# LMIs in Control/Time-Delay Systems/Continuous Time/LMI for Stability of Retarded Differential Equation with Slowly-Varying Delay

LMIs in Control/Time-Delay Systems/Continuous Time/LMI for Stability of Retarded Differential Equation with Slowly-Varying Delay

This page describes an LMI for stability analysis of a continuous-time system with a time-varying delay. In particular, a delay-independent condition is provided to test uniform asymptotic stability of a retarded differential equation 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. Moreover, the delay is assumed to vary only slowly in time, with a temporal derivative bounded by a value less than one. Solving the LMI for a particular value of this bound, uniform asymptotic stability can be shown for any time-delay satisfying this bound.

**The System**Edit

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 time , assuming a value no greater than some . Moreover, we assume that the function is differentiable at any time, with the derivative bounded by some value , assuring the delay to be slowly-varying in time.

**The Data**Edit

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

**The Optimization Problem**Edit

Based on the provided data, uniform asymptotic stability can be determined by testing feasibility of the following LMI:

**The LMI:** Delay-Independent Uniform Asymptotic Stability for Continuous-Time TDSEdit

**Conclusion:**Edit

If the presented LMI is feasible, the system will be uniformly asymptotically stable for any delay function satisfying . That is, independent of the values of the delays and the starting time :

- For any real number , there exists a real number such that:

- There exists a real number such that for any real number , there exists a time such that:

Here, we let for denote the delayed state function at time . The norm of this function is defined as the maximal value of the vector norm assumed by the state over the delayed time interval, given by:

Obtaining a feasible point for the LMI, this result can be proven using a Lyapunov-Krasovkii functional:

Notably, if matrices prove feasibility of the LMI for the pair , these same matrices will also prove feasibility of the LMI for the pair . As such, feasibility of this LMI proves uniform asymptotic stability of both systems:

Moreover, since the result is independent of the value of the delay, it will also hold for a delay . Hence, if the LMI is feasible, the matrices will be Hurwitz.

**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

- [1] - Delay-dependent stability LMI for continuous-time TDS

- [2] - Stability LMI for delayed discrete-time system

## External LinksEdit

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.