Continuous-Discrete Observers for Time-Varying Nonlinear Systems: A Tutorial on Recent Results

Abstract

Continuous-discrete systems can occur when the plant state evolves in continuous time but the output values are only available at discrete instants. Continuous-discrete observers have the valuable property that the observation error between the true state of the system and the observer state converges to zero in a uniform way. The design of continuousdiscrete observers can often be done by building framers, which provide componentwise upper and lower bounds for the plant state. This paper is a tutorial on these approaches, highlighting recent results in the literature, and also providing previously unpublished, original results which are not being simultaneously submitted elsewhere.

Design of continuous–discrete observers for time-varying nonlinear systems

“…See also Ahmed-Ali et al (2013b); Farza et al (2013); Karafyllis and Kravaris (2009) for observers based on output predictors and Andrieu and Nadri (2010); Deza et al (1992); Hammouri et al (2006); Dinh (2013, 2014); Tellez-Anguiano et al (2012); Karafyllis and Kravaris (2012); and see Ahmed-Ali et al (2013a), which presents results that allow delayed and sampled measurements. The work Ahmed-Ali et al (2009) builds continuous-discrete observers for nonlinear systems, where the input acts on the system to satisfy a persistent excitation condition, while Nadri and Hammouri (2003) covers systems with known inputs and which are linear in the state. The work Karafyllis and Kravaris (2009) shows that if a system admits a suitable continuous time observer and the observer satisfies certain robustness properties, then one can augment the observer by a new output predictor system to produce a continuous-discrete observer.…”

Kalman Filter Design for Intermittent Optical Wireless Communication Systems on Time Scales

“…This is because time scales theory allows a mathematical description of continuous-discrete hybrid processes using more general dynamical equations within a unified framework, rather than unilaterally using difference or differential equations. • A conventional approach to solving the problem is the continuous-discrete Kalman filter, which consists of continuous-time state prediction and discrete-time state update [27], [28]. Yet, the limitation of this approach is that the discrete output must be uniformly sampled (i.e., with a fixed step size), which is often not the case for realistic intermittent OWC systems.…”

Lyapunov characterization of uniform exponential stability for nonlinear infinite-dimensional systems

“…This means that the implemented input is, for almost every time, different from the designed controller. Several methods have been developed in the literature of ordinary differential equations for sampled-data observer design under discrete-time measurements (see, e.g., [2,19,24,29]), and for sampled-data control design guaranteeing a globally stable closed-loop system (see, e.g., [1,13]). Apart from time-delays systems (see, e.g., [7,20,39] for sampled-data control and [29,30] for sampled-data observer design), few results exist for infinite-dimensional systems.…”

“…Several methods have been developed in the literature of ordinary differential equations for sampled-data observer design under discrete-time measurements (see, e.g., [2,19,24,29]), and for sampled-data control design guaranteeing a globally stable closed-loop system (see, e.g., [1,13]). Apart from time-delays systems (see, e.g., [7,20,39] for sampled-data control and [29,30] for sampled-data observer design), few results exist for infinite-dimensional systems. The difficulties come from the fact that the developed methods do not directly apply to the infinite-dimensional case, for which even the well-posedness of sampled-data control dynamics is not obvious (see, e.g., [21] for more details).…”