Abstract
The purpose of this paper is to investigate the optimal control of a class of discrete, time-invariant, variable-structure systems. Both deterministic and stochastic problems are considered for unbounded control and the cost functional quadratic in state. Solutions are obtained in closed-form.
In the deterministic case, it is seen that the regular path and singular paths satisfying the functional equation of dynamic programming may exist simultaneously. It is shown then that the regular path is optimal. Both additive and multiplicative plant noise are considered in the stochastic problem. It is shown that the presence of noise considerably simplifies the analysis since the cases with singularities are contained in sets of measure zero.
This research was supported in part by the National Science Foundation Grant GK-36531 and GK-22905A#2.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Swamy, K.N., “Optimal Control of Single-Input Discrete Bilinear Systems,” D.Sc. Dissertation, Washington University, St. Louis, Mo., December 1973.
Swamy, K.N., and Tarn, T.J., “Deterministic Optimal Control of Single-Input Discrete Bilinear Systems,” submitted for publication.
Swamy, K.N., and Tarn, T.J., “Stochastic Optimal Control of Single-Input Discrete Bilinear Systems,” submitted for publication.
Swamy, K.N., and Tarn, T.J., “Optimal Control of Discrete Bilinear Systems,” in “Proceedings of NATO Advanced Study Institute on Geometric and Algebraic Methods for Nonlinear Systems,” Aug.-Sept. 1973, Imperial College, London.
Mohler, R.R., and Ruberti, A., ed., “Theory and Applications of Variable Structure Systems,” Academic Press, New York, 1972.
Goka, T., Tarn, T.J., and Zaborszky, J., “On the Controllability of a Class of Discrete Bilinear Systems,” Automatica, Vol. 9, No. 5, September 1973, pp. 615–622.
Tarn, T.J., Elliott, D.L., and Goka, T., “Controllability of Discrete Bilinear Systems with Bounded Control,” IEEE Transactions on Automatic Control, Vol. AC-18, June 1973, pp. 298–301.
Rink, R.E., and Mohler, R.R., “Completely Controllable Bilinear Systems,” SIAM J. Contr., Vol. 6, No. 3, pp. 477–486, 1968.
d’Alessandro, P., “Structural Properties of Bilinear Discrete-Time Systems,” Ric. Automatica, Vol. 3, Aug. 1972.
Bruni, C., Di Pillo, G., and Koch, G., “Mathematical Models and Identification of Bilinear Systems,” in “Theory and Applications of Variable Structure Systems” (Mohler, R.R. and Ruberti, A., ed.), Academic Press, New York, 1972, pp. 137–152.
Brockett, R.W., “On the Algebraic Structure of Bilinear Systems,” in “Theory and Applications of Variable Structure Systems” (Mohler, R.R., and Ruberti, A., ed.), Academic Press, New York, 1972, pp. 153–168.
Bellman, R., “Adaptive Control Processes: A Guided Tour,” Princeton University Press, Princeton, 1961, p. 56.
Jacobson, D.H., and Mayne, D.Q., “Differential Dynamic Programming,” American Elsevier Publishing Company, Inc., New York, 1970, p. 136.
Graybill, F.A., “Introduction to Matrices with Applications in Statistics,” Wadsworth Publishing Company, Belmont, Calif., 1969, p. 231.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1974 Plenum Press, New York
About this chapter
Cite this chapter
Gupta, S.K., Swamy, K.N., Tarn, TJ., Zaborszky, J. (1974). On a Class of Variable-Structure Systems. In: Fu, K.S., Tou, J.T. (eds) Learning Systems and Intelligent Robots. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-2106-4_11
Download citation
DOI: https://doi.org/10.1007/978-1-4684-2106-4_11
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4684-2108-8
Online ISBN: 978-1-4684-2106-4
eBook Packages: Springer Book Archive