Schulz M. Control Theory in Physics and Other Fields of Science: Concepts, Tools, and Applications
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Preface XI
References
1. S. Sambursky, The physical world of the Greeks (London, 1956).
2. O. Neugebauer, The exact science in Antiquity (Princeton, 1957).
3. S. Sambursky, The physical world of the late antiquity (London, 1962).
4. G. Galilei: Dialogues concerning two New Sciences, translated by H. Crew and
A. de Salvio (Prometheus Books, Buffalo N.Y., 1998).
5. P Costabel, J. Peiffer: Die Gesammelten Werke der Mathematiker und Physiker
der Familie Bernoulli (Birkh¨auser, Basel, 1988).
Contents
1 Introduction ............................................... 1
1.1 The Aim of Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Dynamic State of Classical Mechanical Systems . . . . . . . . . . . . . 3
1.3 Dynamic State of Complex Systems . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.1 What Is a Complex System? . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.2 Relevant and Irrelevant Degrees of Freedom. . . . . . . . . . . 9
1.3.3 Quasi-Deterministic Versus Quasi-Stochastic Evolution . 10
1.4 The Physical Approach to Control Theory . . . . . . . . . . . . . . . . . . 13
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Deterministic Control Theory ............................. 17
2.1 Introduction: The Brachistochrone Problem . . . . . . . . . . . . . . . . . 17
2.2 The Deterministic Control Problem . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.1 Functionals, Constraints, and Boundary Conditions . . . . 19
2.2.2 Weak and Strong Minima . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 The Simplest Control Problem: Classical Mechanics . . . . . . . . . . 22
2.3.1 Euler–Lagrange Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.2 Optimum Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.3 One-Dimensional Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 General Optimum Control Problem . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4.1 Lagrange Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4.2 Hamilton Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4.3 Pontryagin’s Maximum Principle . . . . . . . . . . . . . . . . . . . . 42
2.4.4 Applications of the Maximum Principle . . . . . . . . . . . . . . 45
2.4.5 Controlled Molecular Dynamic Simulations . . . . . . . . . . . 53
2.5 The Hamilton–Jacobi Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
XIV Contents
3 Linear Quadratic Problems ................................ 61
3.1 Introduction to Linear Quadratic Problems . . . . . . . . . . . . . . . . . 61
3.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.1.2 The Performance Functional . . . . . . . . . . . . . . . . . . . . . . . . 62
3.1.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.1.4 The General Solution of Linear Quadratic Problems . . . 71
3.2 Extensions and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.2.1 Modifications of the Performance . . . . . . . . . . . . . . . . . . . . 73
3.2.2 Inhomogeneous Linear Evolution Equations . . . . . . . . . . . 75
3.2.3 Scalar Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.3 The Optimal Regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.3.1 Algebraic Ricatti Equation . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.3.2 Stability of Optimal Regulators . . . . . . . . . . . . . . . . . . . . . 79
3.4 Control of Linear Oscillations and Relaxations . . . . . . . . . . . . . . 81
3.4.1 Integral Representation of State Dynamics . . . . . . . . . . . . 81
3.4.2 Optimal Control of Generalized Linear Evolution
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.4.3 Perturbation Theory for Weakly Nonlinear Dynamics . . 88
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4 Control of Fields .......................................... 93
4.1 Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.1.1 Classical Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.1.2 Hydrodynamic Field Equations . . . . . . . . . . . . . . . . . . . . . 99
4.1.3 Other Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2 ControlbyExternal Sources ..............................103
4.2.1 General Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2.2 Control Without Spatial Boundaries . . . . . . . . . . . . . . . . . 104
4.2.3 Passive Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 114
4.3 Control via Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 116
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5 Chaos Control .............................................123
5.1 Characterization of Trajectories in the Phase Space . . . . . . . . . . 123
5.1.1 General Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.1.2 Conservative Hamiltonian Systems . . . . . . . . . . . . . . . . . . 124
5.1.3 Nonconservative Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.2 Time-Discrete Chaos Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.2.1 Time Continuous Control Versus Time Discrete Control 128
5.2.2 Chaotic Behavior of Time Discrete Systems . . . . . . . . . . . 132
5.2.3 Control of Time Discrete Equations . . . . . . . . . . . . . . . . . . 135
5.2.4 Reachability and Stabilizability . . . . . . . . . . . . . . . . . . . . . 137
5.2.5 Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.3 Time-Continuous Chaos Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.3.1 Delayed Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Contents XV
5.3.2 Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6 Nonequilibrium Statistical Physics .........................149
6.1 Statistical Approach to Phase Space Dynamics . . . . . . . . . . . . . . 149
6.1.1 The Probability Distribution . . . . . . . . . . . . . . . . . . . . . . . . 149
6.2 The Liouville Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.3 Generalized Rate Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.3.1 Probability Distribution of Relevant Quantities. . . . . . . . 153
6.3.2 The Formal Solution of the Liouville Equation . . . . . . . . 155
6.3.3 The Nakajima–Zwanzig Equation . . . . . . . . . . . . . . . . . . . . 156
6.4 Notation of Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.4.1 Measures of Central Tendency . . . . . . . . . . . . . . . . . . . . . . 161
6.4.2 Measure of Fluctuations around the Central Tendency . 162
6.4.3 Moments and Characteristic Functions . . . . . . . . . . . . . . . 162
6.4.4 Cumulants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.5 Combined Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.5.1 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.5.2 Joint Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.6 Markov Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.7 Generalized Fokker–Planck Equation . . . . . . . . . . . . . . . . . . . . . . . 169
6.7.1 Differential Chapman–Kolmogorov Equation . . . . . . . . . . 169
6.7.2 Deterministic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.7.3 Markov Diffusion Processes . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.7.4 Jump Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.8 Correlation and Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.8.1 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.8.2 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.8.3 Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.9 Stochastic Equations of Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.9.1 The Mori–Zwanzig Equation . . . . . . . . . . . . . . . . . . . . . . . . 179
6.9.2 Separation of Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . 182
6.9.3 Wiener Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
6.9.4 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . 185
6.9.5 Ito’s Formula and Fokker–Planck Equation . . . . . . . . . . . 189
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
7 Optimal Control of Stochastic Processes ...................193
7.1 Markov Diffusion Processes under Control . . . . . . . . . . . . . . . . . . 193
7.1.1 Information Level and Control Mechanisms . . . . . . . . . . . 193
7.1.2 Path Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
7.1.3 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
7.2 Optimal Open Loop Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
7.2.1 Mean Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
7.2.2 Tree Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
XVI Contents
7.3 Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
7.3.1 The Control Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
7.3.2 Linear Quadratic Problems . . . . . . . . . . . . . . . . . . . . . . . . . 210
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
8 Filters and Predictors .....................................213
8.1 Partial Uncertainty of Controlled Systems . . . . . . . . . . . . . . . . . . 213
8.2 Gaussian Processes ......................................215
8.2.1 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . 215
8.2.2 Convergence Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
8.3 L´evyProcesses..........................................223
8.3.1 Form-Stable Limit Distributions . . . . . . . . . . . . . . . . . . . . . 223
8.3.2 Convergence to Stable L´evy Distributions . . . . . . . . . . . . 226
8.3.3 Truncated L´evy Distributions . . . . . . . . . . . . . . . . . . . . . . 227
8.4 RareEvents ............................................228
8.4.1 The Cram´erTheorem..............................228
8.4.2 Extreme Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
8.5 Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
8.5.1 Linear Quadratic Problems with Gaussian Noise . . . . . . 232
8.5.2 Estimation of the System State . . . . . . . . . . . . . . . . . . . . . 232
8.5.3 Ljapunov Differential Equation . . . . . . . . . . . . . . . . . . . . . . 237
8.5.4 Optimal Control Problem for Kalman Filters . . . . . . . . . 239
8.6 Filters and Predictors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
8.6.1 General Filter Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
8.6.2 Wiener Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
8.6.3 Estimation of the System Dynamics . . . . . . . . . . . . . . . . . 245
8.6.4 Regression and Autoregression . . . . . . . . . . . . . . . . . . . . . . 246
8.6.5 The Bayesian Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
8.6.6 Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
9 Game Theory ..............................................265
9.1 UnpredictableSystems ...................................265
9.2 Optimal Control and Decision Theory . . . . . . . . . . . . . . . . . . . . . . 267
9.2.1 Nondeterministic and Probabilistic Regime . . . . . . . . . . . 267
9.2.2 Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
9.3 Zero-Sum Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
9.3.1 Two-Player Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
9.3.2 Deterministic Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
9.3.3 Random Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
9.4 Nonzero-Sum Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
9.4.1 Nash Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
9.4.2 Random Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
Contents XVII
10 Optimization Problems ....................................279
10.1 Notations of Optimization Theory . . . . . . . . . . . . . . . . . . . . . . . . . 279
10.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
10.1.2 Convex Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
10.2 Optimization Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
10.2.1 Extremal Solutions Without Constraints . . . . . . . . . . . . . 282
10.2.2 Extremal Solutions with Constraints . . . . . . . . . . . . . . . . . 285
10.2.3 Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
10.2.4 Combinatorial Optimization Problems . . . . . . . . . . . . . . . 287
10.2.5 Evolution Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
1
Introduction
1.1 The Aim of Control Theory
Control theory plays an important role in several fields of economics, tech-
nological sciences, and mathematics. On the other hand, control theory is
not a common expression in natural sciences. That is all the more surprising,
because both, scientific theories and scientific experiments, actually contain
essential features of control theoretical concepts. This appraisal also applies
to physics, although especially in this case many subdisciplines, for example,
mechanics or statistical physics, are strongly related to several ideas of control
theory.
We always speak about control theory in connection with well-defined sys-
tems. In order to be more precise, control theory deals with the behavior of
dynamic systems over time. The controllability of such systems does not nec-
essarily depend on the degree of complexity. From a general point of view, we
should distinguish between external control and intrinsic control mechanisms.
The external control is also denoted as an open loop control. In principle,
this control transmits a certain protocol onto the dynamics of the system. In
this case, it is unimportant how the practical control is achieved. It may be the
result of a personal control by an observer or of a previously fixed program. If
the transmission of the program is ended, the system remains in its last state
or it follows its own free dynamics further.
As an example, consider the flight of an airplane. In this case, the system
is the machine itself. The goal of the control is to bring the airplane from
airport A to another airport B. The pilot itself may be interpreted as an
external controller of the system. Now let us assume that all the activities of
the pilot are recorded in a protocol. A theoretical way to repeat the flight from
A to B under the same control is to implement the protocol in the autopilot of
the airplane. This is a correct conduct if the airplane is driving perfectly under
the same boundary conditions. In reality, the airplane will lose the planned
way because direction and strength of the wind, temperature and air pressure
vary considerable.
M. Schulz: Control Theory in Physics and other Fields of Science
STMP 215,1–15 (2006)
c
Springer-Verlag Berlin Heidelberg 2006
2 1 Introduction
Obviously the main disadvantage of an open-loop control type is the lack
of sensitivity to the dynamics of the controlled system in its time-dependent
environment, because there is no direct connection between the output of
the system and its input. Therefore, the external control plays an important
role especially for systems with a few degrees of freedom and reproducible
boundary conditions.
To avoid the problems of the external control it is necessary to introduce
feedback mechanisms. The output of the system is fed back to any change of
the current dynamics of the system to a desired reference dynamics. environ-
ment. The controller measures the difference between the reference dynamics
and the output, i.e., the current error, to change the inputs to the system.
This kind of control is also denoted as a closed-loop control or feedback con-
trol. In the case of our example, a feedback control may be realized by con-
necting the autopilot with instruments which measure position, altitude, and
flight-direction of the airplane so that each deviation from the course may
be immediately corrected. Another possibility of obtaining a closed-loop con-
trol is to enlarge formally the current system “airplane” to the more complex
system “human pilot and airplane”.
A real dividing line between systems which are favored for an exclusive ex-
ternal control or an exclusive feedback control cannot be defined. The choice
of an appropriate control mechanism especially for technological systems is
the task of control engineering. This discipline focuses on the mathematical
modeling of systems of a diverse nature, analyzing their dynamic behavior,
and using control theory to make a controller that will cause the systems to
behave in a desired manner. The field of control within chemical engineer-
ing is often known as process control. It deals primarily with the control of
variables in a chemical process in a plant. We expect definitely an increasing
number and increasing variety of the intrinsic control mechanisms if the de-
gree of complexity of the system under control increases. The predominant
part of the control mechanisms of extremely complex systems is mostly a
result of a hierarchical self-organization. Typical examples of such more or
less self-controlled systems are biological organisms and social or economic
systems with an enormous number of, partially still nonenlightened, control
mechanisms.
The control of a system may be realized by different methods. Small phys-
ical systems may be sufficiently controlled by a change of boundary condi-
tions and the variation of external fields while more complex systems become
controllable via various flexible system parameters or by the injection and
extraction, respectively, of energy or matter. But we remark that all possible
variable quantities of a system are basically usable for a control. In the frame-
work of control theory all quantities which may be used for a control of the
system are defined as input or control function u(t)={u
1
(t),...,u
n
(t)}.
The control mechanisms alone are not the main topic of control theory.
This theory connects the system under control and especially their control
mechanisms with a certain control aim as an optimalization criterion. In the
1.2 Dynamic State of Classical Mechanical Systems 3
case of our airplane example, the shortest way, the cheapest way and the safest
way between A and B are possible, but not identical control aims. The choice
of one of these criteria or of a weighted composition of these possible aims
depends on the intentions taken into account by the control designer. The
control theory ask for an optimal control in order to find a control law, i.e.,
an optimum input corresponding to the optimalization criterion. The control
aim is often defined by a so-called cost functional which should be minimized
to obtain the optimum input u
∗
(t). It usually takes the form of an integral
over the time of a certain function, plus a final contribution that depends on
the state in which the system ends up.
The difference between an open-loop control and a closed-loop control can
now also be defined within the control function. The optimal input of an
open-loop control can be completely determined before the system starts the
dynamics. Thus, the control has an a priori character. This concept becomes
relevant if the dynamics of the system is deterministic from a theoretical and
an experimental point of view. In contrast to this behavior, the control func-
tion of a closed-loop control is generated during the evolution of the system.
The current state of the system and possibly the history determine the cur-
rent change of the input with respect to minimize the cost functional. We also
denote such behavior an a posteriori control.
1.2 Dynamic State of Classical Mechanical Systems
The determination of an optimum control requires the knowledge of the un-
derlying dynamics of the system under control. In the framework of classical
physics, the mechanical state of the system is completely defined by the set
of the time-dependent degrees of freedom. The mechanical state of a given
system with 2N degrees of freedom consists of N generalized coordinates q
i
(i =1,...,N)andN generalized momenta p
i
conjugate to the q
i
. The dy-
namics can be written in terms of deterministic Hamilton’s equations as
dq
i
dt
=
∂H
∂p
i
and
dp
i
dt
= −
∂H
∂q
i
, (1.1)
where H = H(q, p, u) is the Hamiltonian of the system. The Hamiltonian
depends on the mechanical state given by the complete set of all q
i
(t)and
p
i
(t), and on the input u(t), defined by the current control law. Formally, the
mechanical degrees of freedom can be combined into a 2N-dimensional vector
Γ (t)={q
1
,...,q
N
,p
1
,...,p
N
}. Thus, the whole system can be represented
at time t by a point Γ (t)ina2N-dimensional space, spanned by a reference
frame of 2N axes, corresponding to the degrees of freedom. This space is called
the phase space P. It plays a fundamental role and is the natural framework
of the dynamics of classical many-body systems.