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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]), in control objects, 41
\[Bullet] ( ), in control objects, 41
\[EmptyUpTriangle] (  ), in control objects, 41
\[ScriptCapitalS] (  ), in control objects, 41
\[ScriptCapitalT] (  ), in control objects, 41
\[ScriptK] (  ), in control objects, 43
\[ScriptS] (  ), in control objects, 41
\[ScriptT] (  ), in control objects, 43
\[ScriptU] (  ), in control objects, 43
\[ScriptX] (  ), in control objects, 43
\[ScriptY] (  ), in control objects, 43
\[ScriptZ] (  ), in control objects, 41
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