In this paper, the design of a data-rate constrained observer for a dynamical system is presented. This observer is designed to function both in discrete time and continuous time. The system is connected to a remote location via a communication channel which can transmit limited amounts of data per unit of time. The objective of the observer is to provide estimates of the state at the remote location through messages that are sent via the channel. The observer is designed such that it is robust toward losses in the communication channel. Upper bounds on the required communication rate to implement the observer are provided in terms of the upper box dimension of the state space and an upper bound on the largest singular value of the system’s Jacobian. Results that provide an analytical bound on the required minimum communication rate are then presented. These bounds are obtained by using the Lyapunov dimension of the dynamical system rather than the upper box dimension in the rate. The observer is tested through simulations for the Lozi map and the Lorenz system. For the Lozi map, the Lyapunov dimension is computed. For both systems, the theoretical bounds on the communication rate are compared to the simulated rates.

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In this paper, we consider the synchronization of two linear systems that are subject to unknown perturbations. One of the systems is driven by a reference signal, which is unknown to the second system. Both systems can only use a one-way communication channel to exchange information. The channel is subject to data-rate constraints. The messages are generated by a smart sensor that measures the state and is capable of performing some computations. The messages are received by a controller which interprets them to apply an appropriate control input to the system. The objective is to design a sensor/controller pair, which we will refer to as a communication protocol, such that the distance between the states of both systems is bounded, and as small as possible communication rate is necessary. In this paper, a communication protocol that achieves this objective is presented together with the minimum channel rate to implement it. Simulations of the communication protocol are provided to support the theoretical work. I. INTRODUCTIONThe widespread usage of wireless technologies in industry has created a new area in the field of dynamics and control that takes into account the data-rate constrained channels. All problems studied in this sub-field share common features: one or several dynamical systems, and possibly their sensors, controllers, actuators, are placed at locations remote from one another. The different devices are connected via communication channels which can only transmit limited amounts of data per unit of time, hence the name data-rate constrained communication channels. The many examples of such applications include remote sensors that communicate via Wi-Fi, microelectromechanical systems (where the constraints are due to the size of the components), platoons of connected vehicles, formation control for drones, etc... The fact that these data-rate constrained channels are connected to dynamical systems implies that it is necessary to design specific strategies for that situation, as opposed to simply relying on classical signal theory or control and estimation theory separately. For most problems, the need for persistent communication is due to one or several sources

In this paper, we develop a communication protocol for the observation of discrete time, possibly unstable, dynamical systems over communication channels with limited communication capacity. We develop an observer based on the upper box dimension for one-way communication channels that leads to a certain type of observability. This communication scheme preserves observability under communication losses which makes the communication scheme robust towards communication losses without feedback in the communication channel. Using Lyapunov-like techniques, we provide bounds on the minimum communication rate required to implement this observer. We also use the Lyapunov dimension to provide analytical upper bounds on the communication rate. We compute an analytical upper bound and an exact expression for the Lyapunov dimension of the smoothened Lozi map. This bound is then tested in simulations of the communication protocol for the observation problem of the smoothened Lozi map.