Lyapunov Methods for Time-Invariant Delay Difference Inclusions

Abstract

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Stabilisation of linear delay difference inclusions via time-varying control Lyapunov functions

“…Then, it can be shown that [1] the DDI (3) is GES if and only if the augmented system (4) admits a standard Lyapunov function with exponential upper and lower bounds. When interpreted for the DDI (3), this Lyapunov function is called a Lyapunov-Krasovskii function (LKF) and satisfies the conditions corresponding to the Krasovskii approach.…”

“…This shows that the Krasovskii approach provides non-conservative conditions for GES. Furthermore, the conditions corresponding to the Razumikhin approach can be shown to be sufficient but not necessary [1] for GES and the corresponding Lyapunov function is called a LyapunovRazumikhin function (LRF).…”

“…Linear delay difference inclusions (DDIs) have the ability to model a wide variety of relevant linear processes in the discrete-time domain, such as uncertain systems with delay, systems with time-varying delay [1] and certain types of networked control systems [2][3][4]. Therefore the stabilisation of such systems, possibly subject to constraints, is a frequently studied problem.…”

“…Based on the Razumikhin approach a state feedback controller was obtained for systems with delay in [14] and for uncertain systems with delay in [1]. Furthermore, a control strategy that takes constraints into account was developed for uncertain systems with delay in [15].…”

“…Interestingly, the control scheme proposed in [15] consists of an online optimisation-based component that is tractable for large delays because of an on-line Minkowski addition of sets. Unfortunately, the method also requires the off-line computation of a local static state-feedback controller and thus, as it is also the case for the control schemes proposed in [1,14], remains computationally demanding. Furthermore, as the Razumikhin approach provides [1] sufficient but not necessary conditions for stability, any method based on this approach is inherently conservative.…”