# LMIs in Control/pages/stabilization of second order systems

LMIs in Control/pages/stabilization of second order systems

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## The System

Here, we want to stabilize a second order system of the following form:

{\displaystyle {\begin{aligned}M{\ddot {x}}+D{\dot {x}}+Kx&=Bu,\end{aligned}}}

where ${\displaystyle x\in R^{n}}$ and ${\displaystyle u\in R^{r}}$ are the state vector and the control vector, respectively, and M (called the "mass matrix"), D (called the "structural damping matrix"), K (called the "stiffness matrix"), and B are the system coefficient matrices of appropriate dimensions.

To make the system follow standard convention, we reformulate the system as:

{\displaystyle {\begin{aligned}A_{2}{\ddot {x}}+A_{1}{\dot {x}}+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; ${\displaystyle y_{d}\in R^{m_{d}}}$ and ${\displaystyle y_{p}\in R^{m_{p}}}$ are the derivative output vector and the proportional output vector, respectively; and ${\displaystyle A_{2},A_{1},A_{0},B,C_{d},}$ and ${\displaystyle C_{p}}$ are the system coefficient matrices of appropriate dimension. Note that ${\displaystyle A_{2}}$ must be ${\displaystyle >0}$, and ${\displaystyle A_{0},A_{2}}$ must be ${\displaystyle \in S^{n}}$

## The Data

The matrices ${\displaystyle A_{2},A_{1},A_{0},B,C_{d},C_{p}}$.

## The Optimization Problem

For the system described, we choose the following control law

{\displaystyle {\begin{aligned}u&=K_{p}y_{p}+K_{d}y_{d}\\&=K_{p}C_{p}{\hat {x}}+K_{d}C_{d}x,\end{aligned}}}

with ${\displaystyle K_{p}\in R^{r*m_{p}}}$, we obtain the closed-loop system as follows:

{\displaystyle {\begin{aligned}A_{2}{\ddot {x}}+(A_{1}-BK_{p}C_{p}){\dot {x}}+(A_{0}-BK_{d}C_{d})x&=0.\\\end{aligned}}}

We are tasked to design a state feedback control law such that the above system is hurwitz stable.

First, in order to solve this problem, we need to introduce 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:

{\displaystyle {\begin{aligned}A_{2}>0,A_{1}+A_{1}^{T}>0,A_{0}>0\\\end{aligned}}}

## The LMI: Stabilization of Second Order Systems

There is a solution if there exists matrices ${\displaystyle K_{p}\in R^{r*m_{p}}}$ and ${\displaystyle K_{d}\in R^{r*m_{d}}}$ that satisfy the following LMIs:

{\displaystyle {\begin{aligned}A_{0}-BK_{d}C_{d}>0,\end{aligned}}}

and

{\displaystyle {\begin{aligned}(A_{1}-BK_{p}C_{p})+(A_{1}-BK_{p}C_{p})^{T}>0.\end{aligned}}}

## Conclusion:

Finally, having solved the LMI the optimization will produce two matrices, ${\displaystyle K_{p}}$ and ${\displaystyle K_{d}}$ that can be substituted into the system as

{\displaystyle {\begin{aligned}u&=K_{p}C_{p}{\dot {x}}+K_{d}C_{d}x\end{aligned}}}

to obtain a stabilized second order system.

## Implementation

A link to CodeOcean or other online implementation of the LMI