# LMIs in Control/Controller Synthesis/Continuous Time/LQ Regulation via H2 control

**LQ Regulation via Control**[edit | edit source]

The LQR design problem is to build an optimal state feedback controller for the system such that the following quadratic performance index.

is minimized, where

The following assumptions should hold for a traditional solution.

is stabilizable.

is observable, with .

**Relation to performance**[edit | edit source]

For the system given above an auxiliary system is constructed

where

Where represents an impulse disturbance. Then with state feedback controller the closed loop transfer function from disturbance to output is

Then the LQ problem and the norm of are related as

Then norm minimization leads minimization of .

**Data**[edit | edit source]

The state-representation of the system is given and matrices are chosen for the optimal LQ problem.

**The Problem Formulation:**[edit | edit source]

Let assumptions and hold, then the state feedback control of the form exists such that if and only if there exist , and . Then can be obtained by the following LMI.

**The LMI:**[edit | edit source]

**Conclusion:**[edit | edit source]

In this case, a feedback control law is given as .

**External Links **[edit | edit source]

- LMIs in Control Systems Analysis, Design and Applications - Duan and Yu
- A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.