LMIs in Control/Discrete Time/Stability Condition for Discrete-Time TDS

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LMIs in Control/Discrete Time/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[edit]

The system under consideration is one of the form:

In this description, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle A_1 } are matrices in . The variable denotes a delay in the state at discrete time , assuming a value no greater than some .

The Data[edit]

To determine stability of the system, the following parameters must be known:

The Optimization Problem[edit]

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[edit]

In this notation, the symbols are used to indicate appropriate matrices to assure the overall matrix is symmetric.


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:



An example of the implementation of this LMI in Matlab is provided on the following site:

Note that this implementation requires packages for YALMIP with solver mosek, though a different solver can also be implemented.

Related LMIs[edit]

  • TDSDC – Delay-dependent stability LMI for continuous-time TDS

External Links[edit]

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:

Return to Main Page:[edit]