Error Correcting Codes for Distributed Control

The problem of stabilizing an unstable plant over a noisy communication link is an increasingly important one that arises in applications of networked control systems. Although the work of Schulman and Sahai over the past two decades, and their development of the notions of "tree codes"\phantom{} and "anytime capacity", provides the theoretical framework for studying such problems, there has been scant practical progress in this area because explicit constructions of tree codes with efficient encoding and decoding did not exist. To stabilize an unstable plant driven by bounded noise over a noisy channel one needs real-time encoding and real-time decoding and a reliability which increases exponentially with decoding delay, which is what tree codes guarantee. We prove that linear tree codes occur with high probability and, for erasure channels, give an explicit construction with an expected decoding complexity that is constant per time instant. We give novel sufficient conditions on the rate and reliability required of the tree codes to stabilize vector plants and argue that they are asymptotically tight. This work takes an important step towards controlling plants over noisy channels, and we demonstrate the efficacy of the method through several examples.Comment: 39 page

Tree Codes Improve Convergence Rate of Consensus Over Erasure Channels

We study the problem of achieving average consensus between a group of agents over a network with erasure links. In the context of consensus problems, the unreliability of communication links between nodes has been traditionally modeled by allowing the underlying graph to vary with time. In other words, depending on the realization of the link erasures, the underlying graph at each time instant is assumed to be a subgraph of the original graph. Implicit in this model is the assumption that the erasures are symmetric: if at time t the packet from node i to node j is dropped, the same is true for the packet transmitted from node j to node i. However, in practical wireless communication systems this assumption is unreasonable and, due to the lack of symmetry, standard averaging protocols cannot guarantee that the network will reach consensus to the true average. In this paper we explore the use of channel coding to improve the performance of consensus algorithms. For symmetric erasures, we show that, for certain ranges of the system parameters, repetition codes can speed up the convergence rate. For asymmetric erasures we show that tree codes (which have recently been designed for erasure channels) can be used to simulate the performance of the original "unerased" graph. Thus, unlike conventional consensus methods, we can guarantee convergence to the average in the asymmetric case. The price is a slowdown in the convergence rate, relative to the unerased network, which is still often faster than the convergence rate of conventional consensus algorithms over noisy links

The Kalman Like Particle Filter: Optimal Estimation With Quantized Innovations/Measurements

We study the problem of optimal estimation using quantized innovations, with application to distributed estimation over sensor networks. We show that the state probability density conditioned on the quantized innovations can be expressed as the sum of a Gaussian random vector and a certain truncated Gaussian vector. This structure bears close resemblance to the full information Kalman filter and so allows us to effectively combine the Kalman structure with a particle filter to recursively compute the state estimate. We call the resulting filter the Kalman like particle filter (KLPF) and observe that it delivers close to optimal performance using far fewer particles than that of a particle filter directly applied to the original problem. We also note that the conditional state density follows a, so called, generalized closed skew-normal (GCSN) distribution

Particle filtering for Quantized Innovations

In this paper, we re-examine the recently proposed distributed state estimators based on quantized innovations. It is widely believed that the error covariance of the Quantized Innovation Kalman filter follows a modified Riccati recursion. We present stable linear dynamical systems for which this is violated and the filter diverges. We propose a Particle Filter that approximates the optimal nonlinear filter and observe that the error covariance of the Particle Filter follows the modified Riccati recursion. We also simulate a Posterior Cramer-Rao bound (PCRB) for this filtering problem

Distributed Control and Computing: Optimal Estimation, Error Correcting Codes, and Interactive Protocols

Emerging applications of networked control and distributed computing are characterized by decentralization of information and the need to exchange it over potentially unreliable communication networks. This results in novel interactive communication scenarios that are incompatible with conventional information and coding theoretic approaches. To address this gap, through the early and late 1990's, a new information theoretic notion called anytime reliability and a new coding paradigm called tree codes were proposed. Although the central role of tree codes in several interactive communication problems such as distributed control and computing has been well understood, there have been no practical constructions till date. For the first time, we have an explicit ensemble of linear tree codes with efficient encoding and decoding for the class of erasure channels. In the process, we have developed novel non-asymptotic sufficient conditions on the kind of communication reliability required to stabilize control systems over noisy channels. We also study the application of tree codes to interactive protocols over erasure networks and illustrate their benefits through the example of average consensus

Linear Time-Invariant Anytime Codes for Control Over Noisy Channels

The problem of stabilizing an unstable plant over a noisy communication link is an increasingly important one that arises in problems of distributed control and networked control systems. Although the work of Schulman, and Sahai and Mitter over the past two decades, and their development of the notions of “tree codes” and “anytime capacity” respectively, provides the theoretical framework for studying such problems, there has been scant practical progress in this area because explicit constructions of tree codes with efficient encoding and decoding did not exist. To stabilize an unstable plant driven by bounded noise over a noisy channel one often needs real-time encoding and real-time decoding and a reliability which increases exponentially with delay, which is what tree codes guarantee. We propose an ensemble of random causal linear codes with a time invariant structure and show that they are tree codes with probability one. For erasure channels, we show that the average complexity of maximum likelihood decoding is bounded by a constant for all time if the code rate is smaller than the computational cutoff rate. For rates larger than the computational cutoff rate, we present an alternate way to perform maximum likelihood decoding with a complexity that grows linearly with time. We give novel sufficient conditions on the rate and reliability required of the tree codes to stabilize vector plants and argue that they are asymptotically tight

The Kalman-Like Particle Filter: Optimal Estimation With Quantized Innovations/Measurements

We study the problem of optimal estimation and control of linear systems using quantized measurements. We show that the state conditioned on a causal quantization of the measurements can be expressed as the sum of a Gaussian random vector and a certain truncated Gaussian vector. This structure bears close resemblance to the full information Kalman filter and so allows us to effectively combine the Kalman structure with a particle filter to recursively compute the state estimate. We call the resulting filter the Kalman-like particle filter (KLPF) and observe that it delivers close to optimal performance using far fewer particles than that of a particle filter directly applied to the original problem