This paper addresses the use of multidimensional scaling in the evaluation of controller performance. Several nonlinear systems are analyzed based on the closed loop time response under the action of a reference step input signal. Three alternative performance indices, based on the time response, Fourier analysis, and mutual information, are tested. The numerical experiments demonstrate the feasibility of the proposed methodology and motivate its extension for other performance measures and new classes of nonlinearities.

References

References
1.
Atherton
,
D. P.
,
1975
,
Nonlinear Control Engineering
,
van Nostrand Reinhold
,
London
.
2.
J. E.
Slotine
, and
Li
,
W.
,
1991
,
Applied Nonlinear Control
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
3.
Cox
,
C. S.
,
1987
, “
Algorithms for Limit Cycle Prediction: A Tutorial Paper
,”
Int. J. Electr. Eng. Ed.
,
24
(
2
), pp.
165
182
.
4.
Fernando
B.
, and
Duarte
,
J. T. M.
,
2009
, “
Fractional Describing Function of Systems With Coulomb Friction
,”
Nonlinear Dyn.
,
56
(
4
), pp.
381
387
.10.1007/s11071-008-9405-8
5.
Fernando
B.
, and
Duarte
,
J. T. M.
,
2009
, “
Describing Function of Two Masses With Backlash
,”
Nonlinear Dyn.
,
56
(
4
), pp.
409
413
.10.1007/s11071-008-9410-y
6.
Gelb
,
A.
, and
Vander Velde
,
W. E.
,
1968
,
Multiple-Input Describing Functions and Non-Linear System Design
,
McGraw-Hill
,
New York
.
7.
Charles
,
L.
, and
Phillips
,
R. D. H.
,
1996
,
Feedback Control Systems
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
8.
Torgerson
,
W. S.
,
1958
,
Theory and Methods of Scaling
,
Wiley
,
New York
.
9.
Shepard
,
R. N.
,
1962
, “
The Analysis of Proximities: Multidimensional Scaling With an Unknown Distance Function
,”
Psychometrika
,
27
(
I/II
), pp.
219
246
.10.1007/BF02289621
10.
Kruskal
,
J.
,
1964
, “
Multidimensional Scaling by Optimizing Goodness of Fit to a Nonmetric Hypothesis
,”
Psychometrika
,
29
(
1
), pp.
1
27
.10.1007/BF02289565
11.
Kruskal
,
J. B.
, and
Wish
,
M.
,
1978
,
Multidimensional Scaling
,
Sage Publications
,
Newbury Park
.
12.
Cox
,
T. F.
, and
Cox
,
M. A. A.
,
2001
,
Multidimensional Scaling
,
Chapman and Hall
,
London
.
13.
Borg
,
I.
, and
Groenen
,
P. J.
,
2005
,
Modern Multidimensional Scaling-Theory and Applications
,
Springer-Verlag
,
New York
.
14.
Martinez
,
W. L.
, and
Martinez
,
A. R.
,
2005
,
Exploratory Data Analysis With MATLAB
,
Chapman and Hall
,
London
.
15.
de Leeuw
,
J.
, and
Mair
,
P.
,
2009
, “
Multidimensional Scaling Using Majorization: Smacof in R
,”
J. Stat. Software
,
31
(
3
), pp.
1
30
.
16.
Machado
,
J. A. T.
, and
Galhano
,
A. M. S.
,
2009
, “
Approximating Fractional Derivatives in the Perspective of System Control
,”
Nonlinear Dyn.
,
56
(
4
), pp.
401
407
.10.1007/s11071-008-9409-4
17.
Machado
,
J.
, and
Baleanu
,
D.
,
2011
, “
Characterization Approach to Modified Glassy Carbon Electrode-Nanofilm System Within Multidimensional Scaling
,”
J. Comput. Theor. Nanosci.
,
8
(
2
), pp.
1
6
.10.1166/jctn.2011.1649
18.
Tenreiro
Machado
,
J. A.
, and
Gonçalo
M.
Duarte
,
F. B. D.
,
2011
, “
Identifying Economic Periods and Crisis With the Multidimensional Scaling
,”
Nonlinear Dyn.
,
63
(
4
), pp.
611
622
.10.1007/s11071-010-9823-2
19.
Chang
,
C.-H.
, and
Chang
,
M.-K.
,
1994
, “
Analysis of Gain Margins and Phase Margins of a Nonlinear Reactor Control System
,”
IEEE Trans. Nucl. Sci.
,
41
(
4
), pp.
1686
1691
.10.1109/23.322778
20.
Wu
,
B. F.
,
Perng
,
J.-W.
, and
Chin
,
H.-I.
,
2005
, “
Limit Cycle Analysis of Nonlinear Sampled-Data Systems by Gainphase Margin Approach
,”
J. Franklin Inst.
,
342
(
2
), pp.
175
192
.10.1016/j.jfranklin.2004.10.004
21.
Cha
,
S.-H.
,
2008
, “
Taxonomy of Nominal Type Histogram Distance Measures
,”
Proc. of the American Conference on Applied Mathematics
.
22.
Shannon
,
C. E.
,
1948
, “
A Mathematical Theory of Communication
,”
Bell Syst. Tech. J.
,
27
(
7/10
), pp.
379
423
;623–656.
23.
Jaynes
,
E. T.
,
1957
, “
Information Theory and Statistical Mechanics
,”
Phys. Rev.
,
106
(
4
), pp.
620
630
.10.1103/PhysRev.106.620
24.
Khinchin
,
A. I.
,
1957
,
Mathematical Foundations of Information Theory
,
Dover
,
New York
.
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