# LMIs in Control/pages/stabilization of second order systems

LMIs in Control/pages/stabilization of second order systems

This is a WIP based on the **Template for Creating An LMI Page**

## Contents

**The System**[edit]

Here, we want to stabilize a second order system of the following form:

where and are the state vector and the control vector, respectively, and M (called the "mass matrix"), D (called the "structural damping matrix"), K (called the "stiffness matrix"), and B are the system coefficient matrices of appropriate dimensions.

To make the system follow standard convention, we reformulate the system as:

where: and are the state vector and the control vector, respectively; and are the derivative output vector and the proportional output vector, respectively; and and are the system coefficient matrices of appropriate dimension. Note that must be , and must be

**The Data**[edit]

The matrices .

**The Optimization Problem**[edit]

For the system described, we choose the following control law

with , we obtain the closed-loop system as follows:

We are tasked to design a state feedback control law such that the above system is hurwitz stable.

First, in order to solve this problem, we need to introduce a Lemma. This Lemma comes from Appendix A.6 in "LMI's in Control systems" by Guang-Ren Duan and Hai-Hua Yu. This Lemma states the following:

**The LMI:** Stabilization of Second Order Systems[edit]

There is a solution if there exists matrices and that satisfy the following LMIs:

and

**Conclusion:**[edit]

Finally, having solved the LMI the optimization will produce two matrices, and that can be substituted into the system as

to obtain a stabilized second order system.

**Implementation**[edit]

A link to CodeOcean or other online implementation of the LMI

**Related LMIs**[edit]

Links to other closely-related LMIs

## External Links[edit]

A list of references documenting and validating the LMI.

- 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.