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 SystemEdit

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 DataEdit

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

 

The Optimization ProblemEdit

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.

ImplementationEdit

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 LMIsEdit

  • [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:

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