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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:

https://github.com/asalimil/LMI_for_decentralized_feedback_controller/tree/master

External Links

A list of references documenting and validating the LMI.

• [1] - LMI in Control Systems Analysis, Design and Applications

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