# Control Systems/Glossary

The following is a listing of some of the most important terms from the
book, along with a short definition or description.

## A, B, C

Acceleration Error:The amount of steady state error of the system when stimulated by a unit parabolic input.\
Acceleration Error Constant:A system metric that determines that amount of acceleration error in the system.\
Adaptive Control:A branch of control theory where controller systems are able to change their response characteristics over time, as the input characteristics to the system change.\
Adaptive Gain: when control gain is varied depending on system state or condition, such as a disturbance\
Additivity:A system is additive if a sum of inputs results in a sum of outputs.\
Analog System:A system that is continuous in time and magnitude.\
ARMA: Autoregressive Moving Average, see 1\
ATO: Analog Timed Output. Control loop output is correlated to a timed contact closure.\
A/M: Auto-Manual. Control modes, where auto typically means output is computer-driven, calculated while manual can be field-driven or merely using a static setpoint.

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Bilinear Transform: a variant of the Z-transform, see 2\
Block Diagram:A visual way to represent a system that displays individual system components as boxes, and connections between systems as arrows.\
Bode Plots:A set of two graphs, a \"magnitude\" and a \"phase\" graph, that are both plotted on log scale paper. The magnitude graph is plotted in decibels versus frequency, and the phase graph is plotted in degrees versus frequency. Used to analyze the frequency characteristics of the system.\
Bounded Input, Bounded Output:BIBO. If the input to the system is finite, then the output must also be finite. A condition for **stability**.

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Cascade: When the output of a control loop is fed to/from another loop.\
Causal:A system whose output does not depend on future inputs. All physical systems must be causal.\
Classical Approach:See **Classical Controls**.\
Classical Controls:A control methodology that uses the transform domain to analyze and manipulate the **Input-Output** characteristics of a system.\
Closed Loop:a controlled system using feedback or feedforward\
Compensator:A Control System that augments the shortcomings of another system.\
Condition Number:\
Conditional Stability:A system with variable gain is conditionally stable if it is BIBO stable for certain values of gain, but not BIBO stable for other values of gain.\
Continuous-Time:A system or signal that is defined at all points t.\
Control Rate: the rate at which control is computed and any appropriate output sent. Lower bound is sample rate.\
Control System:A system or device that manages the behavior of another system or device.\
Controller:See **Control System**.\
Convolution:A complex operation on functions defined by the integral of the two functions multiplied together, and time-shifted.\
Convolution Integral:The integral form of the convolution operation.\
CQI: Control Quality Index, $=1-abs(PV-SP)/max[PVmax-SP,SP-PVmin]$, 1 being ideal.\
CV: Controlled variable

## D, E, F

Damping Ratio:A constant that determines the damping properties of a system.\
Deadtime: time shift between the output change and the related effect (typ. at least one control sample). One sees \"Lag\" used for this action sometimes.\
Digital:A system that is both **discrete-time**, and **quantized**.\
Direct action: target output increase is required to bring the process variable (PV) to setpoint (SP) when PV is below SP. Thus, PV increases with output increase directly.\
Discrete magnitude:See **quantized**.\
Discrete time:A system or signal that is only defined at specific points in time.\
Distributed:A system is distributed if it has both an infinite number of states, and an infinite number of state variables. See **Lumped**.\
Dynamic:A system is called dynamic if it doesn\'t have memory. See **Instantaneous**, **Memory**.

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Eigenvalues:Solutions to the characteristic equation of a matrix. If the matrix is itself a function of time, the eigenvalues might be functions of time. In this case, they are frequently called **eigenfunctions**.\
Eigenvectors:The nullspace vectors of the characteristic equation for particular eigenvalues. Used to determine state-transitions, among other things. See 3\
Euler\'s Formula:An equation that relates complex exponentials to complex sinusoids.\
Exponential Weighted Average (EWA): Apportions fractional weight to new and existing data to form a working average. Example EWA=0.70\*EWA+0.30\*latest, see Filtering.\
External Description:A description of a system that relates the input of the system to the output, without explicitly accounting for the internal states of the system.

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Feedback:The output of the system is passed through some sort of processing unit H, and that result is fed into the plant as an input.\
Feedforward: whwn apriori knowledge is used to forecast at least part of the control response.\
Filtering (noise): Use of signal smoothing techniques to reject undesirable components like noise. Can be as simple as using exponential weighted averaging on the input.\
Final Value Theorem:A theorem that allows the steady-state value of a system to be determined from the transfer function.\
FOH:First order hold\
Frequency Response:The response of a system to sinusoids of different frequencies. The Fourier Transform of the impulse response.\
Fourier Transform:An integral transform, similar to the Laplace Transform, that analyzes the frequency characteristics of a system.

See 4

## G, H, I

Game Theory:A branch of study that is related to control engineering, and especially **optimal control**. Multiple competing entities, or \"players\" attempt to minimize their own cost, and maximize the cost of the opponents.\
Gain:A constant multiplier in a system that is typically implemented as an amplifier or attenuator. Gain can be changed, but is typically not a function of time. Adaptive control can use time-adaptive gains that change with time.\
General Description:An external description of a system that relates the system output to the system input, the system response, and a time constant through integration.

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Hendrik Wade Bode:Electrical Engineer, did work in control theory and communications. Is primarily remembered in control engineering for his introduction of the **bode plot**.\
Harry Nyquist:Electrical Engineer, did extensive work in controls and information theory. Is remembered in this book primarily for his introduction of the **Nyquist Stability Criterion**.\
Homogeniety:Property of a system whose scaled input results in an equally scaled output.\
Hybrid Systems:Systems which have both analog and digital components.

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Impulse:A function denoted δ(t), that is the derivative of the unit step.\
Impulse Response:The system output when the system is stimulated by an impulse input. The Inverse Laplace Transform of the transfer function of the system.\
Initial Conditions:The conditions of the system at time $t = t_0$, where t~0~ is the first time the system is stimulated.\
Initial Value Theorem:A theorem that allows the initial conditions of the system to be determined from the Transfer function.\
Input-Output Description:See **external description**.\
Instantaneous:A system is instantaneous if the system doesn\'t have memory, and if the current output of the system is only dependent on the current input. See **Dynamic**, **Memory**.\
Integrated Absolute Error (IAE):absolute error (ideal vs actual performance) is integrated over the analysis period.\
Integrated Squared Error (ISE):squared error (ideal vs actual performance) is integrated over the analysis period.\
Integrators:A system pole at the origin of the S-plane. Has the effect of integrating the system input.\
Inverse Fourier Transform:An integral transform that converts a function from the frequency domain into the time-domain.\
Inverse Laplace Transform:An integral transform that converts a function from the S-domain into the time-domain.\
Inverse Z-Transform:An integral transform that converts a function from the Z-domain into the discrete time domain.

## J, K, L

Lag: The observed process impact from an output is slower than the control rate.\
Laplace Transform:An integral transform that converts a function from the time domain into a complex frequency domain.\
Laplace Transform Domain:A complex domain where the Laplace Transform of a function is graphed. The imaginary part of **s** is plotted along the vertical axis, and the real part of **s** is plotted along the horizontal axis.\
Left Eigenvectors:Left-hand nullspace solutions to the characteristic equation of a matrix for given eigenvalues. The rows of the inverse transition matrix.\
Linear:A system that satisfies the **superposition principle**. See **Additive** and **Homogeneous**.\
Linear Time-Invariant: LTI. See **Linear**, and **Time-Invariant**.\
Low Clamp: User-applied lower bound on control output signal.\
L/R: Local/Remote operation.\
LQR: Linear Quadratic Regulator.\
Lumped:A system with a finite number of states, or a finite number of state variables.

## M, N, O

Magnitude: the gain component of frequency response. This is often all that is considered in saying a discrete filter\'s response is well matched to the analog\'s. It is the DC gain at 0 frequency.\
Marginal Stability:A system has an oscillatory response, as determined by having imaginary poles or imaginary eigenvalues.\
Mason\'s Rule: see 5\
MATLAB: Commercial software having a Control Systems toolbox. Also see Octave.\
Memory:A system has memory if its current output is dependent on previous and current inputs.\
MFAC:Model Free Adaptive Control.\
MIMO:A system with multiple inputs and multiple outputs.\
Modern Approach:see **modern controls**\
Modern Controls:A control methodology that uses the state-space representation to analyze and manipulate the **Internal Description** of a system.\
Modified Z-Transform:A version of the Z-Transform, expanded to allow for an arbitrary processing delay.\
MPC: Model Predictive Control.\
MRAC: Model Reference Adaptive Control.\
MV: can denote Manipulated variable or Measured variable (not the same)

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Natural Frequency:The fundamental frequency of the system, the frequency for which the system\'s frequency response is largest.\
Negative Feedback:A feedback system where the output signal is subtracted from the input signal, and the difference is input to the plant.\
The Nyquist Criteria:A necessary and sufficient condition of stability that can be derived from **Bode plots**.\
Nonlinear Control:A branch of control engineering that deals exclusively with non-linear systems. We do not cover nonlinear systems in this book.

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OCTAVE: Open-source software having a Control Systems toolbox. Also see MATLAB.\
Offset: The discrepancy between desired and actual value after settling. P-only control can give offset.\
Oliver Heaviside:Electrical Engineer, Introduced the Laplace Transform as a tool for control engineering.\
Open Loop: when the system is not closed, its behavior has a free-running component rather than controlled\
Optimal Control:A branch of control engineering that deals with the minimization of system cost, or maximization of system performance.\
Order:The order of a polynomial is the highest exponent of the independent variable in that exponent. The order of a system is the order of the Transfer Function\'s denominator polynomial.\
Output equation:An equation that relates the current system input, and the current system state to the current system output.\
Overshoot:measures the extent of system response against desired (setpoint tracking).

## P, Q, R

Parabolic:A parabolic input is defined by the equation$\frac{1}{2}t^2u(t)$.\
Partial Fraction Expansion:A method by which a complex fraction is decomposed into a sum of simple fractions.\
Percent Overshoot:PO, the amount by which the step response overshoots the reference value, in percentage of the reference value.\
Phase: the directional component of frequency response, not typically well-matched between a discrete filter equivalent to the analog version, especially as frequency approaches the Nyquist limit. The final value in the limit drives system stability, and stems from the poles and zeros of the characteristic equation.\
PID:Proportional-Integral-Derivative\
Plant:A central system which has been provided, and must be analyzed or controlled.\
PLC:Programmable Logic Controller\
Pole:A value for s that causes the denominator of the transfer function to become zero, and therefore causes the transfer function itself to approach infinity.\
Pole-Zero Form:The transfer function is factored so that the locations of all the poles and zeros are clearly evident.\
Position Error:The amount of steady-state error of a system stimulated by a unit step input.\
Position Error Constant:A constant that determines the position error of a system.\
Positive Feedback:A feedback system where the system output is added to the system input, and the sum is input into the plant.\
PSD:The power spectral density which shows the distribution of power in the spectrum of a particular signal.\
Pulse Response:The response of a digital system to a unit step input, in terms of the transfer matrix.\
PV: Process variable

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Quantized:A system is quantized if it can only output certain discrete values.\
Quarter-decay: the time or number of control rates required for process overshoot to be limited to within 1/4 of the maximum peak overshoot (PO) after a SP change. If the PO is 25% at sample time N, this would be time N+k when subsequent PV remains \< SP\*1.0625, presuming the process is settling.

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Raise-Lower: Output type that works from present position rather than as a completely new computed spanned output. For R/L, the % change should be applied to the working clamps i.e. 5%(hi clamp-lo clamp).\
Ramp:A ramp is defined by the function $tu(t)$.\
Reconstructors:A system that converts a digital signal into an analog signal.\
Reference Value:The target input value of a feedback system.\
Relaxed:A system is relaxed if the initial conditions are zero.\
Reverse action: target output decrease is required to bring the process variable (PV) to setpoint (SP) when PV is below SP. Thus, PV decreases with output increase.\
Rise Time:The amount of time it takes for the step response of the system to reach within a certain range of the reference value. Typically, this range is 80%.\
Robust Control:A branch of control engineering that deals with systems subject to external and internal noise and disruptions.

## S, T, U, V

Samplers:A system that converts an analog signal into a digital signal.\
Sampled-Data Systems:See *Hybrid Systems*\'.\
Sampling Time:In a discrete system, the sampling time is the amount of time between samples. Reflects the lower bound for Control rate.\
SCADA: Supervisory Control and Data Acquisition.\
S-Domain:The domain of the Laplace Transform of a signal or system.\
Second-order System;\
Settling Time:The amount of time it takes for the system\'s oscillatory response to be damped to within a certain band of the steady-state value. That band is typically 10%.\
Signal Flow Diagram:A method of visually representing a system, using arrows to represent the direction of signals in the system.\
SISO: Single input, single output.\
Span: the designed operation region of the item,=high range-low range. Working span can be smaller if output clamps are used.\
Stability:Typically \"BIBO Stability\", a system with a well-behaved input will result in a well-behaved output. \"Well-behaved\" in this sense is arbitrary.\
Star Transform:A version of the Laplace Transform that acts on discrete signals. This transform is implemented as an infinite sum.\
State Equation:An equation that relates the future states of a system with the current state and the current system input.\
State Transition Matrix:A coefficient matrix, or a matrix function that relates how the system state changes in response to the system input. In time-invariant systems, the state-transition matrix is the matrix exponential of the system matrix.\
State-Space Equations:A set of equations, typically written in matrix form, that relates the input, the system state, and the output. Consists of the state equation and the output equation. See 6\
State-Variable:A vector that describes the internal state of the system.\
Stability:The system output cannot approach infinity as time approaches infinity. See **BIBO**, **Lyapunov Stability**.\
Step Response:The response of a system when stimulated by a unit-step input. A unit step is a setpoint change for setpoint tracking.\
Steady State:The output value of the system as time approaches infinity.\
Steady State Error:At steady state, the amount by which the system output differs from the reference value.\
Superposition:A system satisfies the condition of superposition if it is both additive and homogeneous.\
System Identification: method of trying to identify the system characterization , typically through least squares analysis of input,output and noise data vectors. May use ARMA type framework.\
System Type:The number of ideal integrators in the system.

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Time-Invariant:A system is time-invariant if an input time-shifted by an arbitrary delay produces an output shifted by that same delay.\
Transfer Function:The ratio of the system output to its input, in the S-domain. The Laplace Transform of the function\'s impulse response.\
Transfer Function Matrix:The Laplace transform of the state-space equations of a system, that provides an external description of a MIMO system.

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Uniform Stability:Also \"Uniform BIBO Stability\", a system where an input signal in the range \[0, 1\] results in a finite output from the initial time until infinite time. See 7.\
Unit Step:An input defined by $u(t)$. Practically, a setpoint change.\
Unity Feedback:A feedback system where the feedback loop element H has a transfer function of 1.

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Velocity Error:The amount of steady-state error when the system is stimulated by a ramp input.\
Velocity Error Constant:A constant that determines that amount of velocity error in a system.

## W, X, Y, Z

W-plane: Reference plane used in the bilinear transform.\
Wind-up: when the numerics of computed control adjustment can \"wind-up\", yielding control correction with an inappropriate component unless prevented. An example is the \"I\" contribution of PID if output has been disconnected during PID calculation

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Zero:A value for s that causes the numerator of the transfer function to become zero, and therefore causes the transfer function itself to become zero.\
Zero Input Response:The response of a system with zero external input. Relies only on the value of the system state to produce output.\
Zero State Response:The response of the system with zero system state. The output of the system depends only on the system input.\
ZOH: Zero order hold.\
Z-Transform:An integral transform that is related to the Laplace transform through a change of variables. The Z-Transform is used primarily with digital systems. See 8

