This paper presents a method for designing state observers with exponential error decay for nonlinear systems whose output measurements are affected by known time-varying delays. A modular approach is followed, where subobservers are connected in cascade to achieve a desired exponential convergence rate (chain observer). When the delay is small, a single-step observer is sufficient to carry out the goal. Two or more subobservers are needed in the the presence of large delays. The observer employs delay-dependent time-varying gains to achieve the desired exponential error decay. The proposed approach allows to deal with vector output measurements, where each output component can be affected by a different delay. Relationships among the error decay rate, the bound on the measurement delays, the observer gains, and the Lipschitz constants of the system are presented. The method is illustrated on the synchronization problem of continuous-time hyperchaotic systems with buffered measurements.

Interval observers are dynamic systems that provide upper and lower bounds of the true state trajectories of systems. In this work we introduce a technique to design interval observers for linear systems affected by state and measurement disturbances, based on the Internal Positive Representations (IPRs) of systems, that exploits the order preserving property of positive systems. The method can be applied to both continuous and discrete time systems.Index Terms-Linear system observers, positive systems, uncertain systems. I. INTRODUCTIONThe properties of positive linear systems have been a subject of study in system theory for long time [11], [14]. Some of these properties, for example that the trajectories of positive systems are ordered with respect to the initial conditions and the forcing input, can be exploited in the problem of state estimation.This technical note proposes a framework for the state observation problem of linear systems in presence of bounded input/output uncertainties which is based on the use of positive systems to design interval observers, that is, a system of observers that provides a confident region that contains the trajectory of the observed system [13], [16]-[18]. A key tool to apply the positive properties to general (i.e., non necessarily positive) systems is the internally positive realization (IPR) [4]-[6], [12]. Some recent works on interval observers based on positive systems is reported in [2], [3], [20], where positive observed systems are considered. The case of stable linear systems is considered in [16], where the authors propose an approach based on a time-varying change of coordinates that transforms an autonomous system into a positive one. This approach is used to design a Luenberger observer with a positive error dynamics, on which an interval observer can be built. This idea has subsequently been extended to complex intervals in [7], and, using a time invariant change of coordinates, to a class of nonlinear systems in [18], to linear time-invariant and time-varying discrete-time systems in [8], [9], and to continuous-time time-varying systems in [10].Essentially, the idea behind these approaches is to design a stable observer by means of an appropriate gain and then to find a coordinate change such that the resulting error dynamics is positive. In this technical note we propose to reverse the approach: a positive observer of the system is initially built, and the observer gain is subsequently chosen so as to have a stable error dynamics. We show that this is always possible for observable systems. The design technique is very Manuscript

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The problem of realizing arbitrary transfer functions as combinations of positive filters, or, more in general, of representing arbitrary systems by means of Internally Positive Representations (IPRs), has been widely investigated in the discrete-time framework. An IPR is a positive state-space representation, endowed with input, state and output transformations, that realizes arbitrary input-state-output dynamics. This technical note investigates the problem of the IPR construction for continuous-time systems, and proposes a technique that provides stable IPRs for a particular class of systems

This paper concerns the problem of the control of linear systems by means of feedback from delayed output, where the delay is known and time-varying. The main advantage of the approach is that it can be applied to systems with any delay bound, i.e. not only small delays. The predictor is based on a combination of finite-dimensional elementary predictors whose number can be suitably chosen to compensate any delay. The single-predictor element is an original proposal, and the class of delays to which the schema can be applied includes, but it is not limited to, continuous delay functions.