Cybernetical physics

Cybernetical physics is a scientific area on the border of Cybernetics and Physics which studies physical systems with cybernetics methods. Cybernetics methods are understood as methods developed within control theory, information theory, systems theory and related areas: control design, estimation, identification, optimization, pattern recognition, signal processing, image processing, etc. Physical systems are also understood in a broad sense: either lifeless or living nature or of artificial (engineering) origin, having reasonably understood dynamics and models suited for posing cybernetical problems. Research objectives in cybernetical physics are frequently formulated as analyses of a class of possible system state changes under external (controlling) actions of a certain class. An auxiliary goal is design of controlling actions required to achieve prespecified property change. Among typical control action classes are functions which are constant in time (bifurcation analysis, optimization), functions which depend only on time (vibration mechanics, spectroscopic studies, program control), and functions whose value depends on measurement results at the same time or previous instants. The last class is of special interest since these functions correspond to system analysis by means of external feedback (feedback control).

Roots of cybernetical physics

Until recently no creative interaction of physics and control theory (cybernetics) has been seen and no control theory methods have been directly used for discovering new physical effects and phenomena. The situation has dramatically changed in the 1990s when two new areas emerged: control of chaos and quantum control.

Control of chaos

In 1990 the paper ^[1] was published in Physical Review Letters by Edward Ott, Celso Grebogi and James Yorke from the University of Maryland discovered that even small feedback action can dramatically change behavior of a nonlinear system, e.g. turn chaotic motions into periodic ones and vice versa. The idea became popular in the physics community almost immediately and since 1990 hundreds of papers were published demonstrating the ability of small control, with or without feedback, to change dynamics of real or model systems significantly. By 2003, the Ott, Grebogi and Yorke's paper^[1] has been quoted over 1300 times whilst the total number of the papers relating to control of chaos exceeded 4000 by the beginning of the 21st century. The number of papers published in peer reviewed journals achieved 300-400 papers per year. The method proposed in ^[1] is now called the OGY-method after the authors' initials.

Later a number of other methods were proposed for transformation of chaotic trajectories into periodic ones? E.g. delayed feedback (Pyragas method) ^[2]. Numerous nonlinear and adaptive control methods were also applied for control of chaos, see surveys in ^[3][4][5][6].

It is important that the results obtained were interpreted as discovering new properties of physical systems. Thousands of papers were published that examine and predict properties of systems based on using control, identification and other cybernetic methods. Notably, an overwhelming part of those papers were published in physical journals, their authors were representing physical departments of the universities. It has become clear that such type of control goals are important not only for control of chaos, but also for control of a broader class of oscillatory processes. This provides evidence for the existence of the new emerging field of research related to both physics and control, that of ‘’’Cybernetical Physics’’’ ^[7][8],

Quantum Control

It is conceivable that molecular physics was the area where ideas of control appeared first. One may trace its roots back to the Middle Ages, where alchemists were seeking ways to change a natural course of chemical reactions whilst attempting to transform lead into gold. The next milestone was set by the famous British physicist James Clerk Maxwell. In 1871 he introduced a hypothetical being, known under the name Maxwell's Demon with ability to measure velocities of gas molecules and to direct fast molecules to one part of the vessel, keeping slow molecules in the other part. It produces a temperature difference between the two parts of the vessel which seems to contradict the Second Law of thermodynamics. Now, after more than a century of fruitful life, Demon is even more active than in the past. In recent papers the issues of experimental implementation of the Maxwell's Demon are discussed, particularly at the quantum-mechanical level ^[9].

In the end of the 1970s the first mathematical results for control of quantum mechanical models were established based on control theory ^[10] In the end of the 1980s—beginning of the 1990s rapid development of laser industry led to appearance of ultrafast, the so called femtosecond lasers. A new generation of lasers have the ability to generate pulses with duration of about a few femtoseconds and even less (1 fs = 10 ^{− 15}sec). The duration of such a pulse is comparable with the period of a molecule's natural oscillation. Therefore, femtosecond laser can be, in principle, used as a mean for control of single molecules and atoms. A consequence of such an application is a possibility of realizing an alchemists' dream to change the natural course of chemical reactions. A new area in chemistry emerged – femtochemistry and new femtotechnologies were developed. Ahmed Zewail from Caltech was awarded with the 1999 Nobel Prize in Chemistry for his work on femtochemistry.

Using the apparatus of modern control theory, new horizons in studying interaction of atoms and molecules may open and new ways and possible limits for intervention into intimate processes of the microworld may be discovered. Besides, control is an important part of many recent nanoscale applications: nanomotors, nanowires, nanochips, nanorobots, etc. The number of publications in the peer reviewed journals per year exceeds 600.

Control Thermodynamics

The basics of thermodynamics were stated by Sadi Carnot in 1824. He considered a heat engine which operates by drawing heat from a source which is at thermal equilibrium at temperature T_hot, and delivering useful work. Carnot saw that, in order to operate continuously, the engine requires also a cold reservoir with the temperature T_cold, to which some heat can be discharged. By simple logic he established the famous ‘’’Carnot's Principle’’’: ‘’No heat engine can be more efficient than a reversible one operating between the same temperatures’’.

In fact it was nothing but the solution to an optimal control problem: maximum work can be extracted by a reversible machine and the value of extracted work depends only on the temperatures of the source and the bath. Later Kelvin introduced his absolute temperature scale (Kelvin scale) and accomplished the next step, evaluating the Carnot's reversible efficiency $\eta_{Carnot}=1-\frac{T_{cold}}{T_{hot}}.$ However, most works were devoted to study of stationary systems over infinite time interval, while for practical purposes it is important to know possibilities and limitations of the system evolution for finite times as well as under other types of constraints caused by a finite amount of resources available.

The pioneer works devoted to evaluation of finite time limitations for heat engines were published by I.Novikov in 1957^[11] and, independently by F.L.Curzon and B.Ahlborn in 1975 ^[12] that the efficiency at maximum power per cycle of a heat engine coupled to its surroundings through a constant heat conductor is$\eta_{NCA}=1-\sqrt(\frac{T_{cold}}{T_{hot}}).$ (Novikov-Curzon-Ahlborn formula). Note the Novikov-Curzon-Ahlborn process is also optimal in the sense of minimal dissipation. Otherwise, if the dissipation degree is given, the process corresponds to the maximum entropy principle. Later the results ^[12][11] were extended and generalized for other criteria and for more complex situations based on the modern optimal control theory. As a result a new direction in thermodynamics arose known under the names ‘’optimization thermodynamics’’, ‘’finite-time thermodynamics’’, Endoreversible thermodynamics or '’control thermodynamics’’ see ^[13].

Subject and Methodology of Cybernetical Physics

By the end of the 1990s it has become clear that a new area in physics dealing with control methods has emerged. The term ‘’’cybernetical physics’’’ was proposed in^[7][8]. Subject and methodology of the field are systematically presented in ^[14][15].

Description of the control problems related to cybernetical physics includes classes of controlled plant models, control objectives (goals) and admissible control algorithms. The methodology of cybernetical physics, comprises typical methods used for solving the problems and typical results in the field.

Models of Controlled Systems

A formal statement of any control problem begins with a model of the system to be controlled (plant) and a model of the control objective (goal). Even if the plant model is not given (like in many real world applications) it should be determined in some way. The system models used in cybernetics are similar to traditional models of physics and mechanics with one difference; the inputs and outputs of the model should be explicitly specified. The following main classes of models are considered in the literature related to control of physical systems: continuous systems with lumped parameters described in state space by differential equations; distributed (spatio-temporal) systems described by partial differential equations; discrete-time state-space models described by difference equations.

Control Goals

It is natural to classify control problems by their control goals. We list here five kind of goals.

Regulation (often called stabilization or positioning) is the most common and simple control goal. Regulation is understood as driving the state vectorx(t) (or the output vectory(t)) to some equilibrium statex * (respectively, y * ).

Tracking. State tracking is driving a solutionx(t) to the prespecified function of timex * (t). Similarly, output tracking is driving the outputy(t) to the desired output functiony * (t). The problem is more complex if the desired equilibrium x * or trajectory x * (t) are unstable in the absence of control action. For example, a typical problem of chaos control can be formulated as tracking of an unstable periodic solution (orbit). The key feature of the control problems for physical systems is that the goal should be achieved by means of sufficiently small control. A limit case is stabilization of a system by an arbitrarily small control. Solvability of this task is not obvious if the trajectory x * (t) is unstable, like for the case of chaotic systems, see ^[1].

Generation (excitation) of oscillations. The third class of control goals corresponds to the problems of ‘’excitation’’ or ‘’generation’’ of oscillations. Here, it is assumed that the system is initially at rest. The problem is to find out if it is possible to drive it into an oscillatory mode with the desired characteristics (energy, frequency, etc.) In this case the goal trajectory of the state vector x * (t) is not prespecified. Moreover, the goal trajectory may be unknown, or may even be irrelevant to the achievement of the control goal. Such problems are well known in electrical, radio engineering, acoustics, laser and vibrational technologies—wherever it is necessary to create an oscillatory mode for the system. Such a class of control goals can be related to problems of dissociation, ionization of molecular systems, escape from a potential well, chaotization and other problems related to growth of the system energy and its possible phase transition. Sometimes such problems can be reduced to tracking, but the reference trajectories x * (t) in these cases are not necessarily periodic and may be unstable. Besides, the goal trajectory x * (t) may be known only partially.

Synchronization. The fourth important class of control goals corresponds to synchronization (more accurately, ‘’controlled synchronization’’ as distinct from ‘’autosynchronization’’ or ‘’self-synchronization’’). Generally speaking, synchronization is understood as concurrent change of the states of two or more systems or, perhaps, concurrent change of some quantities related to the systems, e.g. equalizing of oscillation frequencies. If the required relation is established only asymptotically, one speaks about ‘’asymptotic synchronization’’. If synchronization does not exist in the system without control we may pose the problem as finding the control function which ensures synchronization in the closed-loop system, i.e. synchronization may be a control goal. Synchronization problem differs from the model reference control problem in that some phase shifts between the processes are allowed that are either constant or tend to constant values. Besides, in a number of synchronization problems the links between the systems to be synchronized are bidirectional ones. In such cases the limit mode (synchronous mode) in the overall system is not known in advance.

Modification of the limit sets (attractors) of the systems. The last class of the control goals is related to modification of some quantitative characteristics of the limit behavior of the system. It includes such specific goals as

• changing the type of the equilibrium (e.g. transformation of an unstable equilibrium into a stable one or vice versa);
• changing the type of the limit set (e.g. transformation of a limit cycle into a chaotic attractor or vice versa, changing fractal dimension of the limit set, etc.);
• changing the position or the type of the bifurcation point in the parameter space of the system.

Investigation of the above problems started in the end of the 1980s with the works on bifurcation control and continued in the works on control of chaos. Ott, Grebogi and Yorke ^[1] and their followers introduced a new class of control goals, not requiring any quantitative characteristic of the desired motion. Instead, the desired qualitative type of the limit set (attractor) was specified, e.g. control should provide the system with a chaotic attractor. Additionally, the desired degree of chaoticity may be specified by means of specifying Lyapunov exponent, fractal dimension, entropy, etc. see ^[4][5].

In addition to the main control goal some additional goals or constraints may be specified. A typical example is the ``small control" requirements: control function should have small power or should provide small expenditure of energy. Such a restriction is needed to avoid ``violence" and preserve inherent properties of the system under control. This is important to ensure elimination of artefacts and adequate study of the system. Three types of control are used in physical problems: constant control, feedforward control and feedback control. Implementation of a feedback control requires additional measurement devices working in real time which are often hard to install. Therefore, studying the system may start with application of inferior forms of control: time-constant and then feedforward control. The possibilities of changing the system behavior by means of feedback control can then be studied.

Methodology

The methodology of cybernetical physics is based on control theory. Typically, some parameters of physical systems are unknown and some variables are not available for measurement. From the control viewpoint this means that control design should be performed under significant uncertainty, i.e. methods of robust control or adaptive control should be used. A variety of design methods have been developed by control theorists and control engineers for both linear and nonlinear systems. Methods of partial control, control by weak signals, etc. have also been developed.

Fields of research and Prospects

Currently an interest in application of control methods in physics is still growing. The following areas of research are being developed actively ^[14][15]:

• Control of oscillations
• Control of synchronization
• Control of chaos, bifurcations
• Control of phase transitions, stochastic resonance
• Optimal control in thermodynamics
• Control of micromechanical, molecular and quantum systems

Among most important applications are: control of fusion, control of beams, control in nano- and femtotechnologies.

In order to facilitate information exchange in the area of cybernetical physics the International Physics and Control Society (IPACS) was created The IPACS organizes regular conferences (Physics and Control Conferences), supports electronic library IPACS Electronic Library and information portal Physics and Control Resources.

References

1. ^ ^{a b c d e} Ott E., Grebogi C., Yorke G. Controlling chaos. Phys. Rev. Lett. 1990. V.64. (11) 1196-1199.
2. ^ Pyragas K. Continuous control of chaos by self-controlling feedback. Phys. Lett. A. 1992. V.170. 421-428.
3. ^ Fradkov A.L., Pogromsky A.Yu., Introduction to control of oscillations and chaos. Singapore: World Scientific Publ., 1998.
4. ^ ^{a b} Andrievsky, B. R. \& Fradkov, A. L. 2003. Control of chaos. I. Methods. Autom. Remote Control, Vol. 64, No. 5, 2003, pp. 673--713.
5. ^ ^{a b} Andrievsky, B. R., Fradkov, A. L. Control of Chaos. II. Applications. Autom. Remote Control. Vol. 65, 4, 2004, pp. 505--533.
6. ^ Handbook of Chaos Control, Second completely revised and enlarged edition, Eds: E. Schoell, H.G. Schuster. Wiley-VCH, 2007.
7. ^ ^{a b} Fradkov A.L. Exploring nonlinearity by feedback. Physica D. 1999, V. 128, N 2-4. 159-168.
8. ^ ^{a b} Fradkov AL. Investigation of physical systems by feedback. Autom. Remote Control 60 (3): 471-483, 1999.
9. ^ Leff H.S. and A.F.Rex (Eds). Maxwell's Demon 2: entropy, classical and quantum information, computing: 2nd edition. Institute of Physics. 2003.
10. ^ Butkovskii A.G., Samoilenko Yu.I. Control of Quantum-Mechanical Processes. Dordrecht: Kluwer Acad. Publ., 1990 (In Russian: Moscow: Nauka, 1984,
11. ^ ^{a b} Novikov I.I., The efficiency of atomic power stations, Atomic Energy 3 (11), 409--412, 1957; (English translation: Nuclear Energy II 7. 125–-128, 1958).
12. ^ ^{a b} Curzon F.L., Ahlburn B., Efficiency of a Carnot engine at maximum power output. Am.J. Phys., 43, 22-24, 1975.
13. ^ Berry R.S., Kazakov V.A., Sieniutycz S., Szwast Z., Tsirlin A.M. Thermodynamic Optimization of Finite Time Processes. Wiley. N.Y., 2000.
14. ^ ^{a b} A.L. Fradkov. Application of cybernetical methods in physics. Physics-Uspekhi, Vol. 48 (2), 2005, 103-127.
15. ^ ^{a b} Fradkov A.L. Cybernetical physics: from control of chaos to quantum control. Springer-Verlag, 2007, (Preliminary Russian version: St.Petersburg, Nauka, 2003).

External links

• Portal, Physics and Control Resources
• IPACS Electronic Library
• International Physics and Control Society (IPACS)