LMIs in Control/pages/stabilization of second order systems

LMIs in Control/pages/stabilization of second order systems

Stabilization is a vastly important concept in controls, and is no less important for second order systems. A second order system can be conceptualized most simply by the model of a mass-spring-damper. Velocity and position are of course chosen as the states for this system, and the state space model can be written as it is below. The goal of stabilization in this context is to design a control law that is made up of two controller gain matrices , . These allow the construction of a stabilized closed loop controller.

The SystemEdit

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  .

To further define:   is  and is the state vector,   is   and is the state matrix on   ,   is   and is the state matrix on   ,   is   and is the state matrix on  ,   is   and is the input matrix,   is   and is the input,   and   are   and are the output matrices,   is   and is the output from  , and   is   and is the output from  .


 

The DataEdit

The matrices  .

The Optimization ProblemEdit

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 SystemsEdit

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.

ImplementationEdit

This implementation requires Yalmip and Sedumi. https://github.com/rezajamesahmed/LMImatlabcode/blob/master/stab2ndorder.m

Related LMIsEdit

Robust Stabilization of Second-Order Systems

External LinksEdit

This LMI comes from

  • [1] - "LMIs in Control Systems: Analysis, Design and Applications" by Guang-Ren Duan and Hai-Hua Yu

Other resources:

ReferencesEdit

Duan, G. (2013). LMIs in control systems: analysis, design and applications. Boca Raton: CRC Press, Taylor & Francis Group.

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