Topological Entropy Bounds for Switched Linear Systems with Lie Structure

We study event-triggered control for stabilization of unstable linear plants over rate-limited communication channels subject to unknown, bounded delay. On one hand, the timing of event triggering carries implicit information about the state of the plant. On the other hand, the delay in the communication channel causes information loss, as it makes the state information available at the controller out of date. Combining these two effects, we show a phase transition behavior in the transmission rate required for stabilization using a given event-triggering strategy. For small values of the delay, the timing information carried by the triggering events is substantial, and the system can be stabilized with any positive rate. When the delay exceeds a critical threshold, the timing information alone is not enough to achieve stabilization, and the required rate grows. When the the delay equals the inverse of the entropy rate of the plant, the implicit information carried by the triggering events perfectly compensates the loss of information due to the communication delay, and we recover the rate requirement prescribed by the data-rate theorem. We also provide an explicit construction yielding a sufficient rate for stabilization, as well as results for vector systems. Our results do not rely on any a priori probabilistic model for the delay or the initial conditions. Index TermsData-rate theorem, event-triggered control, control under communication constraints, quantized control I. INTRODUCTIONCyber-physical systems (CPS) are engineering systems that integrate computing, communication, and control. They arise in a wide range of areas such as robotics, energy, civil infrastructure, manufacturing, and transportation [3], [4]. Due to the need for tight integration of different components, requirements and time scales, the modeling, analysis, and design of CPS present new challenges. One key aspect is the presence of finite-rate, digital communication channels in the feedback loop. Data-rate theorems quantify the effect that communication has on stabilization by stating that the communication rate available in the feedback loop should be at least as large as the intrinsic entropy rate of the system (corresponding to the sum of the logarithms of the unstable modes). In this way, the controller can compensate for the expansion of the state occurring during the communication process. Early formulations of data-rate theorems appeared in [5]-[7], followed by the key contributions in [8], [9]. More recent extensions include time-varying rate, Markovian, erasure, additive white and colored Gaussian, and multiplicative noise feedback communication channels [10]-[16], formulations for nonlinear systems [17]-[19], for optimal control [20]-[22], for systems with random parameters [23]-[26], and for switching systems [27]-[29]. Connections with information theory are highlighted in [19], [30]-[33]. Extended surveys of the literature appear in [34], [35] and in the book [36].Another key aspect of CPS to which we pay special attention he...

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“…we can also ensure (7). The condition in (27) can be rewritten as 1 − ρ 0 e −(σ+A)γ ≤ e (A+σ)(ts−q(ts)) ≤ 1 + ρ 0 e −(σ+A)γ .…”

Section: Sufficient Condition On the Transmission Ratementioning

confidence: 99%