LMIs in Control/pages/Hankel Norm 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

$\begin{array}{cc}& \underset{\{Q⪰0,{\gamma }^2,\theta \}}{\min }{\gamma }^2\\ & s.t.D_{zw}(\theta )=0,A^{\mathrm{\top }}Q+QA+C_z(\theta )C_z(\theta )⪯0,{\gamma }^{2}I-W_c^{1/2}Q{W}_c^{1/2}⪰0,\end{array}$

where $W_c$ is the controllability Gramian, i.e., $W_c\triangleq \underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{At}{B}_w{B}_w^{\mathrm{\top }}{e}^{A^{\mathrm{\top }}t}dt$.

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.