LMIs in Control/pages/D stabilization

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There are a wide variety of control design problems that are addressed in a wide variety of different ways. One of the most important control design problem is that of state feedback stabilization. One such state feedback problem, which will be the main focus of this article, is that of -Stabilization, a form of -Stabilization where the closed-loop poles are located on the left-half of the complex plane.

The System[edit | edit source]

For this particular problem, suppose that we were given a linear system in the form of:

where , , and represents either the differential operator (in the continuous-time case) or the one-step forward operator (for the discrete-time system case). Then the LMI for determining the -stabilization case would be obtained as described below.

The Data[edit | edit source]

In order to obtain the LMI, we need the following 2 matrices: .

The Optimization Problem[edit | edit source]

Suppose - for the linear system given above - we were asked to design a state-feedback control law where such that the closed-loop system:

is -stable, then the system would be stabilized as follows.

The LMI: -Stabilization[edit | edit source]

From the given pieces of information, it is clear that the optimization problem only has a solution if there exists a matrix and a symmetric matrix that satisfies the following:

Conclusion:[edit | edit source]

Given the resulting controller matrix , it can be observed that the matrix is -stable.

Implementation[edit | edit source]

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

Related LMIs[edit | edit source]

External Links[edit | edit source]

A list of references documenting and validating the LMI.

Return to Main Page:[edit | edit source]