# 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 ${\displaystyle u=Kx}$  can be divided into ${\displaystyle n}$  sub-controllers ${\displaystyle u_{i}=K_{i}x_{i},\quad i=1,2,...,n}$ .

## The Data

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

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

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

## The Optimization Problem

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

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

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

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

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:

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

## Conclusion:

The controller gain matrix is defined as:

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

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

${\displaystyle F=ZY^{-1}}$ .

## Implementation

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