LMIs in Control/pages/Robust Stabilization of Second-Order Systems

LMIs in Control/pages/Robust Stabilization of Second-Order Systems

Stabilization is a vastly important concept in controls, and is no less important for second order systems with perturbations. Such a second order system can be conceptualized most simply by the model of a mass-spring-damper, with added perturbations. 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 , and . These allow the construction of a stabilized closed loop controller.

The SystemEdit

In this LMI, we have an uncertain second-order linear system, that can be modeled in state space 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   are the system coefficient matrices of appropriate dimensions.

  and   are the perturbations of matrices   and  , respectively, are bounded, and satisfy

 

or

 

where   and   are two sets of given positive scalars,   and   are the i-th row and j-th collumn elements of matrices   and  , respectively. Also, the perturbation notations also satisfy the assumption that   and  .

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   and perturbations   describing the second order system with perturbations.

The Optimization ProblemEdit

For the system defined as shown above, we need to design a feedback control law such that the following system is Hurwitz stable. In other words, we need to find the matrices   and   in the below system.

 

However, to do proceed with the solution, first we need to present 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: if  , then the following is also true for the system described above:

The system is hurwitz stable if

 ,

or

the system is hurwitz stable if

 

, where   are the numbers of nonzero elements in matrices   respectively.

The LMI: Robust Stabilization of Second Order SystemsEdit

This problem is solved by finding matrices   and   that satisfy either of the following sets of LMIs.

 

or

 

Conclusion:Edit

Having solved the above problem, the matrices   and   can be substituted into the input as   to robustly stabilize the second order uncertain system. The new system is stable in closed loop.

ImplementationEdit

This implementation requires Yalmip and Sedumi.

https://github.com/rezajamesahmed/LMImatlabcode/blob/master/ROBstab2ndorder.m

Related LMIsEdit

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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