LMIs in Control/pages/Continuous Time D-Stability Controller

Continuous-Time D-Stability Controller

In some control problems, people are still interested in designing a controller whose poles are located in specific regions of a complex plane while also simultaneously ensuring that the controller is stable. One such method with which this can be possible is called D-stability.


The SystemEdit

Suppose we were given the continuous-time system

 

whose stability was not known, and where  ,  ,  , and   for any  . Then the controller that simultaneously stabilizes the above system while ensuring that the poles are at their desired location can be achieved by the controller  .

The DataEdit

In order to properly define the acceptable region of the poles in the complex plane, we need the following pieces of data:

  • matrices  ,  ,  ,  
  • rise time ( )
  • settling time ( )
  • percent overshoot ( )

Having these pieces of information will now help us in formulating the optimization problem.

The Optimization ProblemEdit

Using the data given above, we can now define our optimization problem. In order to do that, we have to first define the acceptable region in the complex plane that the poles can lie on using the following inequality constraints:

Rise Time:  

Settling Time:  

Percent Overshoot:  

Assume that   is the complex pole location, then:

 

This then allows us to modify our inequality constraints as:

Rise Time:  

Settling Time:  

Percent Overshoot:  

which not only allows us to map the relationship between complex pole locations and inequality constraints, but it also now allows us to easily formulate our LMIs for this problem.

The LMI: The Continuous-Time Controller D-StabilityEdit

Keeping the above inequalities in mind, we observe the following:

Suppose there now exists a symmetric matrix   and matrix  , we can now determine the controller the following LMIs:

 

Conclusion:Edit

Given the resulting controller  , we can now determine that the pole locations   of   satisfies the inequality constraints  ,   and  .

ImplementationEdit

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

Related LMIsEdit

External LinksEdit

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