Index

Absorption column example, AbsorptionColumn, 115

model reduction for, 85

AbsorptionColumn, 85, 115

Aircraft example, 48

AircraftF8, 116

AircraftL1011, 117

Lyapunov equation for, 27

pole assignment for, 52

AircraftF8, 48, 116

AircraftL1011, 27, 52, 117

Algebraic Riccati equations, 29

AmmoniaReactor, 118

Automobile gas turbine example, GasTurbine, 121

Benchmark systems, 115

Binary distillation column example, DistillationColumn, 119

Block Hessenberg forms, 41, 45

Canonical forms, 41

controller-Hessenberg, 41

observer-Hessenberg, 45

CARE, 29

Case study systems, 115

Catalytic cracker example, CatalyticCracker, 126

Catalytic reactor example, CatalyticReactor, 118

CatalyticCracker, 126

CatalyticReactor, 118

CH-47 helicopter example, HelicopterCH47, 122

Cheap control, 16, 39

Chemical plant example, ChemicalPlant, 127

ChemicalPlant, 127

Cholesky factorization, of Gramians, 8, 80-81

CholeskyFactorControllabilityGramian, 8, 80, 82, 86, 95

CholeskyFactorObservabilityGramian, 8, 80, 82, 86, 95

Condenser, 119

Constrained feedback stabilization, 10, 50, 61

of drum boiler, 19

ControlInput, 53, 55

Controllability, 41, 80

and controller-Hessenberg forms, 14, 42, 47

Controllability Gramian, Cholesky factor of, 80

Controllability index, 45

Controllability test, using block Hessenberg form, 9, 47

using controllability Gramian, 9

using Lyapunov equations, 27

ControllabilityGramian, 86

ControllabilityTest, 9, 15, 48, 82

Controllable, 9, 41, 49, 83

Controller design, and LQR problem, 15

using feedback stabilization, 19

Controller-Hessenberg form, 9, 41

ControllerHessenbergForm, 43

in pole assignment, 51

lower, 42

LowerControllerHessenbergForm, 43

unreduced, 42, 47

upper, 42

ControllerHessenbergForm, 9, 14, 43, 49

Damping factor, 61, 65-66

of drum boiler, 17

Damping ratio, 50, 62, 65

DampingFactorRegion, 10, 20, 61, 67-68

DampingRatioRegion, 10, 62

DARE, 29

Deflating subspace, 32

Design constraints, 50, 61

DiscreteLyapunovSolve, 5, 8, 26

DiscreteRiccatiSolve, 8, 26, 29, 34

Distillation column example, DistillationColumn, 119

DistillationColumn, 119

DominantSubsystem, 10, 80, 85-86

Drum boiler case study, 11

Drum boiler example, controller-Hessenberg forms for, 44

DrumBoiler, 12, 120

feedback stabilization for, 65

observer-Hessenberg forms for, 46

Drum pressure, in drum boiler, 12, 120

DrumBoiler, 12, 44, 46, 65, 120

Eigenvalues, generalized, 101

Eigenvector matrix, 101

Eigenvectors, generalized, 101

Electric power system example, PowerSystem, 123

Estimator, reduced-order, 10, 22

ReducedOrderEstimator, 69

Example systems, 115

Explicit QR algorithm, for pole assignment, 9, 53

F-8 aircraft example, AircraftF8, 116

controllability of, 48

controller-Hessenberg form for, 48

Feedback stabilization, 10, 50, 61

for drum boiler, 19

Lyapunov, 10, 27, 66

Feedwater flow, in drum boiler, 12

Flight control system example, FlightControlSystem, 120

pole assignment for, 56

Riccati equation for, 30, 35, 37

FlightControlSystem, 30, 35, 56, 120

Fluid catalytic reactor example, CatalyticReactor, 118

Frequency domain identification, 97

Frequency response, 85, 88, 97

FrequencyResponseIdentify, 11, 89, 98, 100

and ImpulseResponseIdentify, 98

FrequencyWeights, in FrequencyResponseIdentify, 98

FullRankControllerHessenbergBlocks, 9, 15, 41, 48

FullRankObserverHessenbergBlocks, 9, 21, 41, 48

Gas turbine example, GasTurbine, 121

GasTurbine, 121

Gaussian distribution, 95

Generalized eigenvalue problem, 101

Generalized eigenvalues, 11, 101

GeneralizedEigenvalues, 101

Generalized eigenvector method, for Riccati equations, 29, 32

Generalized eigenvectors, 11, 101

GeneralizedEigenvectors, 101

Generalized Schur decomposition, 11, 101, 104

GeneralizedSchurDecomposition, 104

GeneralizedSchurDecompositionOrdered, 104

ordered, 11

Generalized Schur method, for Riccati equations, 29, 32

GeneralizedEigendecomposition, as value of SolveMethod, 8, 33

GeneralizedEigensystem, 11, 33, 101-102

GeneralizedEigenvalues, 11, 101-102

GeneralizedEigenvectors, 11, 101-102

GeneralizedSchurDecomposition, 11, 33, 101, 105

as value of SolveMethod, 8, 33

GeneralizedSchurDecompositionOrdered, 101, 105

Generator axle example, TurbineAxle, 126

norm, and Gramians, 80

using Lyapunov equations, 27

control, and Riccati equations, 29

norm, 84

and Gramians, 80

using Lyapunov equations, 27

Hamiltonian matrix, 30, 34

Hammarling algorithm, for Cholesky factorization, 82, 85

Hankel matrix, 89, 93, 98

Hankel singular values, 80, 82, 84, 86, 91, 95

Heat flow, in drum boiler, 12

Helicopter example, HelicopterCH47, 122

HelicopterCH47, 122

Hessenberg forms, 41

Hessenberg matrix, 76

Hessenberg-Schur method, for Lyapunov equations, 8, 26

HessenbergSchurDecomposition, as value of SolveMethod, 8, 27

Highway example, SmartHighway, 125

Implicit QR algorithm, for pole assignment, 53

Implicit RQ algorithm, for pole assignment, 9, 54

ImplicitRQDecomposition, 9, 54

Impulse response, in system identification, 90

of steam power system, 39

ImpulseResponseIdentify, 10, 89-90

Industrial control, case study, 11

InitialConditions, in OutputResponse, 95

InitialGuess, in DiscreteRiccatiSolve, 8, 37

in RiccatiSolve, 8, 37

Internal balancing, partial, 85

Internally balanced realizations, 80, 85

InternallyBalancedForm, 80

Invariant subspace, 30

Inverse-free methods, for Riccati equations, 8, 29, 32

Iterations, number of, 4

J-100 jet engine example, ServoJ100, 124

Jet engine example, partial pole assignment for, 59

ServoJ100, 124

Kalman filter, 29

Kalman-Bucy filter, 29

KalmanEstimator, 69

Left matrix fraction, 98

Linear quadratic Gaussian problem, 29

Linear quadratic regulator, 29

Lower controller-Hessenberg form, LowerControllerHessenbergForm, 42-43

Lower observer-Hessenberg form, ObserverHessenbergForm, 46

LowerControllerHessenbergForm, 9, 41, 43

LQEstimatorGains, 7

LQG, 29, 38

LQR, 29, 38

LQR controller, for drum boiler, 15

LQRegulatorGains, 7, 16, 39

Lyapunov algorithms, for feedback stabilization, 10

Lyapunov equations, 8, 26, 50

discrete, 5

DiscreteLyapunovSolve, 26

for feedback stabilization, 66

for Gramians, 80

Hessenberg-Schur method for, 8, 26

in Newton method for CARE and DARE, 36

LyapunovSolve, 26

Schur method for, 8, 26

Lyapunov feedback stabilization, 65-66

for drum boiler, 20

Lyapunov shift, LyapunovShift, 67

LyapunovShift, 10, 67

LyapunovSolve, 8, 26

LyapunovSolveOptions, 37

Magnetic tape example, feedback stabilization for, 67

MagneticTape, 122

state estimator for, 70

MagneticTape, 67, 70, 122

Markov parameters, 89, 98

Matrix decompositions, 11

Matrix equations, 26

Matrix pencil, 32

Matrix Sign-function method, 34

for Riccati equations, 8, 29

MatrixSign, as value of SolveMethod, 8, 35, 39

MaxIterations, in Advanced Numerical Methods functions, 4

MaxOrder, in FrequencyResponseIdentify, 98, 100

in ImpulseResponseIdentify, 91

in OutputResponseIdentify, 96

Measurement noise, 117

Method, in ReducedOrderEstimator, 72

in StateFeedbackGains, 51, 56

Minimal realization, 89

Minimum energy control, 16, 39

Model identification, 10, 89

Model reduction, 10, 80, 84

using balanced truncation, 80

using Lyapunov equations, 27

using Schur method, 10, 84

using square-root method, 10, 84

Natural frequency, 63

NaturalFrequencyRegion, 10, 63

Newton, as value of SolveMethod, 8, 16, 37

Newton method, for Riccati equations, 8, 29, 36

in LQRegulatorGains, 16

Noise, sensitivity to, 19

white, 94-95

Observability, 41, 80

and observer-Hessenberg forms, 20, 45, 47

Observability Gramian, Cholesky factor of, 80

Observability index, 45

Observability test, using block Hessenberg form, 9, 47

using Lyapunov equations, 27

using observability Gramian, 9

ObservabilityTest, 9, 21, 48, 82

Observable, 41, 49

Observer, 69

reduced-order, 10, 22

ReducedOrderEstimator, 69

Observer design, for drum boiler, 20

Observer-Hessenberg form, 9, 45

lower, 45

ObserverHessenbergForm, 46

unreduced, 45, 47

upper, 45

UpperObserverHessenbergForm, 46

ObserverHessenbergForm, 9, 21, 41, 46, 49

Ordered Schur decomposition, 30, 56, 58, 85

OutputResponse, 17, 39, 75, 90, 95

OutputResponseIdentify, 11, 89, 94, 96

and ImpulseResponseIdentify, 94

Paper machine example, Cholesky factor of controllability Gramian for, 83

discrete Lyapunov equation for, 5

discrete Riccati equation for, 33

Lyapunov equation for, 28

PaperMachine, 127

system identification for, 99

PaperMachine, 5, 28, 33, 83, 99, 127

Partial Lyapunov shift, PartialLyapunovShift, 67

Partial pole assignment, 50, 58

in state estimator, 71

PartialLyapunovShift, 10, 67

in StateFeedbackGains, 20

Pencil, 32

Persistently exciting input, 94

Pivoting, in controller-Hessenberg forms, 43

in observer-Hessenberg forms, 46

Pole assignment, 9, 50

partial, 9, 58

using explicit QR algorithm, 9, 50, 53

using implicit QR algorithm, 53

using implicit RQ algorithm, 9, 50

using recursive algorithms, 9, 50-51

using Schur method, 9, 50, 56

Pole placement, 9, 50

partial, 9, 58

using explicit QR algorithm, 9, 50, 53

using implicit QR algorithm, 53

using implicit RQ algorithm, 9, 50

using recursive algorithms, 9, 50-51

using Schur method, 9, 50, 56

Poles, and stability, in drum boiler, 12

Position control, 125

PositiveDiagonalCholeskyFactorControllabilityGramian, 9, 82

PositiveDiagonalCholeskyFactorObservabilityGramian, 9, 82

Power plant example, TurbineAxle, 126

Power system example, PowerSystem, 123

PowerSystem, 123

Process noise, 117

Projection technique, for partial pole assignment, 9, 58

QR factorization, 32

QR iteration, 53

QRDecomposition, as value of Method, 9, 53

QZ decomposition, 104

RealBlockForm, in GeneralizedSchurDecomposition, 105

in GeneralizedSchurDecompositionOrdered, 11

in SchurDecompositionOrdered, 11

Reboiler, 119

Recursive, as value of Method, 9, 51

Recursive bidiagonal method, for pole assignment, 51

for reduced-order estimation, 77

Recursive method, for pole assignment, 9, 51

Recursive triangular method, for reduced-order estimation, 10, 77

RecursiveBidiagonal, 10, 77

RecursiveBlockBidiagonal, 10, 21, 77

RecursiveBlockTriangular, 10, 77

RecursiveRQDecomposition, 9, 54

RecursiveTriangular, 10, 77

Reduced-order estimator, 10

via pole assignment, 10, 74

via Sylvester-observer equation, 10, 76

ReducedOrderEstimator, 10, 21, 69, 72

and StateFeedbackGains, 75

RejectionLevel, 87

Residuals, of solution of matrix equations, 5

Resonance frequency, in drum boiler, 13

Riccati equations, 8, 16, 26, 29, 101, 104

and Lyapunov equations, 27

DiscreteRiccatiSolve, 29

generalized eigenvector method for, 8, 32

generalized Schur method for, 8, 32

inverse-free methods for, 8, 32

matrix Sign-function method for, 8, 34

Newton method for, 8, 36

RiccatiSolve, 29

Schur method for, 8, 30

RiccatiSolve, 7-8, 26, 29

Risers, in drum boiler, 12

Robust stability, 27

Robustness, of numerical methods, 3

Rotating axle example, TurbineAxle, 126

RQ decomposition, 51

in recursive method, 9, 54

Sampled, in FrequencyResponseIdentify, 100

in ImpulseResponseIdentify, 91

in OutputResponseIdentify, 96

Schur decomposition, generalized, 104

ordered, 30, 56, 58

Schur method, for Lyapunov equations, 8, 26

for model reduction, 10, 80, 84

for pole assignment, 9, 56, 75

for Riccati equations, 8, 29-30

SchurDecomposition, as value of Method, 9-10, 75

as value of ReductionMethod, 85

as value of SolveMethod, 8, 27, 30

SchurDecompositionOrdered, 11

Separation property, 23

ServoJ100, 59, 124

Servomechanism example, ServoJ100, 124

Settling time, 50, 62

SettlingTimeRegion, 10, 62, 67

Sign-function method, for Riccati equations, 29

SimulationPlot, 39

of drum boiler, 13

Singular values, Hankel, 82

Smart highway example, SmartHighway, 125

SmartHighway, 125

SolveMethod, in LQRRegulatorGains, 29

Square-root method, for model reduction, 10, 80, 84

SquareRoot, as value of Method, 10

as value of ReductionMethod, 85

Stability, of drum boiler, 12

Stability analysis, using Lyapunov equations, 27

Stability region, 61

Stabilization, by state feedback, 50

Stabilizing solution, of Riccati equations, 29

Staircase algorithm, 41, 46

State estimation, 69

and pole assignment, 70

full-order, 69

reduced-order, 71

StateFeedbackConnect, 17

StateFeedbackGains, 9, 20, 50, 52, 54, 57, 60, 68, 75

for feedback stabilization, 64, 67

for partial pole assignment, 58

Steam flow, in drum boiler, 12, 120

Steam power system example, feedback design for, 38

LQ regulator for, 38

reduced-order estimator for, 72, 75, 78

SteamPowerSystem, 128

system identification for, 92, 94

SteamPowerSystem, 38, 72, 92, 94, 128

Strong modes, 95

Subspace system identification, 93

Sylvester equations, 8, 26

Sylvester-observer equation, 72, 76

recursive bidiagonal method for, 77

recursive block diagonal method for, 77

recursive block triangular method for, 77

recursive triangular method for, 77

Symplectic matrix, 30

System identification, 10, 89

in frequency domain, 97

in time domain, 89

using frequency response, 11

using impulse response, 10

using input-output data, 11, 93

using Markov parameters, 89

Tandem-rotor helicopter example, HelicopterCH47, 122

Time-domain identification, 89

Time-domain response, in drum boiler, 12

ToDiscreteTime, 91

Tolerance, in Advanced Numerical Methods functions, 4

in Controllable, 15

in ImpulseResponseIdentify, 91

in Observable, 21

in OutputResponseIdentify, 96

Tubular ammonia reactor example, AmmoniaReactor, 118

Turbine axle example, TurbineAxle, 126

TurbineAxle, 126

Uncontrollable state, distance from, 15

Uncontrollable states, and controller-Hessenberg forms, 42

Uncontrollable system, 15

Unobservable states, and controller-Hessenberg forms, 45

Unreduced controller-Hessenberg form, 42

Unreduced observer-Hessenberg form, 45, 76

Upper controller-Hessenberg form, ControllerHessenbergForm, 43

Upper observer-Hessenberg form, UpperObserverHessenbergForm, 45-46

UpperObserverHessenbergForm, 9, 41, 46

Velocity control, 125

Weak modes, 95

White noise, 94-95

Wind disturbances, 117

WorkingPrecision, in Advanced Numerical Methods functions, 4

in DiscreteLyapunovSolve, 5

$SamplingPeriod, 91