On Stability in the Presence of Analog Erasure Channel Between the Controller and the Actuator

Consider a discrete-time networked control system, in which the controller has direct access to noisy measurements of the output of the plant. However, information flows from the controller to the actuator via a channel that features Bernoulli erasure events. If an erasure occurs, the channel outputs an erasure symbol; otherwise, it transmits a real finite-dimensional vector. We determine necessary and sufficient conditions for the stabilizability of an unstable linear time-invariant finite-dimensional plant. Given a minimal state-space representation for the plant, the necessary and sufficient conditions for stabilizability are expressed in terms of the probability of erasure at the channel and of the spectral radius of the one-step state transition matrix. There are two main results in the technical note. The first result shows that if the actuator has processing capabilities, then the necessary and sufficient conditions for stabilizability remain unchanged with or without acknowledgements from the actuator to the controller. The second result shows that the stabilizability conditions are identical for two types of actuators: (Type I) Processing at the actuator has access to the plant's model; (Type II) Processing at the actuator uses a universal algorithm that does not depend on the model of the plant. Thus, neither the knowledge of the model of the plant at the actuator, nor the presence of acknowledgements from the actuator to the controller, can be used to alter or relax the necessary and sufficient conditions for stabilizability.

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