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.
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
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 matrices and perturbations describing the second order system with perturbations.
The Optimization Problem
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
the system is hurwitz stable if
, where are the numbers of nonzero elements in matrices respectively.
The LMI: Robust Stabilization of Second Order Systems
This problem is solved by finding matrices and that satisfy either of the following sets of LMIs.
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.
This implementation requires Yalmip and Sedumi.
Stabilization of Second-Order Systems
This LMI comes from
-  - "LMIs in Control Systems: Analysis, Design and Applications" by Guang-Ren Duan and Hai-Hua Yu
Duan, G. (2013). LMIs in control systems: analysis, design and applications. Boca Raton: CRC Press, Taylor & Francis Group.
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