# LMIs in Control/Stability Analysis/Continuous Time/Optimization Over Affine Family of Linear Systems

## Optimization over an Affine Family of Linear Systems

Presented in this page is a general framework for optimizing various convex functionals for a system which depends affinely, or linearly, on a parameter using linear matrix inequalities. The optimization problem presented on this page generalizes an LMI which can be applied to various problems within linear systems and control. Some examples of these applications are finding the ${\displaystyle H_{2}}$ and ${\displaystyle H_{\infty }}$ norms, entropy, dissipativity, and the Hankel norm of an affinely parametric system.

## The System

Consider a family of linear systems
${\displaystyle {\dot {x}}=Ax+B_{w}w,}$
${\displaystyle z=C_{z}({\theta })x+D_{zw}({\theta })w}$
with state space realization ${\displaystyle (A,B_{w},C_{z},D_{zw})}$ where ${\displaystyle C_{z}}$ and ${\displaystyle D_{zw}}$ depend affinely on the parameter ${\displaystyle {\theta }\in \mathbb {R} ^{p}}$.

We assume ${\displaystyle A}$ is stable and ${\displaystyle (A,B_{w})}$ is controllable.

The transfer function, ${\displaystyle H_{\theta }(s){\widehat {=}}C_{z}({\theta })(sI-A)^{-1}B_{w}+D_{zw}({\theta })}$ depends affinely on ${\displaystyle {\theta }}$.

## The Data

The transfer function ${\displaystyle H}$, and system matrices ${\displaystyle A}$, ${\displaystyle B_{w}}$, ${\displaystyle C_{z}}$, ${\displaystyle D_{zw}}$ are known. ${\displaystyle {\varphi }}$ represents the convex functionals, ${\displaystyle {\alpha }}$ and ${\displaystyle {\psi }}$ represent some auxiliary variables dependent on the problem being solved.

## The LMI:Generalized Optimization for Affine Linear Systems

Several control theory problems, mentioned earlier, take the following form:
minimize ${\displaystyle {\psi }_{0}(H_{\theta })}$
subject to ${\displaystyle {\psi }_{i}(H_{\theta })<{\alpha }_{i},i=1,...,p}$

Problems of this nature can be formulated as an LMI by representing ${\displaystyle {\psi }_{i}(H_{\theta })<{\alpha }_{i}}$ as an LMI in ${\displaystyle {\theta },{\alpha }_{i},}$ and possibly ${\displaystyle {\psi }_{i}}$ such that ${\displaystyle F_{i}({\theta },{\alpha }_{i},{\psi }_{i})>0}$

Thus, the general optimization problem to be applied to an affine family of linear systems is as follows:
minimize ${\displaystyle {\alpha }_{0}}$
subject to ${\displaystyle F_{i}({\theta },{\alpha }_{i},{\psi }_{i})>0,i=0,1,....,P}$

## Conclusion:

The LMI for this generalized optimization problem may be extended to various convex functionals for affine parametric systems. For extensions of this LMI, see the related LMIs section.

## Implementation

Implementation of LMI's of this form require Yalmip and a linear solver such as Sedumi or SDPT3.

${\displaystyle H_{\infty }}$ Norm for Affine Parametric Systems - MATLAB code for an extension of this generalized LMI.

Entropy Bond for Affine Parametric Systems - MATLAB code for an extension of this generalized LMI.

LMI can be applied to other extensions in stability and controller analysis. Please see the related LMI pages in the section below.