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 SystemEdit

In a decentralized controller design, the state feedback controller   can be divided into   sub-controllers  .

The DataEdit

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

 

where   is a   vector of state variables,   is the input matrix,   is the output matrix, and   is called the feedforward matrix. We assume that all the four matrices,  ,  ,  , and   are given.

The Optimization ProblemEdit

We aim to solve the  -optimal full-state feedback control problem using a controller  .

In a decentralized fashion, the control input   can be divided into sub-controllers   and can be written as  .

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

 

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

The LMI: LMI for decentralized feedback controllerEdit

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

 

where   is a positive definite matrix and   such that the aforemtntioned constraints in LMIs are satisfied.

Conclusion:Edit

The controller gain matrix is defined as:

 

where   can be found after solving the LMIs and obtaining the variables matrices   and  . Thus,

 .

ImplementationEdit

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

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

Related LMIsEdit

External LinksEdit

A list of references documenting and validating the LMI.

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

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