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 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 discrete time  , assuming a value no greater than some  .

The DataEdit

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

 

The Optimization ProblemEdit

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

The LMI: Asymptotic Stability for Discrete-Time TDSEdit

 

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:

 

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

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

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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