# LMIs in Control/pages/H infinity Norm for 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 affinity 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 \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 ${\mathcal {H}}_{\infty }$  norm of the system.

## Remark

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