba3fa76cad1a2f9c49caa

Conclusion

The dissipativity of $H_{\theta }$ (see [Boyd,eq:6.59]) exceeds $\gamma$ if and only if the above LMI holds and the value function returns the minimum provable dissipativity.

Remark

It is worth noticing that passivity corresponds to zero dissipativity.

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.