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Systems and Controls Glossary
Purpose
This website is intended to provide visitors with an evergrowing systems and controls glossary.
Please forward comments and suggestions to Dr. Rodriguez
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I
J
K
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M
N
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P
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X
Y
Z
A
Actuator - a power conversion device (e.g. pneumatic motor, pneumatic valve, hydraulic motor, electric motor, etc.) which is responsible for generating control signals to the plant.
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B
BIBO Stable - a system is bounded-input bounded-output stable if bounded inputs get mapped to bounded outputs. An LTI system is BIBO stable if and only if its impulse response is absolutely integrable.
Big Picture - generally, feedback control system design seeks to design a controller K for a plant P based on a nominal model P_o for P such that the resulting nominal closed loop system (with P_o and K) exhibits the following properties: stability, stability robustness with respect to realistic modeling errors, good low frequency command following (tracking), good low frequency disturbance rejection, and good high frequency noise attenuation, performance robustness with respect to realistic modeling errors.
Block Diagram - a diagram which shows how the subsystems of a system are interconnected.
Bode Plot - defined for an LTI system with transfer function H; refers to two plots: (i) a plot of 20log_10 |H(jw)| versus log_10 w on a semilog graph - called the magnitude response or spectrum and (ii) a plot of angle of H(jw) versus versus log_10 w on a semilog graph - called the phase response or spectrum. Also referred to as the frequency response or spectrum of the system H.
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C
Command Following - refers to the ability of a feedback system to follow (or track) reference commands. Typically, it is desirable for feedback systems to track low frequency reference commands. Also see Tracking Error.
Compensator - refers to a system which uses feedback measurements to generate control signals to a plant under control; the idea is to positively alter the properties of the plant in terms of measures such as command following, disturbance rejection, noise attenuation, sensitivity to uncertainty, etc.. Also called controller.
Complementary Root Locus - the complementary root locus for a negative feedback system with open loop transfer function L(s) = k n(s)/d(s) is a plot in the s-plane of the roots of the closed loop characteristic equation
d(s) + k n(s) = 0
as the scalar constant k is varied from 0 to minus infinity. By so doing, one sees how the closed loop poles depend on the scalar constant - and possibly a design parameter - k. Also see Root Locus.
Complementary Sensitivity Transfer Function - given a negative feedback system with open loop transfer function L, the sensitivity transfer function - denoted T - is defined as follows:
T(s) = 1 - S(s) = L/(1 + L(s))
where S denotes the sensitivity transfer function. See
Sensitivity Transfer Function.
Controllability - a system is said to be controllable at t = t_o if for any initial state
x(to) = x_o and desired final state x_f, there exists a finite time t_f and a
control u(t) - where t lies in [t_o, t_f) - which transfers the state of the
system from x(t_o) = x_o to x(t_f) = x_f.
An LTI system (A,B,C,D) is controllable if and only if the
controllability matrix
C(A,B) = [ B AB A^{2}B A^{3}B :::::: A^{n-1}B ]
has full row rank; i.e. has n linearly independent rows. Here, A is an nxn matrix.
Controllability is very related to the problem of stabilizing a plant via state feedback. See
State Feedback,
Modal Controllability,
Stabilizability. Also see dual concept of
Observability.
Critically Damped Poles - refers to identical real poles which are stable (i.e. lie in the open left half plane). Such poles have a damping ratio zeta of unity and are associated with decaying exponentials in the time domain.
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D
DC Gain - the dc gain of an LTI system with transfer function H refers to the quantity H(0). The utility of this concept can be seen as follows. If the system H is stable and a step is applied then the steady state output of the system is H(o) times the size of the applied step.
Delay Margin - amount of time lag (time delay) which can be introduced into a feedback loop before the closed loop system goes unstable. Also see
Gain Margin and
Phase Margin.
Detectability - a dynamical system is detectable if all of its unstable modes are observable. Also see the dual concept of Stabilizability.
Disturbance Rejection - refers to the ability of a feedback system to attenuate (or reject) disturbances. Typically, it is desirable for feedback systems to attenuate/reject low frequency disturbances.
Duality - refers to the fact that controllability and observability are dual concepts. The pair (A,B) is controllable if and only if the pair (A^H, B^H) is observable. The pair (A,C) is observable if and only if the pair (A^H, C^H) is controllable. Similarly, stabilizability and detectability are dual concepts.
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E
Error Signal - usually refers to the difference between a desired quantity and the actual quantity. Also see
Tracking Error.
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F
Feedback - refers to the the use of feedback measurements to enhance the properties of a physical system called the plant. Feedback may be used for any one of the following reasons:
(i) stabilization of an unstable system,
(ii) altering the natural modes of a system,
(iii) sensitivity reduction to modeling uncertainty,
(iv) command following,
(v) rejection/attenuation of disturbances,
(vi) approximate linearization, and
(vii) approximate inversion. It is arguable that uncertainty - in general - is the main reason for needing feedback control laws. One, of course, must always be extremely careful when closing a feedback loop. This is because - if done improperly - the resulting closed loop system may be unstable - even if the open loop system is stable. Given this, one must conclude that feedback can be very dangerous!
Feedback System - refers to a sytem which uses feedback measurements to enhance the properties of a physical system called the plant.
Frequency Response - see Bode Plot.
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G
Gain Crossovers - consider a negative feedback system with open loop transfer function L(s). The term gain crossovers, or gain crossover frequecies, refers to those frequencies w such that |L(jw)| = 1. Phase margin - by definition - is measured at gain crossover frequencies. Also see Phase Margin.
Gain Margin - the upward gain margin of a feedback system is the amount that the gain of a feedback loop may be increased before going unstable. Upward gain margin is expressed as a fraction which is greater than or equal to unity. The downward gain margin of a feedback system is the amount that the gain of a feedback loop may be reduced before going unstable. Downward gain margin is expressed as a fraction which is less than or equal to unity. See
Phase Margin and
Delay Margin. Also see Phase Crossovers. Gain margin is - by definition - measured at phase crossover frequencies.
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H
High Pass Filter (HPF) - a system which allows high frequency signals to pass through while low frequency signals are comparatively attenuated. An differentiator is an example of a LPF. Practically, the current through a capacitor is obtained by differentiating the voltage across it. The voltage across an inductor is obtained by differentiating the current through the inductor.
Hurwitz Polynomial - a polynomial with its roots in the open left half plane.
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I
Imaginary Crossovers - consider a negative feedback system with open loop transfer function kL(s) = k n(s)/d(s) where k is a scalar parameter. The term imaginary crossovers, or imaginary crossover frequencies, refers to pairs (k, s) on the root locus (k >= 0) or complementary root locus (k <= 0) where s = jw lies on the imaginary axis; i.e. imaginary crossovers refer to pairs (k, s) where s is an imaginary closed loop pole.
Suppose that k is non-zero. In such a case, the corresponding imaginary crossovers are called phase crossovers or phase crossover frequencies. The imaginary crossovers, or phase crossover frequencies, for k > 0 (i.e. on the root locus) correspond to w values such that the angle of the open loop transfer function L(jw) is -180 degrees +/- any integer multiple of 360 degrees. The imaginary crossovers, or phase crossover frequencies, for k < 0 (i.e. on the complementary root locus) correspond to w values such that the angle of the open loop transfer function L(jw) is 0 degrees +/- any integer multiple of 360 degrees.
Impulse Response - the impulse response of a system is the response of the system to a Dirac-delta function (distribution). Unless otherwise specified, a unit delta function input under zero initial conditions is implied.
Internal Model Principle - refers to the idea that to follow a reference command r or reject a disturbance d, one needs a model of that reference command or disturbance within the feedback loop.
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J
Jordan Form of a Matrix - while some nxn matrices are not diagonalizable (i.e. do not have a set of independent eigenvectors), all matrices can be placed in Jordan Form.
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K
Kalman Filter - a model based filter which is used to estimate the state of a system from knowledge of noisy output and input measurements. A least squares error criterion is used to construct optimal linear estimates based on first and second order statistics for each noise source (e.g. sensor noise and process noise). Developed by R.E. Kalman during the early 1960's. Also see
State Estimator and
Observer.
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L
Laplace Transform - useful for solving problems involving systems described by linear ordinary differential equations with constant coefficients. The Unilateral Laplace Transform of a function f is denoted by the sybol F(s) and is defined as the integral from 0- to infinity of f(t)exp(-st). The resulting function F is defined for those values s in the complex plane for which the Laplace integral makes sense; i.e. is finite. Such values of s are said to make up the regions of convergence of the function F. See Region of Convergence.
Linear Function - a function f is said to be linear if it satisfies the relationship
f(a_1 x_1 + a_2 x_2) = a_1 f(x_1) + a_2 f(x_2)
for all scalars a_1, a_2, x_1, x_2.
Linear Operator - an operator T is said to be linear if it satisfies the relationship
T(a_1 x_1 + a_2 x_2) = a_1 T(x_1) + a_2 T(x_2)
for all scalars a_1, a_2 and all functions x_1 and x_2.
Linear Time Invariant (LTI) System - refers to dynamical systems which are linear and time-invariant.
Low Pass Filter (LPF) - a system which allows low frequency signals to pass through while high frequency signals are comparatively attenuated. An integrator is an example of a LPF. Practically, the current through an inductor is obtained by integrating the voltage across it. The voltage across a capacitor is obtained by integrating the current through the inductor.
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M
MIMO - multiple-input multiple-output.
Minimum Phase System - a system whose poles and zeros lie in the open left half plane. The idea is that such poles and zeros are associated with first quadrant phase plots. See Bode Plot.
Method of the Transfer Function (MOTF) - refers to the idea that the sinusoidal component of an LTI system's output may be computed using the systems's transfer function H as follows. If the input is x(t) = sin w_o t, then the sinusoidal component of the system's output is given by
y_sinusoidal = |H(jw_o)| sin (w_o t + Angle of H(jw_o) ).
This method works provided that H(jw_o) is well defined; i.e. a finite complex number. If H(jw_o) is not defined, then the method is not applicable and another method must be used.
More specifically, let H(s) denote the transfer function of a SISO LTI stable system. Suppose
that the input
u(t) = A sin( wt + theta )
is applied to the system H. Given, this, it follows that the steady state
output yss is given by
yss = A |H(jw)| sin( wt + theta + Angle(H(jw)) )
where
H(jw) = |H(jw)| e^{j Angle(H(jw)) }.
A similar expression holds when a "cosine input" is applied; i.e. if the input
u(t) = A cos( wt + theta )
is applied to H, then the steady state output yss is given by
yss = A |H(jw)| cos( wt + theta + Angle(H(jw)) ).
The above result is fundamental for the study of LTI systems.
The idea extends to multiple-input multiple-output (MIMO) LTI systems.
This result is arguably one of the most important ideas which are taught to
engineering students.
Modal Controllability - it can be shown that an LTI system (A,B,C,D) is controllable if and only if each of its modes are controllable. A mode (eigenvalue) is controllable if a control can be constructed to move the system's state anywhere along the associated eigenvector. If this is not possible, we say that the mode is uncontrollable. Also see Controllability and Stabilizability.
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N
Noise Attenuation - refers to the ability of a feedback system to attenuate noise (typically sensor noise). Typically, it is desirable for feedback systems to attenuate/reject high frequency sensor noise.
Nominal Model - refers to a mathematical model which approximates (usually accurately) a given system. An engineer may, for example, neglect high frequency dynamics in situations which require low frequency operation. Care must always be taken in making such approximations.
Non-Minimum Phase System - refers to systems which have at least one pole and/or zero in the closed right half plane. Usually this term is used to refer to systems which possess zeros in the open right half plane.
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O
Open Loop Poles - refers to poles (finite or at infinity) of the open loop transfer function.
Open Loop Zeros - refers to zeros (finite or at infinity) of the open loop transfer function.
Open Right Half Plane - refers to all points in the complex s-plane which possess a real part which is strictly negative.
Operator - a mathematical object which maps functions to functions. A system is an example of an operator. Extends the concept of a function.
Observability - refers to the ability or inability to uniquely reconstruct (estimate) the state of a system in finite time given output and input measurements. If the state can be determined uniquely in finite time, then we say that the system is observable. If not, we say that the system is unobservable. Also see the dual concept of
Controllability.
Observer - a model based filter which is used to reconstruct (or estimate) the state of a system from knowledge of noisy output and input measurements. Also called a State Estimator and Kalman Filter.
Overdamped Poles - refers to distinct real poles which are stable (i.e. lie in the open left half plane). Such poles have a damping ratio zeta which lies in the interval (1, infinity) and are associated with decaying exponentials in the time domain.
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P
Phase Crossovers - see imaginary crossvers.
Phase Margin - amount of phase lag which can be introduced into a feedback loop before the closed loop system goes unstable. See Gain Margin and
Delay Margin. Phase margin - by definition is measured at gain crossovers or gain crossover frequencies.
PID Controller - refers to controllers whose structure is the sum of three terms: a proportional term, an integral term, and a derivative term; e.g.
K(s) = k_1 + k_2/s + k_3 s
Such controllers have found widespread application in the process control industry. Many algorithms exist for the automatic selection of the PID design parameters k_1, k_2, and k_3. Such algorithms are called PID tunning algorithms. Be careful when using them.
Plant - physical system to be controlled; system whose properties we want to alter via feedback.
Pole - value of s in the complex plane at which a real-rational transfer function approaches infinity; refers to natural frequencies of a system. Also called mode or natural mode.
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Q
Quadratic Form - any function of the form f(x) = x^H Q x, where Q is a square matrix and x is a vector of suitable dimension. Here x^H denotes the conjugate transpose of the vector x.
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R
Reference Command - refers to an externally applied signal which represents a command; usually represents the desired output.
Region of Convergence (ROC) - the region of convergence of a transform F(s) refers to those values s in the complex plane for which the Laplace Transform integral yields a finite number; i.e. is well defined. See Laplace Transform.
Resonant Frequency - frequency at which the Bode Plot's magnitude response peaks.
Rise Time - refers to the time required for the output of a system to reach 90% of the final output value in response to a specific input. The rise time depends on the applied signal and on the dynamical properties of the system (e.g. time constants).
Root Locus - the root locus for a negative feedback system with open loop transfer function
L(s) = k n(s)/d(s) is a plot in the s-plane of the roots of the closed loop characteristic equation
d(s) + k n(s) = 0
as the scalar constant k is varied from 0 to infinity. By so doing, one sees how the closed loop poles depend on the scalar constant - and possibly a design parameter - k. Also see Complementary Root Locus.
Routh Table - a table formed from the coefficients of a polynomial. The table is used to determine the number of right half plane roots of the polynomial. It is also useful for determining the location of any roots which are symmetric with respect to the origin.
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S
Sensitivity Transfer Function - given a negative feedback system with open loop transfer function L, the sensitivity transfer function - denoted S - is defined as follows:
S(s) = 1/(1 + L(s)).
See
Complementary Sensitivity Transfer Function.
Series Compensation - refers to a compensator which is in series with the plant. Other forms of compensation include Parallel Compensations and Feedforward Compensation.
SISO - single-input single-output.
Stabilzability - a dynamical system is stabilizable if all of its unstable modes are controllable. Also see the dual concept of Detectability.
Stable System - usually refers to systems whose poles all lie in the open left half plane. Such poles are associated with decaying exponentials and exponential sinusoids in the time domain. Also see Unstable System.
State Estimator - refers to systems which process input and output measurements in order to estimate the state of the system. Also see
Kalman Filter and
Observer.
State Feedback - refers to control laws which generate control signals on the basis of state information alone. Also referred to as a control law which exploits perfect information. Most of the literature todate has focussed on constant gain state feedback control laws; i.e. state feedback control laws which take on the form u = - G x where u denotes an m-dimensional control, x denotes an n-dimensional state, and G is a constant mxn control gain matrix.
State of a System - refers to the minimum information required at a given instant to determine the future evolution (e.g. output) of the system given knowledge of the current and future inputs. Also called initial state.
State Space - modeling technique for dynamical systems which expresses a dynamical system in the following form:
xdot = f(x, u)
y = g(x, u)
where u represents the input or control, x the state, and y the output. In general, each of these are vectors: u is m-dimensional, x is n-dimensional, and y is p-dimensional. For LTI systems the above state space representation takes on the form:
xdot = A x + Bu
y = C x + Du
where A is an nxn matrix, B is an nxm matrix, C is a pxn martrix, and D is a pxm matrix. In such a case we say that the above LTI system is described by the state space quadruple (A,B,C,D). The transfer function matrix associated with this multiple-input multiple-output (MIMO) system is given by:
H(s) = C(sI - A)^{-1} B + D.
State Variables - set of variables which can be used to summarize the state of the system at any given time instant. Usually selected to be variables which are associated with energy storage within the system (e.g. capacitor voltages, inductor currents, spring positions, speed of masses, etc.).
Step Response - the step response of a system is the response of the system to a step function. Unless otherwise specified, a unit step function input under zero initial conditions is implied.
System - an operator which maps input signals u to output signals y. This concept represents a natural extension to the concept of a function.
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T
Time Constant - the time constant tau associated with a decaying exponential exp(-st) is defined by the relationship:
tau = 1/Re(s)
where Re(s) denotes the real part of the complex quantity s. Tau is measured using the same units that t is measured in. The above exponential is said to take approximately 5 tau time units (or 5 time constants) to decay to zero. This terminology is commonly used by engineers. It should be noted that values of s close to (far from) the origin of the complex s-plane (i.e. s = 0) correspond to large (small) time constants.
Time Delay - refers to a system whose output y is always the applied input u delayed (shifted) by a non-negative quantity delta where delta is the size of the time delay; i.e. y(t) = u(t - delta). Such a system is LTI and has a transfer function given by exp(-s delta).
Tracking Error - difference between the commnded output r and the actual output y; e_t = r - y.
Transfer Function - meaningful only for LTI systems; defined as the Laplace transform of the output divided by the Laplace transform of the input assuming all initial conditions are set to zero.
Transmission Zero - a dynamical system has a transmission zero as z_o if their exists an input direction u_o and an initial condition x_o (not both zero), such that if the input is
given by
u(t) = u_o exp(z_o t)
and
x(o-) = x_o,
then
x(t) = x_o exp(z_o t)
and
y(t) = 0
for all t greater than or equal to zero. The column vector [ x_o^H u_o^H ]^H is called the transmission zero direction associated with the transmission zero z_o.
The transmission zeros of a MIMO LTI system described by the state space
quadruple (A,B,C,D) are the roots of the following polynomial:
det(z_oI - A) det( C(z_oI-A)^{-1}B + D ) = 0
The corresponding zero directions (x_o, u_o) are found by solving:
(z_o I - A) x_o - B u_o = 0
C x_o + D u_o = 0.
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U-Y
Uncertainty - refers to the fact that the following is never known exactly: plant dynamics and parameters (e.g. actuator dynamics and parameters), sensor dynamics and parameters,
functional form of disturbances, disturbance parameters, functional form of sensor noise, sensor noise parameters. Such uncertainty is the main reason for needing feedback control laws.
Example. To illustrate the effects of uncertainty, consider that a typical swept-wing transport at high subsonic speeds will experience a
(1) reduction in wing lift-curve slope of about 20%,
(2) reduction in tail pitching moment contribution of about 30%,
(3) reduction in elevator effectiveness of about 50%, and a
(4) 10% MAC (mean aerodynamic chord) forward shift in the wing
aerodynamic center (this affects stability)
due to flexiblity along the longitudinal axis.
Undamped Poles - refers to a set of complex conjugate poles on the imaginary axis. Such poles have a zero damping ratio zeta and are associated with undamped sinusoidal oscillations in the time domain.
Underdamped Poles - refers to a set of complex conjugate poles which are stable (i.e. lie in the open left half plane). Such poles have a damping ratio zeta which lies in the interval (0, 1) and are associated with exponentially decaying sinusoids in the time domain.
Unmodeled Dynamics - dynamics which have not been modeled for any of the following reasons: (i) becuase they are too complex to model, (ii) becuase they are neglible for the physical situation at hand, or (iii) because they are simply unknown to the individual developing the model. In practice, high frequency dynamics are usually not well modeled because of uncertainty associated with them. Such dynamics are therefore sometimes neglected. Control engineers should always strive to quantify how large an error is being made by making such an approximation. They should also understand how the approximation will affect the performance of the final control system design.
Unstable System - usually refers to systems which possess at least one pole in the open right half plane or a double pole at the origin or complex conjugate double poles on the imaginary axis. The first is associated with a rising exponential or a rising exponential sinusoid. The latter two are associated with ramp-like signals in the time domain. Also see
Stable System.
Youla Parameterization - parameterizations which parameterizes the set of all stabilizing controllers for an LTI plant.
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Z
Zeta - greek letter used to denote damping ratio or damping factor. Also see
Underdamped Poles.
Zero - refers to input frequencies which are absorbed by a system.
Zero Input Response - response of a system with the input set to zero; response to an inital condition with no input.
Zero State Response - response of a system with the state set to zero; response to an input with zero initial condition.
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