Abstract
The discrete-continuous systems are considered as the systems with impulsive inputs, which cause the discontinuous behaviour of the system paths. The robustness for nonlinear discrete-continuous systems means the stability of the system path with respect to the approximation of pure impulsive inputs by ordinary ones. Necessary and sufficient conditions of this type of robustness give the opportunity to extract rather narrow class of robust systems. Although robust systems look like very attarctive from theoretical point of view and have many of usefull features, the great part of dynamical system which are of a practical interest are non-robust, and some nontraditional theoretical tools are necessary for their treatment. In this paper the problem of description and path representation in the form of differential equation with a measure is considered. Some specific features of non-robust systems are discussed from the point of controllability. It was shown that non-robust systems can provide the additional controllability opportunities by using the impulsive inputs.
This work was supported in part by National Science Foundation of USA grant CMS 94-1447s and International Association for the Promotion of Cooperation with Scientists from the Independent States of the Former Soviet Union (INTAS) grants 94-697 and 93-2622
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References
A. Bensoussan, S. Sethy, R. Vickson and N. Derzko, “Stochastic production planning with production constraints,” SIAM J. Control Optim., 22, 920–935 (1984).
H. Cartan, Calcul Différentiel. Formses Différentielles, Hermann, Paris (1967).
F. N. Grigor'ev, N. A. Kuznetsov, and A. P. Serebrovskii, The Control of Observations in Automatic Systems [in Russian], Nauka, Moscow (1986).
V. K. Gorbunov and G. U. Nurakhunova, “Processes with controlled discontinuities of phase trajectories and simulation of production with moving basic funds,” Izv. Akad. Nauk SSSR, Tekh. Kibern., No. 6, 55–61 (1975).
V. I. Gurman, “Optimal processes with unbounded derivatives,” Avtomat. Telemekh., No. 12, 14–21 (1972).
V. K. Ivanov, B. M. Miller, P. I. Kitsul, and A. M. Petrovskii, “A mathematical model of control of the healing of an organism damaged by a malignant growth,” in: Biological Aspects of Control Theory [in Russian], Inst. Control Sciences, Moscow, No. 8 (1976), pp. 15–22.
A. N. Kolmogorov and S.V. Fomin, Elements of Theory of Function and Functional Analysis [in Russian], Nauka, Moscow, 1976.
M. A. Krasnosel'skii and A. V. Pokrovskii, “Vibrostable differential equations with continuous right-hand side,” Tr. Mosk. Mat. Obshch., 27, 93–112 (1972).
M. A. Krasnosel'skii and A. V. Pokrovskii, Systems with Hysteresis [in Russian], Nauka, Moscow (1983).
H. J. Kushner, “On the optimal timing of observation for linear control systems with unknown initial states,” IEEE Trans. Automat. Control, AC-9, 144–145 (1964).
D. Lawden, Optimal Trajectories for Space navigation Butterworth, London, 1963.
E. B. Lee and L. Markus, Foundations of Optimal Control Theory John Wiley and Sons, Inc., New York, London, Sydney, 1967.
B. M. Miller, “Stability of solutions of ordinary differential equations with measure,” Uspekhi Mat. Nauk, No. 2, 198 (1978).
B. M. Miller, “The nonlinear sampled-data control problem for systems described by ordinary measure differential equations I, II,” Automat. Remote Control, 39, No. 1, 57–67; No. 3, 338–344 (1978).
B. M. Miller, “Optimization of dynamic systems with a generalized control,” Automat. Remote Control, 50, No. 6, 733–742 (1989).
B. M. Miller, “Generalized optimization in problems of observation control,” Automat. Remote Control, 52, No. 10, 83–92 (1991).
B. M. Miller., “The generalized solutions of nonlinear optimization problems with impulse controls,” SIAM J. Control Optim., 34, No. 4, 1420–1440 (1996)
L. Neustadt, “A general theory of minimum-fuel space trajectories,” J. SIAM. Ser. A, Control, 3, No. 2, 317–356 (1965).
Yu. V. Orlov, Theory of Optimal Systems with Generalized Controls, [in Russian], Nauka, Moscow (1988).
L. C. Young, Lectures on Variational Calculus and the Theory of Optimal Control, W. B. Saunders Company, Philadelphia, Londod, Toronto, 1969.
S. T. Zavalishchin and A. N. Sesekin, Impulsive Processes. Models and Applications [in Russian], Nauka, Moscow (1991).
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© 1997 Springer-Verlag Berlin Heidelberg
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Miller, B.M. (1997). Representation of robust and non-robust solutions of nonlinear discrete-continuous systems. In: Maler, O. (eds) Hybrid and Real-Time Systems. HART 1997. Lecture Notes in Computer Science, vol 1201. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0014728
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DOI: https://doi.org/10.1007/BFb0014728
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