Lyapunov Methods for Time-Invariant Delay Difference Inclusions.

SIAM Journal on Control and Optimization (Impact Factor: 1.38). 01/2012; 50:110-132. DOI: 10.1137/100807065
Source: DBLP
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: Input-to-state stability (ISS) of interconnected systems with each subsystem described by a difference equation subject to an external disturbance is considered. Furthermore, special attention is given to time delay, which gives rise to two relevant problems: (i) ISS of interconnected systems with interconnection delays, which arise in the paths connecting the subsystems, and (ii) ISS of interconnected systems with local delays, which arise in the dynamics of the subsystems. The fact that a difference equation with delay is equivalent to an interconnected system without delay is the crux of the proposed framework. Based on this fact and small-gain arguments, it is demonstrated that interconnection delays do not affect the stability of an interconnected system if a delay-independent small-gain condition holds. Furthermore, also using small-gain arguments, ISS for interconnected systems with local delays is established via the Razumikhin method as well as the Krasovskii approach. A combination of the results for interconnected systems with interconnection delays and local delays, respectively, provides a framework for ISS analysis of general interconnected systems with delay. Thus, a scalable ISS analysis method is obtained for large-scale interconnections of difference equations with delay.
    Mathematics of Control Signals and Systems 01/2012; 24. · 0.42 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: This paper proposes a novel approach to stability analysis of discrete-time nonlinear periodi-cally time-varying systems. The contributions are as follows. Firstly, a relaxation of standard Lyapunov conditions is derived. This leads to a less conservative Lyapunov function that is required to decrease at every period rather than at each time instant. Secondly, for linear periodic systems with constraints, we show that compared to standard Lyapunov theory, the novel stability concept yields a larger estimate of the region of attraction. An example illustrates the effectiveness of the developed theory.
  • [Show abstract] [Hide abstract]
    ABSTRACT: For the stability analysis of time-delay systems, the Razumikhin approach provides (at the cost of some conservatism) a set of conditions that are relatively easy to verify when compared to the Krasovskii approach. Unfortunately, currently, for linear delay difference inclusions (DDIs) verification of these conditions is only possible by solving a bilinear matrix inequality (BMI). To obtain a tractable stability analysis method for DDIs, an alternative set of Razumikhin-type conditions is proposed in this paper, which are based on a technique that was developed for interconnected systems in Willems (1972). In particular, via the proper selection of storage and supply functions, these conditions can be used to establish input-to-state stability (ISS) for general DDIs. When linear DDIs and quadratic functions are considered, ℓ2ℓ2-disturbance attenuation can be established by solving a single linear matrix inequality (LMI). Moreover, this LMI is shown to be less conservative than the BMI corresponding to the existing Razumikhin-type conditions for linear DDIs.
    Automatica. 02/2013; 49(2):619–625.

Full-text (2 Sources)

1 Download
Available from
Jul 8, 2014