LMIs in Control/pages/Entropy Bound for Affine Parametric Varying Systems

The System

$\begin{array}{cc}\dot{x}(t)& =Ax(t)+B_{w}w(t),\\ z(t)& =C_z(\theta )x(t)+D_{zw}(\theta )w(t),\end{array}$

where $C_z$ and $D_{zw}$ depend affinely on parameter $\theta \in {ℝ}^p$.

The Data

The matrices $A,B_w,C_z(.),D_{zw}(.)$.

The Optimization Problem:

Solve the following semi-definite program

$\begin{array}{cc}& \underset{\{P\succ 0,{\gamma }^2,\lambda ,\theta \}}{\min }{\gamma }^2\\ & s.t.D_{zw}(\theta )=0,\left[\begin{array}{ccc}{A}^{\mathrm{\top }}P+PA& P{B}_w& C_z(\theta {)}^{\mathrm{\top }}\\ B_w^{\mathrm{\top }}P& -{\gamma }^{2}I& 0\\ C_z(\theta )& 0& -I\end{array}\right]⪯0,\mathrm{T}\mathrm{r}\mathrm{(}{B}_{\mathrm{w}}^{\mathrm{\top }}\mathrm{P}{\mathrm{B}}_{\mathrm{w}}\mathrm{)}\mathrm{\le }\mathrm{\lambda }\mathrm{.}\end{array}$

Implementation

https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/02f31a2d7a22b2464dfe9212eb76409bda9439b1

Conclusion

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

$\begin{array}{c}{I}_{\gamma }(H_{\theta })\triangleq \{\begin{array}{c}\frac{-{\gamma }^2}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\log \det (I-{\gamma }^2{H}_{\theta }(i\omega )H_{\theta }(i\omega {)}^{*})d\omega ,\text{if}\Vert H_{\theta }{\Vert }_{\mathrm{\infty }}<\gamma \\ \mathrm{\infty },\text{otherwise.}\end{array}\end{array}$

Remark

When it is finite, $I_{\gamma }(H_{\theta })$ is given by $\mathrm{T}\mathrm{r}\mathrm{(}{B}_{\mathrm{w}}^{\mathrm{\top }}\mathrm{P}{\mathrm{B}}_{\mathrm{w}}\mathrm{)}$ where $P$, is asymmetric matrix with the smallest possible maximum singular value among all solutions of a Riccati equation.

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.