Abstract
This paper addresses the compensation of wave actuator dynamics in scalar extremum seeking (ES) for static maps. Infinite-dimensional systems described by partial differential equations (PDEs) of wave type have not been considered so far in the literature of ES. A distributed-parameter-based control law using back-stepping approach and Neumann actuation is initially proposed. Local exponential stability as well as practical convergence to an arbitrarily small neighborhood of the unknown extremum point is guaranteed by employing Lyapunov–Krasovskii functionals and averaging theory in infinite dimensions. Thereafter, the extension for wave equations with Dirichlet actuation, antistable wave PDEs as well as the design for the delay-wave PDE cascade are also discussed. Numerical simulations illustrate the theoretical results.
Issue Section:
Research Papers
References
1.
Krstić
,
M.
, and
Wang
,
H.-H.
, 2000
, “
Stability of Extremum Seeking Feedback for General Nonlinear Dynamic Systems
,” Automatica
,
36
(4
), pp. 595
–601
.10.1016/S0005-1098(99)00183-12.
Ghods
,
N.
, and
Krstic
,
M.
, 2011
, “
Source Seeking With Very Slow or Drifting Sensors
,” ASME J. Dyn. Syst., Meas., Control
,
133
(4
), p. 044504
.10.1115/1.40036393.
Scheinker
,
A.
, and
Krstic
,
M.
, 2014
, “
Non-C2 Lie Bracket Averaging for Nonsmooth Extremum Seekers
,” ASME J. Dyn. Syst., Meas., Control
,
136
(1
), p. 011010
.10.1115/1.40254574.
Frihauf
,
P.
,
Liu
,
S. J.
, and
Krstic
,
M.
, 2014
, “
A Single Forward-Velocity Control Signal for Stochastic Source Seeking With Multiple Nonholonomic Vehicles
,” ASME J. Dyn. Syst., Meas., Control
,
136
(5
), p. 051024
.10.1115/1.40275775.
Bagheri
,
M.
,
Krstic
,
M.
, and
Naseradinmousavi
,
P.
, 2018
, “
Multivariable Extremum Seeking for Joint-Space Trajectory Optimization of a High-Degrees-of-Freedom Robot
,” ASME J. Dyn. Syst., Meas., Control
,
140
(11
), p. 111017
.10.1115/1.40407526.
Oliveira
,
T. R.
,
Krstic
,
M.
, and
Tsubakino
,
D.
, 2017
, “
Extremum Seeking for Static Maps With Delays
,” IEEE Trans. Automat. Control
,
62
(4
), pp. 1911
–1926
.10.1109/TAC.2016.25649587.
Rusiti
,
D.
,
Evangelisti
,
G.
,
Oliveira
,
T. R.
,
Gerdts
,
M.
, and
Krstic
,
M.
, 2019
, “
Stochastic Extremum Seeking for Dynamic Maps With Delays
,” IEEE Control Syst. Lett.
,
3
(1
), pp. 61
–66
.10.1109/LCSYS.2018.28516028.
Feiling
,
J.
,
Koga
,
S.
,
Krstić
,
M.
, and
Oliveira
,
T. R.
, 2018
, “
Gradient Extremum Seeking for Static Maps With Actuation Dynamics Governed by Diffusion PDEs
,” Automatica
,
95
(9
), pp. 197
–206
.10.1016/j.automatica.2018.05.0239.
Oliveira
,
T. R.
,
Feiling
,
J.
,
Koga
,
S.
, and
Krstić
,
M.
, 2018
, “
Scalar Newton-Based Extremum Seeking for a Class of Diffusion PDEs
,” IEEE Conference on Decision and Control
, Miami Beach, FL
, Dec. 17–19, pp. 2926
–2931
.10.1109/CDC.2018.861926010.
Oliveira
,
T. R.
,
Feiling
,
J.
, and
Krstic
,
M.
, 2019
, “
Extremum Seeking for Maximizing Higher Derivatives of Unknown Maps in Cascade With Reaction-Advection-Diffusion PDEs
,” 13th IFAC Workshop on Adaptive and Learning Control Systems
, Winchester, UK
, Dec. 4–6, pp. 210
–215
.10.1016/j.ifacol.2019.12.65111.
Oliveira
,
T. R.
,
Feiling
,
J.
,
Koga
,
S.
, and
Krstic
,
M.
, 2020
, “
Extremum Seeking for Unknown Scalar Maps in Cascade With a Class of Parabolic Partial Differential Equations
,” Int. J. Adaptive Control Signal Process.
, pp. 1
–26
.10.1002/acs.311712.
Krstic
,
M.
, 2011
, “
Dead-Time Compensation for Wave/String PDEs
,” ASME J. Dyn. Syst., Meas., Control
,
133
(3
), p. 031004
.10.1115/1.400363813.
Moura
,
S. J.
,
Chaturvedi
,
N. A.
, and
Krstic
,
M.
, 2014
, “
Adaptive Partial Differential Equation Observer for Battery State-of-Charge/State-of-Health Estimation Via an Electrochemical Model
,” ASME J. Dyn. Syst., Meas., Control
,
136
(1
), p. 011015
.10.1115/1.402480114.
Sezgin
,
A.
, and
Krstic
,
M.
, 2015
, “
Boundary Backstepping Control of Flow-Induced Vibrations of a Membrane at High Mach Numbers
,” ASME J. Dyn. Syst., Meas., Control
,
137
(8
), p. 081003
.10.1115/1.402946815.
Ghaffari
,
A.
,
Moura
,
S.
, and
Krstic
,
M.
, 2015
, “
PDE-Based Modeling, Control, and Stability Analysis of Heterogeneous Thermostatically Controlled Load Populations
,” ASME J. Dyn. Syst., Meas., Control
,
137
(10
), p. 101009
.10.1115/1.403081716.
Wang
,
J.
,
Koga
,
S.
,
Pi
,
Y.
, and
Krstic
,
M.
, 2018
, “
Axial Vibration Suppression in a Partial Differential Equation Model of Ascending Mining Cable Elevator
,” ASME J. Dyn. Syst., Meas., Control
,
140
(11
), p. 111003
.10.1115/1.404021717.
Siranosian
,
A.
,
Krstic
,
M.
,
Smyshlyaev
,
A.
, and
Bement
,
M.
, 2009
, “
Motion Planning and Tracking for Tip Displacement and Deflection Angle for Flexible Beams
,” ASME J. Dyn. Syst., Meas., Control
,
131
(3
), p. 031009
.10.1115/1.307215218.
Gu
,
K.
, and
Niculescu
,
S. I.
, 2003
, “
Survey on Recent Results in the Stability and Control of Time-Delay Systems
,” ASME J. Dyn. Syst., Meas., Control
,
125
(2
), pp. 158
–165
.10.1115/1.156995019.
Evesque
,
S.
,
Annaswamy
,
A. M.
,
Niculescu
,
S.
, and
Dowling
,
A. P.
, 2003
, “
Adaptive Control of a Class of Time-Delay Systems
,” ASME J. Dyn. Syst., Meas., Control
,
125
(2
), pp. 186
–193
.10.1115/1.156775520.
Olgac
,
N.
, and
Sipahi
,
R.
, 2005
, “
The Cluster Treatment of Characteristic Roots and the Neutral Type Time-Delayed Systems
,” ASME J. Dyn. Syst., Meas., Control
,
127
(1
), pp. 88
–97
.10.1115/1.187649421.
Olgac
,
N.
, and
Sipahi
,
R.
, 2003
, “
Analysis of a System of Linear Delay Differential Equations
,” ASME J. Dyn. Syst., Meas., Control
,
125
(2
), pp. 215
–223
.10.1115/1.156812122.
Bekiaris-Liberis
,
N.
, and
Krstic
,
M.
, 2014
, “
Compensation of Wave Actuator Dynamics for Nonlinear Systems
,” IEEE Trans. Automat. Control
,
59
(6
), pp. 1555
–1572
.10.1109/TAC.2014.230905723.
Aarsnes
,
U. J. F.
, Aamo
,
O. M.
, and Krstic
,
M.
, 2019
, “
Extremum Seeking for Real-Time Optimal Drilling Control
,” American Control Conference
, Philadelphia, PA
, July 10–12, pp. 5222
–5227
.10.23919/ACC.2019.881516224.
Krstić
,
M.
, 2009
, “
Compensating a String PDE in the Actuation or Sensing Path of an Unstable ODE
,” IEEE Trans. Automat. Control
,
54
(6
), pp. 1362
–1368
.10.1109/TAC.2009.201555725.
Krstić
,
M.
, and
Smyshlyaev
,
A.
, 2008
, Boundary Control of PDEs: A Course on Backstepping Designs
,
SIAM
, Philadelphia, PA
.26.
Hale
,
J. K.
, and
Lunel
,
S. M. V.
, 1990
, “
Averaging in Infinite Dimensions
,” J. Integral Equations Appl.
,
2
(4
), pp. 463
–494
.10.1216/jiea/118107558327.
Oliveira
,
T. R.
, and
Krstic
,
M.
, 2019
, “
Compensation of Wave PDEs in Actuator Dynamics for Extremum Seeking Feedback
,” 13th IFAC Workshop on Adaptive and Learning Control Systems
, Winchester, UK
, Dec. 4–6, pp. 134
–139
.10.1016/j.ifacol.2019.12.63428.
Khalil
,
H. K.
, 2002
, Nonlinear Systems
,
Prentice Hall
, Upper Saddle River, NJ
.29.
Karafyllis
,
I.
, and
Krstic
,
M.
, 2018
, Input-to-State Stability for PDEs
,
Springer
, Cham, Switzerland
.30.
Susto
,
G. A.
, and
Krstic
,
M.
, 2010
, “
Control of PDE–ODE Cascades With Neumann Interconnections
,” J. Franklin Inst.
,
347
(1
), pp. 284
–314
.10.1016/j.jfranklin.2009.09.00531.
Ghaffari
,
A.
,
Krstić
,
M.
, and
Nešić
,
D.
, 2012
, “
Multivariable Newton-Based Extremum Seeking
,” Automatica
,
48
(8
), pp. 1759
–1767
.10.1016/j.automatica.2012.05.05932.
Krstic
,
M.
, 2009
, Delay Compensation for Nonlinear, Adaptive, and PDE Systems
,
Birkhauser
, Boston, MA
.33.
Datko
,
R.
,
Lagnese
,
J.
, and
Polis
,
M. P.
, 1986
, “
An Example on the Effect of Time Delays in Boundary Feedback Stabilization of Wave Equations
,” SIAM J. Control Optim.
,
24
(1
), pp. 152
–156
.10.1137/032400734.
Liu
,
S.-J.
, and
Krstic
,
M.
, 2010
, “
Stochastic Averaging in Continuous Time and Its Applications to Extremum Seeking
,” IEEE Trans. Automat. Control
,
55
(10
), pp. 2235
–2250
.10.1109/TAC.2010.204329035.
Ariyur
,
K.
, and
Krstić
,
M.
, 2003
, Real Time Optimization by Extremum Seeking Control
,
Wiley
, Hoboken, NJ
.36.
Wang
,
L.
,
Chen
,
S.
, and
Ma
,
K.
, 2016
, “
On Stability and Application of Extremum Seeking Control Without Steady-State Oscillation
,” Automatica
,
68
(6
), pp. 18
–26
.10.1016/j.automatica.2016.01.00937.
Moura
,
S. J.
, and
Chang
,
Y. A.
, 2013
, “
Lyapunov-Based Switched Extremum Seeking for Photovoltaic Power Maximization
,” Control Eng. Pract.
,
21
(7
), pp. 971
–980
.10.1016/j.conengprac.2013.02.00938.
Scheinker
,
A.
, and
Krstić
,
M.
, 2014
, “
Extremum Seeking With Bounded Update Rates
,” Syst. Control Lett.
,
63
(1
), pp. 25
–31
.10.1016/j.sysconle.2013.10.00439.
Durr
,
H. B.
,
Stankovic
,
M. S.
,
Ebenbauer
,
C.
, and
Johansson
,
K. H.
, 2013
, “
Lie Bracket Approximation of Extremum Seeking Systems
,” Automatica
,
49
(6
), pp. 1538
–1552
.10.1016/j.automatica.2013.02.01640.
Grushkovskaya
,
V.
,
Zuyev
,
A.
, and
Ebenbauer
,
C.
, 2018
, “
On a Class of Generating Vector Fields for the Extremum Seeking Problem: Lie Bracket Approximation and Stability Properties
,” Automatica
,
94
(8
), pp. 151
–160
.10.1016/j.automatica.2018.04.02441.
Oliveira
,
T. R.
,
Feiling
,
J.
,
Koga
,
S.
, and
Krstic
,
M.
, 2020
, “
Multivariable Extremum Seeking for PDE Dynamic Systems
,” IEEE Trans. Automat. Control
, pp. 1
–8
.10.1109/TAC.2020.3005177Copyright © 2021 by ASME
You do not currently have access to this content.
