LMIs in Control/Click here to continue/Time-Delay Systems/Stability Condition for Discrete-Time TDS

This page describes an LMI for stability analysis of a discrete-time system with a time-varying delay. In particular, a delay-dependent condition is provided to test asymptotic stability of a discrete-delay system through feasibility of an LMI. The system under consideration pertains a single discrete delay, with the extent of the delay at any time bounded by some known value. Solving the LMI for different values of this bound, a limit on the delay can be attained for which the system remains asymptotically stable. The System The system under consideration is one of the form: In this description, and are matrices in . The variable denotes a delay in the state at discrete time , assuming a value no greater than some . The Data To determine stability of the system, the following parameters must be known: The Optimization Problem Based on the provided data, asymptotic stability can be determined by testing feasibility of the following LMI: The LMI: Asymptotic Stability for Discrete-Time TDS In this notation, the symbols are used to indicate appropriate matrices to assure the overall matrix is symmetric. Conclusion: If the presented LMI is feasible, the system will be asymptotically stable for any sequence of delays within the interval . That is, independent of the values of the delays at any time: For any real number , there exists a real number such that: Obtaining a feasible point for the LMI, this result can be proven using a Lyapunov-Krasovkii functional: where: Implementation An example of the implementation of this LMI in Matlab is provided on the following site: https://github.com/djagt/LMI_Codes/blob/main/D_TDS_DC.m Note that this implementation requires packages for YALMIP with solver mosek, though a different solver can also be implemented. Related LMIs TDSDC – Delay-dependent stability LMI for continuous-time TDS LMI for Stability of Retarded Differential Equation with Slowly-Varying Delay – Delay-independent stability LMI for continuous-time TDS Discrete Time Lyapunov Stability – Stability LMI for non-delayed discrete-time system External Links The presented results have been obtained from: Fridman E. 2014. Introduction to Time-Delay Systems, Analysis and Control. Springer. ISBN: 978-3-319-09392-5. Additional information on LMI's in control theory can be obtained from the following resources: 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. LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd. Return to Main Page: