LMIs in Control/pages/TDSDC
The System
[edit | edit source]The problem is to check the stability of the following linear time-delay system on a delay dependent condition
where
is the initial condition
represents the time-delay
is a known upper-bound of
For the purpose of the delay dependent system we rewrite the system as
The Data
[edit | edit source]The matrices are known
The LMI: The Time-Delay systems (Delay Dependent Condition)
[edit | edit source]From the given pieces of information, it is clear that the optimization problem only has a solution if there exists a symmetric positive definite matrix
and a scalar such that
Here
This LMI has been derived from the Lyapunov function for the system. It follows that the system is asymptotically stable if
This is obtained by replacing with
Conclusion:
[edit | edit source]We can now implement these LMIs to do stability analysis for a Time delay system on the delay dependent condition
Implementation
[edit | edit source]The implementation of the above LMI can be seen here
https://github.com/yashgvd/LMI_wikibooks
Related LMIs
[edit | edit source]Time Delay systems (Delay Independent Condition)
External Links
[edit | edit source]- [1] - LMI in Control Systems Analysis, Design and Applications
- 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.
- D. d. S. Madeira and J. Adamy, "Static output feedback: An LMI condition for stabilizability based on passivity indices," 2016 IEEE Conference on Control Applications (CCA), Buenos Aires, 2016, pp. 960-965.