Girish N. Nair

Feedback Control Under Data Rate Constraints: An Overview

The emerging area of control with limited data rates incorporates ideas from both control and information theory. The data rate constraint introduces quantization into the feedback loop and gives the interconnected system a two-fold nature, continuous and symbolic. In this paper, we review the results available in the literature on data-rate-limited control. For linear systems, we show how fundamental tradeoffs between the data rate and control goals, such as stability, mean entry times, and asymptotic state norms, emerge naturally. While many classical tools from both control and information theory can still be used in this context, it turns out that the deepest results necessitate a novel, integrated view of both disciplines

Stabilization with data-rate-limited feedback : tightest attainable bounds

Abstract This paper investigates the stabilizability of a linear, discrete-time plant with a real-valued output when the controller, which may be nonlinear, receives observation data at a known rate. It is first shown that, under a finite horizon cost equal to the m th output moment, the problem reduces to quantizing the initial output. Asymptotic quantization theory is then applied to directly obtain the limiting coding and control scheme as the horizon approaches infinity. This is proven to minimize a particular infinite horizon cost, the value of which is derived. A necessary and sufficient condition then follows for there to exist a coding and control scheme with the specified data rate that takes the m th output moment to zero asymptotically with time. If the open-loop plant is finite-dimensional and time-invariant, this condition simplifies to an inequality involving the data rate and the unstable plant pole with greatest magnitude. Analagous results automatically hold for the related problem of state estimation with a finite data rate.

Topological feedback entropy and Nonlinear stabilization

It is well known in the field of dynamical systems that entropy can be defined rigorously for completely deterministic open-loop systems. However, such definitions have found limited application in engineering, unlike Shannons statistical entropy. In this paper, it is shown that the problem of communication-limited stabilization is related to the concept of topological entropy, introduced by Adler et al. as a measure of the information rate of a continuous map on a compact topological space. Using similar open cover techniques, the notion of topological feedback entropy (TFE) is defined in this paper and proposed as a measure of the inherent rate at which a map on a noncompact topological space with inputs generates stability information. It is then proven that a topological dynamical plant can be stabilized into a compact set if and only if the data rate in the feedback loop exceeds the TFE of the plant on the set. By taking appropriate limits in a metric space, the concept of local TFE (LTFE) is defined at fixed points of the plant, and it is shown that the plant is locally uniformly asymptotically stabilizable to a fixed point if and only if the data rate exceeds the plant LTFE at the fixed point. For continuously differentiable plants in Euclidean space, real Jordan forms and volume partitioning arguments are then used to derive an expression for LTFE in terms of the unstable eigenvalues of the fixed point Jacobian.

Data Rate Theorem for Stabilization Over Time-Varying Feedback Channels

A data rate theorem for stabilization of a linear, discrete-time, dynamical system with arbitrarily large disturbances, over a rate-limited, time-varying communication channel is presented. Necessary and sufficient conditions for stabilization are derived, their implications and relationships with related results in the literature are discussed. The proof techniques rely on both information-theoretic and control-theoretic tools.

Systems engineering for irrigation systems: Successes and challenges

In Australia, gravity fed irrigation systems are critical infrastructure essential to agricultural production and export. By supplementing these large scale civil engineering systems with an appropriate information infrastructure, sensors, actuators and a communication network it is feasible to use systems engineering ideas to improve the exploitation of the irrigation system. This paper reports how classical ideas from system identification and control can be used to automate irrigation systems to deliver a near on-demand water supply with vastly improved overall distribution efficiency.

Stochastic consensus over noisy networks with Markovian and arbitrary switches

This paper considers stochastic consensus problems over lossy wireless networks. We first propose a measurement model with a random link gain, additive noise, and Markovian lossy signal reception, which captures uncertain operational conditions of practical networks. For consensus seeking, we apply stochastic approximation and derive a Markovian mode dependent recursive algorithm. Mean square and almost sure (i.e., probability one) convergence analysis is developed via a state space decomposition approach when the coefficient matrix in the algorithm satisfies a zero row and column sum condition. Subsequently, we consider a model with arbitrary random switching and a common stochastic Lyapunov function technique is used to prove convergence. Finally, our method is applied to models with heterogeneous quantizers and packet losses, and convergence results are proved.

Communication-limited stabilization of linear systems

This paper investigates the problem of stabilizing a linear, discrete-time plant using a digital link with a finite data rate. The plant model is infinite-dimensional and time-varying, with a real-valued output which is zero at negative times and distributed according to a probability density p at time zero. Finite and infinite horizon costs in terms of the m-th output moment are defined and the equations of the optimal, finite horizon coder-controller derived. Asymptotic quantization theory is then used to obtain the solution as the horizon tends to infinity, without needing to explicitly solve the finite horizon problem. It is shown that this limiting coder-controller is optimal with respect to the infinite horizon cost, provided that p satisfies certain technical conditions. This immediately leads to a necessary and sufficient condition for the existence of a coder-controller that takes the m-th output moment to zero asymptotically with time. If the open-loop plant is finite-dimensional and time-invariant, this condition simplifies to an inequality involving the data rate and the open-loop pole with greatest magnitude. Analogous results automatically hold for the related problem of state estimation with a finite data rate.

Networked sensor management and data rate control for tracking maneuvering targets

This paper presents sensor and data rate control algorithms for tracking maneuvering targets. The manuevering target is modeled as a jump Markov linear system. We present novel extensions of the Interacting Multiple Model (IMM), Particle filter tracker, and Probabilistic Data Association (PDA) algorithms to handle sensor and data rate control. Numerical studies illustrate the performance of these sensor and data rate control algorithms.

Infimum data rates for stabilising Markov jump linear systems

In the past years, the problem of stabilising linear dynamical systems with low feedback data rates has been intensively investigated. A particular focus has been the characterisation of the infimum data rate for stabilisability, which specifies the smallest rate, in bits per unit time, at which information can circulate in a stable feedback loop. This paper extends this line of research to the case of fully-observed, finite-dimensional, linear systems without process noise but with control-independent, Markov parameters. Unlike previous formulations, the coding alphabet is permitted to be random and time-varying via a possible dependence on the observed Markov modes. Using quantisation techniques and real Jordan forms, it is shown that the smallest asymptotic mean data rate for stabilisability in r-th absolute output moment, over all coding and control schemes, is given by an exponent which measures the asymptotic mean growth rate of unstable eigenspace volumes. An explicit formula for it is obtained in the case of antistable dynamics. For scalar systems, this expression is quite different from an earlier one derived assuming a constant alphabet, in particular being independent of the moment order.

State estimation under bit-rate constraints

This paper considers the problem of estimating the state of a dynamic system from measurements obtained via a digital link with finite data rate R. The structures of the optimal coder and estimator for Markovian systems are derived. In particular, it is shown that the optimal coder for a Gauss-Markov system consists of a Kalman filter, followed by a stage which encodes the current Kalman estimate according to the symbols previously transmitted. A new suboptimal coder-estimator for linear systems is then constructed. Provided that a certain inequality linking the data rate to the dynamical parameters is satisfied, and under very mild assumptions on the noise distributions, this coder-estimator yields an expected absolute estimation error of the same order as in the classical situation with no data rate constraint. Hence if the classical estimation error approaches zero, then the rate-constrained error goes to zero at exactly the same speed.