# 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**[edit | edit source]

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

**The Data**[edit | edit source]

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 Problem**[edit | edit source]

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 controller[edit | edit source]

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 | edit source]

The controller gain matrix is defined as:

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

.

**Implementation**[edit | edit source]

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

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

**Related LMIs**[edit | edit source]

## External Links[edit | edit source]

A list of references documenting and validating the LMI.

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