LMIs in Control/pages/Delay Independent Time-Delay Stabilization

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Stabilization of Time-Delay Systems - Delay Independent Case

Suppose, for instance, there was a system where a time-delay was introduced. In that instance, stabilization would have to be done in a different manner. The following example demonstrates how one can stabilize such a system independent of the delay.

The System[edit | edit source]

For this particular problem, suppose that we were given the time-delayed system in the form of:


Then the LMI for determining the Time-Delay System for the Delay-Independent case would be obtained as described below.

The Data[edit | edit source]

In order to obtain the LMI, we need the following 3 matrices: and .

The Optimization Problem[edit | edit source]

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

is simultaneously both uniform and asymptotically stable, then the system would be stabilized as follows.

The LMI: The Delay-Independent Stabilization of Time-Delay Systems[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 2 symmetric matrices and that satisfy the following:

Conclusion:[edit | edit source]

Given the resulting feedback gain matrix , it can be observed that the matrix is asymptotically stable while simultaneously ensuring that the solution is delay-independent from the time-delay system where this gain matrix was derived.

Implementation[edit | edit source]

  • Example Code - A GitHub link that contains code (titled "DelayIndependentTimeDelay.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]