# LMIs in Control/pages/Entropy Bound 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 _{\{P\succ 0,\gamma ^{2},\lambda ,\theta \}}\gamma ^{2}\\&\quad s.t.\quad D_{zw}(\theta )=0,{\begin{bmatrix}A^{\top }P+PA&PB_{w}&C_{z}(\theta )^{\top }\\B_{w}^{\top }P&-\gamma ^{2}I&0\\C_{z}(\theta )&0&-I\end{bmatrix}}\preceq 0,\quad {\rm {{Tr}(B_{w}^{\top }PB_{w})\leq \lambda .}}\end{aligned}}}

## Conclusion

The value function of the above semi-definite program returns a bound for ${\displaystyle \gamma }$ -entropy of the system, which is defined as

{\displaystyle {\begin{aligned}I_{\gamma }(H_{\theta })\triangleq {\begin{cases}{\frac {-\gamma ^{2}}{2\pi }}\int _{-\infty }^{\infty }\log \det(I-\gamma ^{2}H_{\theta }(i\omega )H_{\theta }(i\omega )^{*})d\omega ,\quad {\text{if}}\ \|H_{\theta }\|_{\infty }<\gamma \\\infty ,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad {\text{otherwise.}}\end{cases}}\end{aligned}}}

## Remark

When it is finite, ${\displaystyle I_{\gamma }(H_{\theta })}$  is given by ${\displaystyle {\rm {{Tr}(B_{w}^{\top }PB_{w})}}}$  where ${\displaystyle P}$ , is asymmetric matrix with the smallest possible maximum singular value among all solutions of a Riccati equation.