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

The System

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

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

The Data

The matrices $A,B_{w},C_{z}(.),D_{zw}(.)$.

The Optimization Problem:

Solve the following semi-definite program

${\begin{aligned}&\min _{\{P\succ 0,\gamma ,\theta \}}\gamma \\&\quad s.t.\quad {\begin{bmatrix}A^{\top }P+PA&PB_{w}-C_{z}(\theta )^{\top }\\B_{w}^{\top }P-C_{z}(\theta )&2\gamma I-D_{zw}(\theta )-D_{zw}(\theta )^{\top }\end{bmatrix}}\preceq 0.\end{aligned}}$

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.