An information-theoretical approach to regulation

Abstract

A new approach to the analysis of control systems by means of informationtheoretical concepts and techniques is presented which is based on a formulation of the regulation problem in terms of the minimization of a cost function. This approach is much more oriented towards real control problems than the usual method based on entropy functions. Using rate-distortion theory, the requisite variety of the controller and the requisite information from the measurements which are necessary for a certain performance are derived. The results are applied to the disturbance decoupling problem in linear systems. Principal aspects of causecontrolled regulation and error-controlled regulation are discussed. IntroductionOn a heuristic level, it is,obvious that control and the transmission of information within the control system are intimately related. The basic property of closed-loop control is the use of information about the outputs to be controlled and perhaps the internal state of the system for the generation of control signals. On a theoretical level, however, this fact has not yet gained much attention. Only very few attempts have been made to analyse control systems by means of information-theoretical concepts and techniques.The first step in this direction is due to Ashby (1956) who formulated the 'principle of requisite variety' which states that only variety in the actions of the controller can reduce the undesired variety of the controlled variables. The work of Ashby has been continued in the same spirit by other authors (Conant 1969, Weidemann 1.969). The common feature of these approaches is that the performance otjthe control loop is measured by the entropy of the residual error at the output. This explains also why Ashby and Conant confined their work to systems and signals with a finite number of discrete states, for the measurement of the variety of continuous signals is much more complicated than in the discrete case (Engell 1984).The use of the entropy of the residual error as a performance criterion, however, is completely artificial in nearly all practical situations, because there is hardly any direct relevance of this quantity. Moreover, in many cases it is doubtful whether minimization of the entropy of the error leads to an acceptable behaviour of the

Towards quantum cybernetics

“…We will discuss the classical limitations to successful regulation in Section 2, which are summarized by the surprisingly simple law of requisite variety, originally introduced by Ashby [2] and then rediscovered and extended in more recent years [17][18][19]30]. In Section 3, we comment on potential quantization strategies for the regulation protocol of Fig.…”

“…The question is to set general prescriptions to determine the minimal requirements for a successful state transformation and regulation and to explore the limits imposed by the fundamental laws of physics, in particular the second law of thermodynamics. In fact, an information-theoretic analysis of classical state transformation and regulation has been provided [17][18][19], but a full treatment in the quantum mechanical scenario is missing. In a parlance which is familiar to information theorists, we may then ask what is the resource for quantum regulation.…”

“…This has been independently rederived in the contexts of control theory [17][18][19] and computational mechanics [30]. In Eq.…”

“…In general, even if the goal is to achieve a pure state of the system (and this is not always true), the focus on entropy reduction both in the ergodic and non-ergodic case does not distinguish between different pure states. Therefore, while focusing on entropy reduction seems the standard approach to employ information theory to control problems [17,18,44], this is arguably not universally sufficient for control even in the classical case [19,45]. Optimal regulation should then be benchmarked by the minimisation of an appropriate, experimentally appealing cost function.…”