Nové paradigmy teórie riadenia
NEW PARADIGMS OF CONTROL THEORY

Jan SARNOVSKY

Department of Cybernetics and Artificial Intelligence, TU Kosice, email: sarnovj@tuke.sk

ABSTRACT: The dominant paradigms in control theory are described in the contribution. The evolution of the control theory as a part of science is based on the classical cumulativistic development of paradigm where an old paradigm is replaced by a new one. From this point of view it is very important to apply the correspondence principle in the control theory. The analysis of paradigm of control theory enables the anticipation of possible development of the control theory in the sense of determining or prognosing the new paradigm. In connection with the development of the control theory it is necessary to take into account evolutionary cybernetics and interdisciplinary connections.

Key words: control theory, paradigm, evolution

1.      INTRODUCTION

The problems of science development and the development of scientific theories especially, are one of the most discussed questions in present methodology of sciences. The problem of the relation between logical structure of science knowledge and its history is particularly expressive in scientific revolution, which is proceeding in natural sciences of our age. Niels Bohr, when he was building up the theory of atom, he was formulating “the correspondence principle”, which is regarded as the most important general principle of methodology of sciences. It has become the central point of interests of philosophers, logicians, and methodologist examining the problems and science development, and the crucial point of various philosophical and methodological conceptions. The Bohr’s principle having expressed the correspondence between the old theory T₁ and new theory T₂ suppose:

1.      using to the extreme the idea of old theory at the constructing of theoretical model of new theory,

2.      introducing the new postulates in the case of rising of antinomies, which enable its overcoming,

3.      building up the new theory T₂ from which it is possible to deduce theory T₁^’ corresponding with the kernel of original theory T₁, and T₁^’ needn’t be identical with T_1.

The Bohr’s principle contradicts with the viewpoint of classical cumulativism, which the development of science cognition understands as the gradual growth of knowledge. This principle, however, contradict with extreme anticumulativism.

2.      PHILOSOPHICAL AND METHODOLOGICAL CONCEPTIONS

The problems, which have been given above, are solved by various philosophical and methodological conceptions.

a.      Logical positivism comes out from the principles of the extreme empirism and verificionism and incline to classical cumulativism.

b.      The Popper philosophy of science comes out from some antagonistic attitudes against positivism, for example it pose inductionism versus deductionism, empirism versus critical rationalism and especially versus so called methodological falsificationism. The growth of scientific cognition is possible to express by scheme:P₁ - TT - EE - P₂, where P₁ are problems to be solved, TT are the tentative theories, EE is elimination of errors, and P₂ are new problems.

c.      Historical school (Kuhn, Lakatos, Feyerabend) – the representatives of this school acutely condemn empirism, ahistorism, cumulativism, etc.

The basic notions of Kuhn’s conception are “paradigm” and  “scientific community”. The structure of paradigm is certain metaphysic (ontological) assertions, values and models modes of solving of concrete tasks. The scientific community is the group (groups) of scientists which confess and “believe” in a certain paradigm. From there we have the notion of “normal science”, which is based on the classical cumulativistic development of paradigm and the notion “scientific revolution” which means the replacing of an old paradigm by a new one and has anticumulativistic character. The Kuhn’s cycle of the scientific knowledge has the period of normal science: (the using paradigm to the known facts, the extending of its limits, to the solving of “puzzles”, the interpretation in the textbooks). The next is the period of crisis (the scientific community is ceasing “to believe” in old paradigm and is begin “to believe in a new one). The scientific revolution is the solving of the qualitative changes in the development of scientific knowledge.

Lakatos had attempted about the definite synthesis of Popper’s and Kuhn’s conceptions. He has introduced the notion “research programs”, whereby the function of paradigm has so called “firm kernel” of research program, which is commonly accepted and as such is irrefutable. The source of development, according to Lakatos, is the concurrency of research programs.

The most radical wing of this school represent P.K. Feyerabend, which in the essence deny the correspondence principle and offers the conception of “pluralistic methodology”, respectively, “the anarchistic methodology” of knowledge. In the sense of this conception, the philosophy of science is changing to the philosophy of history of science.

3.      THE PARADIGMS IN CONTROL THEORY

In the following we will characterise the development of control theory on the basis of cited above notions, we will define its paradigms, and on the basis of such analysis we will determine the destination of control theory in the next period.

The paradigm of classic control theory (CT)

The development of CT, its history, is the subject of many works in which are introduced the fundamental stages of CT development, the contributions of individual scientists, the prognosis of the following development. Although the evaluating and approaches of individual authors are various, yet it is possible to follow the main evolution line of CT. As in the encyclopaedia of cybernetics is stated, the foundations of CT as a science are set in the works of English physicist G. Maxwell, the Russian scientist I. A. Vyshnegradskij and A. A. Lyapunov, the Slovak scientist A. Stodola. This first evolution stage very good elaborated by M. Tolle in the book “Regulung der Kraft- machinen, Berlin, Springer, 1905), had ended on the beginning of 20^th century. According to A. A.  Andropov, A. Stodola had very important role in the development of CT, while the works of Tolle evaluate as a pedagogical and compiling.

The introduced stage we can name in the sense of T. Kuhn as the paradigm of classical CT. In the next period is CT unceasingly unfolding, improved and completed (the stage of “normal science”), for example diagrams of Nyquist (1932)   Michajlov (1938) and Nejmark( 1947), stability criteria of Routh-Schur, Hurwitz (by the stimulus of A. Stodola); there are solved the problems of linearization of non-linear system, problems of autonomous and invariant systems, and the problems connected with the acting of stochastic disturbances. The difficulties arising at the solving of the more complex tasks of the synthesis of multivariable systems had necessitated to a certain accumulating of problems, which we can comprehend in the sense of T. Kuhn, as a certain “crisis period”.

The paradigm of optimal and adaptive CT

The solving of big part of these problems was reached in the next period of the development of CT, in the fiftieth years. This stage can be characterised as a certain “revolution” and the change of paradigm, which can be named as “the paradigm of optimal and adaptive CT” (in the sixtieth had been used the name “the modern control theory”). The change of paradigm had had the typical signs of “scientific revolution”.

Let us introduce at least some of them: establishing the notion “the state”(the state space); consistent using state description in the form of vector and matrix differential equation (instead of classical scalar one); using exact mathematical methods (variation calculus, Bellman’s dynamic programming. Pontrjagin’s maximum principle, Kalman’s filtration), instead of various intuitive graphical-computing ones for the determining of regulator parameters. At the same time is worked out theory of adaptive and learning system and begin the using of computers for the design and control of introduced system. From today point of view there had been really “revolution change of paradigm” (from the time viewpoint too; the period of classical paradigm had been some decades, the change of paradigm several years).

The paradigm of large-scale dynamic system control

At the end of sixtieth years each time the more frequently had been creating the claims to solve tasks, which had been characterised by the more complexity. In the realm of large-scale control systems arise the next leap-the next change of paradigm. It follows the paradigm of control of “extra-ordinary complicated systems”.

The all-classical procedures consist in common assumption of centralisation (the information about system and the calculation based on this information are centralised too. It is necessary to stress, that from the theoretical point of view, the notion of centrality is common for both, the classical CT and for modern CT. The assumption of centrality we use e.g. at the SISO systems as well as at the MIMO systems.

In the large-scale control system, the assumption of centrality failures, because of absence of the centralised information, or for the lack of central computing system capacity. The examples of these systems are e.g.: energetic system, transport networks, communicating networks, ecological and economic systems, and the large-scale technological processes. The massive using of microcomputers provokes the need of producing distributed computer control systems. The creating of various information systems with distributed databases represents a new challenge for decentralised decision and control systems.

There is an effort to produce decentralised decision and control systems, distributed computing control systems and hierarchical control system from the economical and reliable reasons. The requiring goals - the producing of distributed computing control structure - are not possible to reach by help of known centralised methods and procedures, connected with classical and modern control theory.

The decentralised control had arisen as a response to these difficulties. The base characteristic of decentralised control is the existence of the interaction among subsystems.   The basic problem of decentralised control methods is the decomposition. The large-scale system is necessary to divide (decompose) to the set of simpler subsystems. The every subsystem has its goal and its activity is in harmony with these goals. The goals of individual subsystems can be contradictory. That is the reason for the coordination of subsystems. The coordinator (the decision unit which perform the coordination) control the activity of individual subsystems in order to attain the global goals.

The paradigm of DEDS

At the beginning of eightieth begin the rapid development of automation of the discrete manufacturing processes. The big research teams were solving the problems of robot control and so-called flexible manufacturing systems (FMS). The control methods, which have been given above, had termed as inconvenient. This crisis situation had necessitated to the arising of “the paradigm of DEDS”(discrete events dynamic system).

At present time in modern technological systems new classes of model need to be developed, because traditional models are not appropriate. An example of these systems is FMS, computer networks, traffic systems, etc. The state of these systems, the evolution of the system is governed by the intricate interactions of discrete events. For these systems does not exist elegant and succinct models and there are no control methods rivalling the economy and the power of these for continuous systems.

The second group of systems, so-called hybrid dynamical systems (HDS), in which the state of systems include both continuous and discrete variables, but also nonnumeric symbolic variable (manipulators with set of sensors, subsystems of technological processes, systems with many failure modes such as the large-scale power systems.

The research of control methods for DEDS and HDS should integrate the results of classical control theory with methods and models such as formal language theory, formal logic, queuing networks, markov chains, Petri nets, and with the development in computer science and principles and methods of artificial intelligence.

The correspondence principle in CT

Let’s analyse how the principle of correspondence is asserted in the development of CT. For the simplicity and shortness we will showe some examples. The classical control theory (theory T₁) was not able to solve the optimal control of multivariable dynamic systems completely and unambiguously. The results which were reached on the base of variational calculus in analytical design of regulators (Letov, Kalman), is expressed by the known relations u = Kx (theory T₂). From the theory T₂ is possible to derive relations for SISO on the base of introduced relations (theory T₁^’), which partially or wholly correspond with the classical theory T₁.

The analogical examples we can find between the paradigm of large-scale dynamic systems and the paradigm of optimal an adaptive CT. At the very high number of state variables arise the problems of stability and quality of system control, because there is needed to solve the big number of equation (theory T₁). The starting point is the decomposition and decentralised control (theory T₂). Such decentralised control correspond, and even, n some cases, is identical with the original centralised design (theory T₁^’).

The more interesting results would bring the analysis of the asserting of correspondence principle between the paradigm of DEDS and the preceding paradigms. With regards to the considerably differences (different mathematical apparatus, difference to defining of control goals, the physical difference of control objects, etc.), makes valuable the principle of correspondence only on the more abstract level, especially in the acceptance of notions and terminology from preceding paradigms of CT.

4.      THE ANTICIPATIONS OF THE NEXT POSSIBLE EVOLUTION

The analysis of paradigm of control theory enables the anticipation of possible development of CT in the sense of determining or prognosing the new paradigm. It is indicated that there are many problems, which arises at the control of large-scale systems in the original cybernetics sense.

The complexity and heterogeneity of considering systems require using the modelling and simulation methods for design and realisation of complex systems. In the connection with simulation the experimentation (including the simulating experiments) is the necessary part of analysis and design. It is particularly necessary especially in the system (e.g. DEDS), where there is not the long tradition in modelling.

To the considerable measure, in the control of complex cybernetics system can contribute the methods of artificial intelligence; especially in the creation of mathematical models of complex systems, in the formalisation of process control and decision processes. Next in the using of programming tools of artificial intelligence for control of cybernetics systems, and in the creation of expert systems for the control and decision.

From the introduced knowledge, as has been given above, results the determining of these objects of CT, which are indicated as a complex systems, multilevel systems, hierarchical systems, etc. It is necessary to stress that the notion of complex systems is one of the base cybernetics categories, and the author had been busy with its in some works referring to philosophical and methodological aspects of cybernetics complex systems and artificial intelligence. The really complex (as S. Beer says-the extraordinary complex systems) can be described by help of some formalism only in the isolated cases.

In the majority definition of cybernetics is stressed that it is the science about control. From these reasons there is necessary to examine the notion of control and its task in the contemporary science knowledge. This notion is reaching, at the present time, far beyond the frame of cybernetics, and in the defining direction serve as a connecting factor of sciences examining the various kinds of control.

The cybernetics system are characteristic by high level of organisation, by ability to secure the processes of development, selforganization and selfreproducing. These systems introduce the highest degree of control systems.

The generally feature of these system are the superiority (qualitative and quantitative) of internal connections over external ones; the presence of integral qualities. In the connections with the fact, that the main content of control is its purposefulness, Ashby and Beer consider as a goal of control the homeostasis. That is, the homeostasis is this qualitative edge, which separate the control processes from among the all processes of material world.

However, the homeostasis does not cover the all essence of control. It is, so saying, the substance of first order. In the structural aspect is characterised by negative feedback and in the functional aspect by goal of keeping of its existence by adapting to its environment.

In the literature as a control goal often occur the keeping or raising of system organisation, the transition from the more probably state to the less probably one, respective by lowering of entropy. But it may occur to some contradictions. For example how to enlist not rare cases when the control goal is the lowering of its level of the organisation, nay the aware of destroying of control object.

The more complicated is the system and its functions, the more complicated are the manifestation of its control. Absolutely particular difficulties arise at the analysis of complex man-machine systems and social systems.

In the connection with the growth of mankind arises many problems which substantially changes the conditions of human existence on the Earth as also the traditional ideas about the technical but also about the social and cultural evolutions; there are creating so-called the global problems of the world evolution. The system approach is used for its study. The most used method is the cybernetics modelling for the creating of so-called “model of the world”. The base goal of global modelling is the studying of various alternatives of the future development of mankind. For such approach to the knowledge of the future is characteristic the original understanding of the present, of the substance and the direction of the future changes.

One of the base principles of the existence of the activity of the complex systems is its controllability. For the time being we don’t know whether exist some limit of the complexity of the system structure. It is not excluding that the exceeding of the defining threshold of the number of the elements, and its interactions, can necessitate to possible disintegration of the system. S. Lem note that the development of the civilisation does not necessitate automatically to the growth of the all-individual rights. On the contrary, the development can be accompanied by establishing of the new “prohibitions”.

The notions which has been given above from the realm of cybernetics and complex systems has the modest goal: to direct attention to the facts, that these science areas can render some methodological starting point at the solving of the intricate problems of the present time. In this way is opened the new paradigm in the control theory-the paradigm of “extraordinary complex (cybernetics) systems”. This future stage of control theory in the harmony with the correspondence principle must include all results of control theory development, and related science realms (e.g. the methods of modelling and simulation, the methods of artificial intelligence, etc. This introduced paradigm is opening the new possibilities in the research programs and its using in the pedagogical process.

At the same time, the ebb-and-tide of scientific fashions had drawn researchers interested in more concrete models from the original “first order” cybernetics to artificial intelligence, and from there to neural networks, autonomous robots, artificial life, and complex adaptive systems. Although all these approaches are closely related, many of the important cybernetic insights seem to have been forgotten in the process. As a result, for many people nowadays “cybernetics” only survives as a fancy label for various technological hypes. The time seems ripe to revive and renew the cybernetic tradition by integrating its ideas with the newly developed insights into evolution and complexity.

Evolutionary cybernetics

Evolutionary cybernetics can be defined as the study of how the processes of variation and selection give rise to organization. This means, first of all, a study of the dynamics of distinctions, connections, variety, closure and constraint, that is, the fundamental aspects of organized complexity. This will allow us to better understand how systems emerge out of unstructured aggregates of components, and how variation and selection take place at different system levels and between different, co-evolving systems.  Then, we need to study the evolution of goal-directedness, i.e. control systems. This problem has two aspects, one qualitative or discrete (how does a control level emerge?), one more quantitative or continuous (how does a control system develop?). The evolutionary emergence of control can be called a metasystem transition, and is the subject of the accordingly named “metasystem transition theory”.  Darwinian reasoning readily explains why control systems tend to emerge: natural selection prefers systems that can survive in spite of a variety of perturbations, and thus control systems that have survival as their basic goal will tend to proliferate. As to the mechanism producing control, it is relatively easy to imagine variation processes closing in on themselves in a negative feedback loop constrained by a particular value of a variable (e.g.  autocatalytic cycles of chemical reactions that depend on the availability of one particular scarce molecule, such as DNA).  Once a control loop has arisen, a cybernetic analysis of the different properties and components of this loop can help us understand how variation and selection can make control more efficient. For example, a greater variety of possible actions will allow the control system to cope with more diverse problems (Ashby’s law of requisite variety), while a greater sensitivity in perception will allow it to distinguish more accurately between different situations. A better internal model of its environment will allow it to more reliably anticipate the results of its actions, while a more balanced hierarchy of goals and values will guide it in choosing those actions that are most likely to contribute to its long-term survival.

5.      Interdisciplinary connections

Evolutionary cybernetics is clearly related to a number of other new disciplines and approaches. It stands to the domain of evolutionary systems (also called “systems theory of evolution”), as cybernetics stands to its sister discipline of systems science: in principle the subject domains mostly overlap, in practice the cybernetics focus is more on functions and goals, while the systems focus is more on structures. This means that evolutionary cybernetics will pay more attention to issues such as values, ethics, future development, and technological applications. Both evolutionary cybernetics and systems are closely related to biosemiotics, which, however, has the more restricted focus of the emergence of closed sign systems or “meaning” in organisms.

The presently popular approach of complex adaptive systems too studies many of the same issues, but lacks a focus on hierarchical organization and especially goal-directedness, preferring the use of methods imported from mathematics, physics and chemistry, where the notion of function or “purpose” is absent. The cybernetic paradigm is more influential in the related disciplines of artificial life and especially autonomous agents, which try to model the development of autonomous, goal-directed systems. However, the emphasis here is purely on implementations of models in hardware or software, without much attention to their necessary theoretical and philosophical foundations, or their applications to wider social and psychological issues. Some of these philosophical, psychological and social issues have become the subject of an array of new disciplines: evolutionary epistemology, ethics, metaphysics, psychology, economics, memetics, etc. However, apart from their common base in Darwinian theory, these approaches stand largely on their own, lacking the transdisciplinary integration that has always been the hallmark of cybernetics and systems science.

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