# LMIs in Control/pages/LMI for Decentralized Feedback Control

LMI for Decentralized Feedback Control

In large-scale systems like a multi-agent robotic system, national economies, or chemical refineries, an actuator should act based on local information, which necessitates a decentralized or distributed control strategy. In a decentralized control framework, the controllers are distributed and each controller has only access to a subset of local measurements. We describe LMI formulations for a general decentralized control framework and then provide an illustrative example of a decentralized control design.

## The System

In a decentralized controller design, the state feedback controller $u=Kx$  can be divided into $n$  sub-controllers $u_{i}=K_{i}x_{i},\quad i=1,2,...,n$ .

## The Data

A general state space representation of a linear time-invariant system is as follows:

{\begin{aligned}&{\dot {x}}=Ax+Bu\\&y=Cx+Du\end{aligned}}

where $x$  is a $n\times n$  vector of state variables, $B$  is the input matrix, $C$  is the output matrix, and $D$  is called the feedforward matrix. We assume that all the four matrices, $A$ , $B$ , $C$ , and $D$  are given.

## The Optimization Problem

We aim to solve the $H_{\infty }$ -optimal full-state feedback control problem using a controller $u=Kx$ .

In a decentralized fashion, the control input $u$  can be divided into sub-controllers $u_{1},u_{2},...,u_{j}$  and can be written as $u=[u_{1}\quad u_{2}\quad ...\quad u_{j}]_{1\times n}^{\text{T}}$ .

For instance, let $j=3$  and $n=6$ . Thus, there are three control inputs $u_{1}$ , $u_{2}$ , and $u_{3}$ . We also assume that u_{1} only depends on the first and the second states, while $u_{2}$  and $u_{3}$  only depend on thrid to sixth states. For this example, the controller gain matrix can be described by:

$K={\begin{bmatrix}k_{1}&k_{2}&0&0&0&0\\0&0&k_{3}&k_{4}&k_{5}&k_{6}\\0&0&k_{7}&k_{8}&k_{9}&k_{10}\\\end{bmatrix}}$

Thus, the decentralized controller gain consists of sub-matrices of gains.

## The LMI: LMI for decentralized feedback controller

The mathematical description of the LMI formulation for a decentralised optimal full-state feedback controller can be described by:

{\begin{aligned}&{\text{min}}\quad \gamma \\&{\begin{bmatrix}YA^{\text{T}}+AY+Z^{\text{T}}B_{2}^{\text{T}}+B_{2}Z&*^{T}&*^{T}\\B_{1}^{T}&-\gamma I&*^{T}\\YC_{1}^{T}+Z^{T}D_{12}&D_{11}&-\gamma I\end{bmatrix}}\end{aligned}}

where $Y>0$  is a positive definite matrix and $Z$  such that the aforemtntioned constraints in LMIs are satisfied.

## Conclusion:

The controller gain matrix is defined as:

$K={\begin{bmatrix}0&0\\0&F\end{bmatrix}}$

where $F$  can be found after solving the LMIs and obtaining the variables matrices $Y$  and $Z$ . Thus,

$F=ZY^{-1}$ .

## Implementation

A link to Matlab codes for this problem in the Github repository: