# 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:

{\displaystyle {\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 ${\displaystyle x\in R^{n}}$ and ${\displaystyle u\in R^{r}}$ are the state vector and the control vector, respectively, ${\displaystyle y_{d}\in R^{m_{p}}}$ and ${\displaystyle y_{d}\in R^{m_{p}}}$ are the derivative output vector and the proportional output vector, respectively, and ${\displaystyle A_{2},A_{1},A_{0},B,C_{d},C_{p}}$ are the system coefficient matrices of appropriate dimensions.

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

{\displaystyle {\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

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

## The Data

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

{\displaystyle {\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 ${\displaystyle A_{2}>0,A_{1}+A_{1}^{T}>0,andA_{0}>0}$, then the following is also true for the system described above:

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

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

## The LMI: Robust Stabilization of Second Order Systems

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

{\displaystyle {\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

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

## Implementation

A link to CodeOcean or other online implementation of the LMI