This review presents a framework for the input-output analysis, model reduction, and control design for fluid dynamical systems using examples applied to the linear complex Ginzburg–Landau equation. Major advances in hydrodynamics stability, such as global modes in spatially inhomogeneous systems and transient growth of non-normal systems, are reviewed. Input-output analysis generalizes hydrodynamic stability analysis by considering a finite-time horizon over which energy amplification, driven by a specific input (disturbances/actuator) and measured at a specific output (sensor), is observed. In the control design the loop is closed between the output and the input through a feedback gain. Model reduction approximates the system with a low-order model, making modern control design computationally tractable for systems of large dimensions. Methods from control theory are reviewed and applied to the Ginzburg–Landau equation in a manner that is readily generalized to fluid mechanics problems, thus giving a fluid mechanics audience an accessible introduction to the subject.

1.
Butler
,
K. M.
, and
Farrell
,
B. F.
, 1992, “
Three-Dimensional Optimal Perturbations in Viscous Shear Flow
,”
Phys. Fluids A
0899-8213,
4
, pp.
1637
1650
.
2.
Farrell
,
B. F.
, 1988, “
Optimal Excitation of Perturbations in Viscous Shear Flow
,”
Phys. Fluids
0031-9171,
31
, pp.
2093
2102
.
3.
Reddy
,
S. C.
, and
Henningson
,
D. S.
, 1993, “
Energy Growth in Viscous Channel Flows
,”
J. Fluid Mech.
0022-1120,
252
, pp.
209
238
.
4.
Reddy
,
S. C.
Schmid
,
P. J.
, and
Henningson
,
D. S.
, 1993, “
Pseudospectra of the Orr–Sommerfeld Operator
,”
SIAM J. Appl. Math.
0036-1399,
53
(
1
), pp.
15
47
.
5.
Farrell
,
B. F.
, and
Ioannou
,
P. J.
, 1996, “
Generalized Stability Theory. Part I. Autonomous Operators
,”
J. Atmos. Sci.
0022-4928,
53
, pp.
2025
2040
.
6.
Schmid
,
P. J.
, 2007, “
Nonmodal Stability Theory
,”
Annu. Rev. Fluid Mech.
0066-4189,
39
, pp.
129
162
.
7.
Schmid
,
P. J.
, and
Henningson
,
D. S.
, 2001,
Stability and Transition in Shear Flows
,
Springer
,
New York
.
8.
Trefethen
,
L. N.
,
Trefethen
,
A. E.
,
Reddy
,
S. C.
, and
Driscoll
,
T.
, 1993, “
Hydrodynamic Stability Without Eigenvalues
,”
Science
0036-8075,
261
, pp.
578
584
.
9.
Jovanovic
,
M. R.
, and
Bamieh
,
B.
, 2005, “
Componentwise Energy Amplification in Channel Flows
,”
J. Fluid Mech.
0022-1120,
534
, pp.
145
183
.
10.
Åkervik
,
E.
,
Hœpffner
,
J.
,
Ehrenstein
,
U.
, and
Henningson
,
D. S.
, 2007, “
Optimal Growth, Model Reduction and Control in a Separated Boundary-Layer Flow Using Global Eigenmodes
,”
J. Fluid Mech.
0022-1120,
579
, pp.
305
314
.
11.
Bewley
,
T. R.
, and
Liu
,
S.
, 1998, “
Optimal and Robust Control and Estimation of Linear Paths to Transition
,”
J. Fluid Mech.
0022-1120,
365
, pp.
305
349
.
12.
Chevalier
,
M.
,
Hœpffner
,
J.
,
Åkervik
,
E.
, and
Henningson
,
D. S.
, 2007, “
Linear Feedback Control and Estimation Applied to Instabilities in Spatially Developing Boundary Layers
,”
J. Fluid Mech.
,
588
, pp.
163
187
. 0022-1120
13.
Chevalier
,
M.
,
Hœpffner
,
J.
,
Bewley
,
T. R.
, and
Henningson
,
D. S.
, 2006, “
State Estimation in Wall-Bounded Flow Systems. Part 2: Turbulent Flows
,”
J. Fluid Mech.
0022-1120,
552
, pp.
167
187
.
14.
Hœpffner
,
J.
,
Chevalier
,
M.
,
Bewley
,
T. R.
, and
Henningson
,
D. S.
, 2005, “
State Estimation in Wall-Bounded Flow Systems. Part I: Laminar Flows
,”
J. Fluid Mech.
0022-1120,
534
, pp.
263
294
.
15.
Högberg
,
M.
,
Bewley
,
T. R.
, and
Henningson
,
D. S.
, 2003, “
Linear Feedback Control and Estimation of Transition in Plane Channel Flow
,”
J. Fluid Mech.
0022-1120,
481
, pp.
149
175
.
16.
Högberg
,
M.
,
Bewley
,
T. R.
, and
Henningson
,
D. S.
, 2003, “
Relaminarization of Reτ=100 Turbulence Using Gain Scheduling and Linear State-Feedback Control Flow
,”
Phys. Fluids
1070-6631,
15
, pp.
3572
3575
.
17.
Joshi
,
S. S.
,
Speyer
,
J. L.
, and
Kim
,
J.
, 1997, “
A Systems Theory Approach to the Feedback Stabilization of Infinitesimal and Finite-Amplitude Disturbances in Plane Poiseuille Flow
,”
J. Fluid Mech.
,
332
, pp.
157
184
. 0022-1120
18.
Lee
,
K. H.
,
Cortelezzi
,
L.
,
Kim
,
J.
, and
Speyer
,
J.
, 2001, “
Application of Reduced-Order Controller to Turbulent Flow for Drag Reduction
,”
Phys. Fluids
1070-6631,
13
, pp.
1321
1330
.
19.
Monokrousos
,
A.
,
Brandt
,
L.
,
Schlatter
,
P.
, and
Henningson
,
D. S.
, 2008, “
DNS and Less of Estimation and Control of Transition in Boundary Layers Subject to Free-Stream Turbulence
,”
Int. J. Heat Fluid Flow
0142-727X,
29
(
3
), pp.
841
855
.
20.
Anderson
,
B.
, and
Moore
,
J.
, 1990,
Optimal Control: Linear Quadratic Methods
,
Prentice-Hall
,
New York
.
21.
Kwakernaak
,
H.
, and
Sivan
,
R.
, 1972,
Linear Optimal Control Systems
,
Wiley Interscience
,
New York
.
22.
Lewis
,
F. L.
, and
Syrmos
,
L. V.
, 1995,
Optimal Control
,
Wiley
,
New York
.
23.
Zhou
,
K.
,
Doyle
,
J. C.
, and
Glover
,
K.
, 2002,
Robust and Optimal Control
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
24.
Bewley
,
T. R.
, 2001, “
Flow Control: New challenges for a New Renaissance
,”
Prog. Aerosp. Sci.
0376-0421,
37
, pp.
21
58
.
25.
Gal-El-Hak
,
M.
, 1996, “
Modern Developments in Flow Control
,”
Appl. Mech. Rev.
,
49
, pp.
365
379
. 0003-6900
26.
Kim
,
J.
, 2003, “
Control of Turbulent Boundary Layers
,”
Phys. Fluids
1070-6631,
15
, pp.
1093
1105
.
27.
Kim
,
J.
, and
Bewley
,
T. R.
, 2007, “
A Linear Systems Approach to Flow Control
,”
Annu. Rev. Fluid Mech.
0066-4189,
39
, pp.
383
417
.
28.
Hill
,
D. C.
, 1995, “
Adjoint Systems and Their Role in the Receptivity Problem for Boundary Layers
,”
J. Fluid Mech.
0022-1120,
292
, pp.
183
204
.
29.
Luchini
,
P.
, and
Bottaro
,
A.
, 1998, “
Gortler Vortices: A Backward-in-Time Approach to the Receptivity Problem
,”
J. Fluid Mech.
0022-1120,
363
, pp.
1
23
.
30.
Chomaz
,
J. M.
, 2005, “
Global Instabilities in Spatially Developing Flows: Non-Normality and Nonlinearity
,”
Annu. Rev. Fluid Mech.
0066-4189,
37
, pp.
357
392
.
31.
Huerre
,
P.
, and
Monkewitz
,
P. A.
, 1990, “
Local and Global Instabilities in Spatially Developing Flows
,”
Annu. Rev. Fluid Mech.
0066-4189,
22
, pp.
473
573
.
32.
Cohen
,
K.
,
Siegel
,
S.
,
McLaughlin
,
T.
,
Gillies
,
E.
, and
Myatt
,
J.
, 2005, “
Closed-Loop Approaches to Control of a Wake Flow Modeled by the Ginzburg–Landau Equation
,”
Comput. Fluids
,
34
, pp.
927
949
. 0045-7930
33.
Lauga
,
E.
, and
Bewley
,
T. R.
, 2003, “
The Decay of Stabilizability With Reynolds Number in a Linear Model of Spatially Developing Flows
,”
Proc. R. Soc. London, Ser. A
0950-1207,
459
, pp.
2077
2095
.
34.
Lauga
,
E.
, and
Bewley
,
T. R.
, 2004, “
Performance of a Linear Robust Control Strategy on a Nonlinear Model of Spatially Developing Flows
,”
J. Fluid Mech.
,
512
, pp.
343
374
. 0022-1120
35.
Monkewitz
,
P.
, 1989, “
Feedback Control of Global Oscillations in Fluid Systems
,” AIAA Paper No. 89-0991.
36.
Park
,
D. S.
,
Ladd
,
D. M.
, and
Hendricks
,
E. W.
, 1993, “
Feedback Control of a Global Mode in Spatially Developing Flows
,”
Phys. Lett. A
0375-9601,
182
, pp.
244
248
.
37.
Briggs
,
R. J.
, 1964,
Electron-Stream Interaction With Plasmas
,
MIT
,
Cambridge, MA
.
38.
Huerre
,
P.
, 2000, “
Open Shear Flow Instabilities
,”
Perspectives in Fluid Dynamics
,
Cambridge University Press
,
Cambridge, England
, pp.
159
229
.
39.
Chomaz
,
J. M.
,
Huerre
,
P.
, and
Redekopp
,
L. G.
, 1991, “
A Frequency Selection Criterion in Spatially Developing Flows
,”
Stud. Appl. Math.
,
84
, pp.
119
144
. 0022-2526
40.
Le Dizes
,
S.
,
Huerre
,
P.
,
Chomaz
,
J. M.
, and
Monkewitz
,
P. A.
, 1996, “
Linear Global Modes in Spatially Developing Media
,”
Philos. Trans. R. Soc. London, Ser. A
,
354
, pp.
169
212
. 0080-4614
41.
Monkewitz
,
P.
, 1990, “
The Role of Absolute and Convective Instability in Predicting the Behavior of Fluid Systems
,”
Eur. J. Mech. B/Fluids
,
9
, pp.
395
413
. 0997-7546
42.
Davies
,
E. B.
, 2002, “
Non-Self-Adjoint Differential Operators
,”
Bull. London Math. Soc.
0024-6093,
34
, pp.
513
532
.
43.
Trefethen
,
L. N.
, 1997, “
Pseudospectra of Linear Operators
,”
SIAM Rev.
0036-1445,
39
(
3
), pp.
383
406
.
44.
Trefethen
,
L. N.
, and
Embree
,
M.
, 2005,
Spectra and Pseudospectra: The Behavior Of Nonnormal Matrices and Operators
,
Princeton University Press
,
Princeton, NJ
.
45.
Chomaz
,
J. M.
,
Huerre
,
P.
, and
Redekopp
,
L. G.
, 1991, “
The Effect of Nonlinearity and Forcing on Global Modes
,”
New Trends in Nonlinear Dynamics and Pattern-Forming Phenomena
(
NATO Advanced Series Institute, Series B: Physics
),
P.
Coullet
and
P.
Huerre
, eds.,
Springer
,
New York
, pp.
259
274
.
46.
Cossu
,
C.
, and
Chomaz
,
J. M.
, 1997, “
Global Measures of Local Convective Instabilities
,”
Phys. Rev. Lett.
0031-9007,
78
, pp.
4387
4390
.
47.
Henningson
,
D. S.
, and
Åkervik
,
E.
, 2008, “
The Use of Global Modes to Understand Transition and Perform Flow Control
,”
Phys. Fluids
1070-6631,
20
, p.
031302
.
48.
Hunt
,
R. E.
, and
Crighton
,
D. G.
, 1991, “
Instability of Flows in Spatially Developing Media
,”
Proc. R. Soc. London, Ser. A
1364-5021,
435
, pp.
109
128
.
49.
Chomaz
,
J. M.
,
Huerre
,
P.
, and
Redekopp
,
L. G.
, 1988, “
Bifurcations to Local and Global Modes in Spatially Developing Flows
,”
Phys. Rev. Lett.
0031-9007,
60
, pp.
25
28
.
50.
Chomaz
,
J. M.
,
Huerre
,
P.
, and
Redekopp
,
L. G.
, 1987, “
Models of Hydrodynamic Resonances in Separated Shear Flows
,”
Sixth Symposium on Turbulent Shear Flows
, Toulouse, France.
51.
Åkervik
,
E.
,
Ehrenstein
,
U.
,
Gallaire
,
F.
, and
Henningson
,
D. S.
, 2008, “
Global Two-Dimensional Stability Measures of the Flat Plate Boundary-Layer Flow
,”
Eur. J. Mech. B/Fluids
0997-7546,
27
, pp.
501
513
.
52.
Ehrenstein
,
U.
, and
Gallaire
,
F.
, 2005, “
On Two-Dimensional Temporal Modes in Spatially Evolving Open Flows: The Flat-Plate Boundary Layer
,”
J. Fluid Mech.
0022-1120,
536
, pp.
209
218
.
53.
Trefethen
,
L. N.
, and
Bau
,
D.
, 1997,
Numerical Linear Algebra
,
SIAM
,
Philadelphia
.
54.
Andersson
,
P.
,
Berggren
,
M.
, and
Henningson
,
D. S.
, 1999, “
Optimal Disturbances and Bypass Transition in Boundary Layers
,”
Phys. Fluids
1070-6631,
11
, pp.
134
150
.
55.
Corbett
,
P.
, and
Bottaro
,
A.
, 2001, “
Optimal Linear Growth in Swept Boundary Layers
,”
J. Fluid Mech.
0022-1120,
435
, pp.
1
23
.
56.
Luchini
,
P.
, 2000, “
Reynolds-Number-Independent Instability of the Boundary Layer Over a Flat Surface: Optimal Perturbations
,”
J. Fluid Mech.
0022-1120,
404
, pp.
289
309
.
57.
Biau
,
D.
, and
Bottaro
,
A.
, 2004, “
Transient Growth and Minimal Defects: Two Possible Initial Paths of Transition to Turbulence in Plane Shear Flows
,”
Phys. Fluids
1070-6631,
16
, pp.
3515
3529
.
58.
Giannetti
,
F.
, and
Luchini
,
P.
, 2007, “
Structural Sensitivity of the First Instability of the Cylinder Wake
,”
J. Fluid Mech.
0022-1120,
581
, pp.
167
197
.
59.
Pier
,
B.
, 2002, “
On the Frequency Selection of Finite-Amplitude Vortex Shedding in the Cylinder Wake
,”
J. Fluid Mech.
0022-1120,
458
, pp.
407
417
.
60.
Provansal
,
M.
,
Mathis
,
C.
, and
Boyer
,
L.
, 1987, “
Bénard–von Kármán Instability: Transient and Forcing Regimes
,”
J. Fluid Mech.
0022-1120,
182
, pp.
1
22
.
61.
Albarède
,
P.
, and
Monkewitz
,
P. A.
, 1992, “
A Model for the Formation of Oblique Shedding and “Chevron” Patterns in Cylinder Wakes
,”
Phys. Fluids A
0899-8213,
4
, pp.
744
756
.
62.
Monkewitz
,
P. A.
,
Williamson
,
C. H. K.
, and
Miller
,
G. D.
, 1996, “
Phase Dynamics of Kármán Vortices in Cylinder Wakes
,”
Phys. Fluids
1070-6631,
8
, pp.
91
96
.
63.
Roussopoulos
,
K.
, and
Monkewitz
,
P.
, 1996, “
Nonlinear Modeling of Vortex Shedding Control in Cylinder Wakes
,”
Physica D
0167-2789,
97
, pp.
264
273
.
64.
Lesshafft
,
L.
,
Huerre
,
P.
,
Sagaut
,
P.
, and
Terracol
,
M.
, 2006, “
Nonlinear Global Modes in Hot Jets
,”
J. Fluid Mech.
0022-1120,
554
, pp.
393
409
.
65.
Nichols
,
J. W.
,
Schmid
,
P. J.
, and
Riley
,
J. J.
, 2007, “
Self-Sustained Oscillations in Variable-Density Round Jets
,”
J. Fluid Mech.
0022-1120,
582
, pp.
341
376
.
66.
Marquillie
,
M.
, and
Ehrenstein
,
U.
, 2002, “
On the Onset of Nonlinear Oscillations in a Separating Boundary-Layer Flow
,”
J. Fluid Mech.
0022-1120,
458
, pp.
407
417
.
67.
Ho
,
C. M.
, and
Huerre
,
P.
, 1984, “
Perturbed Free Shear Layers
,”
Annu. Rev. Fluid Mech.
0066-4189,
16
, pp.
365
424
.
68.
Kailath
,
T.
, 1980,
Linear Systems
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
69.
Farrell
,
B. F.
, and
Ioannou
,
P. J.
, 1993, “
Stochastic Forcing of the Linearized Navier–Stokes Equations
,”
Phys. Fluids A
0899-8213,
5
, pp.
2600
2609
.
70.
van der Schaft
,
A. J.
, 1991, “
Duality for Linear Systems: External and State Space Characterization of the Adjoint Problem
,”
Analysis of Controlled Dynamical Systems
,
B.
Bonnard
B.
Bride
,
J. P.
Gauthier
, and
I.
Kupka
, eds.,
Birkhäuser
,
Boston
, pp.
393
403
.
71.
Hœpffner
,
J.
, 2006, “
Stability and Control of Shear Flows Subject to Stochastic Excitations
,” Ph.D. thesis, KTH Stockholm, Stockholm.
72.
Bamieh
,
B.
, and
Dahleh
,
M.
, 2001, “
Energy Amplification in Channel Flows With Stochastic Excitation
,”
Phys. Fluids
1070-6631,
13
, pp.
3258
3269
.
73.
Hœpffner
,
J.
, and
Brandt
,
L.
, 2008, “
Stochastic Approach to the Receptivity Problem Applied to Bypass Transition in Boundary Layers
,”
Phys. Fluids
1070-6631,
20
, pp.
024108
.
74.
Datta
,
B.
, 2003,
Numerical Methods for Linear Control Systems Design and Analysis
,
Elsevier
,
New York
.
75.
Zhou
,
K.
,
Salomon
,
G.
, and
Wu
,
E.
, 1999, “
Balanced Realization and Model Reduction for Unstable Systems
,”
Int. J. Robust Nonlinear Control
1049-8923,
9
, pp.
183
198
.
76.
Antoulas
,
C. A.
, 2005,
Approximation of Large-Scale Dynamical Systems
,
SIAM
,
Philadelphia
.
77.
Lumley
,
J. L.
, 1970,
Stochastic Tools in Turbulence
,
Academic
,
New York
.
78.
Obintata
,
G.
, and
Andersson
,
B. D.
, 2001,
Model Reduction for Control System Design
,
Springer
,
New York
.
79.
Green
,
M.
, and
Limebeer
,
J. N.
, 1995,
Linear Robust Control
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
80.
Skogestad
,
S.
, and
Postlethwaite
,
I.
, 2005,
Multivariable Feedback Control: Analysis and Design
,
2nd ed.
,
Wiley
,
New York
.
81.
Moore
,
B.
, 1981, “
Principal Component Analysis in Linear Systems: Controllability, Observability, and Model Reduction
,”
IEEE Trans. Autom. Control
0018-9286,
26
(
1
), pp.
17
32
.
82.
Antoulas
,
A. C.
,
Sorensen
,
D. S.
, and
Gugercin
,
S.
, 2001, “
A Survey of Model Reduction Methods for Large-Scale Systems
,”
Contemp. Math.
,
280
, pp.
193
219
. 0271-4132
83.
C. W.
Rowley
, 2005, “
Model Reduction for Fluids Using Balanced Proper Orthogonal Decomposition
,”
Int. J. Bifurcation Chaos Appl. Sci. Eng.
0218-1274,
15
(
3
), pp.
997
1013
.
84.
Willcox
,
K.
, and
Peraire
,
J.
, 2002, “
Balanced Model Reduction Via the Proper Orthogonal Decomposition
,”
AIAA J.
,
40
(
11
), pp.
2323
2330
. 0098-3500
85.
Sirovich
,
L.
, 1987, “
Turbulence and the Dynamics of Coherent Structures. Parts I–III
,”
Q. Appl. Math.
,
45
, pp.
561
590
. 0033-569X
86.
Ilak
,
M.
, and
Rowley
,
C. W.
, 2008, “
Modeling of Transitional Channel Flow Using Balanced Proper Orthogonal Decomposition
,”
Phys. Fluids
1070-6631,
20
, p.
034103
.
87.
Ahuja
,
S.
,
Rowley
,
C. W.
,
Kevrekidis
,
I. G.
, and
Wei
,
M.
, 2007, “
Low-Dimensional Models for Control of Leading-Edge Vortices: Equilibria and Linearized Models
,”
45th AIAA Aerospace Sciences Meeting and Exhibit
, AIAA Paper No. 2007-709.
88.
Joslin
,
R. D.
, 1998, “
Aircraft Laminar Flow Control
,”
Annu. Rev. Fluid Mech.
0066-4189,
30
, pp.
1
29
.
89.
Moin
,
P.
, and
Bewley
,
T. R.
, 1994, “
Feedback Control of Turbulence
,”
Appl. Mech. Rev.
,
47
(
6
), pp.
S3
S113
. 0022-1120
90.
Choi
,
H.
,
Moin
,
P.
, and
Kim
,
J.
, 1994, “
Active Turbulence Control for Drag Reduction in Wall-Bounded Flows
,”
J. Fluid Mech.
0022-1120,
262
, pp.
75
110
.
91.
Bewley
,
T. R.
, and
Moin
,
P.
, 1994, “
Optimal Control of Turbulent Channel Flows
,”
Active Control of Vibration and Noise
, ASME DE-Vol. No.
75
,
E. W.
Hendricks
,
K. W.
Wang
,
A. H.
Von Flotow
,
R.
Shoureshi
, and
T. W.
Farabee
, eds.,
ASME
,
New York, NY
.
92.
Bewley
,
T. R.
,
Moin
,
P.
, and
Temam
,
R.
, 2001, “
DNS-Based Predictive Control of Turbulence: An Optimal Benchmark for Feedback Algorithms
,”
J. Fluid Mech.
,
447
, pp.
179
225
. 0022-1120
93.
Guégan
,
A.
,
Schmid
,
P. J.
, and
Huerre
,
P.
, 2006, “
Optimal Energy Growth and Optimal Control in Swept Hiemenz Flow
,”
J. Fluid Mech.
,
566
, pp.
11
45
. 0022-1120
94.
Zuccher
,
P.
,
Luchini
,
S.
, and
Bottaro
,
A.
, 2006, “
Algebraic Growth in a Blasius Boundary Layer: Optimal and Robust Control by Mean Suction in the Nonlinear Regime
,”
J. Fluid Mech.
,
556
, pp.
189
216
. 0022-1120
95.
Farrell
,
B. F.
, and
Ioannou
,
P. J.
, 2000,
Flow Control: Passive, Active and Reactive Flow Management
,
Cambridge University Press
,
London
.
96.
Gunzburger
,
M. D.
, 1995,
Flow Control
,
Springer
,
Berlin
.
97.
Skelton
,
R.
,
Iwasaki
,
T.
, and
Grigoriadis
,
K.
, 1998,
A Unified Algebraic Approach to Linear Control Design
,
Taylor & Francis
,
London
.
98.
Kalman
,
R. E.
, 1960, “
A New Approach to Linear Filtering and Prediction Problems
,”
Trans. ASME Ser. D. J. Basic Eng.
,
82
, pp.
24
45
.
99.
Laub
,
A. J.
, 1991, “
Invariant Subspace Methods for the Numerical Solution of Riccati Equations
,”
The Riccati Equation
,
S.
Bittanti
,
A. J.
Laub
, and
J. C.
Willems
, eds.,
Springer
,
Berlin
, pp.
163
196
.
100.
Högberg
,
M.
, 2001, “
Optimal Control of Boundary-Layer Transition
,” Ph.D. thesis, KTH Stockholm, Stockholm.
101.
Doyle
,
J.
, 1978, “
Guaranteed Margins for LQG Regulators
,”
IEEE Trans. Autom. Control
0018-9286,
23
, pp.
756
757
.
102.
Doyle
,
J. C.
,
Glover
,
K.
,
Khargonekar
,
P. P.
, and
Francis
,
B. A.
, 1989, “
State-Space Solutions to Standard H2
and H∞ Control Problems,”
IEEE Trans. Autom. Control
0018-9286,
34
, pp.
831
847
.
103.
Kailath
,
T.
, 1973, “
Some New Algorithms for Recursive Estimation in Constant Linear Systems
,”
IEEE Trans. Inf. Theory
,
19
, pp.
750
760
. 0018-9448
104.
Gillies
,
E. A.
, 1998, “
Low-Dimensional Control of the Circular Cylinder Wake
,”
J. Fluid Mech.
0022-1120,
371
, pp.
157
178
.
105.
Ahuja
,
S.
, and
Rowley
,
C. W.
, 2008, “
Low-Dimensional Models for Feedback Stabilization of Unstable Steady States
,”
46th AIAA Aerospace Sciences Meeting and Exhibit
, AIAA Paper No. 2008-553.
106.
Weideman
,
J. A. C.
, and
Reddy
,
S. C.
, 2000, “
A MATLAB Differentiation Matrix Suite
,”
ACM Trans. Math. Softw.
0098-3500,
26
, pp.
465
519
.
107.
Tang
,
T.
, 1993, “
The Hermite Spectral Method for Gaussian-Type Functions
,”
J. Sci. Comput.
,
14
, pp.
594
606
. 0885-7474
108.
Abramowitz
,
M.
, and
Stegun
,
I. E.
, 1964,
Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables
,
National Bureau of Standards
,
Washington, DC
.
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