Abstract

The requirement of satisfying an integral constraint imposed on a linear system's transient step-response is considered in this paper. The problem is first analyzed to determine the specific structure of a system's transfer function that helps satisfy such constraints. Analytical results are derived for a class of second-order systems with an additional zero. The results are extended to higher order transfer functions. Subsequently, a standard compensation consisting of a combination of feedforward and feedback actions is proposed to transform a given transfer function to the desired structure. Necessary and sufficient conditions to guarantee stability of the resulting closed-loop system are derived. Next, the problem of satisfying integral constraints in the presence of parametric uncertainty is addressed by augmenting adaptive estimation strategies to the feedforward and feedback compensation structure. Simulation results are provided for validation. The theory presented here is an abstraction from power management algorithms for hybrid power systems, such as a fuel cell hybridized with an ultracapacitor. Further work is ongoing to extend the theory to nonlinear systems.

References

References
1.
Franklin
,
G. F.
,
Powell
,
J. D.
, and
Emami-Naeini
,
A.
,
2018
,
Feedback Control of Dynamic Systems
, 8th ed.,
Pearson
,
New York
.
2.
Corrington
,
M. S.
,
1965
, “
Simplified Calculation of Transient Response
,”
Proc. IEEE
,
53
(
3
), pp.
287
292
.10.1109/PROC.1965.3686
3.
Liou
,
M. L.
,
1966
, “
A Novel Method of Evaluating Transient Response
,”
Proc. IEEE
,
54
(
1
), pp.
20
23
.10.1109/PROC.1966.4569
4.
Aaron
,
M. R.
, and
Kaiser
,
J. F.
,
1965
, “
On the Calculation of Transient Response
,”
Proc. IEEE
,
53
(
9
), pp.
1269
1269
.10.1109/PROC.1965.4217
5.
Hauksdóttir
,
A. S.
,
1996
, “
Analytic Expressions of Transfer Function Responses and Choice of Numerator Coefficients (Zeros)
,”
IEEE Trans. Autom. Control
,
41
(
10
), pp.
1482
1488
.10.1109/9.539429
6.
Moore
,
K. L.
, and
Bhattacharyya
,
S. P.
,
1990
, “
A Technique for Choosing Zero Locations for Minimal Overshoot
,”
IEEE Trans. Autom. Control
,
35
(
5
), pp.
577
580
.10.1109/9.53527
7.
Moore
,
K. L.
,
1989
, “
Design Techniques for Transient Response Control
,” Ph.D. thesis,
Texas A&M University
, College Station, TX.
8.
El-Khoury
,
M.
,
Crisalle
,
O. D.
, and
Longchamp
,
R.
,
1993
, “
Influence of Zero Locations on the Number of Step-Response Extrema
,”
Automatica
,
29
(
6
), pp.
1571
1574
.10.1016/0005-1098(93)90023-M
9.
Mohsenizadeh
,
N.
,
Darbha
,
S.
, and
Bhattacharyya
,
S. P.
,
2011
, “
Synthesis of PID Controllers With Guaranteed Non-Overshooting Transient Response
,” 50th IEEE Conference on Decision and Control and European Control Conference (
CDC-ECC
), Orlando, FL, Dec. 12–15, pp.
447
452
.10.1109/CDC.2011.6161429
10.
Bechlioulis
,
C. P.
, and
Rovithakis
,
G. A.
,
2010
, “
Prescribed Performance Adaptive Control for Multi-Input Multi-Output Affine in the Control Nonlinear Systems
,”
IEEE Trans. Autom. Control
,
55
(
5
), pp.
1220
1226
.10.1109/TAC.2010.2042508
11.
Fan
,
B.
,
Yang
,
Q.
,
Jagannathan
,
S.
, and
Sun
,
Y.
,
2018
, “
Asymptotics Tracking Controller Design for Nonlinear Systems With Guaranteed Performance
,”
IEEE Trans. Vybernetics
,
48
(
7
), pp.
2001
2011
.10.1109/TCYB.2017.2726039
12.
Haack
,
B.
, and
Tomizuka
,
M.
,
1991
, “
The Effect of Adding Zeroes to Feedforward Controllers
,”
ASME J. Dyn. Syst., Meas., Control
,
113
(
1
), pp.
6
10
.10.1115/1.2896362
13.
Croft
,
D.
,
Shedd
,
G.
, and
Devasia
,
S.
,
2000
, “
Creep, Hysteresis, and Vibration Compensation for Piezoactuators: Atomic Force Microscopy Application
,”
Proceedings of the American Control Conference
(
ACC
), Vol.
3
, Chicago, IL, June 28–30, pp.
2123
2128
.10.1109/ACC.2000.879576
14.
Zhao
,
Y.
, and
Jayasuriya
,
S.
,
1994
, “
Feedforward Controllers and Tracking Accuracy in the Presence of Plant Uncertainties
,”
Proceedings of the American Control Conference
(
ACC
), Vol.
1
, Baltimore, MD, June 29–July 1, pp.
360
364
.10.1109/ACC.1994.751759
15.
Devasia
,
S.
,
2002
, “
Should Model-Based Inverse Inputs Be Used as Feedforward Under Plant Uncertainty?
,”
IEEE Trans. Autom. Control
,
47
(
11
), pp.
1865
1871
.10.1109/TAC.2002.804478
16.
Hunt
,
L. R.
,
Ramakrishna
,
V.
, and
Meyer
,
G.
,
1998
, “
Stable Inversion and Parameter Variations
,”
Syst. Control Lett.
,
34
(
4
), pp.
203
207
.10.1016/S0167-6911(98)00026-7
17.
Wu
,
Y.
, and
Zou
,
Q.
,
2009
, “
Robust Inversion-Based 2-DOF Control Design for Output Tracking: Piezoelectric-Actuator Example
,”
IEEE Trans. Control Syst. Technol.
,
17
(
5
), pp.
1069
1082
.10.1109/TCST.2008.2005111
18.
Clayton
,
G. M.
,
Tien
,
S.
,
Leang
,
K. K.
,
Zou
,
Q.
, and
Devasia
,
S.
,
2009
, “
A Review of Feedforward Control Approaches in Nanopositioning for High-Speed SPM
,”
ASME J. Dyn. Syst., Meas., Control
,
131
(
6
), p.
061101
.10.1115/1.4000158
19.
Boerlage
,
M.
,
Steinbuch
,
M.
,
Lambrechts
,
P.
, and
van de Wal
,
M.
,
2003
, “
Model-Based Feedforward for Motion Systems
,”
Proceedings of the IEEE Conference on Control Applications
, Vol.
2
, Istanbul, Turkey, June 25, pp.
1158
1163
.10.1109/CCA.2003.1223174
20.
Graichen
,
K.
,
Hagenmeyer
,
V.
, and
Zeitz
,
M.
,
2005
, “
A New Approach to Inversion-Based Feedforward Control Design for Nonlinear Systems
,”
Automatica
,
41
(
12
), pp.
2033
2041
.10.1016/j.automatica.2005.06.008
21.
Kim
,
Y. C.
,
Keel
,
L. H.
, and
Bhattacharyya
,
S. P.
,
2003
, “
Transient Response Control Via Characteristic Ratio Assignment
,”
IEEE Trans. Autom. Control
,
48
(
12
), pp.
2238
2244
.10.1109/TAC.2003.820153
22.
Singhose
,
W. E.
, and
Pao
,
L. Y.
,
1996
, “
Comparison of Input Shaping Techniques for Speed-Critical Multi-Mode Flexible Systems
,”
Proceeding of the IEEE International Conference on Control Applications
, Dearborn, MI, Sept. 15–Nov. 18, pp.
379
384
.10.1109/CCA.1996.558816
23.
Jones
,
S. D.
, and
Ulsoy
,
A. G.
,
1999
, “
An Approach to Control Input Shaping With Application to Coordinate Measuring Machines
,”
ASME J. Dyn. Syst., Meas., Control
,
121
(
2
), pp.
242
247
.10.1115/1.2802461
24.
Singhose
,
W.
,
Crain
,
E.
, and
Seering
,
W.
,
1997
, “
Convolved and Simultaneous Two-Mode Input Shapers
,”
IEE Proc.-Control Theory Appl.
,
144
(
6
), pp.
515
520
.10.1049/ip-cta:19971439
25.
Al Hamidi
,
Y.
, and
Rakotondrabe
,
M.
,
2016
, “
Comparison Study of Input Shaping Techniques to Control an Underactuated Flexible Link System
,”
SPIE Paper No. 9859
.
26.
Singh
,
T.
, and
Singhose
,
W.
,
2002
, “
Tutorial on Input Shaping/Time Delay Control of Maneuvering Flexible Structures
,”
Proceedings of the American Control Conference
, IEEE, Vol.
3
, Anchorage, AK, May 8–10, pp.
1717
1731
.http://code.eng.buffalo.edu/tdf/papers/acc_tut.pdf
27.
Tuttle
,
T. D.
, and
Seering
,
W. P.
,
1994
, “
A Zero-Placement Technique for Designing Shaped Inputs to Suppress Multiple-Mode Vibration
,”
Proceedings of the American Control Conference
(
ACC'94
), Vol.
3
, IEEE, Baltimore, MD, June 29–July 1, pp.
2533
2537
.10.1109/ACC.1994.735015
28.
Bruijnen
,
D.
, and
van Dijk
,
N.
,
2012
, “
Combined Input Shaping and Feedforward Control for Flexible Motion Systems
,”
American Control Conference
(
ACC
), IEEE, Montreal, QC, Canada, June 27–29, pp.
2473
2478
.10.1109/ACC.2012.6315055
29.
Conker
,
C.
,
Yavuz
,
H.
, and
Bilgic
,
H. H.
,
2016
, “
A Review of Command Shaping Techniques for Elimination of Residual Vibrations in Flexible-Joint Manipulators
,”
J. Vibroeng.
,
18
(
5
), pp.
2947
2958
.10.21595/jve.2016.16725
30.
Allag
,
T.
, and
Das
,
T.
,
2011
, “
Robust Control of Solid Oxide Fuel Cell Ultracapacitor Hybrid System
,”
IEEE Trans. Control Syst. Technol.
,
20
(
1
), pp.
1
10
.10.1109/TCST.2010.2098478
31.
Das
,
T.
, and
Snyder
,
S.
,
2013
, “
Robust Adaptive Control of Solid Oxide Fuel Cell Ultra-Capacitor Hybrid Systems
,”
IEEE Trans. Control Syst. Technol.
,
21
(
2
), pp.
372
383
.10.1109/TCST.2011.2181514
32.
Madani
,
O.
,
Bhattacharjee
,
A.
, and
Das
,
T.
,
2016
, “
Decentralized Power Management in a Hybrid Fuel Cell Ultracapacitor System
,”
IEEE Trans. Control Syst. Technol.
,
24
(
3
), pp.
765
778
.10.1109/TCST.2015.2464295
33.
Thomson
,
W. T.
, and
Dahleh
,
M. D.
,
1998
,
Theory of Vibration With Applications
, 5th ed.,
Prentice Hall
,
Upper Saddle River, NJ
.
34.
Morari
,
M.
,
1985
, “
Robust Stability of Systems With Integral Control
,”
IEEE Trans. Autom. Control
,
30
(
6
), pp.
574
577
.10.1109/TAC.1985.1104012
35.
Grosdidier
,
P.
,
Morari
,
M.
, and
Holt
,
B. R.
,
1985
, “
Closed-Loop Properties From Steady-State Gain Information
,”
Ind. Eng. Chem. Fundam.
,
24
(
2
), pp.
221
235
.10.1021/i100018a015
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