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

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 ${\displaystyle K_{p}}$, and ${\displaystyle K_{d}}$. These allow the construction of a stabilized closed loop controller.

## 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 ${\displaystyle A_{2}+\Delta A_{2}>0}$.

To further define: ${\displaystyle x}$ is${\displaystyle \in R^{n}}$ and is the state vector, ${\displaystyle A_{0}}$ is ${\displaystyle \in R^{n*n}}$ and is the state matrix on ${\displaystyle x}$ , ${\displaystyle A_{1}}$ is ${\displaystyle \in R^{n*n}}$ and is the state matrix on ${\displaystyle {\dot {x}}}$ , ${\displaystyle A_{2}}$ is ${\displaystyle \in R^{n*n}}$ and is the state matrix on ${\displaystyle {\ddot {x}}}$, ${\displaystyle B}$ is ${\displaystyle \in R^{n*r}}$ and is the input matrix, ${\displaystyle u}$ is ${\displaystyle \in R^{r}}$ and is the input, ${\displaystyle C_{d}}$ and ${\displaystyle C_{p}}$ are ${\displaystyle \in R^{m*n}}$ and are the output matrices, ${\displaystyle y_{d}}$ is ${\displaystyle \in R^{m}}$ and is the output from ${\displaystyle C_{d}}$, and ${\displaystyle y_{p}}$ is ${\displaystyle \in R^{m}}$ and is the output from ${\displaystyle C_{p}}$.

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

The system is hurwitz stable if

{\displaystyle {\begin{aligned}\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}\end{aligned}}},

or

the system is hurwitz stable if

{\displaystyle {\begin{aligned}\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}\|\}\end{aligned}}}

, 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 robustly stabilize the second order uncertain system. The new system is stable in closed loop.

## Implementation

This implementation requires Yalmip and Sedumi.