Every optimization based engineering design produces a performance metric which as a function of the tolerance parameters, is structurally stable but not generic. This can lead to a severe loss of performance (catastrophes) under a smooth change in the tolerance parameters. This paper gives an exposition of how norm based optimal control methods naturally lead to performance catastrophes, as well as an alternative systematic loop shaping design methodology which corrects the problem.

1.
Arnold, V. I., 1981, Singularity Theory, Cambridge University Press, Cambridge.
2.
Arnold, V. I., Gusein-Zade, S. M., and Varchenko, A. N., 1985, Singularities of Differentiable Maps, Vol. 1, Birkhauser, Boston, MA.
3.
Arnold, V. I., 1986, Catastrophe Theory, Springer-Verlag, Berlin.
4.
Arnold, V. I., 1988, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, Berlin.
5.
Bauer
L.
,
Keller
H.
,
Reiss
E.
,
1975
, “
Multiple Eigenvalues Lead to Secondary Bifurcations
,”
SIAM J. Appl. Math.
, Vol.
17
, pp.
101
123
.
6.
Borisovich, Y., Bliznyakov, N., Israilevich, Y., and Fomenko, T., 1985, Introduction to Topology, MIR Publishers, Moscow.
7.
Bryzgalova
L. N.
,
1977
, “
Singularities of the Maximum of a Parametrically Dependent Function
,”
Funct. Analy. Applic.
, Vol.
11
, pp.
59
60
.
8.
Bryzgalova
L. N.
,
1978
, “
Maximum Functions of a Family of Functions Depending on Parameters
,”
Funct. Analy. Applic.
, Vol.
12
, pp.
66
67
.
9.
Gilmore, R., 1981, Catastrophe Theory for Scientists and Engineers, Wiley, New York.
10.
Horn, R. A., and Johnson, C. R., 1985, Matrix Analysis, Cambridge University Press, Cambridge.
11.
Horowitz, I. M., 1963, Synthesis of Feedback Systems, Academic Press, Orlando, FL.
12.
Horowitz
I. M.
,
1973
, “
Optimum Loop Transfer Function in Minimum Phase Feedback Systems
,”
Int. J. Control
, Vol.
18
, No.
1
, pp.
97
113
.
13.
Keel, L. H., and Bhattacharrya, S. P., 1997, “Robust, Fragile or Optimal?,” Proceedings, American Control Conference, Albuquerque, NM, June.
14.
Keener
J. P.
,
1973
, “
Perturbed Bifurcation Theory at Multiple Eigenvalues
,”
Acta. Rational Mech. Analy.
, Vol.
56
, pp.
384
391
.
15.
Kucera, V., 1974, Discrete Linear Control: The Polynomial Equation Approach, Wiley, New York.
16.
Lu, Y. C., 1976, Singularity Theory, and an Introduction to Catastrophe Theory, Springer-Verlag, Berlin.
17.
Matov
V. I.
,
1982
, “
The Topological Classification of Germs of the Maximum and Minimax Functions of a Family of Functions in General Position
,”
Russian Math. Surveys
, Vol.
4
, pp.
127
128
.
18.
Nordgren
R. E.
,
Franchek
M. A.
, and
Nwokah
O. D. I.
,
1995
, “
A Design Procedure for the Exact H SISO-Robust Performance Problem
,”
Int. J. Robust and Nonlinear Control
, Vol.
5
, pp.
107
118
.
19.
Nwokah, O. D. I., Ukpai, U. I., Gasteneau, Z., and Happawana, G. S., 1997, “Catastrophes in Modern Optimal Controllers,” Proceedings, American Control Conference, Albuquerque, NM, June.
20.
Nwokah, O. D. I., and Afolabi, D., 1997, “Erosion of Optimum Feedback Designs by Spectral Singularities,” Proceedings, American Control Conference, Albuquerque, NM, June.
21.
Ostrowski, A. S., 1973, Solutions of Equations in Euclidian and Banach Spaces, Academic Press, Orlando, FL.
22.
Poston, T., and Stewart, I., 1978, Catastrophe Theory and Its Applications, Pitman, London.
23.
Robinson, E. A., 1962, Random Wavelets and Cybernetic Systems, Haffner Publishers, New York.
24.
Suh, N. P., 1990, The Principles of Design, Oxford University Press, Oxford.
25.
Thom, R., 1975, Structural Stability and Morphogenesis, Benjamin-Addison Wesley, New York.
26.
Thompson, D. F., 1998, “Gain-Bandwidth Optimal Design for the New Formulation QFT,” ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL, Vol. 120, Sept.
27.
Thompson
J. M. T.
,
1973
, “
Erosion of Optimum Designs by Compound Branching Phenomena
,”
J. Mech. Phys. Solids
, Vol.
24
, pp.
135
141
.
28.
Thompson
J. M. T.
, and
Hunt
G. W.
,
1974
, “
Dangers of Structural Optimization
,”
Engrg. Optimization
, Vol.
1
, pp.
99
108
.
29.
Thompson
J. M. T.
, and
Hunt
G. W.
,
1977
, “
A Bifurcation Theory for the Instabilities of Optimization and Design
,”
Synthese
, Vol.
36
, pp.
315
323
.
30.
Tyurina
G. N.
,
1970
, “
Resolution of Singularities of Plane Deformations of Rational Double Points
,”
Funct. Analy. Applic.
, Vol.
41
, pp.
68
73
.
31.
Wookcock, A. E. R., and Poston, T., 1974, “A Geometrical Study of the Elementary Catastrophes,” Lecture Notes in Mathematics #373, Springer-Verlag, Berlin.
This content is only available via PDF.
You do not currently have access to this content.