Lyapunov Methods for Time-Invariant Delay Difference Inclusions

SIAM Journal on Control and Optimization (Impact Factor: 1.46). 01/2012; 50(1):110-132. DOI: 10.1137/100807065
Source: DBLP


Motivated by the fact that delay difference inclusions (DDIs) form a rich modeling class that includes, for example, uncertain time-delay systems and certain types of networked control systems, this paper provides a comprehensive collection of Lyapunov methods for DDIs. First, the Lyapunov-Krasovskii approach, which is an extension of the classical Lyapunov theory to time-delay systems, is considered. It is shown that a DDI is KL-stable if and only if it admits a Lyapunov- Krasovskii function (LKF). Second, the Lyapunov-Razumikhin method, which is a type of smallgain approach for time-delay systems, is studied. It is proved that a DDI is KL-stable if it admits a Lyapunov-Razumikhin function (LRF). Moreover, an example of a linear delay difference equation which is globally exponentially stable but does not admit an LRF is provided. Thus, it is established that the existence of an LRF is not a necessary condition for KL-stability of a DDI. Then, it is shown that the existence of an LRF is a sufficient condition for the existence of an LKF and that only under certain additional assumptions is the converse true. Furthermore, it is shown that an LRF induces a family of sets with certain contraction properties that are particular to time-delay systems. On the other hand, an LKF is shown to induce a type of contractive set similar to those induced by a classical Lyapunov function. The class of quadratic candidate functions is used to illustrate the results derived in this paper in terms of both LKFs and LRFs, respectively. Both stability analysis and stabilizing controller synthesis methods for linear DDIs are proposed.

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Available from: R.H. Gielen, Jul 08, 2014
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    • "The Razumikhin approach enjoys computational practicability since the underlying computations can be executed with respect to the original (non–augmented) state space R m . Unfortunately, the Razumikhin approach provides only, possibly conservative, sufficient conditions for stability, e.g., in [14] a stable scalar linear time–delay system was presented for which the Razumikhin approach is unable to verify stability. Therefore, for such systems the Razumikhin approach is unable to provide a PI set other than {0} h+1 . "
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    ABSTRACT: Methods for the construction of positively invariant (PI) sets, both for interconnected and time-delay systems, either suffer from computational intractability or come with considerable conservatism, with respect to their ability to provide a nontrivial PI set as well as their ability to recover or even approximate the maximal positively invariant (mPI) set. Therefore, we apply the notion of PI families of sets to interconnected and time-delay systems. We show that for such systems, this notion enjoys computational practicability and, at the same time, is nonconservative, both in terms of the type of sets it produces and its ability to approximate and recover the mPI set. Moreover, this technique also provides a tractable stability analysis tool. In this paper, for both interconnected and time-delay systems, the properties of PI families of sets are analyzed, their use is illustrated via several examples, and their construction via convex optimization algorithms is discussed.
    Full-text · Article · Jul 2014 · SIAM Journal on Control and Optimization
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    • "Theorems 2 and 3 can be extended to delay difference inclusions, which form a more general modeling class that includes uncertain systems and systems with time– varying delay, using the techniques developed in [6] "
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    ABSTRACT: This technical note considers stability analysis of time-delay systems described by delay difference equations (DDEs). All existing analysis methods for DDEs that rely on the Razumikhin approach provide sufficient, but not necessary conditions for asymptotic stability. Nevertheless, Lyapunov-Razumikhin functions are of interest because they induce invariant sets in the underlying state space of the dynamics. Therefore, we propose a relaxation of the Razumikhin conditions and prove that the relaxed conditions are necessary and sufficient for asymptotic stability of DDEs. For linear DDEs, it is shown that the developed conditions can be verified by solving a linear matrix inequality. Moreover, it is indicated that the proposed relaxation of Lyapunov-Razumikhin functions has an important implication for the construction of invariant sets for linear DDEs.
    Full-text · Article · Oct 2013 · IEEE Transactions on Automatic Control
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    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.
    Full-text · Article · Apr 2012 · Mathematics of Control Signals and Systems
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