LMIs in Control/pages/Hankel 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 affinely 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

${\begin{aligned}&\min _{\{Q\succeq 0,\gamma ^{2},\theta \}}\gamma ^{2}\\&\quad s.t.\quad D_{zw}(\theta )=0,\quad A^{\top }Q+QA+C_{z}(\theta )C_{z}(\theta )\preceq 0,\quad \gamma ^{2}I-W_{c}^{1/2}QW_{c}^{1/2}\succeq 0,\end{aligned}}$

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

Implementation

https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/0faedcdd9fba92bc27a318d80159c04a0b342f35

Conclusion

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

Remark

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

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.