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 $u_i=K_i{x}_i,i=1,2,...,n$.

The Data

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

$\begin{array}{cc}& \dot{x}=Ax+Bu\\ & y=Cx+Du\end{array}$

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 $H_{\mathrm{\infty }}$-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 $u=[u_1{u}_2...u_j{]}_{1\times n}^{\text{T}}$.

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:

$K=\left[\begin{array}{cccccc}{k}_1& k_2& 0& 0& 0& 0\\ 0& 0& k_3& k_4& k_5& k_6\\ 0& 0& k_7& k_8& k_9& k_{10}\end{array}\right]$

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:

$\begin{array}{cc}& \text{min}\gamma \\ & \left[\begin{array}{ccc}Y{A}^{\text{T}}+AY+Z^{\text{T}}{B}_2^{\text{T}}+B_{2}Z& {*}^T& {*}^T\\ B_1^T& -\gamma I& {*}^T\\ Y{C}_1^T+Z^T{D}_{12}& D_{11}& -\gamma I\end{array}\right]\end{array}$

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:

$K=\left[\begin{array}{cc}0& 0\\ 0& F\end{array}\right]$

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

$F=Z{Y}^{-1}$.

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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