LMIs in Control/pages/Full-State Feedback Optimal Control Hinf LMI

From Wikibooks, open books for an open world
Jump to navigation Jump to search

Full State Feedback Optimal Control[edit | edit source]

Full State Feedback is a control technique that attempts to place the system's closed loop system poles in specified locations based off of performance specifications given. methods formulate this task as an optimization problem and attempt to minimize the norm of the system. In a single-input single-output (SISO) system this norm represents the maximum gain on a magnitude Bode plot. In the case of multi-input multi-output (MIMO) systems it can be interpreted as maximum response to a perturbation introduced to the system. In either, by minimizing the we are minimizing the worst case effect of a disturbance to the system, whether it is noise or another perturbation.

The System[edit | edit source]

The system is represented using the 9-matrix notation shown below.

where is the state, is the regulated output, is the sensed output, is the exogenous input, and is the actuator input, at any .

The lower linear fractional transformation (LFT) is used to implement a controller into the system. The lower LFT is denoted as and is formed by with . For full-state feedback we consider a controller of the form . This is a special case where and results in a controller of the form .

The Data[edit | edit source]

, , , , , , , , are known.

The LMI:Full State Feedback Optimal Control LMI[edit | edit source]

The following are equivalent.

1) There exists a such that

2) There exists and such that


Then .

Conclusion:[edit | edit source]

The above LMI, if feasible, will determine the bound on the norm of the system. In addition to this is also determined allowing the closed loop system to be determined using the controller found during the optimization.

Implementation[edit | edit source]

This implementation requires Yalmip and Sedumi. https://github.com/eoskowro/LMI/blob/master/FSF_Hinf.m

Related LMIs[edit | edit source]

Full State Feedback Optimal H2 LMI

External Links[edit | edit source]

Return to Main Page:[edit | edit source]