Control theory is a field of mathematics that deals with the
control
Control may refer to:
Basic meanings Economics and business
* Control (management), an element of management
* Control, an element of management accounting
* Comptroller (or controller), a senior financial officer in an organization
* Controllin ...
of
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
s in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a desired state, while minimizing any ''delay'', ''overshoot'', or ''steady-state error'' and ensuring a level of control
stability
Stability may refer to:
Mathematics
*Stability theory, the study of the stability of solutions to differential equations and dynamical systems
** Asymptotic stability
** Linear stability
** Lyapunov stability
** Orbital stability
** Structural sta ...
; often with the aim to achieve a degree of
optimality
Optimality may refer to:
* Mathematical optimization
* Optimality Theory in linguistics
* optimality model In biology, optimality models are a tool used to evaluate the costs and benefits of different organismal features, traits, and characterist ...
.
To do this, a controller with the requisite corrective behavior is required. This controller monitors the controlled
process variable
In control theory, a process variable (PV; also process value or process parameter) is the current measured value of a particular part of a process which is being monitored or controlled. An example of this would be the temperature of a furnace ...
(PV), and compares it with the reference or
set point (SP). The difference between actual and desired value of the process variable, called the ''error'' signal, or SP-PV error, is applied as feedback to generate a control action to bring the controlled process variable to the same value as the set point. Other aspects which are also studied are
controllability Controllability is an important property of a control system, and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control.
Controllability and observabil ...
and
observability
Observability is a measure of how well internal states of a system can be inferred from knowledge of its external outputs.
In control theory, the observability and controllability of a linear system are mathematical duals.
The concept of observ ...
. Control theory is used in
control system engineering
Control engineering or control systems engineering is an engineering discipline that deals with control systems, applying control theory to design equipment and systems with desired behaviors in control environments. The discipline of controls o ...
to design automation that have revolutionized manufacturing, aircraft, communications and other industries, and created new fields such as
robotics
Robotics is an interdisciplinary branch of computer science and engineering. Robotics involves design, construction, operation, and use of robots. The goal of robotics is to design machines that can help and assist humans. Robotics integrat ...
.
Extensive use is usually made of a diagrammatic style known as the
block diagram
A block diagram is a diagram of a system in which the principal parts or functions are represented by blocks connected by lines that show the relationships of the blocks. . In it the
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, theoretically models the system's output for ...
, also known as the system function or network function, is a mathematical model of the relation between the input and output based on the
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s describing the system.
Control theory dates from the 19th century, when the theoretical basis for the operation of governors was first described by
James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
. Control theory was further advanced by
Edward Routh
Edward John Routh (; 20 January 18317 June 1907), was an English mathematician, noted as the outstanding coach of students preparing for the Mathematical Tripos examination of the University of Cambridge in its heyday in the middle of the ninet ...
in 1874,
Charles Sturm and in 1895,
Adolf Hurwitz, who all contributed to the establishment of control stability criteria; and from 1922 onwards, the development of
PID control
A proportional–integral–derivative controller (PID controller or three-term controller) is a control loop mechanism employing feedback that is widely used in industrial control systems and a variety of other applications requiring continuous ...
theory by
Nicolas Minorsky.
Although a major application of
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
control theory is in
control systems engineering
Control engineering or control systems engineering is an engineering discipline that deals with control systems, applying control theory to design equipment and systems with desired behaviors in control environments. The discipline of controls o ...
, which deals with the design of
process control
An industrial process control in continuous production processes is a discipline that uses industrial control systems to achieve a production level of consistency, economy and safety which could not be achieved purely by human manual control. I ...
systems for industry, other applications range far beyond this. As the general theory of feedback systems, control theory is useful wherever feedback occurs - thus control theory also has applications in life sciences, computer engineering, sociology and
operations research
Operations research ( en-GB, operational research) (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a discipline that deals with the development and application of analytical methods to improve deci ...
.
History
![Boulton and Watt centrifugal governor-MJ](https://upload.wikimedia.org/wikipedia/commons/c/c7/Boulton_and_Watt_centrifugal_governor-MJ.jpg)
Although control systems of various types date back to antiquity, a more formal analysis of the field began with a dynamics analysis of the
centrifugal governor
A centrifugal governor is a specific type of governor with a feedback system that controls the speed of an engine by regulating the flow of fuel or working fluid, so as to maintain a near-constant speed. It uses the principle of proportional cont ...
, conducted by the physicist
James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
in 1868, entitled ''On Governors''.
A centrifugal governor was already used to regulate the velocity of windmills. Maxwell described and analyzed the phenomenon of
self-oscillation
Self-oscillation is the generation and maintenance of a periodic motion by a source of power that lacks any corresponding periodicity. The oscillator itself controls the phase with which the external power acts on it. Self-oscillators are therefor ...
, in which lags in the system may lead to overcompensation and unstable behavior. This generated a flurry of interest in the topic, during which Maxwell's classmate,
Edward John Routh
Edward John Routh (; 20 January 18317 June 1907), was an English mathematician, noted as the outstanding coach of students preparing for the Mathematical Tripos examination of the University of Cambridge in its heyday in the middle of the ninet ...
, abstracted Maxwell's results for the general class of linear systems.
Independently,
Adolf Hurwitz analyzed system stability using differential equations in 1877, resulting in what is now known as the
Routh–Hurwitz theorem
In mathematics, the Routh–Hurwitz theorem gives a test to determine whether all root of a function, roots of a given polynomial lie in the left half-plane. Polynomials with this property are called stable polynomial, Hurwitz stable polynomials. ...
.
A notable application of dynamic control was in the area of crewed flight. The
Wright brothers made their first successful test flights on December 17, 1903, and were distinguished by their ability to control their flights for substantial periods (more so than the ability to produce lift from an airfoil, which was known). Continuous, reliable control of the airplane was necessary for flights lasting longer than a few seconds.
By
World War II
World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the vast majority of the world's countries—including all of the great powers—forming two opposin ...
, control theory was becoming an important area of research.
Irmgard Flügge-Lotz
Irmgard Flügge-Lotz, née Lotz (16 July 1903 – 22 May 1974) was a German-American mathematician and aerospace engineer. She was a pioneer in the development of the theory of discontinuous automatic control, which has found wide application ...
developed the theory of discontinuous automatic control systems, and applied the
bang-bang principle to the development of
automatic flight control equipment for aircraft. Other areas of application for discontinuous controls included
fire-control system
A fire-control system (FCS) is a number of components working together, usually a gun data computer, a director, and radar, which is designed to assist a ranged weapon system to target, track, and hit a target. It performs the same task as a ...
s,
guidance system
A guidance system is a virtual or physical device, or a group of devices implementing a controlling the movement of a ship, aircraft, missile, rocket, satellite, or any other moving object. Guidance is the process of calculating the changes in po ...
s and
electronics
The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using electronic devices. Electronics uses active devices to control electron flow by amplification ...
.
Sometimes, mechanical methods are used to improve the stability of systems. For example,
ship stabilizers are fins mounted beneath the waterline and emerging laterally. In contemporary vessels, they may be gyroscopically controlled active fins, which have the capacity to change their angle of attack to counteract roll caused by wind or waves acting on the ship.
The
Space Race
The Space Race was a 20th-century competition between two Cold War rivals, the United States and the Soviet Union, to achieve superior spaceflight capability. It had its origins in the ballistic missile-based nuclear arms race between the tw ...
also depended on accurate spacecraft control, and control theory has also seen an increasing use in fields such as economics and artificial intelligence. Here, one might say that the goal is to find an
internal model that obeys the
good regulator theorem. So, for example, in economics, the more accurately a (stock or commodities) trading model represents the actions of the market, the more easily it can control that market (and extract "useful work" (profits) from it). In AI, an example might be a chatbot modelling the discourse state of humans: the more accurately it can model the human state (e.g. on a telephone voice-support hotline), the better it can manipulate the human (e.g. into performing the corrective actions to resolve the problem that caused the phone call to the help-line). These last two examples take the narrow historical interpretation of control theory as a set of differential equations modeling and regulating kinetic motion, and broaden it into a vast generalization of a
regulator interacting with a
plant
Plants are predominantly photosynthetic eukaryotes of the kingdom Plantae. Historically, the plant kingdom encompassed all living things that were not animals, and included algae and fungi; however, all current definitions of Plantae exclud ...
.
Open-loop and closed-loop (feedback) control
![Feedback loop with descriptions](https://upload.wikimedia.org/wikipedia/commons/2/24/Feedback_loop_with_descriptions.svg)
Fundamentally, there are two types of control loops: open loop control and closed loop (feedback) control.
In open loop control, the control action from the controller is independent of the "process output" (or "controlled process variable" - PV). A good example of this is a central heating boiler controlled only by a timer, so that heat is applied for a constant time, regardless of the temperature of the building. The control action is the timed switching on/off of the boiler, the process variable is the building temperature, but neither is linked.
In closed loop control, the control action from the controller is dependent on feedback from the process in the form of the value of the process variable (PV). In the case of the boiler analogy, a closed loop would include a thermostat to compare the building temperature (PV) with the temperature set on the thermostat (the set point - SP). This generates a controller output to maintain the building at the desired temperature by switching the boiler on and off. A closed loop controller, therefore, has a feedback loop which ensures the controller exerts a control action to manipulate the process variable to be the same as the "Reference input" or "set point". For this reason, closed loop controllers are also called feedback controllers.
The definition of a closed loop control system according to the British Standard Institution is "a control system possessing monitoring feedback, the deviation signal formed as a result of this feedback being used to control the action of a final control element in such a way as to tend to reduce the deviation to zero."
Likewise; "A ''Feedback Control System'' is a system which tends to maintain a prescribed relationship of one system variable to another by comparing functions of these variables and using the difference as a means of control."
Other examples
An example of a control system is a car's
cruise control
Cruise control (also known as speed control, cruise command, autocruise, or tempomat) is a system that automatically controls the speed of a motor vehicle. The system is a servomechanism that takes over the throttle of the car to maintain a ste ...
, which is a device designed to maintain vehicle speed at a constant ''desired'' or ''reference'' speed provided by the driver. The ''controller'' is the cruise control, the ''plant'' is the car, and the ''system'' is the car and the cruise control. The system output is the car's speed, and the control itself is the engine's
throttle
A throttle is the mechanism by which fluid flow is managed by constriction or obstruction.
An engine's power can be increased or decreased by the restriction of inlet gases (by the use of a throttle), but usually decreased. The term ''throttle'' ...
position which determines how much power the engine delivers.
A primitive way to implement cruise control is simply to lock the throttle position when the driver engages cruise control. However, if the cruise control is engaged on a stretch of non-flat road, then the car will travel slower going uphill and faster when going downhill. This type of controller is called an ''
open-loop controller
In control theory, an open-loop controller, also called a non-feedback controller, is a control system in which the control action is independent of the "process output", which is the process variable that is being controlled."Feedback and cont ...
'' because there is no
feedback
Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause-and-effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. The notion of cause-and-effect has to be handled ...
; no measurement of the system output (the car's speed) is used to alter the control (the throttle position.) As a result, the controller cannot compensate for changes acting on the car, like a change in the slope of the road.
In a ''
closed-loop control system'', data from a sensor monitoring the car's speed (the system output) enters a controller which continuously compares the quantity representing the speed with the reference quantity representing the desired speed. The difference, called the error, determines the throttle position (the control). The result is to match the car's speed to the reference speed (maintain the desired system output). Now, when the car goes uphill, the difference between the input (the sensed speed) and the reference continuously determines the throttle position. As the sensed speed drops below the reference, the difference increases, the throttle opens, and engine power increases, speeding up the vehicle. In this way, the controller dynamically counteracts changes to the car's speed. The central idea of these control systems is the ''feedback loop'', the controller affects the system output, which in turn is measured and fed back to the controller.
Classical control theory
To overcome the limitations of the
open-loop controller
In control theory, an open-loop controller, also called a non-feedback controller, is a control system in which the control action is independent of the "process output", which is the process variable that is being controlled."Feedback and cont ...
, control theory introduces
feedback
Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause-and-effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. The notion of cause-and-effect has to be handled ...
.
A
closed-loop controller
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
uses feedback to control
states or
outputs of a
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
. Its name comes from the information path in the system: process inputs (e.g.,
voltage
Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
applied to an
electric motor
An electric motor is an Electric machine, electrical machine that converts electrical energy into mechanical energy. Most electric motors operate through the interaction between the motor's magnetic field and electric current in a Electromagneti ...
) have an effect on the process outputs (e.g., speed or torque of the motor), which is measured with
sensor
A sensor is a device that produces an output signal for the purpose of sensing a physical phenomenon.
In the broadest definition, a sensor is a device, module, machine, or subsystem that detects events or changes in its environment and sends ...
s and processed by the controller; the result (the control signal) is "fed back" as input to the process, closing the loop.
Closed-loop controllers have the following advantages over
open-loop controller
In control theory, an open-loop controller, also called a non-feedback controller, is a control system in which the control action is independent of the "process output", which is the process variable that is being controlled."Feedback and cont ...
s:
* disturbance rejection (such as hills in the cruise control example above)
* guaranteed performance even with
model
A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure.
Models c ...
uncertainties, when the model structure does not match perfectly the real process and the model parameters are not exact
*
unstable
In numerous fields of study, the component of instability within a system is generally characterized by some of the outputs or internal states growing without bounds. Not all systems that are not stable are unstable; systems can also be mar ...
processes can be stabilized
* reduced sensitivity to parameter variations
* improved reference tracking performance
In some systems, closed-loop and open-loop control are used simultaneously. In such systems, the open-loop control is termed
feedforward and serves to further improve reference tracking performance.
A common closed-loop controller architecture is the
PID controller
A proportional–integral–derivative controller (PID controller or three-term controller) is a control loop mechanism employing feedback that is widely used in industrial control systems and a variety of other applications requiring continuou ...
.
Closed-loop transfer function
The output of the system ''y''(''t'') is fed back through a sensor measurement ''F'' to a comparison with the reference value ''r''(''t''). The controller ''C'' then takes the error ''e'' (difference) between the reference and the output to change the inputs ''u'' to the system under control ''P''. This is shown in the figure. This kind of controller is a closed-loop controller or feedback controller.
This is called a single-input-single-output (''SISO'') control system; ''MIMO'' (i.e., Multi-Input-Multi-Output) systems, with more than one input/output, are common. In such cases variables are represented through
vectors instead of simple
scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
values. For some
distributed parameter systems
In control theory, a distributed-parameter system (as opposed to a lumped-parameter system) is a system whose state space is infinite-dimensional. Such systems are therefore also known as infinite-dimensional systems. Typical examples are system ...
the vectors may be infinite-
dimensional
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
(typically functions).
![simple feedback control loop2](https://upload.wikimedia.org/wikipedia/commons/9/90/Simple_feedback_control_loop2.svg)
If we assume the controller ''C'', the plant ''P'', and the sensor ''F'' are
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
and
time-invariant
In control theory, a time-invariant (TIV) system has a time-dependent system function that is not a direct function of time. Such systems are regarded as a class of systems in the field of system analysis. The time-dependent system function is ...
(i.e., elements of their
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, theoretically models the system's output for ...
''C''(''s''), ''P''(''s''), and ''F''(''s'') do not depend on time), the systems above can be analysed using the
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
on the variables. This gives the following relations:
:
:
:
Solving for ''Y''(''s'') in terms of ''R''(''s'') gives
:
The expression
is referred to as the ''closed-loop transfer function'' of the system. The numerator is the forward (open-loop) gain from ''r'' to ''y'', and the denominator is one plus the gain in going around the feedback loop, the so-called loop gain. If
, i.e., it has a large
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
with each value of ''s'', and if
, then ''Y''(''s'') is approximately equal to ''R''(''s'') and the output closely tracks the reference input.
PID feedback control
![PID en](https://upload.wikimedia.org/wikipedia/commons/4/43/PID_en.svg)
A proportional–integral–derivative controller (PID controller) is a
control loop
A control loop is the fundamental building block of industrial control systems. It consists of all the physical components and control functions necessary to automatically adjust the value of a measured process variable (PV) to equal the value of ...
feedback mechanism
Feedback occurs when outputs of a system are routed back as inputs as part of a Signal chain (signal processing chain), chain of Causality, cause-and-effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. ...
control technique widely used in control systems.
A PID controller continuously calculates an ''error value'' as the difference between a desired
setpoint and a measured
process variable
In control theory, a process variable (PV; also process value or process parameter) is the current measured value of a particular part of a process which is being monitored or controlled. An example of this would be the temperature of a furnace ...
and applies a correction based on
proportional,
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
, and
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
terms. ''PID'' is an initialism for ''Proportional-Integral-Derivative'', referring to the three terms operating on the error signal to produce a control signal.
The theoretical understanding and application dates from the 1920s, and they are implemented in nearly all analogue control systems; originally in mechanical controllers, and then using discrete electronics and later in industrial process computers.
The PID controller is probably the most-used feedback control design.
If is the control signal sent to the system, is the measured output and is the desired output, and is the tracking error, a PID controller has the general form
:
The desired closed loop dynamics is obtained by adjusting the three parameters , and , often iteratively by "tuning" and without specific knowledge of a plant model. Stability can often be ensured using only the proportional term. The integral term permits the rejection of a step disturbance (often a striking specification in
process control
An industrial process control in continuous production processes is a discipline that uses industrial control systems to achieve a production level of consistency, economy and safety which could not be achieved purely by human manual control. I ...
). The derivative term is used to provide damping or shaping of the response. PID controllers are the most well-established class of control systems: however, they cannot be used in several more complicated cases, especially if
MIMO
In radio, multiple-input and multiple-output, or MIMO (), is a method for multiplying the capacity of a radio link using multiple transmission and receiving antennas to exploit multipath propagation. MIMO has become an essential element of wir ...
systems are considered.
Applying
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
ation results in the transformed PID controller equation
:
:
with the PID controller transfer function
:
As an example of tuning a PID controller in the closed-loop system , consider a 1st order plant given by
:
where and are some constants. The plant output is fed back through
:
where is also a constant. Now if we set
, , and
, we can express the PID controller transfer function in series form as
:
Plugging , , and into the closed-loop transfer function , we find that by setting
:
. With this tuning in this example, the system output follows the reference input exactly.
However, in practice, a pure differentiator is neither physically realizable nor desirable due to amplification of noise and resonant modes in the system. Therefore, a
phase-lead compensator type approach or a differentiator with low-pass roll-off are used instead.
Linear and nonlinear control theory
The field of control theory can be divided into two branches:
* ''
Linear control theory
A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large industrial co ...
'' – This applies to systems made of devices which obey the
superposition principle
The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So tha ...
, which means roughly that the output is proportional to the input. They are governed by
linear differential equation
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
:a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b( ...
s. A major subclass is systems which in addition have parameters which do not change with time, called ''
linear time invariant
In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly define ...
'' (LTI) systems. These systems are amenable to powerful
frequency domain
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a signa ...
mathematical techniques of great generality, such as the
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
,
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
,
Z transform
In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation.
It can be considered as a discrete-tim ...
,
Bode plot,
root locus
In control theory and stability theory, root locus analysis is a graphical method for examining how the roots of a system change with variation of a certain system parameter, commonly a gain within a feedback system. This is a technique used as a ...
, and
Nyquist stability criterion
In control theory and stability theory, the Nyquist stability criterion or Strecker–Nyquist stability criterion, independently discovered by the German electrical engineer at Siemens in 1930 and the Swedish-American electrical engineer Harry ...
. These lead to a description of the system using terms like
bandwidth
Bandwidth commonly refers to:
* Bandwidth (signal processing) or ''analog bandwidth'', ''frequency bandwidth'', or ''radio bandwidth'', a measure of the width of a frequency range
* Bandwidth (computing), the rate of data transfer, bit rate or thr ...
,
frequency response
In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of sy ...
,
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s,
gain
Gain or GAIN may refer to:
Science and technology
* Gain (electronics), an electronics and signal processing term
* Antenna gain
* Gain (laser), the amplification involved in laser emission
* Gain (projection screens)
* Information gain in de ...
,
resonant frequencies
Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillati ...
,
zeros and poles
In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable. In some sense, it is the simplest type of singularity. Technically, a point is a pole of a function if i ...
, which give solutions for system response and design techniques for most systems of interest.
* ''
Nonlinear control theory
Nonlinear control theory is the area of control theory which deals with systems that are nonlinear, time-variant, or both. Control theory is an interdisciplinary branch of engineering and mathematics that is concerned with the behavior of dyn ...
'' – This covers a wider class of systems that do not obey the superposition principle, and applies to more real-world systems because all real control systems are nonlinear. These systems are often governed by
nonlinear differential equation
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
s. The few mathematical techniques which have been developed to handle them are more difficult and much less general, often applying only to narrow categories of systems. These include
limit cycle
In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity ...
theory,
Poincaré map
In mathematics, particularly in dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection of a periodic orbit in the state space of a continuous dynamical system with a certain lower-dimensiona ...
s,
Lyapunov stability theorem, and
describing function In control systems theory, the describing function (DF) method, developed by Nikolay Mitrofanovich Krylov and Nikolay Bogoliubov in the 1930s, and extended by Ralph Kochenburger is an approximate procedure for analyzing certain nonlinear control ...
s. Nonlinear systems are often analyzed using
numerical method
In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm.
Mathem ...
s on computers, for example by
simulating their operation using a
simulation language
A computer simulation language is used to describe the operation of a simulation on a computer.Fritzson, Peter, and Vadim Engelson.Modelica—A unified object-oriented language for system modeling and simulation" European Conference on Object-Orie ...
. If only solutions near a stable point are of interest, nonlinear systems can often be
linearized by approximating them by a linear system using
perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
, and linear techniques can be used.
Analysis techniques - frequency domain and time domain
Mathematical techniques for analyzing and designing control systems fall into two different categories:
* ''
Frequency domain
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a signa ...
'' – In this type the values of the
state variable
A state variable is one of the set of variables that are used to describe the mathematical "state" of a dynamical system. Intuitively, the state of a system describes enough about the system to determine its future behaviour in the absence of a ...
s, the mathematical
variables representing the system's input, output and feedback are represented as functions of
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
. The input signal and the system's
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, theoretically models the system's output for ...
are converted from time functions to functions of frequency by a
transform
Transform may refer to:
Arts and entertainment
* Transform (scratch), a type of scratch used by turntablists
* ''Transform'' (Alva Noto album), 2001
* ''Transform'' (Howard Jones album) or the title song, 2019
* ''Transform'' (Powerman 5000 album ...
such as the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
,
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
, or
Z transform
In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation.
It can be considered as a discrete-tim ...
. The advantage of this technique is that it results in a simplification of the mathematics; the ''
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s'' that represent the system are replaced by ''
algebraic equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
s'' in the frequency domain which is much simpler to solve. However, frequency domain techniques can only be used with linear systems, as mentioned above.
* ''
Time-domain state space representation
In control engineering, a state-space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations or difference equations. State variables are variables w ...
'' – In this type the values of the
state variable
A state variable is one of the set of variables that are used to describe the mathematical "state" of a dynamical system. Intuitively, the state of a system describes enough about the system to determine its future behaviour in the absence of a ...
s are represented as functions of time. With this model, the system being analyzed is represented by one or more
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s. Since frequency domain techniques are limited to
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
systems, time domain is widely used to analyze real-world nonlinear systems. Although these are more difficult to solve, modern computer simulation techniques such as
simulation language
A computer simulation language is used to describe the operation of a simulation on a computer.Fritzson, Peter, and Vadim Engelson.Modelica—A unified object-oriented language for system modeling and simulation" European Conference on Object-Orie ...
s have made their analysis routine.
In contrast to the frequency domain analysis of the classical control theory, modern control theory utilizes the time-domain
state space
A state space is the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory.
For instance, the toy ...
representation, a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations. To abstract from the number of inputs, outputs, and states, the variables are expressed as vectors and the differential and algebraic equations are written in matrix form (the latter only being possible when the dynamical system is linear). The state space representation (also known as the "time-domain approach") provides a convenient and compact way to model and analyze systems with multiple inputs and outputs. With inputs and outputs, we would otherwise have to write down Laplace transforms to encode all the information about a system. Unlike the frequency domain approach, the use of the state-space representation is not limited to systems with linear components and zero initial conditions. "State space" refers to the space whose axes are the state variables. The state of the system can be represented as a point within that space.
System interfacing - SISO & MIMO
Control systems can be divided into different categories depending on the number of inputs and outputs.
*
Single-input single-output
In control engineering, a single-input and single-output (SISO) system is a simple single variable control system with one input and one output. In radio it is the use of only one antenna both in the transmitter and receiver.
Details
SISO sys ...
(SISO) – This is the simplest and most common type, in which one output is controlled by one control signal. Examples are the cruise control example above, or an
audio system
Sound recording and reproduction is the electrical, mechanical, electronic, or digital inscription and re-creation of sound waves, such as spoken voice, singing, instrumental music, or sound effects. The two main classes of sound recording te ...
, in which the control input is the input audio signal and the output is the sound waves from the speaker.
*
Multiple-input multiple-output
In radio, multiple-input and multiple-output, or MIMO (), is a method for multiplying the capacity of a radio link using multiple transmission and receiving antennas to exploit multipath propagation. MIMO has become an essential element of wir ...
(MIMO) – These are found in more complicated systems. For example, modern large
telescope
A telescope is a device used to observe distant objects by their emission, absorption, or reflection of electromagnetic radiation. Originally meaning only an optical instrument using lenses, curved mirrors, or a combination of both to observe ...
s such as the
Keck and
MMT have mirrors composed of many separate segments each controlled by an
actuator
An actuator is a component of a machine that is responsible for moving and controlling a mechanism or system, for example by opening a valve. In simple terms, it is a "mover".
An actuator requires a control device (controlled by control signal) a ...
. The shape of the entire mirror is constantly adjusted by a MIMO
active optics
Active optics is a technology used with reflecting telescopes developed in the 1980s, which actively shapes a telescope's mirrors to prevent deformation due to external influences such as wind, temperature, and mechanical stress. Without active op ...
control system using input from multiple sensors at the focal plane, to compensate for changes in the mirror shape due to thermal expansion, contraction, stresses as it is rotated and distortion of the
wavefront
In physics, the wavefront of a time-varying ''wave field'' is the set (locus) of all points having the same ''phase''. The term is generally meaningful only for fields that, at each point, vary sinusoidally in time with a single temporal freque ...
due to turbulence in the atmosphere. Complicated systems such as
nuclear reactor
A nuclear reactor is a device used to initiate and control a fission nuclear chain reaction or nuclear fusion reactions. Nuclear reactors are used at nuclear power plants for electricity generation and in nuclear marine propulsion. Heat from nu ...
s and human
cells are simulated by a computer as large MIMO control systems.
Topics in control theory
Stability
The ''stability'' of a general
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
with no input can be described with
Lyapunov stability
Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. ...
criteria.
*A
linear system
In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator.
Linear systems typically exhibit features and properties that are much simpler than the nonlinear case.
As a mathematical abstraction o ...
is called
bounded-input bounded-output (BIBO) stable if its output will stay
bounded for any bounded input.
*Stability for
nonlinear system
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
s that take an input is
input-to-state stability Input-to-state stability (ISS)Eduardo D. Sontag. Mathematical Control Theory: Finite-Dimensional Systems. Springer-Verlag, London, 1998Hassan K. Khalil. Nonlinear Systems. Prentice Hall, 2002.[poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in Ce ...]
of its
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, theoretically models the system's output for ...
must have negative-real values, i.e. the real part of each pole must be less than zero. Practically speaking, stability requires that the transfer function complex poles reside
* in the open left half of the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
for continuous time, when the
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
is used to obtain the transfer function.
* inside the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
for discrete time, when the
Z-transform
In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation.
It can be considered as a discrete-tim ...
is used.
The difference between the two cases is simply due to the traditional method of plotting continuous time versus discrete time transfer functions. The continuous Laplace transform is in
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
where the
axis is the real axis and the discrete Z-transform is in
circular coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point (mathematics), point on a plane (mathematics), plane is determined by a distance from a reference point and an angle from a reference direction ...
where the
axis is the real axis.
When the appropriate conditions above are satisfied a system is said to be
asymptotically stable
Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. T ...
; the variables of an asymptotically stable control system always decrease from their initial value and do not show permanent oscillations. Permanent oscillations occur when a pole has a real part exactly equal to zero (in the continuous time case) or a modulus equal to one (in the discrete time case). If a simply stable system response neither decays nor grows over time, and has no oscillations, it is
marginally stable; in this case the system transfer function has non-repeated poles at the complex plane origin (i.e. their real and complex component is zero in the continuous time case). Oscillations are present when poles with real part equal to zero have an imaginary part not equal to zero.
If a system in question has an
impulse response
In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an Dirac delta function, impulse (). More generally, an impulse ...
of
: