# LMIs in Control/pages/H infinity 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 affinity 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 \geq 0\}}\gamma \\&\quad s.t.{\begin{bmatrix}A^{\top }P+PA&PB_{w}\\B_{w}^{\top }P&-\gamma ^{2}I\end{bmatrix}}+{\begin{bmatrix}C_{z}^{\top }(\theta )\\D_{zw}^{\top }(\theta )\end{bmatrix}}{\begin{bmatrix}C_{z}(\theta )&D_{zw}(\theta )\end{bmatrix}}\preceq 0.\end{aligned}}}

## Conclusion

The value function of the above semi-definite program returns the ${\displaystyle {\mathcal {H}}_{\infty }}$  norm of the system.

## Remark

It is assumed that ${\displaystyle A}$  is stable and ${\displaystyle (A,B_{w})}$  is controllable and the semi-infinite convex constraint ${\displaystyle \|H_{\theta }(j\omega )\|<\gamma }$  for all ${\displaystyle \omega \in \mathbb {R} }$ , is converted into a finite-dimensional convex LMI.