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

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.

**Conclusion:**[edit]

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**[edit]

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

- 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[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:

- 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.