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Singular perturbation methods for a class of initial and boundary value problems in multi-parameter classical digital control systems

Published online by Cambridge University Press:  17 February 2009

M. S. Krishnarayalu
M. S. Krishnarayalu
Affiliation:
1/19 Meadow Crescent, Meadowbank, NSW 2114, Australia; e-mail: [email protected].
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Abstract

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A stable linear time-invariant classical digital control system with several widely different small coefficients multiplying the lowest functions is considered. It is formulated as a multi-parameter singularly perturbed system. Perturbation methods are developed for both initial and boundary value problems based on asymptotic expansions of the perturbation parameters. The approximate solution consists of an outer solution and a number of boundary layer correction solutions equal to the number of initial conditions lost in the process of degeneration. An example is provided for illustration.

Type
Research Article
Information
The ANZIAM Journal , Volume 46 , Issue 1 , July 2004 , pp. 67 - 77
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Hoppensteadt, F., “Properties of solutions of ordinary differential equations with small parameters”, Comm. Pure Appl. Math. 24 (1971) 807–840.CrossRefGoogle Scholar
[2]Kailasa Rao, A. and Naidu, D. S., “Singular perturbation method applied to the open-loop discrete optimal control problem”, Opt. Control Appl. Methods 3 (1982) 121–131.Google Scholar
[3]Kando, H. and Iwazumi, T., “Design of observers of stabilizing feedback controllers for singularly perturbed discrete systems”, Proc. IEE D 132 (1985) 1–10.CrossRefGoogle Scholar
[4]Kokotovic, P. V., O'Malley, R. E. Jr and Sannuti, P., “Singular perturbations and order reduction”, Automatica J. IFAC 12 (1976) 123–132.CrossRefGoogle Scholar
[5]Krishnarayalu, M. S., “Singular perturbation methods for one-point, two-point and multi-point boundary value problems in multi-parameter digital control systems”, J. EEE Australia 19 (1999) 97–110.Google Scholar
[6]Krishnarayalu, M. S., “Singular perturbation analysis of a class of initial and boundary value problems in multi-parameter digital control systems”, Control Theory Adv. Technology 10 (1994) 465–477.Google Scholar
[7]Krishnarayalu, M. S., “Singular perturbation method applied to the closed-loop discrete optimal control problem”, Optimal Control Appl. Methods 11 (1990) 75–83.Google Scholar
[8]Krishnarayalu, M. S., “Singular perturbation method applied to the open-loop discrete optimal control problem with two small parameters”, Int. J. Systems Sci. 20 (1989) 793–809.CrossRefGoogle Scholar
[9]Krishnarayalu, M. S. and Naidu, D. S., “Singular perturbation method for boundary value problems in two-parameter discrete control systems”, Int. J. Systems Sci. 19 (1988) 2131–2143.CrossRefGoogle Scholar
[10]Krishnarayalu, M. S. and Naidu, D. S., “Singular perturbation method for initial value problems in two-parameter discrete control systems”, Int. J. Systems Sci. 18 (1987) 2197–2208.Google Scholar
[11]Naidu, D. S., “Singular perturbations and time scales in control theory and applications: An overview”, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 9 (2002) 233–278.Google Scholar
[12]Naidu, D. S. and Kailasa Rao, A., “Singular perturbation method for initial-value problems with inputs in discrete control systems”, Int. J. Control 33 (1981) 953–965.CrossRefGoogle Scholar
[13]Naidu, D. S. and Price, D. B., “Singular perturbations and time scales in the design of digital flight control systems”, NASA Technical paper 2844, 1988.CrossRefGoogle Scholar
[14]O'Malley, R. E. Jr, Introduction to singular perturbations (Academic Press, New York, 1974).Google Scholar
[15]Rajagopalan, P. K. and Naidu, D. S., “A singular perturbation method for discrete control systems”, Int. J. Control 32 (1980) 925–936.Google Scholar
[16]Roberts, S. M. and Shipman, J. S., Two-point boundary value problems (Elsevier, New York, 1972).Google Scholar
[17]Sage, A. P. and White, G. C. III, Optimum system control (Prentice Hall, Englewood Cliffs, 1977).Google Scholar
[18]Saksena, V. R., O'Reilly, J. and Kokotovic, P. V., “Singular perturbations and time-scale methods”, Automatica J. IFAC 20 (1984) 273–293.Google Scholar
[19]Syrcos, G. P. and Sannuti, P., “Singular perturbation modelling of continuous and discrete physical systems”, Int. J. Control 37 (1983) 1007–1022.CrossRefGoogle Scholar