LMIs in Control/pages/Robust Stabilization of Second-Order Systems
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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 .
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 , or
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 stabilize the second order uncertain system.
A link to CodeOcean or other online implementation of the LMI
Links to other closely-related LMIs
A list of references documenting and validating the LMI.
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