Some comments on cybernetics and control Citation for published version (APA): Kickert, W. J. M., Bertrand, J. W. M., & Praagman, J. (1978). Some comments on cybernetics and control. IEEE Transactions on Systems, Man and Cybernetics, 8(11), 805-809. https://doi.org/10.1109/TSMC.1978.4309868 DOI: 10.1109/TSMC.1978.4309868 Document status and date: Published: 01/01/1978 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. 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AND JAAP PRAAGMAN The law of requisite variety states that the capacity of a device Abstract The theory of cybernetics as introduced by Ashby and as a regulator cannot exceed its capacity as a channel of communi- developed by Ashby and Conant will be analyzed and commented cation, or to put it in Ashby's words: "only variety in the regula- upon. Ashby's law of requisite variety and, in particular, the underly- tor can force down the variety due to the disturbances; only ing measure of optimality the quantity of entropy are examined. variety can destroy variety." Imagine a system composed of a set Next the cybernetic theorem of error control and cause control is D of disturbances, a set R of control actions, and a set Z of observed, and finally the cybernetic theorem of the necessity of outcomes, defined by a mapping 4: D x R -+ Z. This obviously modeling for regulation is studied. In all three cases several practical represents a control system. By taking finite discrete sets D, R, and conditions and restrictions for the applicability of the theorems to control engineering are pointed out. Z and by visualizing 4 as a table, it can easily be shown that the goal of keeping the outcome Zk E Z constant, that is, of decreas- I. INTRODUCTION ing the variety in the outcomes, can only be met by a correspond- Since the first introduction of cybernetics by N. Wiener [1], as a ing increase in the variety of R. "science of communication and control," numerous contributions The relation between information and control is essentially the to this field have been made. One of the most outstanding contri- following. As the criterion for the success of the regulator (the goal butions, in our opinion, has been made by the late W. R. Ashby, is constancy of the outcomes Z) Shannon's measure of selective who laid a basis for the link between information and control with information in a signal, the entropy H(Z) is introduced: his law Both he and R. C. well-known of requisite variety [2]. H(Z) = - 3 p(zi) log2 P(ZJ) Conant further developed this theory. Some important and well- zi e Z known elements of this development are their views on error- controlled regulators versus cause-controlled regulators [3], and Ashby states that optimality of a regulator R is equivalent to the their views on modeling as a necessary part of regulation [4]. minimization of the entropy H(Z) of the outcomes. One of the These elements only represent part of their work, especially in advantages of this measure of optimality is that it does not view of their recent achievements in the field of complex and presume numerical variables; entropy also applies to variables hierarchical systems [5]-[8]. The analysis and comments in this that can only be classified (nominal). correspondence, however, will be restricted to the above- Strictly speaking, the use of this entropy measure implies that mentioned points, mainly because of their relevance for control. the variables involved (outcomes) are stochastic. (Notice that the There are three -reasons which emphasize the necessity of requirement of stochastic system variables does not imply that the analysis, comments, and criticism. First is the obvious importance system relations be stochastic; on the contrary, most of this cyber- of this branch of cybernetics for control theory. It will be clear netic theory leads to the necessity of deterministic systems rela- that information and communication play an important role in tions.) Most frequently, however, the entropy measure is applied control and that, more specifically, the above-mentioned issues as a measure of variety without strict probability density func- are certainly very important for control theory. The second reason tions. It then serves as a measure of the number of possible alter- is the incomprehensible ignorance of this theory on the part of natives. The assumption behind this use of entropy is that all control engineers. It is astonishing how little attention the theory alternatives have the same probability: of communication, and in particular, the theory ofcybernetics has n n 1 1 received in control theory, though its importance is quite clear. H P(zi)logP(Zi)= log -=log n. The third reason is the unshakable popularity of this theory among system theorists. One might hope that some "cross talk This probability assumption is often omitted. Hence, in fact, the between cyberneticians and control engineers" as Porter calls it entropy measure is not only used in case of stochastic variables [9] will result in a greater appreciation by control engineers of but also with (varying) deterministic variables. Ashby's link between information and control, and in a somewhat It should be remarked that although the use of entropy has an restricted but better founded popularity among system theorists. advantage over classical control theory in that it incorporates stochastic variables, it does not solve the kind of problems that II. THE LAW OF REQUISITE VARIETY [2], [3], [10] are solved by classical control theory, simply because entropy The most famous law ofcybernetics is undoubtedly Ashby's law does not deal with them. Although stability analysis, transfer func- of requisite variety. This very general law, which contrary to the tion theory, etc., do not exclude the existence ofstochastic signals, usual control theory does not presuppose linearity, low-order those theories just do not consider it; they deal with analytical structure, etc., gives an upper limit to the degree of controllability functions in time (pulse, step, ramp, sinusoid, transient response, that a regulator can possibly achieve. In view of the generality of steady state, etc.). In contrast, entropy only considers aggregates of variables, such as density functions and varieties. Hence, a great deal of control theory is not covered by this theory of cybernetics. Manuscript received August 31, 1977; revised July 6, 1978. The authors are with the Department of Industrial Engineering, Technological Secondly, it seems questionable to equate optimality with mini- University of Eindhoven, Eindhoven, The Netherlands. mal entropy. A well-known fact of information theory is that the 0018-9472/78/1100-0805$00.75 ) 1978 IEEE 806 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. SMC-8, NO. 11, NOVEMBER 1978 p(Z) the variance as a measure of variability, it will be clear that con- trol strategy 2) is far better than strategy 1). The variance of (1, 4, 9) is much larger than that of (4, 5, 6), irrespective of any desired mean value. Thus entropy is neither a measure for the mean value of the fluctuating variable, nor a measure for the variance around this mean value. Entropy is a measure for variety, that is, the number of possibilities weighted according to their probabilities of Fig. 1. Comparison of entropies. occurrence. This remark represents quite a serious objection against Ashby's measure of entropy in case of a numerical output variable, so let us examine this somewhat more formally. Entropy entropy H(Z) is minimal when there is one i such that p(Zi) = 1 is defined as and p(zj) = 0, Vj ¢ i, and that H(Z) is maximal when all p(zj), H(Z) Z p(z = zi) log p(z zi), zj E Z are equal.