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 SystemEdit

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 DataEdit

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

The Optimization ProblemEdit

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 ControlEdit

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.

ImplementationEdit

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

Related LMIsEdit

External LinksEdit

A list of references documenting and validating the LMI.

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