Control theory is an interdisciplinary branch of
engineering and
mathematics, that deals with the behavior of
dynamical systems. The desired
output of a system is called the
reference. When one or
more output variables of a system need to follow a certain
reference over time, a
controller manipulates the
inputs to a system to obtain the desired effect on the output of
the system.
Overview
Control theory is
An example
Consider an automobile's
cruise
control, which is a device designed to maintain a constant
vehicle speed; the
desired or
reference speed,
provided by the driver. The
system in this case is the
vehicle. The system output is the vehicle speed, and the control
variable is the engine's
throttle position
which influences engine
torque output.
A primitive way to implement cruise control is simply to lock the
throttle position when the driver engages cruise control. However,
on mountain terrain, the vehicle will slow down going uphill and
accelerate going downhill. In fact, any parameter different than
what was assumed at design time will translate into a proportional
error in the output velocity, including exact mass of the vehicle,
wind resistance, and tire pressure. This type of controller is
called an
open-loop controller
because there is no direct connection between the output of the
system (the vehicle's speed) and the actual conditions encountered;
that is to say, the system does not and can not compensate for
unexpected forces.
In a
closed-loop control system, a sensor monitors
the output (the vehicle's speed) and feeds the data to a computer
which continuously adjusts the control input (the throttle) as
necessary to keep the control error to a minimum (that is, to
maintain the desired speed). Feedback on how the system is actually
performing allows the controller (vehicle's on board computer) to
dynamically compensate for disturbances to the system, such as
changes in slope of the ground or wind speed. An ideal feedback
control system cancels out all errors, effectively mitigating the
effects of any forces that may or may not arise during operation
and producing a response in the system that perfectly matches the
user's wishes.
History
Centrifugal governor in a Boulton
& Watt engine of 1788
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. This
described and analyzed the phenomenon of "hunting", in which lags
in the system can lead to overcompensation and unstable behavior.
This generated a flurry of interest in the topic, during which
Maxwell's classmate
Edward John
Routh generalized the results of Maxwell for the general class
of linear systems. Independently,
Adolf
Hurwitz analyzed system stability using differential equations
in 1877. This result is called 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). Control of the airplane was necessary
for safe flight.
By
World War II, control theory was an
important part of
fire-control
systems,
guidance systems and
electronics. The
Space Race also depended on accurate spacecraft
control. However, control theory also saw an increasing use in
fields such as
economics.
People in systems and control
Many active and historical figures made significant contribution to
control theory, including, for example:
Classical control theory
To avoid the problems of the open-loop controller, control theory
introduces
feedback.A closed-loop
controller uses feedback to
control
states or
outputs 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. velocity or torque of the motor), which is
measured with
sensors and processed by the
controller; the result (the control signal) is used as input to the
process, closing the loop.
Closed-loop controllers have the following advantages over
open-loop controllers:
- disturbance rejection (such as unmeasured friction in a
motor)
- guaranteed performance even with model uncertainties, when the model
structure does not match perfectly the real process and the model
parameters are not exact
- unstable 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 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 (typically
functions).
If we assume the controller
C, the plant
P, and
the sensor
F are
linear and
time-invariant (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:
- Y(s) = P(s) U(s)\,\!
- U(s) = C(s) E(s)\,\!
- E(s) = R(s) - F(s)Y(s).\,\!
Solving for
Y(
s) in terms of
R(
s) gives:
- Y(s) = \left( \frac{P(s)C(s)}{1 + F(s)P(s)C(s)} \right) R(s) =
H(s)R(s).
The expression H(s) = \frac{P(s)C(s)}{1 + F(s)P(s)C(s)} 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 |P(s)C(s)|
\gg 1, i.e. it has a large
norm
with each value of
s, and if |F(s)| \approx 1, then
Y(s) is approximately equal to
R(s). This means
simply setting the reference controls the output.
PID controller
The
PID controller is probably the
most-used feedback control design. "PID" means
Proportional-Integral-Derivative, referring to the three terms
operating on the error signal to produce a control signal. If
u(t) is the control signal sent to the system,
y(t) is the measured output and
r(t) is the
desired output, and tracking error e(t)=r(t)- y(t), a PID
controller has the general form
- u(t) = K_P e(t) + K_I \int e(t)\text{d}t + K_D
\frac{\text{d}}{\text{d}t}e(t).
The desired closed loop dynamics is obtained by adjusting the three
parameters K_P, K_I and K_D, 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
- u(s) = K_P e(s) + K_I \frac{1}{s} e(s) + K_D s e(s)
- u(s) = (K_P + K_I \frac{1}{s} + K_D s) e(s)
with the PID controller transfer function
- C(s) = (K_P + K_I \frac{1}{s} + K_D s).
Modern control theory
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 vector within that
space.
Topics in control theory
Stability
The
stability of a general
dynamical system with no input can be
described with
Lyapunov stability
criteria. A
linear system that takes
an input is called
bounded-input
bounded-output stable if its output will stay
bounded for any bounded input. Stability
for
nonlinear systems 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 satisfy
some criteria depending on whether a continuous or discrete time
analysis is used:
- In continuous time, the Laplace
transform is used to obtain the transfer function. A system is
stable if the poles of this transfer function lie strictly in the
closed left half of the complex plane
(i.e. the real part of all the poles is less than zero).
- In discrete time the Z-transform is
used. A system is stable if the poles of this transfer function lie
strictly inside the unit circle. i.e.
the magnitude of the poles is less than one).
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 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.
Differences between the two cases are not a contradiction. The
Laplace transform is in
Cartesian
coordinates and the Z-transform is in
circular coordinates, and it can be
shown that
- the negative-real part in the Laplace domain can map onto the
interior of the unit circle
- the positive-real part in the Laplace domain can map onto the
exterior of the unit circle
If a system in question has an
impulse
response of
- \ x[n] = 0.5^n u[n]
then the
Z-transform (see
this example), is given
by
- \ X(z) = \frac{1}{1 - 0.5z^{-1}}\
which has a pole in z = 0.5 (zero
imaginary part). This system is BIBO
(asymptotically) stable since the pole is
inside the unit
circle.
However, if the impulse response was
- \ x[n] = 1.5^n u[n]
then the Z-transform is
- \ X(z) = \frac{1}{1 - 1.5z^{-1}}\
which has a pole at z = 1.5 and is not BIBO stable since the pole
has a modulus strictly greater than one.
Numerous tools exist for the analysis of the poles of a system.
These include graphical systems like the
root
locus,
Bode plots or the
Nyquist plots.
Mechanical changes can make equipment (and control systems) more
stable. Sailors add ballast to improve the stability of ships.
Cruise ships use antiroll fins that extend transversely from the
side of the ship for perhaps 30 feet (10 m) and are continuously
rotated about their axes to develop forces that oppose the
roll.
Controllability and observability
Controllability and
observability are main issues in the analysis
of a system before deciding the best control strategy to be
applied, or whether it is even possible to control or stabilize the
system. Controllability is related to the possibility of forcing
the system into a particular state by using an appropriate control
signal. If a state is not controllable, then no signal will ever be
able to control the state. If a state is not controllable, but its
dynamics are stable, then the state is termed Stabilizable.
Observability instead is related to the possibility of "observing",
through output measurements, the state of a system. If a state is
not observable, the controller will never be able to determine the
behaviour of an unobservable state and hence cannot use it to
stabilize the system. However, similar to the stabilizability
condition above, if a state cannot be observed it might still be
detectable.
From a geometrical point of view, looking at the states of each
variable of the system to be controlled, every "bad" state of these
variables must be controllable and observable to ensure a good
behaviour in the closed-loop system. That is, if one of the
eigenvalues of the system is not both
controllable and observable, this part of the dynamics will remain
untouched in the closed-loop system. If such an eigenvalue is not
stable, the dynamics of this eigenvalue will be present in the
closed-loop system which therefore will be unstable. Unobservable
poles are not present in the transfer function realization of a
state-space representation, which is why sometimes the latter is
preferred in dynamical systems analysis.
Solutions to problems of uncontrollable or unobservable system
include adding actuators and sensors.
Control specifications
Several different control strategies have been devised in the past
years. These vary from extremely general ones (
PID controller), to others devoted to very
particular classes of systems (especially
robotics or
aircraft cruise
control).
A control problem can have several specifications. Stability, of
course, is always present: the controller must ensure that the
closed-loop system is stable, regardless of the open-loop
stability. A poor choice of controller can even worsen the
stability of the open-loop system, which must normally be avoided.
Sometimes it would be desired to obtain particular dynamics in the
closed loop: i.e. that the poles have Re[\lambda]
-\overline{\lambda}, where \overline{\lambda} is a fixed value
strictly greater than zero, instead of simply ask that
Re[\lambda]<0<></0<>math>.
Another typical specification is the rejection of a step
disturbance; including an
integrator in
the open-loop chain (i.e. directly before the system under control)
easily achieves this. Other classes of disturbances need different
types of sub-systems to be included.
Other "classical" control theory specifications regard the
time-response of the closed-loop system: these include the
rise time (the time needed by the control system
to reach the desired value after a perturbation), peak
overshoot (the highest value reached by
the response before reaching the desired value) and others
(
settling time, quarter-decay).
Frequency domain specifications are usually related to
robustness (see after).
Modern performance assessments use some variation of integrated
tracking error (IAE,ISA,CQI).
Model identification and robustness
A control system must always have some robustness property. A
robust controller is such that its
properties do not change much if applied to a system slightly
different from the mathematical one used for its synthesis. This
specification is important: no real physical system truly behaves
like the series of differential equations used to represent it
mathematically. Typically a simpler mathematical model is chosen in
order to simplify calculations, otherwise the true system dynamics
can be so complicated that a complete model is impossible.
- System identification
The process of determining the equations that govern the model's
dynamics is called
system
identification. This can be done off-line: for example,
executing a series of measures from which to calculate an
approximated mathematical model, typically its
transfer function or matrix. Such
identification from the output, however, cannot take account of
unobservable dynamics. Sometimes the model is built directly
starting from known physical equations: for example, in the case of
a
mass-spring-damper
system we know that m \ddot (t) = - K x(t) - \Beta \dot{x}(t). Even
assuming that a "complete" model is used in designing the
controller, all the parameters included in these equations (called
"nominal parameters") are never known with absolute precision; the
control system will have to behave correctly even when connected to
physical system with true parameter values away from nominal.
Some advanced control techniques include an "on-line"
identification process (see later). The parameters of the model are
calculated ("identified") while the controller itself is running:
in this way, if a drastic variation of the parameters ensues (for
example, if the robot's arm releases a weight), the controller will
adjust itself consequently in order to ensure the correct
performance.
- Analysis
Analysis of the robustness of a SISO control system can be
performed in the frequency domain, considering the system's
transfer function and using
Nyquist
and
Bode diagrams. Topics include
gain and phase
margin and amplitude margin. For MIMO and, in general, more
complicated control systems one must consider the theoretical
results devised for each control technique (see next section):
i.e., if particular robustness qualities are needed, the engineer
must shift his attention to a control technique including them in
its properties.
- Constraints
A particular robustness issue is the requirement for a control
system to perform properly in the presence of input and state
constraints. In the physical world every signal is limited. It
could happen that a controller will send control signals that
cannot be followed by the physical system: for example, trying to
rotate a valve at excessive speed. This can produce undesired
behavior of the closed-loop system, or even break actuators or
other subsystems. Specific control techniques are available to
solve the problem:
model
predictive control (see later), and
anti-wind up systems. The
latter consists of an additional control block that ensures that
the control signal never exceeds a given threshold.
System classifications
Linear control
For MIMO systems, pole placement can be performed mathematically
using a
state space
representation of the open-loop system and calculating a
feedback matrix assigning poles in the desired positions. In
complicated systems this can require computer-assisted calculation
capabilities, and cannot always ensure robustness. Furthermore, all
system states are not in general measured and so observers must be
included and incorporated in pole placement design.
Nonlinear control
Processes in industries like
robotics and
the
aerospace industry typically
have strong nonlinear dynamics. In control theory it is sometimes
possible to linearize such classes of systems and apply linear
techniques, but in many cases it can be necessary to devise from
scratch theories permitting control of nonlinear systems. These,
e.g.,
feedback linearization,
backstepping,
sliding mode control, trajectory
linearization control normally take advantage of results based on
Lyapunov's theory.
Differential geometry has been widely
used as a tool for generalizing well-known linear control concepts
to the non-linear case, as well as showing the subtleties that make
it a more challenging problem.
Main control strategies
Every control system must guarantee first the stability of the
closed-loop behavior. For
linear
systems, this can be obtained by directly placing the poles.
Non-linear control systems use specific theories (normally based on
Aleksandr Lyapunov's Theory) to
ensure stability without regard to the inner dynamics of the
system. The possibility to fulfill different specifications varies
from the model considered and the control strategy chosen. Here a
summary list of the main control techniques is shown:
- Adaptive control
- Adaptive control uses on-line
identification of the process parameters, or modification of
controller gains, thereby obtaining strong robustness properties.
Adaptive controls were applied for the first time in the aerospace industry in the 1950s, and have
found particular success in that field.
- Hierarchical control
- A Hierarchical control
system is a type of Control
System in which a set of devices and governing software is
arranged in a hierarchical tree. When the links in the tree are
implemented by a computer network,
then that hierarchical control system is also a form of Networked control system.
- Intelligent control
- Intelligent control use
various AI computing approaches like neural networks, Bayesian probability, fuzzy logic, machine
learning, evolutionary
computation and genetic
algorithms to control a dynamic
system.
- Optimal control
- Optimal control is a particular
control technique in which the control signal optimizes a certain
"cost index"
- Robust control
- Robust control deals explicitly
with uncertainty in its approach to controller design. Controllers
designed using robust control methods tend to be able to
cope with small differences between the true system and the nominal
model used for design. The early methods of Bode and others were fairly robust; the state-space
methods invented in the 1960s and 1970s were sometimes found to
lack robustness. A modern example of a robust control
technique is H-infinity
loop-shaping developed by Duncan
McFarlane and Keith Glover of
Cambridge
University, United
Kingdom. Robust methods aim to achieve robust
performance and/or stability in the
presence of small modeling errors.
- Stochastic control
- Stochastic control deals with
control design with uncertainty in the model. In typical stochastic
control problems, it is assumed that there exist random noise and
disturbances in the model and the controller, and the control
design must take into account these random deviations.
See also
- Examples of control systems
- Topics in control theory
- Other related topics
References
Further reading
External links