View source for LMIs in Control/Click here to continue/Controller synthesis/Quadratic Schur Satbilization

<big><big>'''LMI for Quadratic Schur Stabilization'''</big></big>
 
A discrete-time system is said to be stable if all roots of its characteristic equation lie in the open unit disk. This provides a condition for the stability of discrete-time linear systems with polytopic uncertainties and a linear time-invariant system with this property is called a Schur stable system.


== '''The System''' ==
Consider discrete time system
:<math>
\begin{align}
x_{k+1}=Ax_k+Bu_k,\\
\end{align}</math>
where <math>x_k\in \R^n</math>, <math>u_k\in \R^m</math>, at any <math>t\in \R</math>. <br>
The system consist of uncertainties of the following form
:<math>
\begin{align}
\Delta_{A(t)} = A_1 \delta_1 (t)+....+A_k 
\delta_k (t)\\
\Delta_{B(t)} = B_1 \delta_1 (t)+....+B_k 
\delta_k (t)\\
\end{align}</math>

where <math>x\in\mathbb{R}^{m}</math>,<math>u\in\mathbb{R}^{n}</math>,<math>A\in\mathbb{R}^{mxm}</math> and <math>B\in\mathbb{R}^{mxn}</math>

== '''The Data''' ==

The matrices necessary for this LMI are <math>A</math>,<math>\Delta_{A(t)} \, ie \, A_i </math> ,<math>B</math> and <math>\Delta_{B(t)} \, ie \, B_i </math>

== '''The LMI:'''==
There exists some X > 0 and Z such that <br>
::<math>
\begin{align}
 \begin{bmatrix} X  && AX+BZ
\\ (AX+BZ)^T && X \end{bmatrix} +

\begin{bmatrix} 0 && A_iX + B_iZ
\\(A_iX + B_iZ)^T&&0 \end{bmatrix} > 0 \quad i = 1,......,k

\end{align}</math>

== '''The Optimization Problem''' ==

The optimization problem is to find a matrix <math>\begin{align} K \in \mathbb{R}^{r\times n} \end{align}</math> such that:

<math>\begin{align} ||A + BK||_{2} < \gamma \end{align}</math>

According to the definition of the spectral norms of matrices, this condition becomes equivalent to:

<math>\begin{align} (A + BK)^{T}(A + BK) < \gamma^{2}I \end{align}</math>

Using the Lemma 1.2 in [https://www.crcpress.com/LMIs-in-Control-Systems-Analysis-Design-and-Applications/Duan-Yu/p/book/9781466582996 LMI in Control Systems Analysis, Design and Applications] (page 14), the aforementioned inequality can be converted into: 

<math>\begin{align} \begin{bmatrix} -\gamma I & (A + BK) \\ (A + BK)^{T} & -\gamma I\end{bmatrix} < 0\\
\end{align}</math>

== '''Conclusion:''' ==

The Controller gain matrix is extracted as <math> F = ZX^{-1} </math> <br>
<math> u_k = Fx_k </math>
:<math>
\begin{align}
x_{k+1}=Ax_k+Bu_k,\\
\quad\quad\quad = Ax_k + BFx_k\\
\quad\quad = (A+BF)x_k
\end{align}</math>

It follows that the trajectories of the closed-loop system (A+BK) are stable for any <math> \,  \Delta \, \in \, C_0(\Delta_1,...,\Delta_k) </math>

== '''Implementation''' ==
https://github.com/JalpeshBhadra/LMI/blob/master/quadratic_schur_stabilization.m

== '''Related LMIs''' ==
[https://en.wikibooks.org/wiki/LMIs_in_Control/pages/Schur_Complement Schur Complement]<br>
[https://en.wikibooks.org/w/index.php?title=LMIs_in_Control/pages/SchurStabilization&stable=0#External_Links Schur Stabilization]

== '''External Links ''' ==
* [http://control.asu.edu/MAE598_frame.htm LMI Methods in Optimal and Robust Control] - A course on LMIs in Control by Matthew Peet.
* [https://https://arxiv.org/abs/1903.08599/ LMI Properties and Applications in Systems, Stability, and Control Theory] - A List of LMIs by Ryan Caverly and James Forbes.
* [https://web.stanford.edu/~boyd/lmibook/ LMIs in Systems and Control Theory] - A downloadable book on LMIs by Stephen Boyd.
* [https://www.crcpress.com/LMIs-in-Control-Systems-Analysis-Design-and-Applications/Duan-Yu/p/book/9781466582996 LMI in Control Systems Analysis, Design and Applications] 

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