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Abstract:In this paper we study conditions under which an unstable stochastic scalar linear plant with unbounded noise can be internally stabilised using 'zooming'-like coding and control schemes having dynamic, finite-dimensional internal states. Such structures are known to be needed in communicationconstrained control when no bound on the plant noise is available. However, previous schemes were based on coders and controllers starting with identical internal states. In this paper, we remove this assumption and explicitly construct a finite-dimensional coding and control policy that yields mean square stability of all state variables, for a random initial plant state and arbitrary initial encoder and controller states. This holds for any bit rate down to the universal minimum of the Data Rate Theorem. Furthermore, we show that despite the unbounded noise, the error and proportional errors between the scaling factors of the encoder and controller tend to zero in mean square and almost sure senses respectively. This suggests that the policy will still maintain mean square internal stability in the presence of channel bit errors, provided the bit error rate is sufficiently low. We support these conclusions with simulations.

In this paper, we look into performance analysis of bit-rate-limited stochastic control systems with quantized state feedback. We propose a new quantization scheme, and a new functional as a design tool, for computing an a priori bound on the mean square state. I. INTRODUCTIONControl systems employing multiple sensors and actuators that are geographically distributed, are an area of research which attracted much attention in the last decade. When we wish to control systems consisting of many components connected by a digital channel, communication channel constraints have to be taken into account. The reason for this is that classical control theory usually assumes that the plant and feedback controller are either collocated or they can communicate with each other over a channel with infinite capacity, whereas the core of the problem here is that the plant and the feedback controller communicate over a digital channel with finite capacity. Some important developments have been made in recent years in the area of control under communication constraints. Beginning with the seminal paper by [4] and continuing with [1], [2], [5], [8], [16], various schemes have been proposed and proven to asymptotically or practically stabilize linear-time invariant (LTI) systems at sufficiently high data rates. The first rigorous results on minimum data rate were in [1] and [16], where it was shown that a discretized scalar plant with parameter a was stabilizable iff the data rate was not less than log 2 |a| bits per sample interval. These papers by [1], [16] had the first result which basically provided a direct link between the data rate and the dynamics of the system. Similar tight bounds were subsequently obtained for noiseless autoregressive moving average [10] and linear state-space systems [6], [11], [14], using different formulations and techniques. With regard to stochastic plants, separation principles, causal rate-distortion theorems [3], [9], [15] and the notion of feedback capacity [13] have been introduced. In [12]a key lemma, which stochastically bounds the quantization errors in an iterative manner, was provided and by following the same approach [7] also derived similar bounds on the mean-square state of the systems to be stabilized by considering the same functional, but with a slightly modified quantization scheme. In this paper, we introduce a new quantization scheme in contrast to the one proposed by [7], and a new functional as a design tool to obtain tight performance bounds on the MSS (Mean Square State). The paper is organized as