Self-Triggered and Event-Triggered Set-Valued Observers

Abstract

This paper addresses the problem of reducing the required network load and computational power for the implementation of Set-Valued Observers (SVOs) in Networked Control System (NCS). Event-and self-triggered strategies for NCS, modeled as discrete-time Linear Parameter-Varying (LPV) systems, are studied by showing how the triggering condition can be selected. The methodology provided can be applied to determine when it is required to perform a full ("classical") computation of the SVOs, while providing low-complexity state overbounds for the remaining time, at the expenses of temporarily reducing the estimation accuracy. As part of the procedure, an algorithm is provided to compute a suitable centrally symmetric polytope that allows to find hyper-parallelepiped and ellipsoidal overbounds to the exact set-valued state estimates calculated by the SVOs. By construction, the proposed triggering techniques do not influence the convergence of the SVOs, as at some subsequent time instants, set-valued estimates are computed using the conventional SVOs. Results are provided for the triggering frequency of the self-triggered strategy and two interesting cases: distributed systems when the dynamics of all nodes are equal up to a reordering of the matrix; and when the probability distribution of the parameters influencing the dynamics is known. The performance of the proposed algorithm is demonstrated in simulation by using a time-sensitive example.

Event-Triggered Observer Design for Delayed Output-Sampled Systems

“…Define the state observation error as e =x − x, with equations (1) (4) and (6), one obtains the following dynamical equations:…”

“…The problems of event-triggered control systems with full state available have been abundantly studied in the literature [1]- [5], including observer-based control techniques. However, most of these studies are limited only to linear systems [6]- [13], and few papers deal with nonlinear systems [14]- [22]. Precisely, [14], [15] studied an event-triggered mechanism in which the data are sent when the error induced by the event-triggered sampling exceeds a prescribed threshold, and this threshold was replaced by a time-varying one in [16]- [22].…”

Distinguishability of discrete‐time linear systems

“…Moreover, the vector serves as a certificate for the distinguishability. Notice that fault detection has also been posed as a distinguishability problem between the fault‐free and the real models with the techniques described in the works 33,38‐40 requiring testing a similar empty set condition as proposed in Theorem 4 as to declare a guaranteed detection. In using the above formulation, a certificate can be found for the detection while avoiding numerical errors from solvers in dealing with polytopes that are close to a singleton set.…”

Finite‐gain event‐triggered interval observers design for continuous‐time linear systems

“…In the event‐based context, point‐valued estimation approaches have been proposed for discrete‐time systems 15,16 and continuous‐time systems, 17,18 to cite a few. The set‐valued (set‐membership) estimation has also been investigated 19,20 . With discrete‐time systems, Shi et al 19 have proposed an event‐based estimation approach that uses ellipsoids and the set‐valued Kalman filter for systems with multiple outputs, where the set‐valued measurements, the initial state and the uncertainties are assumed Gaussian.…”