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

## The System

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

## The Data

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

## The Optimization Problem:

Solve the following semi-definite program

{\displaystyle {\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}}}

## Conclusion

The dissipativity of ${\displaystyle H_{\theta }}$  (see [Boyd,eq:6.59]) exceeds ${\displaystyle {\gamma }}$  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.