# LMIs in Control/pages/LQ Regulation via H2 Control

**LQ Regulation via H2 Control**

Suppose people were interested in quadratic optimal regulation problem, where instead of the Ricatti equation approach (which is traditionally used in such situations), it is approached using LMI's instead (thereby making it a Linear Quadratic (LQ) problem). Such an approach would be possible by converting the LQ problem into a standard problem.

**The System**[edit]

Consider, for example, constant linear multi-variable system in the form of:

where

Then the LQ optimal regulation problem for the given system is stated as described below.

**The Data**[edit]

In order to obtain the LMI, we need the following 3 matrices: , , , and (the latter two of which are obtained as follows).

**The Optimization Problem**[edit]

Using the multi-variable system as described above, we can see that the optimal state feedback controller is obtained where

is minimized where and . However, it is to be noted that in order for the problem to have a solution, two assumptions are made - both of which must be held true at all times:

- .

Relating this to performance, let us now consider the auxiliary system:

- ,

where represents an impulse disturbance, and

Using the state feedback controller and applying it on the above auxiliary system results in the closed-loop system:

- ,

and the resulting transfer function from disturbance to output being:

thereby resulting in .

**The LMI:** LQ Regulation via H2 Control[edit]

From the given pieces of information, and letting the 2 assumptions as described above hold, then there exist matrices , and satisfying:

**Conclusion:**[edit]

From the LMI, it can be seen that the state feedback control in the form of (where ) exists such that if and only if the matrices are of the appropriate matrix sizes.

**Implementation**[edit]

- Example Code - A GitHub link that contains code (titled "LQRegH2.m") that demonstrates how this LMI can be implemented using MATLAB-YALMIP.

**Related LMIs**[edit]

- LMI for System H2 Norm - LMI to determine the -norm of a system.

## External Links[edit]

A list of references documenting and validating the LMI.

- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMIs in Control Systems: Analysis, Design and Applications - A book co-authored by Guang-Ren Duan and Hai-Hua Yu.