## Packet based Inference and Control

##### Abstract

Communication constraints in Networked Control systems are frequently limits on data packet rates. To efficiently use the available packet rate budgets, we have to resort to event-triggered packet transmission. We have to sample signal waveforms and transmit packets not at deterministic times but at random times adapted to the signals measured. This thesis poses and solves some basic design problems we face in reaching for the extra efficiency.
We start with an estimation problem involving a single sensor. A sensor makes observations of a diffusion process, the state signal, and has to transmit samples of this process to a supervisor which maintains an estimate of the state. The objective of the sensor is to transmit samples strategically to the supervisor to minimize the distortion of the supervisor's estimate while respecting sampling rate constraints. We solve this problem over both finite and infinite horizons when the state is a scalar linear system.
We describe the relative performances of the optimal sampling scheme, the best deterministic scheme and of the suboptimal but simple to implement level-triggered sampling scheme. Apart from the utility of finding the optimal sampling strategies and their performances, we also learnt of some interesting properties of the level-triggered sampling scheme.
The control problem is harder to solve for the same setting with a single sensor. In the estimation problem for the linear state signal, the estimation error is also a linear diffusion and is reset to zero at sampling times. In the control problem, there is no equivalent to the error signal. We pay attention to an infinite horizon average cost problem where, the sampling strategy is chosen to be level-triggered. We design piece-wise constant controls by translating the problem to one for discrete-time Markov chain formed by the sampled state. Using results on the average cost control of Markov chains, we are able to derive optimality equations and iteratively compute solutions.
The last chapter tackles a binary sequential hypothesis testing problem with two sensors. The special feature of the problem is the ability of each sensor to hear the transmissions of the other towards the supervisor. Each sensor is afforded on transmission of a sample of its likelihood ratio process. We restrict attention to level-triggered
sampling. Although we are unable to demonstrate overall optimality of the asynchronous scheme we pursue, we are able to describe its advantages over other level-triggered schemes and of course the deterministic one. The chief merits of this thesis are the formulation and solution of some basic problems in multi-agent estimation and control. In the problems we have attacked, we have been able to deal with the differences in information patterns at sensors and supervisors. The main demerits are the ignoring of packet losses and of variable delays in packet transmissions. The situation of packet losses can however be handled at the expense of additional computations. To summarize, this thesis provides valuable generalizations of the works of Astrom and Bernhardsson and of Kushner on timing of Control actions and of Sampling observations respectively.

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