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 $$ K_{p}$$, and $$ K_{d}$$. 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:



\begin{align} (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{align}$$

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{align} \end{align}$$
 * \Delta A_{2}|_{2} \leq \epsilon_{2}, |\Delta A_{1}|_{2} \leq \epsilon_{1}, |\Delta A_{0}|_{2} \leq \epsilon_{0},

or



\begin{align} max\{\|\Delta a_{2ij}\|\} \leq \eta_{2}, max\{\|\Delta a_{1ij}\|\} \leq \eta_{1}, max\{\|\Delta a_{0ij}\|\} \leq \eta_{0}, \end{align}$$

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

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

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{align} (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{align}$$

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

The system is hurwitz stable if

\begin{align} \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{align}$$,

or

the system is hurwitz stable if



\begin{align}\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{align}$$

, 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{align} 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{align}$$

or



\begin{align} 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{align}$$

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

Implementation
This implementation requires Yalmip and Sedumi.

https://github.com/rezajamesahmed/LMImatlabcode/blob/master/ROBstab2ndorder.m

Related LMIs
Stabilization of Second-Order Systems