Control theory deals with the control of dynamical system
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
; often with the aim to achieve a degree of optimality
To do this, a controller with the requisite corrective behavior is required. This controller monitors the controlled process variable (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
. This is the basis for the advanced type of automation that revolutionized manufacturing, aircraft, communications and other industries. This is ''feedback control'', which involves taking measurements using a sensor
and making calculated adjustments to keep the measured variable within a set range by means of a "final control element", such as a control valve
Extensive use is usually made of a diagrammatic style known as the block diagram
. In it the transfer function
, 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
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
. Control theory was further advanced by Edward Routh
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
theory by Nicolas Minorsky
Although a major application of mathematical
control theory is in control systems engineering
, which deals with the design of process control
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 operation research.
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
, conducted by the physicist James Clerk Maxwell
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
, 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
, 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
A notable application of dynamic control was in the area of manned 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
, control theory was becoming an important area of research. Irmgard Flügge-Lotz
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
s, guidance system
s and electronics
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
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
Open-loop and closed-loop (feedback) control
thumb|400px|Example of a single industrial control loop; showing continuously modulated control of process flow.
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."
An example of a control system is a car's cruise control
, 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
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
'' because there is no feedback
; 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
, control theory introduces feedback
A closed-loop controller
uses feedback to control states
of a dynamical system
. Its name comes from the information path in the system: process inputs (e.g., voltage
applied to an electric motor
) have an effect on the process outputs (e.g., speed or torque of the motor), which is measured with sensor
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
* disturbance rejection (such as hills in the cruise control example above)
* guaranteed performance even with model
uncertainties, when the model structure does not match perfectly the real process and the model parameters are not exact
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
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
values. For some distributed parameter systems
the vectors may be infinite-dimensional
If we assume the controller ''C'', the plant ''P'', and the sensor ''F'' are linear
(i.e., elements of their transfer function
''C''(''s''), ''P''(''s''), and ''F''(''s'') do not depend on time), the systems above can be analysed using the Laplace transform
on the variables. This gives the following relations:
Solving for ''Y''(''s'') in terms of ''R''(''s'') gives
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
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
A proportional–integral–derivative controller (PID controller) is a control loop feedback mechanism
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
and applies a correction based on proportional
, and derivative
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
). 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
systems are considered.
Applying Laplace transformation
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
'' – This applies to systems made of devices which obey the superposition principle
, which means roughly that the output is proportional to the input. They are governed by linear differential equation
s. A major subclass is systems which in addition have parameters which do not change with time, called ''linear time invariant
'' (LTI) systems. These systems are amenable to powerful frequency domain
mathematical techniques of great generality, such as the Laplace transform
, Fourier transform
, Z transform
, Bode plot
, root locus
, and Nyquist stability criterion
. These lead to a description of the system using terms like bandwidth
, frequency response
, resonant frequencies
, zeros and poles
, which give solutions for system response and design techniques for most systems of interest.
* ''Nonlinear control theory
'' – 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
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
theory, Poincaré map
s, Lyapunov stability theorem
, and describing function
s. Nonlinear systems are often analyzed using numerical method
s on computers, for example by simulating
their operation using a simulation language
. 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
, 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 this type the values of the state variable
s, the mathematical variables
representing the system's input, output and feedback are represented as functions of frequency
. The input signal and the system's transfer function
are converted from time functions to functions of frequency by a transform
such as the Fourier transform
, Laplace transform
, or Z transform
. The advantage of this technique is that it results in a simplification of the mathematics; the ''differential equation
s'' that represent the system are replaced by ''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 this type the values of the state variable
s are represented as functions of time. With this model, the system being analyzed is represented by one or more differential equation
s. Since frequency domain techniques are limited to linear
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
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
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
(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
, in which the control input is the input audio signal and the output is the sound waves from the speaker.
* Multiple-input multiple-output
(MIMO) – These are found in more complicated systems. For example, modern large telescope
s such as the Keck
have mirrors composed of many separate segments each controlled by an actuator
. The shape of the entire mirror is constantly adjusted by a MIMO active optics
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
due to turbulence in the atmosphere. Complicated systems such as nuclear reactor
s and human cells
are simulated by a computer as large MIMO control systems.
Topics in control theory
The ''stability'' of a general dynamical system
with no input can be described with Lyapunov stability
*A linear system
is called bounded-input bounded-output (BIBO) stable
if its output will stay bounded
for any bounded input.
*Stability for nonlinear system
s that take an input is input-to-state stability
(ISS), which combines Lyapunov stability and a notion similar to BIBO stability.
For simplicity, the following descriptions focus on continuous-time and discrete-time linear systems.
Mathematically, this means that for a causal linear system to be stable all of the poles
of its transfer function
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
for continuous time, when the Laplace transform
is used to obtain the transfer function.
* inside the unit circle
for discrete time, when the Z-transform
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
axis is the real axis and the discrete Z-transform is in circular coordinates
axis is the real axis.
When the appropriate conditions above are satisfied a system is said to be asymptotically stable
; 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