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