Mathematica 9 is now available

 Documentation /  Control Systems Professional /

SpecialPointsRegistration

Index

noise, 182

Ackermann, 152, 155

robustness of solutions, 159

Ackermann's formula, 17

Ackermann, 154

Active wrappers, 3, 32

Actuators, 25, 175

A/D conversion, 44

Admissible controls, 165

Admissible error, AdmissibleError, 152

Admissible trajectories, 165

AdmissibleError, 152

Ailerons, 156

effectiveness of, 175

of a missile, 175

roll-time constant of, 175

Aircraft example, 156

Algebraic Riccati equation, 172

discrete, 173

All, in DeleteSubsystem, 115

in Subsystem, 115

Amplifiers, 112

Analog simulation, 65

Analog systems, 33

Analog-to-digital converters, 44

Analytic, 92

Angle of attack, of submarine, 72

Antenna example, 182

discrete estimator for, 185

Kalman filter for, 188

ARE, 172

Attitude control, of satellite, 151

of spacecraft, 95

Azimuth control, of antenna, 182

Backward rectangular rule, BackwardRectangularRule, 48

BackwardRectangularRule, 48

Bilinear transformation, BilinearTransform, 48

BilinearTransform, 48, 55

Block diagrams, 98

Bode plot, BodePlot, 85

BodePlot, 50, 85

collecting several plots in one, 144

displaying several objects together, 211

gain and phase margins in, 89

phase unwrapping in, 87

Bridged-T network, 66

Calculus`Pade`, 202

Canonical forms, controllable, 140

controllable companion, 36

Jordan, 138

Kalman controllable, 136

Kalman observable, 136

modal, 138

observable companion, 36

Cascade compensation, 119

Cascade connection, SeriesConnect, 98

Center of gravity, of pendulum, 12

Center of mass, of satellite, 151

Characteristic equation, 80

Characteristic polynomial, 154, 179

Chemical mixture, control of, 167

Classical control, 80

Closed loop, construction of, 105

Companion realizations, controllable, 36

observable, 36

Compensator, 95

Complex exponentials, reducing to trigonometric functions, 62

Complex numbers, notation for, 9

Complex plane, s- vs. z-plane, 4

ComplexRootProbability, 213

ComplexVariables, 38, 131

Composite systems, 98

Concentration control, 167

Condition number, of controllability matrix, 155

Consistency check, ConsistentQ, 210

ConsistentQ, 38, 210

Continuous-time systems, 33, 39, 54

Continuous-time to discrete-time conversion, 4, 44

Continuous-time vs. discrete-time systems, 39

ContinuousTimeQ, 40

Control design, optimal, 165

using pole assignment, 150

Control effort, 165

Control Format palette, 7

Control input, selection in single-input algorithms, ControlInput, 155

Control inputs, for time-domain response, ControlInputs, 60

Control matrix, 34

Control objects, 3

constructing from ODEs, 16

continuous-time vs. discrete-time, 4, 33, 39

converting between, 4, 31-32, 35

default domain of, 39

domain identification of, 4, 32, 39, 45

traditional notations for, 41

ControlInput, 155

ControlInputs, 60

Controllability, 122

Controllability Gramian, 140, 142

ControllabilityGramian, 124, 128

Controllability matrix, 136, 154

ControllabilityMatrix, 124, 126

Controllability test, Controllable, 122

ControllabilityGramian, 129

ControllabilityMatrix, 126

ControllabilityTest, 124

Controllable, 122, 134

Controllable canonical form, 140

Controllable companion realization, ControllableCompanion, 36

Controllable states, 136

Controllable subspace, size of, ControllableSpaceSize, 125

ControllableCompanion, 36

ControllableSpaceSize, 125

ControllableSubsystem, 132-133

vs. KalmanControllableForm, 138

Controlled signal, 81

Controller, current, 192

design of, 119, 150

in output feedback, 105

in state feedback, 114

optimal, 192

Controller, 192

Controller design, 15

Correlation, between process and measurement noises, 181

Cost function, 165, 171

equivalent, 178, 184

quadratic, 166

CountInputs, 210

CountOutputs, 210

CountStates, 210

Covariance matrix, 181

CriticalFrequency, 48

Critically damped, 74

Cross-covariance matrix, 181

Crossover frequency, 91

Current estimator, 192

D/A conversion, 54

Damping ratio, 74, 82, 88, 179

of dominant poles, 81

of third-order system, 76

DARE, 173

DecompositionMethod, in InternallyBalancedForm, 141

in KalmanControllableForm, 137

in KalmanObservableForm, 137

DefaultInputPort, 113

Deflection, of ailerons, 175

Deflection rate, of ailerons, 175

Delay, 53

in heat exchanger, 203

Padé approximation of, 203

DeleteSubsystem, 98, 115-116

Depth control, of submarine, 72

Derivative controller, 119

Determinant expansion formula, DeterminantExpansion, 33

DeterminantExpansion, 33

Deterministic, inputs, 182, 185, 187, 193

state reconstruction, 181

Differential equations, StateSpace model from, 15

Digital systems, 34

Digital-to-analog converters, 54

Dirac delta function, approximating in analog simulation, 79

DiracDelta, in impulse response simulation, 70, 78

DiracDelta, in impulse response simulation, 70, 78

Direct transmission matrix, 34

DirectSolve, as value of SolveMethod, 209

Discrete emulation of continuous design, 178, 184

Discrete simulation, 65

Discrete-time systems, 34, 39, 44

Discrete-time to continuous-time conversion, 4

DiscreteDelta, in impulse response simulation, 169

DiscreteLQRegulatorGains, 178, 184

DiscreteLyapunovSolve, 129, 207

DiscreteRiccatiSolve, 171, 173

DiscreteTimeQ, 40

Displacement, of pendulum, 12

DisplayTogether, vs. DisplayTogetherGraphicsArray, 211

DisplayTogetherGraphicsArray, 52, 144, 211

Domain identification of control objects, 4, 39

Dominant subsystem, DominantSubsystem, 142

DominantSubsystem, 132-133, 142

Double integrator, 80, 151

PID controller for, 120

Dryden Flight Research Center, 156

Dual systems, DualSystem, 130

DualSystem, 130, 135

Dynamic systems, 27

Economized rational approximation, 202

Effectiveness, of ailerons, 175

Eigendecomposition, as value of SolveMethod, 209

Eigensystem, as value of DecompositionMethod, 141

Eigenvalues, and poles, 17, 150, 179

in SchurDecomposition, 204

multiple, 139, 177

of closed-loop system, 17, 150

of Hamiltonian matrix, 177

specifying order of, 204

Eigenvectors, 147

Electrical Engineering Examples, plotting routines from, 80

Emulation, of continuous design, 178, 184

EquationForm, 6, 43

Equilibrium, of magnetic ball, 200

of pendulum, 15

Equivalent cost, 178, 184

Error signal, 81

Estimator, 161

current, 192

discrete by emulation of continuous, 184

gain matrix for, EstimatorGains, 161

Kalman, 182, 185

optimal, 181

predictor, 192

EstimatorGains, 162

Evolution matrix, 34

Exact, 213

ExactConversionFunction, 213

Expanding transfer functions, ExpandRational, 28

ExpandRational, 28

F-8 aircraft example, 156

Factoring transfer functions, FactorRational, 28

FactorRational, 28, 145

pole-zero cancellation in, 145

Feed-forward path, 112

Feedback, negative, 105

path, 112

positive, 105

Feedback connection, FeedbackConnect, 105

Feedback controller design, 15, 150, 201

using Lyapunov's method, 208

Feedback loop, forming with FeedbackConnect, 105

FeedbackConnect, 83, 98, 105, 191

Feedthrough matrix, 34

First-order hold, FirstOrderHold, 48

First-order lag, 94

FirstOrderHold, 48, 56

Flicker noise, 182

Flight control example, 156

Forced response, 57

Forward rectangular rule, ForwardRectangularRule, 48

ForwardRectangularRule, 48

Frames, 84

Free response, 57

Frequency prewarping, CriticalFrequency, 48, 55

Frequency response, 85, 93, 95-96

FullRankControllabilityMatrix, 125

FullRankObservabilityMatrix, 125

Gain band, 96

Gain margin, GainPhaseMargins, 89

GainPhaseMargins, 89

Gaussian distribution, 182, 192

NormalDistribution, 191

Gaussian noise, 188

General rational approximation, 202

GenericConnect, 98, 110

Gramian, 124, 128

GraphicsArray, displaying several objects together, 211

Hamiltonian matrix, 177

Heat exchanger example, 203

Hold equivalence methods, 48

Homogeneous response, 57

Horizon, infinite, 166

Imaginary unit, notation for, 9

ImaginaryToRealRatio, 213

Impulse response, 74

simulating with DiracDelta, 70

simulating with DiscreteDelta, 169

simulation of, 70

Infinite-horizon problem, 166, 182

Infinite-time-to-go problem, 166

Inflow control, 167

Inflow rate, 167

Initial conditions, InitialConditions, 60

InitialConditions, 19, 60

Input matrix, 34

Inputs, count of, 210

deterministic, 182, 185, 193

stochastic, 182, 187

InputVariables, 43

Installation, 1

Integral controller, 119

Integrator, 4, 80, 101

Interconnections, 98

arbitrary, 110

elementary, 98

Internally balanced realizations, InternallyBalancedForm, 140

InternallyBalancedForm, 140

InterpolatingFunction, in time-domain simulations, 65

InterpolatingPolynomial, 202

Interpolation, as value of GainPhaseMargins, 92

Inventory control, 60

Inventory level, control of, 60

Inverted pendulum example, 11, 163

controller for, 15

optimal controller for, 17

InvertedTransformMatrix, 147

Irreducible realization, MinimalRealization, 133

Jacobian matrix, 197

Jordan canonical form, JordanCanonicalForm, 138

JordanCanonicalForm, 138

Kalman, 135

Kalman canonical forms, 136

Kalman decomposition, Kalman, 135

Kalman estimator, 182

KalmanEstimator, 185

Kalman filter, 182, 185

example of, 188

Kalman gain matrix, 184

KalmanControllableForm, 132, 136, 149

KalmanEstimator, 182, 185

KalmanObservableForm, 136

Kautsky-Nichols-Van Dooren algorithm, KNVD, 158

KNVD, 152, 158

Lag system, 49, 52, 94

Laplace transform, LaplaceTransform, 44

LaplaceTransform, 44

Levitation system example, 199

Linear quadratic Gaussian problem, 182

Linear quadratic regulator, 17

LQRegulatorGains, 166

Linearization, 16, 197

Linearize, 197

Linearize, 16, 198

LinearSpacing, 87

LogSpacing, 87

LQ regulator, 17

LQEstimatorGains, 182, 184, 192

LQG controller, Controller, 192

LQG problem, LQEstimatorGains, 182

LQOutputRegulatorGains, 171

LQRegulatorGains, 17-18, 167, 174, 177, 179, 182, 192

vs. DiscreteLQRegulatorGains, 180

vs. LQOutputRegulatorGains, 172

LTI systems, 27, 202

Lyapunov equations, 206

continuous, 206

discrete, 207

for controllability and observability Gramians, 129

Lyapunov function, 208

LyapunovSolve, 129, 206

Magnetic ball suspension example, 199

Magnitude response, 85

MappingFunction, 213

Margins, 87

MarginStyle, 87, 89

MaxIterations, in StateFeedbackGains, 159

Measurement noise, 181, 191-192

MergeSystems, 98, 117

Method, in Rank, 210

in StateFeedbackGains, 151

in ToContinuousTime, 54

in ToDiscreteTime, 48

MIMO systems, 28

frequency response of, 96

Minimal realization, MinimalRealization, 133

MinimalRealization, 38, 132-133

Minimax approximation, 202

Minimum-time response, 208

Missile example, 175

Mixing tank example, 167

controllability Gramian of, 129

LQ regulator for, 168

output regulator for, 172

singular value plot for, 97

Modal realization, JordanCanonicalForm, 138

Model reduction, DominantSubsystem, 142

MinimalRealization, 133

PoleZeroCancel, 144

Modern control, 27

Moment of inertia, of pendulum, 12

Monic polynomial, 121

Most likelihood, 182

Multiple-input, multiple-output (MIMO) systems, 28

minimal realization of, 133

MultipleRootProbability, 213

Multivariable control, 28

NASA, Dryden Flight Research Center, 156

Natural frequency, 74, 82, 179

of third-order system, 76

Natural response, 57

NDSolve, in time-domain simulations, 65

Negative, in FeedbackConnect, 107

in GenericConnect, 111

Negative feedback, 105

Nichols plot, NicholsPlot, 95

NicholsPlot, 95

Noise, , 182

additive, 188

covariance matrix of, 181, 184

Gaussian, 182

high-frequency cutoff of, 182

measurement, 181

process, 181

spectrum of, 181

stationary, 182

white, 181

Nominal solution, 197

None, in DeleteSubsystem, 115

in Subsystem, 115

Nonlinear state-space models, 15

linearization of, 16, 197

simulation of, 21

Nonlinear systems, 197

local linearization of, 197

rational polynomial approximations for, 202

Nonpolynomial transfer functions, 94

NonSingularControllabilityGramian, 125

NonSingularObservabilityGramian, 125

Normal distribution, 182

NormalDistribution, in simulations, 191

NullSpace, as value of DecompositionMethod, 137

in Rank, 210

Numerical errors, control of, 152

Numerical integration methods, 48

NumericalMath`Approximations`, 202

Nyquist frequency, 51

Nyquist plot, NyquistPlot, 93

NyquistPlot, 93

Observability, 122

Observability Gramian, 140, 142

ObservabilityGramian, 124, 128

Observability matrix, 136

ObservabilityMatrix, 124, 126

Observability test, Observable, 122

ObservabilityGramian, 129

ObservabilityMatrix, 126

ObservabilityTest, 124

Observable, 122, 134

Observable canonical form, 36

Observable companion realization, ObservableCompanion, 36

Observable states, 136

Observable subspace, size of, ObservableSpaceSize, 125

ObservableCompanion, 36

ObservableSpaceSize, 125

ObservableSubsystem, 133

vs. KalmanObservableForm, 138

Observation matrix, 34

Observer, 161

Kalman, 182

optimal, 181

Optimal control, 165

Optimal controller, 192

Optimal estimation, 181

Ordered Schur decomposition, SchurDecompositionOrdered, 204

Orthogonal basis, 137, 159

reconstruction of, 130

Orthogonal complement, 130, 137, 159

OrthogonalTransformMatrix, 148-149

Outflow control, 167

Outflow rate, 167

Output controllability matrix, OutputControllabilityMatrix, 126

Output matrix, 34

Output regulator, optimal, LQOutputRegulatorGains, 171

Output response, OutputResponse, 57

OutputControllabilityMatrix, 126

OutputControllable, 122

OutputResponse, 19, 57-58

dummy variable in, 60

initial conditions in, 19, 60

number of input signals in, 60, 65

polling inputs in, 60

simulations with, 65

symbolic solution, 57

Outputs, count of, 210

OutputVariables, 43

Overdamped response, 75

Overshoot, 83

Padé approximation, 202

Parallel connection, ParallelConnect, 103

ParallelConnect, 98, 103, 110, 190

for adding inputs to system, 189

Particular response, 57

Penalty function, 165

Pendulum, inverted, 11, 163

Performance criterion, 165

of a missile, 175

Performance index, 165

of a missile, 175

Period, 40

Phase band, 96

Phase margin, GainPhaseMargins, 89

Phase response, 85

Phase unwrapping, adjusting PlotPoints for, 144

PhaseRange, 87

PhaseRange, 87, 92, 96

PID controller, 95, 119

Pivoting, in SchurDecompositionOrdered, 206

Plant, 27

PlotPoints, 81, 84, 87, 93, 95-96, 144

PlotSampling, 87, 95-96

Pole assignment, 17

controlling numerical errors in, 152

robust, 158

StateFeedbackGains, 150

using Ackermann's formula, 154

Pole placement, 17

StateFeedbackGains, 150

Pole-zero cancellation, PoleZeroCancel, 144

Poles, 18, 150

multiple, 213

of closed-loop system, 17

of transfer function, Poles, 32

of transfer function, ZeroPoleGain, 30

Poles, 32

PoleStyle, 81

PoleZeroCancel, 133, 144

Polynomial approximations, 202

Positive, in FeedbackConnect, 107, 191

in GenericConnect, 111

Positive definite, 166

Positive feedback, 105

Positive semidefinite, 166

Predictor estimator, 192

Prewarping, 55

CriticalFrequency, 48, 55

Process noise, 181, 191-192

Production and inventory control model, 60

Production rate, control of, 60

Proper transfer function, state-space realization of, 36

Proportional controller, 119

QRDecomposition, as value of DecompositionMethod, 137

Quadratic cost function, 166

Ramp response, 77

Random systems, RandomSystem, 211

RandomOrthogonalComplement, 130

in KalmanControllableForm, 137

in KalmanObservableForm, 137

in StateFeedbackGains, 159

RandomSystem, 211

Rank, of matrix, Rank, 210

Rational polynomial approximations, 202

Rational polynomials, as TransferFunction objects, 27

RealBlockForm, in SchurDecompositionOrdered, 206

Realizations, 132

converting between, 132

RealRootProbability, 213

Reduced-order model, 133, 140

ReductionMethod, in ControllableSpaceSize, 126

in ControllableSubsystem, 135

in DominantSubsystem, 143

in MinimalRealization, 135

in ObservableSpaceSize, 126

in ObservableSubsystem, 135

in TransferFunction, 33

Reference signal, 81

Regulator, linear quadratic, LQRegulatorGains, 166

output, 171

RejectionLevel, 142-143

Resolvent matrix, 33

Response, in frequency domain, 85

in time domain, 57

minimum-time, 208

ResponseVariable, 60

Riccati equations, 18, 171-172, 182

RiccatiSolve, 171, 173

Robust pole assignment, KNVD, 158

Roll angle, 156

Roll attitude control, of a missile, 175

Roll rate, 156

Roll time constant, of ailerons, 175

Root loci, evolution of, 84

RootLocusPlot, 80

RootLocusAnimation, 84

RootLocusPlot, 80

RowReduce, as value of DecompositionMethod, 137

Rudder, 156

Sampled, 4, 39

Sampling, 44

Sampling period, choice of, 179

SamplingPeriod, 40

Sampling rate, 40

SamplingPeriod, 40

Satellite attitude control example, 151

controller for, 196

PID controller for, 119

Schur decomposition, ordered, SchurDecompositionOrdered, 204

SchurDecomposition, as value of SolveMethod, 177

vs. SchurDecompositionOrdered, 204

SchurDecompositionOrdered, 204

Second-order system, frequency response of, 85, 88

impulse response of, 78

root loci of, 82

sampling rate for, 179

step response of, 74

Separation principle, 192

Serial connection, SeriesConnect, 98

Series compensation, 119

SeriesConnect, 82, 91, 98, 110

Servo mechanism, 188

Servo motor, 183

SetControlFormat, 7

SetStandardFormat, 7

Sideslip angle, 156

Similarity transformation, for Kalman forms, 136

SimilarityTransform, 146

SimilarityTransform, 132, 146, 149

Simulation, analog, 65

discrete, 65

SimulationPlot, 57, 69, 83, 169

plotting state response with, 72

Single-input, single-output (SISO) systems, 28

minimal realization of, 133

Singular-value plot, SingularValuePlot, 96

SingularValuePlot, 96

SingularValues, as value of DecompositionMethod, 137, 141

in Rank, 210

SISO systems, 28

Solution, nominal, 197

SolveMethod, in DiscreteRiccatiSolve, 177

in RiccatiSolve, 177

Spacecraft, attitude control of, 95

SpecialPointProbability, 213

SpecialPoints, 213

Spectrum, of noise, 181

Stabilizable system, 170

Stable system, 171

in Lyapunov's sense, 208

State equations, 33

nonlinear, 15, 197

State estimation, deterministic, 161

optimal, 181

stochastic, 181

State feedback, StateFeedbackConnect, 114

StateFeedbackGains, 150

State matrix, 34

State reconstruction, deterministic, 161, 181

optimal, 181

stochastic, 181

State response, plotting with SimulationPlot, 72, 201

StateResponse, 57

State trajectories, 165

State-space data structure, StateSpace, 33

State-space models, nonlinear, 15, 197

State-space realizations, 132

State-space systems, 33

StateFeedbackConnect, 19, 98, 114, 169, 179-180, 201

StateFeedbackGains, 17, 150, 201

StateResponse, 57-58

dummy variable in, 60

initial conditions in, 60

number of input signals in, 60, 65

polling inputs in, 60

simulations with, 65

symbolic solution, 57

States, count of, 210

StateSpace, 3, 34

traditional notations for, 6, 41

with RandomSystem, 211

StateVariables, 43

Stationary noise, in wide sense, 182

Steady-state error, 77

Step response, 74, 202

Stern plane, deflection of, 72

Stochastic, inputs, 182, 187

state reconstruction, 181

system, 181, 192

Strictly proper transfer function, and RandomSystem, 213

state-space realization of, 36

Submarine, depth control of, 72

Subspace, controllable, 125

observable, 125

Subsystem, 98, 115, 132

Suspension system example, 199

SV plot, SingularValuePlot, 96

System, analog, 33

composite, 98

continuous-time, 33

digital, 34

discrete-time, 34

dual, 130

dynamic, 27

nonlinear, 197

random, 211

state-space realization of, 33

stochastic, 181

time-varying, 57

T-bridge network, 66

TargetForm, 35

Temperature sensor, 203

Terminal state error, 165

Third-order system, step response of, 76

Time-domain response, 57

Time-domain simulation, 65

Time-varying systems, response of, 57

TimeVariable, 43

ToContinuousTime, 4, 54

ToDiscreteTime, 4, 44, 151, 168, 174, 178, 180, 184

Tolerance, in PoleZeroCancel, 145

Torque, 183

Traditional notations, 5, 41

TraditionalForm, of control objects, 5, 41

Transfer function matrix, data structure for, TransferFunction, 27

Transfer matrix, data structure for, TransferFunction, 27

TransferFunction, 3, 27

as a pure function object, 27

expanding, 28

factoring, 28

nonpolynomial, 94

traditional notations for, 5, 41

variable in, 32

with RandomSystem, 211

Transformation matrix, recovering of, TransformationMatrix, 148

TransformationMatrix, 148

Transient response, 57

Transient state error, 165

Transport lag, 94

Triangle hold, FirstOrderHold, 48, 56

Tustin transformation, BilinearTransform, 48, 55

Uncontrollable states, 136, 158, 171

Undamped response, 75

Underdamped response, 75

Unforced response, 57

Unobservable, states, 136

VerifyPoles, 152

Weak modes, 142

Weak subsystem, 142

Yaw rate, 156

z-transform, modified, 53

ZTransform, 44

Zero-input response, 57

Zero-order hold, ZeroOrderHold, 48

Zero-pole mapping, ZeroPoleMapping, 48

Zero-pole-gain data structure, ZeroPoleGain, 30

Zero-state response, 57

ZeroOrderHold, 48

ZeroPoleGain, 3, 30

variable in, 32

with RandomSystem, 211

ZeroPoleMapping, 48

Zeros, of transfer function, ZeroPoleGain, 30

of transfer function, Zeros, 32

Zeros, 32

ZeroStyle, 81

ZTransform, 44

$ContinuousTimeComplexPlaneVariable, 41

$ContinuousTimeToken, 41

$DiscreteTimeComplexPlaneVariable, 41

$DiscreteTimeToken , 41

$RandomOrthogonalComplement, 130

$Sampled, 39

$SamplingPeriod, 45

Bullet (\[Bullet]), in control objects, 41

\[Bullet] (), in control objects, 41

\[EmptyUpTriangle] ( EmptyUpTriangle), in control objects, 41

\[ScriptCapitalS] ( ScriptCapitalS), in control objects, 41

\[ScriptCapitalT] ( ScriptCapitalT), in control objects, 41

\[ScriptK] ( ScriptK), in control objects, 43

\[ScriptS] ( ScriptS), in control objects, 41

\[ScriptT] ( ScriptT), in control objects, 43

\[ScriptU] ( ScriptU), in control objects, 43

\[ScriptX] ( ScriptX), in control objects, 43

\[ScriptY] ( ScriptY), in control objects, 43

\[ScriptZ] ( ScriptZ), in control objects, 41

SpecialPointsRegistration



Any questions about topics on this page? Click here to get an individual response.Buy NowMore Information
THIS IS DOCUMENTATION FOR AN OBSOLETE PRODUCT. CONTROL
SYSTEMS FUNCTIONALITY
IS NOW AVAILABLE IN MATHEMATICA.