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

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

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## The System

In this LMI, we have an uncertain second-order linear system, that can be modeled in state space as:

{\begin{aligned}(A_{2}+\Delta A_{2}){\ddot {x}}+(A_{1}+\Delta A_{1}){\dot {x}}+(A_{0}+\Delta A_{0})x&=Bu)\\y_{d}&=C_{d}{\dot {x}}\\y_{p}&=C_{p}x\end{aligned}} where $x\in R^{n}$ and $u\in R^{r}$ are the state vector and the control vector, respectively, $y_{d}\in R^{m_{p}}$ and $y_{d}\in R^{m_{p}}$ are the derivative output vector and the proportional output vector, respectively, and $A_{2},A_{1},A_{0},B,C_{d},C_{p}$ are the system coefficient matrices of appropriate dimensions.

$\Delta A_{2},\Delta A_{1},$ and $\Delta A_{0}$ are the perturbations of matrices $A_{2},A_{1},$ and $A_{0}$ , respectively, are bounded, and satisfy

{\begin{aligned}|\Delta A_{2}|_{2}\leq \epsilon _{2},|\Delta A_{1}|_{2}\leq \epsilon _{1},|\Delta A_{0}|_{2}\leq \epsilon _{0},\end{aligned}} or

{\begin{aligned}max\{\|\Delta a_{2ij}\|\}\leq \eta _{2},max\{\|\Delta a_{1ij}\|\}\leq \eta _{1},max\{\|\Delta a_{0ij}\|\}\leq \eta _{0},\end{aligned}} where $\epsilon _{2},\epsilon _{1},\epsilon _{0}$ and $\eta _{2},\eta _{1},\eta _{0}$ are two sets of given positive scalars, $\Delta a_{2ij},\Delta a_{1ij},$ and $\Delta a_{0ij}$ are the i-th row and j-th collumn elements of matrices $\Delta A_{2},\Delta A_{1},$ and $\Delta A_{0},$ , respectively. Also, the perturbation notations also satisfy the assumption that $\Delta A_{2},\Delta A_{0}\in S^{n}$ and $A_{2}+\Delta A_{2}>0$ .

## The Data

The matrices $A_{2},A_{1},A_{0},B,C_{d},C_{p}$ and perturbations $\Delta A_{2},\Delta A_{1},\Delta A_{0},$ 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 $K_{p}$ and $K_{d}$ in the below system.

{\begin{aligned}(A_{2}+\Delta A_{2}){\ddot {x}}+(A_{1}-BK_{p}C_{p}+\Delta A_{1}){\dot {x}}+(A_{0}-BK_{d}C_{d}+\Delta A_{0})x&=0\\\end{aligned}} 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 $A_{2}>0,A_{1}+A_{1}^{T}>0,andA_{0}>0$ , then the following is also true for the system described above:

The system is hurwitz stable if $\lambda _{min}(A_{2})>|\Delta A_{2}|_{2},\lambda _{min}(A_{1}+A_{1}^{T})>|\Delta A_{1}|_{2},\lambda _{min}(A_{0})>|\Delta A_{0}|_{2}$ , or

the system is hurwitz stable if $\lambda _{min}(A_{2})>{\sqrt {l_{2}}}max\{\|\Delta a_{2ij}\|\},\lambda _{min}(A_{1}+A_{1}^{T})>{\sqrt {l_{1}}}max\{\|\Delta a_{1ij}\|\},\lambda _{min}(A_{0})>{\sqrt {l_{0}}}max\{\|\Delta a_{0ij}\|\}$ , where $l_{2},l_{1},l_{0}$ are the numbers of nonzero elements in matrices $\Delta A_{2},\Delta A_{1},\Delta A_{0},$ respectively.

## The LMI: Robust Stabilization of Second Order Systems

This problem is solved by finding matrices $K_{p}\in R^{r*m_{p}}$ and $K_{d}\in R^{r*m_{d}}$ that satisfy either of the following sets of LMIs.

{\begin{aligned}A_{0}-BK_{d}C_{d}&>\epsilon _{0}I,\\(A_{1}-BK_{p}C_{p})+(A_{1}-BK_{p}C_{p})^{T}&>\epsilon _{1}I.\end{aligned}} or

{\begin{aligned}A_{0}-BK_{d}C_{d}&>\eta _{0}{\sqrt {l_{0}}}I,\\(A_{1}-BK_{p}C_{p})+(A_{1}-BK_{p}C_{p})^{T}&>\eta _{1}{\sqrt {l_{1}}}I.\end{aligned}} ## Conclusion:

Having solved the above problem, the matrices $K_{p}$ and $K_{d}$ can be substituted into the input as $u=K_{p}C_{p}{\dot {x}}+K_{d}C_{d}x$ to stabilize the second order uncertain system.

## Implementation

A link to CodeOcean or other online implementation of the LMI