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

**Optimization over an Affine Family of Linear Systems**[edit | edit source]

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 and norms, entropy, dissipativity, and the Hankel norm of an affinely parametric system.

**The System**[edit | edit source]

Consider a family of linear systems

with state space realization where and depend affinely on the parameter .

We assume is stable and is controllable.

The transfer function, depends affinely on .

**The Data**[edit | edit source]

The transfer function , and system matrices , , , are known. represents the convex functionals, and represent some auxiliary variables dependent on the problem being solved.

**The LMI:**Generalized Optimization for Affine Linear Systems[edit | edit source]

Several control theory problems, mentioned earlier, take the following form:

minimize

subject to

Problems of this nature can be formulated as an LMI by representing as an LMI in and possibly such that

Thus, the general optimization problem to be applied to an affine family of linear systems is as follows:

minimize

subject to

**Conclusion:**[edit | edit source]

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**[edit | edit source]

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

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.

**Related LMIs**[edit | edit source]

Norm for Affine Parametric Systems

Entropy Bond for Affine Parametric Systems

Dissipativity for Affine Parametric Systems

**External Links**[edit | edit source]

- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013.

**Return to Main Page:**[edit | edit source]

LMIs in Control: https://en.wikibooks.org/wiki/LMIs_in_Control