Exploiting timing information in event-triggered stabilization of linear systems with disturbances

In the same way that subsequent pauses in spoken language are used to convey information, it is also possible to transmit information in communication networks not only by message content, but also with its timing. This paper presents an event-triggering strategy that utilizes timing information by transmitting in a state-dependent fashion. We consider the stabilization of a continuous-time, time-invariant, linear plant over a digital communication channel with bounded delay and subject to bounded plant disturbances and establish two main results. On the one hand, we design an encoding-decoding scheme that guarantees a sufficient information transmission rate for stabilization. On the other hand, we determine a lower bound on the information transmission rate necessary for stabilization by any control policy

Algorithms for Optimal Control with Fixed-Rate Feedback

We consider a discrete-time linear quadratic Gaussian networked control setting where the (full information) observer and controller are separated by a fixed-rate noiseless channel. The minimal rate required to stabilize such a system has been well studied. However, for a given fixed rate, how to quantize the states so as to optimize performance is an open question of great theoretical and practical significance. We concentrate on minimizing the control cost for first-order scalar systems. To that end, we use the Lloyd-Max algorithm and leverage properties of logarithmically-concave functions and sequential Bayesian filtering to construct the optimal quantizer that greedily minimizes the cost at every time instant. By connecting the globally optimal scheme to the problem of scalar successive refinement, we argue that its gain over the proposed greedy algorithm is negligible. This is significant since the globally optimal scheme is often computationally intractable. All the results are proven for the more general case of disturbances with logarithmically-concave distributions and rate-limited time-varying noiseless channels. We further extend the framework to event-triggered control by allowing to convey information via an additional "silent symbol", i.e., by avoiding transmitting bits; by constraining the minimal probability of silence we attain a tradeoff between the transmission rate and the control cost for rates below one bit per sample

Analysis of Inter-Event Times in Linear Systems under Region-Based Self-Triggered Control

This paper analyzes the evolution of inter-event times (IETs) in linear systems under region-based self-triggered control (RBSTC). In this control method, the state space is partitioned into a finite number of conic regions and each region is associated with a fixed IET. In this framework, studying the steady state behavior of the IETs is equivalent to studying the existence of a conic subregion that is positively invariant under the map that gives the evolution of the state from one event to the next. We provide necessary conditions and sufficient conditions for the existence of a positively invariant subregion (PIS). We also provide necessary and sufficient conditions for a PIS to be asymptotically stable. Indirectly, they provide necessary and sufficient conditions for local convergence of IETs to a constant or to a given periodic sequence. We illustrate the proposed method of analysis and results through numerical simulations.Comment: arXiv admin note: text overlap with arXiv:2201.0209

On Asymptotic Behavior of Inter-Event Times in Planar Systems under Event-Triggered Control

This paper analyzes the asymptotic behavior of inter-event times in planar linear systems, under event-triggered control with a general class of scale-invariant event triggering rules. In this setting, the inter-event time is a function of the ``angle'' of the state at an event. This allows us to analyze the inter-event times by studying the fixed points of the ``angle'' map, which represents the evolution of the ``angle'' of the state from one event to the next. We provide a sufficient condition for the convergence or non-convergence of inter-event times to a steady state value under a scale-invariant event-triggering rule. With the help of ergodic theory, we provide a sufficient condition for the asymptotic average inter-event time to be a constant for all non-zero initial states of the system. Then, we consider a special case where the ``angle'' map is an orientation-preserving homeomorphism. Using rotation theory, we comment on the asymptotic behavior of the inter-event times, including on whether the inter-event times converge to a periodic sequence. We also analyze the asymptotic average inter-event time as a function of the ``angle'' of the initial state of the system. We illustrate the proposed results through numerical simulations.Comment: The previous version of the paper now has been split into two separate papers, one on event-triggered control and one on self-triggered control. The updated part on self-triggered control may be accessed at arXiv:2212.1427