This note concerns with the problem of achieving minimum settling time in linear systems with one or two real right half plane (RHP) zeros subject to the condition that undershoot does not exceed a given threshold. Such problem is of great practical significance, but it has not been formally addressed, to our knowledge. Time optimal control solutions for such systems are readily available based on the well known optimal control theory, but it does not address the practical consideration that the “wrong way response,” i.e., undershoot, must be limited. To be sure, the relationship between the minimum settling time and the undershoot constraint for systems with one or two real RHP zeros has already been given in the literature. How to find the control signal that achieves the minimum settling time, however, is still an open question. In this paper, such control signal is obtained constructively and, combined with feedback, is shown to be rather effective in controlling the system in the presence of model uncertainties and external disturbances, as shown in simulation.

References

References
1.
Hoagg
,
J. B.
, and
Bernstein
,
D. S.
,
2007
, “
Nonminimum-Phase Zeros: Much to Do About Nothing
,”
IEEE Control Syst. Mag.
,
27
(
3
), pp.
45
57
.10.1109/MCS.2007.365003
2.
Skogestad
,
S.
, and
Postlethwaite
,
I.
,
2005
,
Multivariable Feedback Control: Analysis and Design
,
2nd ed.
,
John Wiley
,
Hoboken, NJ
, pp.
173
180
.
3.
Rigney
,
B. P.
,
Pao
,
L. Y.
, and
Lawrence
,
D. A.
,
2009
, “
Nonminimum Phase Dynamic Inversion for Settle Time Applications
,”
IEEE Trans. Control Syst. Technol.
,
17
(
5
), pp.
989
1005
.10.1109/TCST.2008.2002035
4.
Middleton
,
R. H.
,
1991
, “
Tradeoffs in Linear Control System Design
,”
Automatica
,
27
(
2
), pp.
281
292
.10.1016/0005-1098(91)90077-F
5.
Lau
,
K.
,
Middleton
,
R. H.
, and
Braslavsky
,
J. H.
,
2003
, “
Undershoot and Settling Time Tradeoffs for Nonminimum Phase Systems
,”
IEEE Trans. Autom. Control
,
48
(
8
), pp.
1389
1393
.10.1109/TAC.2003.815025
6.
Widder
,
D. V.
,
1934
, “
The Inversion of the Laplace Integral and the Related Moment Problem
,”
Trans. Am. Math. Soc.
,
36
(
1
), pp.
107
200
.10.1090/S0002-9947-1934-1501737-7
7.
Åström
,
K. J.
, and
Hägglund
,
T.
,
1995
,
PID Controllers: Theory, Design and Tuning
,
2nd ed.
,
Instrument Society of America
,
Research Triangle Park, NC
, pp.
284
287
.
8.
Zhang
,
Y.
,
2009
, “
Load Frequency Control of Multiple Area Power Systems
,” M.S. thesis,
Cleveland State University
,
Cleveland, OH
.
You do not currently have access to this content.