Reprinted by permission from Chapter 1: Introduction to Modern Control
Theory, in:

*F.L. Lewis, Applied Optimal Control and Estimation,
Prentice-Hall, 1992. *

# A BRIEF HISTORY OF FEEDBACK CONTROL

## Contents

- Outline
- A Brief History of Automatic Control
- Water Clocks of the Greeks and Arabs
- The Industrial Revolution
- The Millwrights
- Temperature Regulators
- Float Regulators
- Pressure Regulators
- Centrifugal Governors
- The Pendule Sympathique
- The Birth of Mathematical Control Theory
- Differential Equations
- Stability Theory
- System Theory
- Mass Communication and The Bell Telephone System
- Frequency-Domain Analysis
- The World Wars and Classical Control
- Ship Control
- Weapons Development and Gun Pointing
- M.I.T. Radiation Laboratory
- Stochastic Analysis
- The Classical Period of Control Theory
- The Space/Computer Age and Modern Control
- Time-Domain Design For Nonlinear Systems
- Sputnik - 1957
- Navigation
- Optimality In Natural Systems
- Optimal Control and Estimation Theory
- Nonlinear Control Theory
- Computers in Controls Design and Implementation
- The Development of Digital Computers
- Digital Control and Filtering Theory
- The Personal Computer
- The Union of Modern and Classical Control

- The Philosophy of Classical Control
- The Philosophy of Modern Control
- References

*Outline*

*
*In this chapter we introduce modern control theory by two approaches.
First, a short history of automatic control theory is provided. Then, we
describe the philosophies of classical and modern control theory.
Feedback control is the basic mechanism by which systems, whether
mechanical, electrical, or biological, maintain their equilibrium or
homeostasis. In the higher life forms, the conditions under which life can
continue are quite narrow. A change in body temperature of half a degree is
generally a sign of illness. The homeostasis of the body is maintained through
the use of feedback control [Wiener 1948]. A primary contribution of C.R. Darwin
during the last century was the theory that feedback over long time periods is
responsible for the evolution of species. In 1931 V. Volterra explained the
balance between two populations of fish in a closed pond using the theory of
feedback.
Feedback control may be defined as the use of difference signals,
determined by comparing the actual values of system variables to their desired
values, as a means of controlling a system. An everyday example of a feedback
control system is an automobile speed control, which uses the difference between
the actual and the desired speed to vary the fuel flow rate. Since the system
output is used to regulate its input, such a device is said to be a
*closed-loop control system*.
In this book we shall show how to use *modern control theory* to
design feedback control systems. Thus, we are concerned not with natural control
systems, such as those that occur in living organisms or in society, but with
man-made control systems such as those used to control aircraft, automobilies,
satellites, robots, and industrial processes.
Realizing that the best way to understand an area is to examine its
evolution and the reasons for its existence, we shall first provide a short
history of automatic control theory. Then, we give a brief discussion of the
philosophies of classical and modern control theory.
The references for Chapter 1 are at the end of this chapter. The
references for the remainder of the book appear at the end of the book.
**1.1 A BRIEF HISTORY OF AUTOMATIC CONTROL
**There have been many developments in automatic control theory during
recent years. It is difficult to provide an impartial analysis of an area while
it is still developing; however, looking back on the progress of feedback
control theory it is by now possible to distinguish some main trends and point
out some key advances.
Feedback control is an engineering discipline. As such, its progress is
closely tied to the practical problems that needed to be solved during any phase
of human history. The key developments in the history of mankind that affected
the progress of feedback control were:
1. The preoccupation of the Greeks and Arabs with keeping accurate track
of time. This represents a period from about 300 BC to about 1200 AD.
2. The Industrial Revolution in Europe. The Industrial Revolution is
generally agreed to have started in the third quarter of the eighteenth century;
however, its roots can be traced back into the 1600's.
3. The beginning of mass communication and the First and Second World
Wars. This represents a period from about 1910 to 1945.
4. The beginning of the space/computer age in 1957.
One may consider these as phases in the development of man, where he
first became concerned with understanding his place in space and time, then with
taming his environment and making his existence more comfortable, then with
establishing his place in a global community, and finally with his place in the
cosmos.
At a point between the Industrial Revolution and the World Wars, there
was an extremely important development. Namely, control theory began to acquire
its written language- the language of mathematics. J.C. Maxwell provided the
first rigorous mathematical analysis of a feedback control system in 1868. Thus,
relative to this written language, we could call the period before about 1868
the *prehistory* of automatic control.
Following Friedland [1986], we may call the period from 1868 to the early
1900's the *primitive period* of automatic control. It is standard to
call the period from then until 1960 the *classical period*, and the
period from 1960 through present times the *modern period*.
Let us now progress quickly through the history of automatic controls. A
reference for the period -300 through the Industrial Revolution is provided by
[Mayr 1970], which we shall draw on and at times quote. See also [Fuller 1976].
Other important references used in preparing this section included [M. Bokharaie
1973] and personal discussions with J.D. Aplevich of the University of Waterloo,
K.M. Przyluski of the Polish Academy of Sciences, and W. Askew, a former Fellow
at LTV Missiles and Space Corporation and vice-president of E-Systems.
*Water Clocks of the Greeks and Arabs
*The primary motivation for feedback control in times of
antiquity was the need for the accurate determination of time. Thus, in about
-270 the Greek Ktesibios invented a *float regulator* for a water clock.
The function of this regulator was to keep the water level in a tank at a
constant depth. This constant depth yielded a constant flow of water through a
tube at the bottom of the tank which filled a second tank at a constant rate.
The level of water in the second tank thus depended on time elapsed.
The regulator of Ktesibios used a float to control the inflow of water
through a valve; as the level of water fell the valve opened and replenished the
reservoir. This float regulator performed the same function as the ball and cock
in a modern flush toilet.
A float regulator was used by Philon of Byzantium in -250 to keep a
constant level of oil in a lamp.
During the first century AD Heron of Alexandria developed float
regulators for water clocks. The Greeks used the float regulator and similar
devices for purposes such as the automatic dispensing of wine, the design of
syphons for maintaining constant water level differences between two tanks, the
opening of temple doors, and so on. These devices could be called "gadgets"
since they were among the earliest examples of an idea looking for an
application.
In 800 through 1200 various Arab engineers such as the three brothers
Musa, Al-Jazari_, and Ibn al-Sa_'a_ti_ used float regulators for water clocks
and other applications. During this period the important feedback principle of
"on/off" control was used, which comes up again in connection with minimum-time
problems in the 1950's.
When Baghdad fell to the Mongols in 1258 all creative thought along these
lines came to an end. Moreover, the invention of the mechanical clock in the
14th century made the water clock and its feedback control system obsolete. (The
mechanical clock is not a feedback control system.) The float regulator does not
appear again until its use in the Industrial Revolution.
Along with a concern for his place in time, early man had a concern for
his place in space. It is worth mentioning that a pseudo-feedback control system
was developed in China in the 12th century for navigational purposes. The
*south-pointing chariot* had a statue which was turned by a gearing
mechanism attached to the wheels of the chariot so that it continuously pointed
south. Using the directional information provided by the statue, the charioteer
could steer a straight course. We call this a "pseudo-feedback" control system
since it does not technically involve feedback unless the actions of the
charioteer are considered as part of the system. Thus, it is not an automatic
control system.
*The Industrial Revolution
*The Industrial Revolution in Europe followed the introduction of
*prime movers*, or self-driven machines. It was marked by the invention
of advanced grain mills, furnaces, boilers, and the steam engine. These devices
could not be adequately regulated by hand, and so arose a new requirement for
automatic control systems. A variety of control devices was invented, including
float regulators, temperature regulators, pressure regulators, and speed control
devices.
J. Watt invented his steam engine in 1769, and this date marks the
accepted beginning of the Industrial Revolution. However, the roots of the
Industrial Revolution can be traced back to the 1600's or earlier with the
development of grain mills and the furnace.
One should be aware that others, primarily T. Newcomen in 1712, built the
first steam engines. However, the early steam engines were inefficient and
regulated by hand, making them less suited to industrial use. It is extremely
important to realize that the Industrial Revolution did not start until the
invention of improved engines and automatic control systems to regulate them.
*The Millwrights
*The millwrights of Britain developed a variety of feedback
control devices. The fantail, invented in 1745 by British blacksmith E. Lee,
consisted of a small fan mounted at right angles to the main wheel of a
windmill. Its function was to point the windmill continuously into the wind.
The mill-hopper was a device which regulated the flow of grain in a mill
depending on the speed of rotation of the millstone. It was in use in a fairly
refined form by about 1588.
To build a feedback controller, it is important to have *adequate
measuring devices*. The millwrights developed several devices for sensing
speed of rotation. Using these sensors, several speed regulation devices were
invented, including self-regulating windmill sails. Much of the technology of
the millwrights was later developed for use in the regulation of the steam
engine.
*Temperature Regulators
*Cornelis Drebbel of Holland spent some time in England and a
brief period with the Holy Roman Emperor Rudolf II in Prague, together with his
contemporary J. Kepler. Around 1624 he developed an automatic temperature
control system for a furnace, motivated by his belief that base metals could be
turned to gold by holding them at a precise constant temperature for long
periods of time. He also used this *temperature regulator* in an
incubator for hatching chickens.
Temperature regulators were studied by J.J. Becher in 1680, and used
again in an incubator by the Prince de Conti and R.-A.F. de Réaumur in 1754. The
"sentinel register" was developed in America by W. Henry around 1771, who
suggested its use in chemical furnaces, in the manufacture of steel and
porcelain, and in the temperature control of a hospital. It was not until 1777,
however, that a temperature regulator suitable for industrial use was developed
by Bonnemain, who used it for an incubator. His device was later installed on
the furnace of a hot-water heating plant.
*Float Regulators
*Regulation of the level of a liquid was needed in two main areas
in the late 1700's: in the boiler of a steam engine and in domestic water
distribution systems. Therefore, the float regulator received new interest,
especially in Britain.
In his book of 1746, W. Salmon quoted prices for ball-and-cock float
regulators used for maintaining the level of house water reservoirs. This
regulator was used in the first patents for the flush toilet around 1775. The
flush toilet was further refined by Thomas Crapper, a London plumber, who was
knighted by Queen Victoria for his inventions.
The earliest known use of a float valve regulator in a steam boiler is
described in a patent issued to J. Brindley in 1758. He used the regulator in a
steam engine for pumping water. S.T. Wood used a float regulator for a steam
engine in his brewery in 1784. In Russian Siberia, the coal miner I.I. Polzunov
developed in 1765 a float regulator for a steam engine that drove fans for blast
furnaces.
By 1791, when it was adopted by the firm of Boulton and Watt, the float
regulator was in common use in steam engines.
*Pressure Regulators
*Another problem associated with the steam engine is that of
steam-pressure regulation in the boiler, for the steam that drives the engine
should be at a constant pressure. In 1681 D. Papin invented a safety valve for a
pressure cooker, and in 1707 he used it as a regulating device on his steam
engine. Thereafter, it was a standard feature on steam engines.
The pressure regulator was further refined in 1799 by R. Delap and also
by M. Murray. In 1803 a pressure regulator was combined with a float regulator
by Boulton and Watt for use in their steam engines.
*Centrifugal Governors
*The first steam engines provided a reciprocating output motion
that was regulated using a device known as a cataract, similar to a float valve.
The cataract originated in the pumping engines of the Cornwall coal mines.
J. Watt's steam engine with a rotary output motion had reached maturity
by 1783, when the first one was sold. The main incentive for its development was
evidently the hope of introducing a prime mover into milling. Using the rotary
output engine, the Albion steam mill began operation early in 1786.
A problem associated with the rotary steam engine is that of regulating
its speed of revolution. Some of the speed regulation technology of the
millwrights was developed and extended for this purpose.
In 1788 Watt completed the design of the centrifugal flyball governor for
regulating the speed of the rotary steam engine. This device employed two
pivoted rotating flyballs which were flung outward by centrifugal force. As the
speed of rotation increased, the flyweights swung further out and up, operating
a steam flow throttling valve which slowed the engine down. Thus, a constant
speed was achieved automatically.
The feedback devices mentioned previously either remained obscure or
played an inconspicuous role as a part of the machinery they controlled. On the
other hand, the operation of the flyball governor was clearly visible even to
the untrained eye, and its principle had an exotic flavor which seemed to many
to embody the nature of the new industrial age. Therefore, the governor reached
the consciousness of the engineering world and became a sensation throughout
Europe. This was the first use of feedback control of which there was popular
awareness.
It is worth noting that the Greek word for governor is k u ße
r n a
n . In 1947, Norbert Wiener at MIT was searching for a
name for his new discipline of automata theory- control and communication in man
and machine. In investigating the flyball governor of Watt, he investigated also
the etymology of the word k u
ße r n a n and
came across the Greek word for steersman, k u ße r
n t V
. Thus, he selected the name *cybernetics* for his fledgling field.
Around 1790 in France, the brothers Périer developed a float regulator to
control the speed of a steam engine, but their technique was no match for the
centrifugal governor, and was soon supplanted.
*The Pendule Sympathique
*Having begun our history of automatic control with the water
clocks of ancient Greece, we round out this portion of the story with a return
to mankind's preoccupation with time.
The mechanical clock invented in the 14th century is not a closed-loop
feedback control system, but a precision open-loop oscillatory device whose
accuracy is ensured by protection against external disturbances. In 1793 the
French-Swiss A.-L. Breguet, the foremost watchmaker of his day, invented a
closed-loop feedback system to synchronize pocket watches.
The pendule sympathique of Breguet used a special case of speed
regulation. It consisted of a large, accurate precision chronometer with a mount
for a pocket watch. The pocket watch to be synchronized is placed into the mount
slightly before 12 o'clock, at which time a pin emerges from the chronometer,
inserts into the watch, and begins a process of automatically adjusting the
regulating arm of the watch's balance spring. After a few placements of the
watch in the pendule sympathique, the regulating arm is adjusted automatically.
In a sense, this device was used to transmit the accuracy of the large
chronometer to the small portable pocket watch.
*The Birth of Mathematical Control Theory
*The design of feedback control systems up through the Industrial
Revolution was by trial-and-error together with a great deal of engineering
intuition. Thus, it was more of an art than a science. In the mid 1800's
mathematics was first used to analyze the stability of feedback control systems.
Since mathematics is the formal language of automatic control theory, we could
call the period before this time the *prehistory* of control theory.
*Differential Equations
*In 1840, the British Astronomer Royal at Greenwich, G.B. Airy,
developed a feedback device for pointing a telescope. His device was a speed
control system which turned the telescope automatically to compensate for the
earth's rotation, affording the ability to study a given star for an extended
time.
Unfortunately, Airy discovered that by improper design of the feedback
control loop, wild oscillations were introduced into the system. He was the
first to discuss the *instability* of closed-loop systems, and the first
to use *differential equations* in their analysis [Airy 1840]. The theory
of differential equations was by then well developed, due to the discovery of
the infinitesimal calculus by I. Newton (1642-1727) and G.W. Leibniz
(1646-1716), and the work of the brothers Bernoulli (late 1600's and early
1700's), J.F. Riccati (1676-1754), and others. The use of differential equations
in analyzing the motion of dynamical systems was established by J.L. Lagrange
(1736-1813) and W.R. Hamilton (1805-1865).
*Stability Theory
*The early work in the mathematical analysis of control systems
was in terms of differential equations. J.C. Maxwell analyzed the stability of
Watt's flyball governor [Maxwell 1868]. His technique was to linearize the
differential equations of motion to find the *characteristic equation* of
the system. He studied the effect of the system parameters on stability and
showed that the system is stable if the roots of the characteristic equation
have *negative real parts*. With the work of Maxwell we can say that the
theory of control systems was firmly established.
E.J. Routh provided a *numerical technique* for determining when a
characteristic equation has stable roots [Routh 1877].
The Russian I.I. Vishnegradsky [1877] analyzed the stability of
regulators using differential equations independently of Maxwell. In 1893, A.B.
Stodola studied the regulation of a water turbine using the techniques of
Vishnegradsky. He modeled the actuator dynamics and included the delay of the
actuating mechanism in his analysis. He was the first to mention the notion of
the *system time constant*. Unaware of the work of Maxwell and Routh, he
posed the problem of determing the stability of the characteristic equation to
A. Hurwitz [1895], who solved it independently.
The work of A.M. Lyapunov was seminal in control theory. He studied the
stability of nonlinear differential equations using a generalized notion of
energy in 1892 [Lyapunov 1893]. Unfortunately, though his work was applied and
continued in Russia, the time was not ripe in the West for his elegant theory,
and it remained unknown there until approximately 1960, when its importance was
finally realized.
The British engineer O. Heaviside invented operational calculus in
1892-1898. He studied the transient behavior of systems, introducing a notion
equivalent to that of the *transfer function*.
*System Theory
*It is within the study of *systems* that feedback control
theory has its place in the organization of human knowledge. Thus, the concept
of a system as a dynamical entity with definite "inputs" and "outputs" joining
it to other systems and to the environment was a key prerequisite for the
further development of automatic control theory. The history of system theory
requires an entire study on its own, but a brief sketch follows.
During the eighteenth and nineteenth centuries, the work of A. Smith in
economics [*The Wealth of Nations*, 1776], the discoveries of C.R. Darwin
[*On the Origin of Species By Means of Natural Selection* 1859], and
other developments in politics, sociology, and elswehere were having a great
impact on the human consciousness. The study of Natural Philosophy was an
outgrowth of the work of the Greek and Arab philosophers, and contributions were
made by Nicholas of Cusa (1463), Leibniz, and others. The developments of the
nineteenth century, flavored by the Industrial Revolution and an expanding sense
of awareness in global geopolitics and in astronomy had a profound influence on
this Natural Philosophy, causing it to change its personality.
By the early 1900's A.N. Whitehead [1925], with his philosophy of
"organic mechanism", L. von Bertalanffy [1938], with his hierarchical principles
of organization, and others had begun to speak of a "general system theory". In
this context, the evolution of control theory could proceed.
*Mass Communication and The Bell Telephone System
*At the beginning of the 20th century there were two important
occurences from the point of view of control theory: the development of the
telephone and mass communications, and the World Wars.
*Frequency-Domain Analysis
*The mathematical analysis of control systems had heretofore been
carried out using differential equations in the *time domain*. At Bell
Telephone Laboratories during the 1920's and 1930's, the *frequency
domain* approaches developed by P.-S. de Laplace (1749-1827), J. Fourier
(1768-1830), A.L. Cauchy (1789-1857), and others were explored and used in
communication systems.
A major problem with the development of a mass communication system
extending over long distances is the need to periodically amplify the voice
signal in long telephone lines. Unfortunately, unless care is exercised, not
only the information but also the noise is amplified. Thus, the design of
suitable repeater amplifiers is of prime importance.
To reduce distortion in repeater amplifiers, H.S. Black demonstrated the
usefulness of *negative feedback* in 1927 [Black 1934]. The design
problem was to introduce a phase shift at the correct frequencies in the system.
Regeneration Theory for the design of stable amplifiers was developed by H.
Nyquist [1932]. He derived his *Nyquist stability criterion* based on the
polar plot of a complex function. H.W. Bode in 1938 used the magnitude and phase
*frequency response plots* of a complex function [Bode 1940]. He
investigated closed-loop stability using the notions of *gain and phase
margin*.
*The World Wars and Classical Control
*As mass communications and faster modes of travel made the world
smaller, there was much tension as men tested their place in a global society.
The result was the World Wars, during which the development of feedback control
systems became a matter of survival.
*Ship Control
*An important military problem during this period was the control
and navigation of ships, which were becoming more advanced in their design.
Among the first developments was the design of sensors for the purpose of
closed-loop control. In 1910, E.A. Sperry invented the *gyroscope*, which
he used in the stabilization and steering of ships, and later in aircraft
control.
N. Minorsky [1922] introduced his three-term controller for the steering
of ships, thereby becoming the first to use the
*proportional-integral-derivative (PID)* controller. He considered
nonlinear effects in the closed-loop system.
*Weapons Development and Gun Pointing
*A main problem during the period of the World Wars was that of
the accurate pointing of guns aboard moving ship and aircraft. With the
publication of "Theory of Servomechanisms" by H.L. Házen [1934], the use of
mathematical control theory in such problems was initiated. In his paper, Házen
coined the word *servomechanisms*, which implies a master/slave
relationship in systems.
The Norden bombsight, developed during World War II, used synchro
repeaters to relay information on aircraft altitude and velocity and wind
disturbances to the bombsight, ensuring accurate weapons delivery.
*M.I.T. Radiation Laboratory
*To study the control and information processing problems
associated with the newly invented radar, the Radiation Laboratory was
established at the Massachusetts Institute of Technology in 1940. Much of the
work in control theory during the 1940's came out of this lab.
While working on an M.I.T./Sperry Corporation joint project in 1941, A.C.
Hall recognized the deleterious effects of ignoring noise in control system
design. He realized that the frequency-domain technology developed at Bell Labs
could be employed to confront noise effects, and used this approach to design a
control system for an airborne radar. This success demonstrated conclusively the
importance of frequency-domain techniques in control system design [Hall 1946].
Using design approaches based on the transfer function, the block
diagram, and frequency-domain methods, there was great success in controls
design at the Radiation Lab. In 1947, N.B. Nichols developed his *Nichols
Chart* for the design of feedback systems. With the M.I.T. work, the theory
of linear servomechanisms was firmly established. A summary of the M.I.T.
Radiation Lab work is provided in *Theory of Servomechanisms* [James,
Nichols, and Phillips, 1947].
Working at North American Aviation, W.R. Evans [1948] presented his
*root locus* technique, which provided a direct way to determine the
closed-loop pole locations in the s-plane. Subsequently, during the 1950's, much
controls work was focused on the s-plane, and on obtaining desirable closed-loop
step-response characterictics in terms of rise time, percent overshoot, and so
on.
*Stochastic Analysis
*During this period also, *stochastic techniques* were
introduced into control and communication theory. At M.I.T in 1942, N. Wiener
[1949] analyzed information processing systems using models of stochastic
processes. Working in the frequency domain, he developed a *statistically
optimal filter* for stationary continuous-time signals that improved the
signal-to-noise ratio in a communication system. The Russian A.N. Kolmogorov
[1941] provided a theory for discrete-time stationary stochastic processes.
*The Classical Period of Control Theory
*By now, automatic control theory using frequency-domain
techniques had come of age, establishing itself as a paradigm (in the sense of
Kuhn [1962]). On the one hand, a firm mathematical theory for servomechanisms
had been established, and on the other, engineering design techniques were
provided. The period after the Second World War can be called the *classical
period* of control theory. It was characterized by the appearance of the
first textbooks [MacColl 1945; Lauer, Lesnick, and Matdon 1947; Brown and
Campbell 1948; Chestnut and Mayer 1951; Truxall 1955], and by straightforward
design tools that provided great intuition and guaranteed solutions to design
problems. These tools were applied using hand calculations, or at most slide
rules, together with graphical techniques.
*The Space/Computer Age and Modern Control
*With the advent of the space age, controls design in the United
States turned away from the frequency-domain techniques of classical control
theory and back to the differential equation techniques of the late 1800's,
which were couched in the *time domain*. The reasons for this development
are as follows.
*Time-Domain Design For Nonlinear Systems
*The paradigm of classical control theory was very suitable for
controls design problems during and immediately after the World Wars. The
frequency-domain approach was appropriate for *linear time-invariant*
systems. It is at its best when dealing with *single-input/single-output*
systems, for the graphical techniques were inconvenient to apply with multiple
inputs and outputs.
Classical controls design had some successes with nonlinear systems.
Using the noise-rejection properties of frequency-domain techniques, a control
system can be designed that is *robust* to variations in the system
parameters, and to measurement errors and external disturbances. Thus, classical
techniques can be used on a linearized version of a nonlinear system, giving
good results at an equilibrium point about which the system behavior is
approximately linear.
Frequency-domain techniques can also be applied to systems with simple
types of nonlinearities using the *describing function* approach, which
relies on the Nyquist criterion. This technique was first used by the Pole J.
Groszkowski in radio transmitter design before the Second World War and
formalized in 1964 by J. Kudrewicz.
Unfortunately, it is not possible to design control systems for advanced
nonlinear multivariable systems, such as those arising in aerospace
applications, using the assumption of linearity and treating the
single-input/single-output transmission pairs one at a time.
In the Soviet Union, there was a great deal of activity in nonlinear
controls design. Following the lead of Lyapunov, attention was focused on
time-domain techniques. In 1948, Ivachenko had investigated the principle of
*relay control*, where the control signal is switched discontinuously
between discrete values. Tsypkin used the phase plane for nonlinear controls
design in 1955. V.M. Popov [1961] provided his *circle criterion* for
nonlinear stability analysis.
*Sputnik - 1957
*Given the history of control theory in the Soviet Union, it is
only natural that the first satellite, Sputnik, was launched there in 1957. The
first conference of the newly formed International Federation of Automatic
Control (IFAC) was fittingly held in Moscow in 1960.
The launch of Sputnik engendered tremendous activity in the United States
in automatic controls design. On the failure of any paradigm, a return to
historical and natural first principles is required. Thus, it was clear that a
return was needed to the time-domain techniques of the "primitive" period of
control theory, which were based on differential equations. It should be
realized that the work of Lagrange and Hamilton makes it straightforward to
write nonlinear equations of motion for many dynamical systems. Thus, a control
theory was needed that could deal with such nonlinear differential equations.
It is quite remarkable that in almost exactly 1960, major developments
occurred independently on several fronts in the theory of communication and
control.
*Navigation
*In 1960, C.S. Draper invented his inertial navigation system,
which used gyroscopes to provided accurate information on the position of a body
moving in space, such as a ship, aircraft, or spacecraft. Thus, the sensors
appropriate for navigation and controls design were developed.
*Optimality In Natural Systems
*Johann Bernoulli first mentioned the *Principle of
Optimality* in connection with the Brachistochrone Problem in 1696. This
problem was solved by the brothers Bernoulli and by I. Newton, and it became
clear that the quest for optimality is a fundamental property of motion in
natural systems. Various optimality principles were investigated, including the
minimum-time principle in optics of P. de Fermat (1600's), the work of L. Euler
in 1744, and Hamilton's result that a system moves in such a way as to minimize
the time integral of the difference between the kinetic and potential energies.
These optimality principles are all *minimum principles*.
Interestingly enough, in the early 1900's, A. Einstein showed that, relative to
the 4-D space-time coordinate system, the motion of systems occurs in such as
way as to *maximize* the time.
*Optimal Control and Estimation Theory
*Since naturally-occurring systems exhibit optimality in their
motion, it makes sense to design man-made control systems in an optimal fashion.
A major advantage is that this design may be accomplished in the time domain. In
the context of modern controls design, it is usual to minimize the time of
transit, or a quadratic generalized energy functional or *performance
index*, possibly with some constraints on the allowed controls.
R. Bellman [1957] applied *dynamic programming* to the optimal
control of discrete-time systems, demonstrating that the natural direction for
solving optimal control problems is *backwards in time*. His procedure
resulted in closed-loop, generally nonlinear, feedback schemes.
By 1958, L.S. Pontryagin had developed his *maximum principle*,
which solved optimal control problems relying on the *calculus of
variations* developed by L. Euler (1707-1783). He solved the minimum-time
problem, deriving an on/off relay control law as the optimal control
[Pontryagin, Boltyansky, Gamkrelidze, and Mishchenko 1962]. In the U.S. during
the 1950's, the calculus of variations was applied to general optimal control
problems at the University of Chicago and elsewhere.
In 1960 three major papers were published by R. Kalman and coworkers,
working in the U.S. One of these [Kalman and Bertram 1960], publicized the vital
work of Lyapunov in the time-domain control of nonlinear systems. The next
[Kalman 1960a] discussed the optimal control of systems, providing the design
equations for the *linear quadratic regulator (LQR)*. The third paper
[Kalman 1960b] discussed optimal filtering and estimation theory, providing the
design equations for the *discrete Kalman filter*. The *continuous
Kalman filter* was developed by Kalman and Bucy [1961].
In the period of a year, the major limitations of classical control
theory were overcome, important new theoretical tools were introduced, and a new
era in control theory had begun; we call it the era of *modern control*.
The key points of Kalman's work are as follows. It is a *time-domain
approach*, making it more applicable for time-varying linear systems as well
as nonlinear systems. He introduced *linear algebra and matrices*, so
that systems with multiple inputs and outputs could easily be treated. He
employed the concept of the *internal system state*; thus, the approach
is one that is concerned with the internal dynamics of a system and not only its
input/output behavior.
In control theory, Kalman formalized the notion of *optimality in
control theory* by minimizing a very general quadratic generalized energy
function. In estimation theory, he introduced stochastic notions that applied to
nonstationary *time-varying systems*, thus providing a recursive
solution, the Kalman filter, for the least-squares approach first used by C.F.
Gauss (1777-1855) in planetary orbit estimation. The Kalman filter is the
natural extension of the Wiener filter to nonstationary stochastic systems.
Classical frequency-domain techniques provide formal tools for control
systems design, yet the design phase itself remained very much an art and
resulted in nonunique feedback systems. By contrast, the theory of Kalman
provided *optimal solutions* that yielded control systems with
*guaranteed performance*. These controls were directly found by solving
*formal matrix design equations* which generally had unique solutions.
It is no accident that from this point the U.S. space program blossomed,
with a Kalman filter providing navigational data for the first lunar landing.
*Nonlinear Control Theory
*During the 1960's in the U.S., G. Zames [1966], I.W. Sandberg
[1964], K.S. Narendra [Narendra and Goldwyn 1964], C.A. Desoer [1965], and
others extended the work of Popov and Lyapunov in nonlinear stability. There was
an extensive application of these results in the study of nonlinear distortion
in bandlimited feedback loops, nonlinear process control, aircraft controls
design, and eventually in robotics.
*Computers in Controls Design and Implementation
*Classical design techniques could be employed by hand using
graphical approaches. On the other hand, modern controls design requires the
solution of complicated nonlinear matrix equations. It is fortunate that in 1960
there were major developments in another area- digital computer technology.
Without computers, modern control would have had limited applications.
*The Development of Digital Computers
*In about 1830 C. Babbage introduced modern computer principles,
including memory, program control, and branching capabilities. In 1948, J. von
Neumann directed the construction of the IAS stored-program computer at
Princeton. IBM built its SSEC stored-program machine. In 1950, Sperry Rand built
the first commercial data processing machine, the UNIVAC I. Soon after, IBM
marketed the 701 computer.
In 1960 a major advance occurred- the second generation of computers was
introduced which used *solid-state technology*. By 1965, Digital
Equipment Corporation was building the PDP-8, and the *minicomputer*
industry began. Finally, in 1969 W. Hoff invented the *microprocessor*.
*Digital Control and Filtering Theory
*Digital computers are needed for two purposes in modern
controls. First, they are required to *solve the matrix design equations*
that yield the control law. This is accomplished off-line during the design
process. Second, since the optimal control laws and filters are generally
time-varying, they are needed to *implement* modern control and filtering
schemes on actual systems.
With the advent of the microprocessor in 1969 a new area developed.
Control systems that are implemented on digital computers must be formulated in
*discrete time*. Therefore, the growth of *digital control theory*
was natural at this time.
During the 1950's, the theory of *sampled data systems* was being
developed at Columbia by J.R. Ragazzini, G. Franklin, and L.A. Zadeh [Ragazzini
and Zadeh 1952, Ragazzini and Franklin 1958]; as well as by E.I. Jury [1960],
B.C. Kuo [1963], and others. The idea of using digital computers for
*industrial process control* emerged during this period [Ĺström and
Wittenmark 1984]. Serious work started in 1956 with the collaborative project
between TRW and Texaco, which resulted in a computer-controlled system being
installed at the Port Arthur oil refinery in Texas in 1959.
The development of *nuclear reactors* during the 1950's was a
major motivation for exploring industrial process control and instrumentation.
This work has its roots in the control of chemical plants during the 1940's.
By 1970, with the work of K. Ĺström [1970] and others, the importance of
digital controls in process applications was firmly established.
The work of C.E. Shannon in the 1950's at Bell Labs had revealed the
importance of sampled data techniques in the processing of signals. The
applications of *digital filtering theory* were investigated at the
Analytic Sciences Corporation [Gelb 1974] and elsewhere.
*The Personal Computer
*With the introduction of the PC in 1983, the design of modern
control systems became possible for the individual engineer. Thereafter, a
plethora of software control systems design packages was developed, including
ORACLS, Program CC, Control-C, PC-Matlab, MATRIX_{x}, Easy5, SIMNON, and
others.
*The Union of Modern and Classical Control
*With the publication of the first textbooks in the 1960's,
modern control theory established itself as a paradigm for automatic controls
design in the U.S. Intense activity in research and implementation ensued, with
the I.R.E. and the A.I.E.E. merging, largely through the efforts of P. Haggerty
at Texas Instruments, to form the Institute of Electrical and Electronics
Engineers (I.E.E.E) in the early 1960's.
With all its power and advantages, modern control was lacking in some
aspects. The guaranteed performance obtained by solving matrix design equations
means that it is often possible to design a control system that works in theory
without gaining any *engineering intuition* about the problem. On the
other hand, the frequency-domain techniques of classical control theory impart a
great deal of intuition.
Another problem is that a modern control system with any compensator
dynamics can *fail to be robust* to disturbances, unmodelled dynamics,
and measurement noise. On the other hand, robustness is built in with a
frequency-domain approach using notions like the gain and phase margin.
Therefore, in the 1970's, especially in Great Britain, there was a great
deal of activity by H.H. Rosenbrock [1974], A.G.J. MacFarlane and I.
Postlethwaite [1977], and others to extend classical frequency-domain techniques
and the root locus to multivariable systems. Successes were obtained using
notions like the characteristic locus, diagonal dominance, and the inverse
Nyquist array.
A major proponent of classical techniques for multivariable systems was
I. Horowitz, whose *quantitative feedback theory* developed in the early
1970's accomplishes robust design using the Nichols chart.
In 1981 seminal papers appeared by J. Doyle and G. Stein [1981] and M.G.
Safonov, A.J. Laub, and G.L. Hartmann [1981]. Extending the seminal work of
MacFarlane and Postlethwaite [1977], they showed the importance of the
*singular value* plots versus frequency in robust multivariable design.
Using these plots, many of the classical frequency-domain techniques can be
incorporated into modern design. This work was pursued in aircraft and process
control by M. Athans [1986] and others. The result is a *new control
theory* that blends the best features of classical and modern techniques. A
survey of this *robust modern control theory* is provided by P. Dorato
[1987].
**1.2 THE PHILOSOPHY OF CLASSICAL CONTROL
**Having some understanding of the history of automatic control theory,
we may now briefly discuss the philosophies of classical and modern control
theory.
Developing as it did for feedback amplifier design, classical control
theory was naturally couched in the *frequency domain and the s-plane*.
Relying on transform methods, it is primarily applicable for *linear
time-invariant systems*, though some extensions to nonlinear systems were
made using, for instance, the describing function.
The system description needed for controls design using the methods of
Nyquist and Bode is the magnitude and phase of the frequency response. This is
advantageous since the frequency response can be experimentally measured. The
transfer function can then be computed. For root locus design, the transfer
function is needed. The block diagram is heavily used to determine transfer
functions of composite systems. *An exact description of the internal system
dynamics is not needed* for classical design; that is, only the input/output
behavior of the system is of importance.
The design may be carried out *by hand using graphical
techniques*. These methods *impart a great deal of intuition* and
afford the controls designer with a range of design possibilities, so that the
resulting *control systems are not unique*. The design process is an
engineering art.
A real system has disturbances and measurement noise, and may not be
described exactly by the mathematical model the engineer is using for design.
Classical theory is natural for designing control systems that are
*robust* to such disorders, yielding good closed-loop performance in
spite of them. Robust design is carried out using notions like the gain and
phase margin.
*Simple compensators* like proportional-integral-derivative (PID),
lead-lag, or washout circuits are generally used in the control structure. The
effects of such circuits on the Nyquist, Bode, and root locus plots are easy to
understand, so that a suitable compensator structure can be selected. Once
designed, the compensator can be easily tuned on line.
A fundamental concept in classical control is the ability to *describe
closed-loop properties in terms of open-loop properties*, which are known or
easy to measure. For instance, the Nyquist, Bode, and root locus plots are in
terms of the open-loop transfer function. Again, the closed-loop disturbance
rejection properties and steady-state error can be described in terms of the
return difference and sensitivity.
Classical control theory is *difficult to apply in
multi-input/multi-output (MIMO), or multi-loop systems*. Due to the
interaction of the control loops in a multivariable system, each
single-input/single-output (SISO) transfer function can have acceptable
properties in terms of step response and robustness, but the coordinated control
motion of the system can fail to be acceptable.
Thus, classical MIMO or multiloop design requires painstaking effort
using the approach of *closing one loop at a time* by graphical
techniques. A root locus, for instance, should be plotted for each gain element,
taking into account the gains previously selected. This is a
*trial-and-error* procedure that may require multiple iterations, and it
*does not guarantee good results, or even closed-loop stability*.
The multivariable frequency-domain approaches developed by the British
school during the 1970's, as well as quantitative feedback theory, overcome many
of these limitations, providing an effective approach for the design of many
MIMO systems.
**1.3 THE PHILOSOPHY OF MODERN CONTROL
**Modern controls design is fundamentally a time-domain technique. An
exact *state-space model* of the system to be controlled, or plant, is
required. This is a first-order *vector* differential equation of the
form
dx/dt = Ax + Bu
y = Cx
where x(t) is a vector of *internal variables* or system states,
u(t) is a vector of control inputs, and y(t) is a vector of measured outputs. It
is possible to add noise terms to represent process and measurement noises. Note
that the plant is described in the time-domain.
The power of modern control has its roots in the fact that the
state-space model can as well represent a MIMO system as a SISO system. That is,
u(t) and y(t) are generally vectors whose entries are the individual scalar
inputs and outputs. Thus, A, B, C are *matrices* whose elements describe
the system dynamical interconnections.
Modern controls techniques were first firmly established for linear
systems. Extensions to nonlinear systems can be made using the Lyapunov
approach, which extends easily to MIMO systems, dynamic programming, and other
techniques. Open-loop optimal controls designs can be determined for nonlinear
systems by solving nonlinear two-point boundary-value problems.
Exactly as in the classical case, some fundamental questions on the
performance of the closed-loop system can be attacked by investigating
*open-loop properties*. For instance, the open-loop properties of
controllability and observability of (0 (Chapter 2) give insight on what it is
possible to achieve using feedback control. The difference is that, to deal with
the state-space model, *a good knowledge of matrices and linear algebra is
required*.
To achieve suitable closed-loop properties, a feedback control of the
form
u = -Kx
may be used. The feedback gain K is a *matrix* whose elements are
the individual control gains in the system. Since all the states are used for
feedback, this is called *state-variable feedback*. Note that multiple
feedback gains and large systems are easily handled in this framework. Thus, if
there are n state components (where n can be very large in an aerospace or power
distribution system) and m scalar controls, so that u(t) is an m-vector, then K
is an mxn matrix with mn entries, corresponding to mn control loops.
In the standard linear quadratic regulator (LQR), the feedback gain K is
chosen to minimize a quadratic time-domain *performance index (PI)* like
oo
J = / (x^{T}Qx + u^{T}Ru) dt
o

The minimum is sought over all state trajectories. This is an extension
to MIMO systems of the sorts of PIs (ITSE, ITAE, etc.) that were used in
classical control. Q and R are weighting matrices that serve as *design
parameters*. Their elements can be selected to provide suitable performance.
The key to LQR design is the fact that, if the feedback gain matrix K can
be successfully chosen to make J finite, then the integral (0 involving the
norms of u(t) and x(t) is bounded. If Q and R are correctly chosen, well-known
mathematical principles then ensure that x(t) and u(t) go to zero with time.
This *guarantees closed-loop stability* as well as bounded control
signals in the closed-loop system.
It can be shown (see Chapter 3), that the value of K that minimizes the
PI is given by
K = R^{-1}B^{T}S
where S is an nxn matrix satisfying the *Riccati equation
*
0 = A^{T}S + SA - SBR^{-1}B^{T}S + Q.
Within this LQ framework, several points can be made. First, as long as
the system (0 is controllable and Q and R are suitably chosen, the K given by
these equations *guarantees the stability of the closed-loop system
*
dx/dt = (A-BK)x + Bu.
Second, this technique is easy to apply even for multiple-input plants,
since u(t) can be a vector having many components.
Third, the LQR solution relies on the solution of the *matrix design
equation* (0, and so is unsuited to hand calculations. Fortunately, many
design packages are by now available on digital computers for solving the
Riccati design equation for S, and hence for obtaining K. Thus,
*computer-aided design* is an essential feature of modern controls.
The LQR solution is a formal one that gives a *unique answer* to
the feedback control problem once the design parameter Q has been selected. In
fact, *the engineering art in modern design lies in the selection of the PI
weighting matrices Q and R*. A body of theory on this selection process has
evolved. Once Q is properly selected, the matrix design equation is formally
solved for the unique K that guarantees stability.
Observe that *K is computed in terms of the open-loop quantities*
A, B, Q, so that modern and classical design have this feature of determining
closed-loop properties in terms of open-loop quantities in common. However, in
modern control, all the entries of K are determined at the same time by using
the matrix design equations. This corresponds to *closing all the feedback
control loops simultaneously*, which is in complete contrast to the
one-loop-at-a-time procedure of classical controls design.
Unfortunately, formal LQR design gives very *little intuition on the
nature or properties of the closed-loop system*. In recent years, this
deficiency has been addressed from a variety of standpoints.
Although LQR design using state feedback guarantees closed-loop
stability, all the state components are seldom available for feedback purposes
in a practical design problem. Therefore, *output feedback* of the form
u = -Ky
is more useful. LQR design equations for output feedback are more
complicated than (0, but are easily derived (see Chapter 4).
Modern output-feedback design allows one to *design controllers for
complicated systems with multiple inputs and outputs* by formally solving
matrix design equations on a digital computer.
Another important factor is the following. While the state feedback (0
involves feedback from all states to all inputs, offering no structure in the
control system, the output feedback control law (0 can be used to design a
*compensator with a desired dynamical structure*, regaining much of the
intuition of classical controls design.
Feedback laws like (0 and (0 are called *static*, since the
control gains are constants, or at most time-varying. An alternative to static
output feedback is to use a dynamic compensator of the form
dz/dt = Fz + Gy + Eu
u = Hz + Dy.
The inputs of this compensator are the system inputs and outputs. This
yields a closed-loop and is called *dynamic output feedback*. The design
problem is to select the matrices F, G, E, H, D for good closed-loop
performance. An important result of modern control is that closed-loop stability
can be guaranteed by selecting F = A-LC for some matrix L which is computed
using a Riccati design equation similar to (0. The other matrices in (0 are then
easily determined. This design is based on the vital *separation
principle* (Chapter 10).
A disadvantage with design using F = A-LC is that then the dynamic
compensator has the same number of internal states as the plant. In complicated
modern aerospace and power plant applications, this dimension can be very large.
Thus, various techniques for *controller reduction* and *reduced-order
design* have been developed.
In standard modern control, the system is assumed to be exactly described
by the mathematical model (0. In actuality, however, this model may be only an
approximate description of the real plant. Moreover, in practice there can be
disturbances acting on the plant, as well as measurement noise in determining
y(t).
The LQR using full state feedback has some important robustness
properties to such disorders, such as an infinite gain margin, 60° of phase margin, and robustness to some nonlinearities in
the control loops (Chapter 10). On the other hand, the LQR using static or
dynamic output feedback design has no guaranteed robustness properties. With the
work on robust modern control in the early 1980's, there is now a technique
(LQG/LTR, Chapter 10) for designing robust multivariable control systems.
LQG/LTR design incorporates rigorous treatments of the effects of modelling
uncertainties on closed-loop stability, and of disturbance effects on
closed-loop performance.
With the work on robust modern design, *much of the intuition of
classical controls techniques can now be incorporated in modern multivariable
design*.
With modern developments in *digital control theory* and
*discrete-time systems*, modern control is very suitable for the design
of control systems that can be implemented on microprocessors (Part III of the
book). This allows the implementation of controller dynamics that are more
complicated as well as more effective than the simple PID and lead-lag
structures of classical controls.
With recent work in *matrix-fraction descriptions* and
*polynomial equation design*, a MIMO plant can be described not in
state-space form, but in input/output form. This is a direct extension of the
classical transfer function description and, for some applications, is more
suitable than the internal description (0.
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