Section: New Results
NonLinear, SampledData And TimeDelay Systems


The problem of delay estimation for a class of nonlinear timedelay systems is considered in [82]. The theory of noncommutative rings is used to analyze the identifiability. Sliding mode technique is utilized in order to estimate the delay showing the possibility to have a local (or global) delay estimation for periodic (or aperiodic) delay signals.

In [14] we give sufficient conditions guaranteeing the observability of singular linear systems with commensurable delays affected by unknown inputs appearing in both the state equation and the output equation. These conditions allow for the reconstruction of the entire state vector using past and actual values of the system output. The obtained conditions coincide with known necessary and sufficient conditions of singular linear systems without delays.

[67] presents a finitetime observer for linear timedelay systems. In contrast to many observers, which normally estimate the system state in an asymptotic fashion, this observer estimates the exact system state in predetermined finite time. The finitetime observer proposed is achieved by updating the observer state based on actual and pass data of the observer. Simulation results are also presented to illustrate the convergence behavior of the finitetime observer.

The backward observability (BO) of a part of the vector of trajectories of the system state is tackled in [57] for a general class of linear timedelay descriptor systems with unknown inputs. By following a recursive algorithm, we present easy testable sufficient conditions ensuring the BO of descriptor timedelay systems.

Motivated by the problem of phaselocking in droopcontrolled inverterbased microgrids with delays, in [23], the recently developed theory of inputtostate stability (ISS) for multistable systems is extended to the case of multistable systems with delayed dynamics. Sufficient conditions for ISS of delayed systems are presented using LyapunovRazumikhin functions. It is shown that ISS multistable systems are robust with respect to delays in a feedback. The derived theory is applied to two examples. First, the ISS property is established for the model of a nonlinear pendulum and delaydependent robustness conditions are derived. Second, it is shown that, under certain assumptions, the problem of phaselocking analysis in droopcontrolled inverterbased microgrids with delays can be reduced to the stability investigation of the nonlinear pendulum. For this case, corresponding delaydependent conditions for asymptotic phaselocking are given.

Causal and noncausal observability are discussed in [68] for nonlinear time delay systems. By extending the Lie derivative for timedelay systems in the algebraic framework introduced by Xia et al. (2002), we present a canonical form and give sufficient condition in order to deal with causal and noncausal observations of state and unknown inputs of timedelay systems.

[83] presents a finitetime observer for linear timedelay systems with commensurate delay. Unlike the existing observers in the literature which converges asymptotically, the proposed observer provides a finitetime estimation. This is realized by using the wellknown sliding mode technique. Simulation results are also presented in order to illustrate the feasibility of the proposed method.



[104] presents basic concepts and recent research directions about the stability of sampleddata systems with aperiodic sampling. We focus mainly on the stability problem for systems with arbitrary timevarying sampling intervals which has been addressed in several areas of research in Control Theory. Systems with aperiodic sampling can be seen as timedelay systems, hybrid systems, Input/Output interconnections, discretetime systems with timevarying parameters, etc. The goal of the article is to provide a structural overview of the progress made on the stability analysis problem. Without being exhaustive, which would be neither possible nor useful, we try to bring together results from diverse communities and present them in a unified manner. For each of the existing approaches, the basic concepts, fundamental results, converse stability theorems (when available), and relations with the other approaches are discussed in detail. Results concerning extensions of Lyapunov and frequency domain methods for systems with aperiodic sampling are recalled, as they allow to derive constructive stability conditions. Furthermore, numerical criteria are presented while indicating the sources of conservatism, the problems that remain open and the possible directions of improvement. At last, some emerging research directions, such as the design of stabilizing sampling sequences, are briefly discussed.

In [31] we investigate the stability analysis of nonlinear sampleddata systems, which are affine in the input. We assume that a stabilizing controller is designed using the emulation technique. We intend to provide sufficient stability conditions for the resulting sampleddata system. This allows to find an estimate of the upper bound on the asynchronous sampling intervals, for which stability is ensured. The main idea of the paper is to address the stability problem in a new framework inspired by the dissipativity theory. Furthermore, the result is shown to be constructive. Numerically tractable criteria are derived using linear matrix inequality for polytopic systems and using sum of squares technique for the class of polynomial systems.

[76] deals with the sampleddata control problem based on state estimation for linear sampleddata systems. An impulsive system approach is proposed based on a vector Lyapunov function method. Observerbased control design conditions are expressed in terms of LMIs. Some examples illustrate the feasibility of the proposed approach.
