Roger W. Brockett

Quantized feedback stabilization of linear systems

This paper addresses feedback stabilization problems for linear time-invariant control systems with saturating quantized measurements. We propose a new control design methodology, which relies on the possibility of changing the sensitivity of the quantizer while the system evolves. The equation that describes the evolution of the sensitivity with time (discrete rather than continuous in most cases) is interconnected with the given system (either continuous or discrete), resulting in a hybrid system. When applied to systems that are stabilizable by linear time-invariant feedback, this approach yields global asymptotic stability.

Systems with finite communication bandwidth constraints. II. Stabilization with limited information feedback

For part I, see ibid., vol.42, p.1294-8, 1997. In this paper a new class of feedback control problems is introduced. Unlike classical models, the systems considered here have communication channel constraints. As a result, the issue of coding and communication protocol becomes an integral part of the analysis. Since these systems cannot be asymptotically stabilized if the underlying dynamics are unstable, a weaker stability concept called containability is introduced. A key result connects containability with an inequality equation involving the communication data rate and the rate of change of the state.

Systems with finite communication bandwidth constraints. I. State estimation problems

In this paper, we investigate a state estimation problem involving finite communication capacity constraints. Unlike classical estimation problems where the observation is a continuous process corrupted by additive noises, there is a constraint that the observations must be coded and transmitted over a digital communication channel with finite capacity. This problem is formulated mathematically, and some convergence properties are defined. Moreover, the concept of a finitely recursive coder-estimator sequence is introduced. A new upper bound for the average estimation error is derived for a large class of random variables. Convergence properties of some coder-estimator algorithms are analyzed. Various conditions connecting the communication data rate with the rate of change of the underlying dynamics are established for the existence of stable and asymptotically convergent coder-estimator schemes.

Time optimal control in spin systems

In this paper, we study the design of pulse sequences for nuclear magnetic resonance spectroscopy as a problem of time optimal control of the unitary propagator. Radio-frequency pulses are used in coherent spectroscopy to implement a unitary transfer between states. Pulse sequences that accomplish a desired transfer should be as short as possible in order to minimize the effects of relaxation and to optimize the sensitivity of the experiments. Here, we give an analytical characterization of such time optimal pulse sequences applicable to coherence transfer experiments in multiple-spin systems. We have adopted a general mathematical formulation, and present many of our results in this setting, mindful of the fact that new structures in optimal pulse design are constantly arising. From a general control theory perspective, the problems we want to study have the following character. Suppose we are given a controllable right invariant system on a compact Lie group. What is the minimum time required to steer the system from some initial point to a specified final point? In nuclear magnetic resonance (NMR) spectroscopy and quantum computing, this translates to, what is the minimum time required to produce a unitary propagator? We also give an analytical characterization of maximum achievable transfer in a given time for the two-spin system.

System theory on group manifolds and coset spaces.

The purpose of this paper is to study questions regarding controllability, observability, and realization theory for a particular class of systems for which the state space is a differentiable manifold which is simultaneously a group or, more generally, a cosec space. We show that it is possible to give rather explicit expressions for the reachable set and the set of indistinguishable states in the case of autonomous systems. We also establish a type of state space isomorphism theorem. These results parallel, and in part specialize to, results available for the familiar case described by

Hybrid Models for Motion Control Systems

\dot x(t) = Ax(t) + Bu(t)

Control Theory and Singular Riemannian Geometry

,

Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems

y(t) = Cx(t)

Paper: Volterra series and geometric control theory

. Our objective is to reduce all questions about the system to questions about Lie algebras generated from the coefficient matrices entering in the description of the system and in that way arrive at conditions which are easily visualized and tested.

Future directions in control in an information-rich world

This paper presents a mathematical framework for modeling motion control systems and other families of computer controlled devices that are guided by symbol strings as well as real valued functions. It discusses a particular method of organizing the lower level structure of such systems and argues that this method is widely applicable. Because the models used here capture important aspects of control and computer engineering relevant to the real-time performance of symbol/signal systems, they can be used to explore questions involving the joint optimization of instruction set and speed of execution. Finally, some comparisons are made between engineering and biological motion control systems, suggesting that the basic ideas advanced here are consistent with some aspects of motor control in animals.