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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:

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

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:

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

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

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