LMIs in Control/Controller Synthesis/Discrete Time/State Feedback/Closed-Loop Robust Stability and Controller synthesis of Discrete-Time System with Polytopic Uncertainty

Closed-Loop Robust Stability and Controller synthesis of Discrete-Time System with Polytopic Uncertainty
Stability is an important property, stability analysis is necessary for control theory. For robust control, an LMI criterion was developed for the discrete case with closed-loop and open-loop system. The open-loop system is obtained by replacing $$ B = 0 $$..

The System


\begin{align} x_{k+1}&=A_d(\alpha) x_k + B_d(\beta)u_k\\ u_k &= K x_k \end{align}$$

The Data
The matrices $$ A \in R^{n\times n}\; B\in R^{n\times m} \; K\in R^{m\times n}$$.

In the case of robust polytopic uncertainty control we have:

\begin{align}

A &\in \{\sum_{k = 1}^{n} \alpha_k A_k |\text{ } \;\sum_{k = 1}^{n} \alpha_k = 1\}\\ B &\in \{\sum_{k = 1}^{m} \beta_k B_k  |\text{ } \;\sum_{k = 1}^{m} \beta_j= 1\}

\end{align}$$

The Optimization Problem
The following feasibility problem should be solved:

\begin{align} \text{Find} \; &P_{ij}>0, G\text{ is invertible}, L:\\ \begin{bmatrix} P_{ij} & A_iG - B_jL \\ G^T A_i^T - L^T B_j^T &  G + G^T + P_{ij}\end{bmatrix}&>0 & \text{  for   }i = 1,...,n \text{  and   }j = 1,...,m  \\ \end{align}$$

Where $$ L \in R^{m, n}$$, $$ P_{ij}, G \in R^{n,n} = 0 $$ for $$ i = 1,...,n $$ and $$ j = 1,...,m $$ Then our controller $$ K = - LG^{-1}$$

Conclusion:
This LMI allows us to investigate stability for the robust control problem in the case of polytopic uncertainty and gives on the controller for this case

Implementation:

 * - Matlab implementation using the YALMIP framework and SeDuMi solver

Related LMIs:

 * - Discrete Time Stabilizability.
 * - Polytopic stability for continuous time case
 * - Quadratic polytopic stabilization
 * -Quadratic Polytopic Hinf- Optimal State Feedback Control

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