LMIs in Control/pages/stabilization of second order systems

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LMIs in Control/pages/stabilization of second order systems


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


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.


Return to Main Page:[edit]