Sekhar Tatikonda

Control over noisy channels

Communication is an important component of distributed and networked controls systems. In our companion paper, we presented a framework for studying control problems with a digital noiseless communication channel connecting the sensor to the controller. Here, we generalize that framework by applying traditional information theoretic tools of source coding and channel coding to the problem. We present a general necessary condition for observability and stabilizability for a large class of communication channels. Then, we study sufficiency conditions for Internet-like channels that suffer erasures.

Stochastic linear control over a communication channel

We examine linear stochastic control systems when there is a communication channel connecting the sensor to the controller. The problem consists of designing the channel encoder and decoder as well as the controller to satisfy some given control objectives. In particular, we examine the role communication has on the classical linear quadratic Gaussian problem. We give conditions under which the classical separation property between estimation and control holds and the certainty equivalent control law is optimal. We then present the sequential rate distortion framework. We present bounds on the achievable performance and show the inherent tradeoffs between control and communication costs. In particular, we show that optimal quadratic cost decomposes into two terms: A full knowledge cost and a sequential rate distortion cost.

The Capacity of Channels With Feedback

In this paper, we introduce a general framework for treating channels with memory and feedback. First, we prove a general feedback channel coding theorem based on Masseys concept of directed information. Second, we present coding results for Markov channels. This requires determining appropriate sufficient statistics at the encoder and decoder. We give a recursive characterization of these sufficient statistics. Third, a dynamic programming framework for computing the capacity of Markov channels is presented. Fourth, it is shown that the average cost optimality equation (ACOE) can be viewed as an implicit single-letter characterization of the capacity. Fifth, scenarios with simple sufficient statistics are described. Sixth, error exponents for channels with feedback are presented.

Efficient and dynamic routing topology inference from end-to-end measurements

Inferring the routing topology and link performance from a node to a set of other nodes is an important component in network monitoring and application design. In this paper, we propose a general framework for designing topology inference algorithms based on additive metrics. The framework can flexibly fuse information from multiple measurements to achieve better estimation accuracy. We develop computationally efficient (polynomial-time) topology inference algorithms based on the framework. We prove that the probability of correct topology inference of our algorithms converges to one exponentially fast in the number of probing packets. In particular, for applications where nodes may join or leave frequently such as overlay network construction, application-layer multicast, and peer-to-peer file sharing/streaming, we propose a novel sequential topology inference algorithm that significantly reduces the probing overhead and can efficiently handle node dynamics. We demonstrate the effectiveness of the proposed inference algorithms via Internet experiments.

On the Feedback Capacity of Power-Constrained Gaussian Noise Channels With Memory

For a stationary additive Gaussian-noise channel with a rational noise power spectrum of a finite-order L, we derive two new results for the feedback capacity under an average channel input power constraint. First, we show that a very simple feedback-dependent Gauss-Markov source achieves the feedback capacity, and that Kalman-Bucy filtering is optimal for processing the feedback. Based on these results, we develop a new method for optimizing the channel inputs for achieving the Cover-Pombra block-length- n feedback capacity by using a dynamic programming approach that decomposes the computation into n sequentially identical optimization problems where each stage involves optimizing O(L 2) variables. Second, we derive the explicit maximal information rate for stationary feedback-dependent sources. In general, evaluating the maximal information rate for stationary sources requires solving only a few equations by simple nonlinear programming. For first-order autoregressive and/or moving average (ARMA) noise channels, this optimization admits a closed-form maximal information rate formula. The maximal information rate for stationary sources is a lower bound on the feedback capacity, and it equals the feedback capacity if the long-standing conjecture, that stationary sources achieve the feedback capacity, holds

Some scaling properties of large distributed control systems

Understanding the role of communication in networked control systems has become an important topic of study. In this paper we examine conditions on a communication network that insure observability and stabilizability of a control system. We examine both wired networks and wireless networks. For the wired network we give general necessary and sufficient conditions on the link capacities to insure the control objectives can be achieved. For the wireless network we discuss scaling properties and their effects on large control systems.

Control of LQG systems under communication constraints

We consider the control performance of an LQG system with a noisy analog feedback channel between the state-observation and the controller. To bound the performance, we use the sequential rate distortion function and the assumption of equi-memory. We then discuss the trade-offs between control and communication costs and how to relax the equi-memory assumption.

Convergence of the sum-product algorithm

We address the question of convergence in the sum-product algorithm. Specifically, we relate convergence of the sum-product algorithm to the existence of a weak limit for a sequence of Gibbs measures defined on the associated computation tree. Using tools from the theory of Gibbs measures we develop easily testable sufficient conditions for convergence. The failure of convergence of the sum-product algorithm implies the existence of multiple phases for the associated Gibbs specification. These results give new insight into the mechanics of the algorithm.

Network Routing Topology Inference from End-to-End Measurements

Inference of the routing topology and link performance from a node to a set of other nodes is an important component of network monitoring and application design. In this paper we propose a general framework for designing topology inference algorithms based on additive metrics. Our framework allows the integration of both end-to-end packet probing measurements and traceroute type measurements. Based on this framework we design several computationally efficient topology inference algorithms. In particular, we propose a novel sequential topology inference algorithm to address the probing scalability problem and handle dynamic node joining and leaving. We provide sufficient conditions for the correctness of our algorithms and derive lower bounds on the probability of correct topology inference. We conduct Internet experiments to evaluate and demonstrate the effectiveness of our algorithms.

Lossy compression via sparse linear regression: Computationally efficient encoding and decoding

We propose computationally efficient encoders and decoders for lossy compression using a Sparse Regression Code. Codewords are structured linear combinations of columns of a design matrix. The proposed encoding algorithm sequentially chooses columns of the design matrix to successively approximate the source sequence. It is shown to achieve the optimal distortion-rate function for i.i.d Gaussian sources with squared-error distortion. For a given rate, the parameters of the design matrix can be varied to trade off distortion performance with encoding complexity. An example of such a trade-off is: computational resource (space or time) per source sample of O((n/ log n)2) and probability of excess distortion decaying exponentially in n/ log n, where n is the block length. The Sparse Regression Code is robust in the following sense: for any ergodic source, the proposed encoder achieves the optimal distortion-rate function of an i.i.d Gaussian source with the same variance. Simulations show that the encoder has very good empirical performance, especially at low and moderate rates.