# LMIs in Control/pages/Hankel Norm for 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 _{\{Q\succeq 0,\gamma ^{2},\theta \}}\gamma ^{2}\\&\quad s.t.\quad D_{zw}(\theta )=0,\quad A^{\top }Q+QA+C_{z}(\theta )C_{z}(\theta )\preceq 0,\quad \gamma ^{2}I-W_{c}^{1/2}QW_{c}^{1/2}\succeq 0,\end{aligned}}}

where ${\displaystyle W_{c}}$ is the controllability Gramian, i.e., ${\displaystyle W_{c}\triangleq \int _{0}^{\infty }e^{At}B_{w}B_{w}^{\top }e^{A^{\top }t}dt}$.

## Conclusion

The Hanakel norm (i.e., the square root of the maximum eigenvalue) of ${\displaystyle H_{\theta }}$ is less than ${\displaystyle {\gamma }}$ if and only if the above LMI holds and the value function returns the maximum provable Hankel norm.

## Remark

${\displaystyle D_{zw}}$ is assumed to be zero.