ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2014. Vol. 8

An improvement method for hierarchical model with network structure

V. I. Gurman, I. V. Rasina, O. V. Fesko, O. V. Usenko

The systems of heterogeneous structure are widespread in practice, currently such systems are the subject of intense study by the representatives of different scientific schools and directions. These systems include systems with variable structure, discrete-continuous, logic-dynamic, hybrid and heterogeneous dynamic systems. In this article the systems of heterogeneous network structure are considered. For modelling and research the hierarchical approach is used: two-level model is created, the lower the level of which presents different controlled differential systems of homogeneous structure and the upper — network of operators, providing purposeful interaction of continuous subsystems. This model can be seen as a further development of the discrete-continuous model, proposed and investigated in a number of works of the authors. The optimal control problem is formulated, the sufficient conditions of optimality are derived — analogues of known the Krotov’s sufficient conditions of optimality, which involve resolving functions type of Krotov for each level. On the basis of these conditions and the localization principle a method of monotone iterative improvements with linear with respect to the state of the Krotov-type functions is constructed. The involvement of the second derivatives on control variables in its structure allows to take into account ravine surface structure of functional. The method like the model has a two-level structure. On the lower level appears traditional conjugated system of differential equations for the coefficients of resolving functions, whereas on the upper level, conjugated variables are determined from the linear algebraic system of equations. As an example it is considered the optimization of water protection measures in the river basin for a simplified model with an operator tree. The prototype is the lower flows of the Selenga river. For this problem a two-level network model is built and the proposed algorithm is applied. The results of calculations are represented.

control improvement, hierarchical model, network of operators

1. Anohin Ju.A., Gorstko A.B. et al. Matematicheskie modeli i metody upravlenija krupnomasshtabnym vodnym ob’ektom (in Russian). Novosibirsk, Nauka, 1987.

2. Bortakovskij A.S. Dostatochnye uslovija optimal’nosti upravlenija determinirovannymi logiko-dinamicheskimi sistemami (in Russian). Informatika. Ser. Avtomatizacija proektirovanija, 1992, vol. 2–3, pp. 72–79.

3. Vasil’ev S.N. Teorija i primenenie logiko-upravljaemyh system (in Russian). Trudy 2-oj Mezhdunarodnoj konferencii «Identifikacija sistem i zadachi upravlenija» (SICPRO’03)», 2003, pp. 23–52.

4. Gurman V.I. Optimizacija diskretnyh sistem: uchebnoe posobie (in Russian). Irkutsk, Izd-vo Irkut. un-ta, 1976, 121 p.

5. Gurman V.I., Rasina I.V. Sufficient optimality conditions in hierarchical models of nonuniform systems (in Russian). Avtomat. i telemeh., 2013, no. 12, pp. 15–30.

6. Gurman V.I. K teorii optimal’nyh diskretnyh processov (in Russian). Avtomat. i telemeh., 1973, no. 6, pp. 53–58.

7. Gurman V.I., Rasina I.V. O prakticheskih prilozhenijah dostatochnyh uslovij sil’nogo otnositel’nogo minimuma (in Russian). Avtomat. i telemeh., 1979, no. 10, pp. 12–18.

8. Gurman V.I, Rasina I.V. Discrete-continuous representations of impulsive processes in the controllable systems (in Russian). Avtomat. i telemeh., 2012, no. 8, pp. 16–29.

9. Gurman V.I., Fesko O.V., Rasina I.V. Modelirovanie vodoohrannyh meroprijatij v bassejne reki (in Russian). Vestnik BGU. Matematika, informatika, 2013, no. 3, pp. 4–15.

10. Emel’janov S.V. Teorija sistem s peremennoj strukturoj (in Russian). Moscow, Nauka, 1970.

11. Krotov V.F., Gurman V.I. Metody i zadachi optimal’nogo upravlenija [Methods and problems of optimal control]. Moscow, Nauka, 1973, 446 p.

12. Miller B.M., Rubinovich E. Ja. Optimizacija dinamicheskih sistem s impul’snymi upravlenijami [Optimization of dynamic systems with pulse control]. Moscow, Nauka, 2005, 429 p.

13. Rasina I.V. Iterative optimization algorithms for discrete-continuous processes (in Russian). Avtomat. i telemeh., 2012, no. 10, pp. 3–17.

14. Lygeros J. Lecture Notes on Hyrid Systems. University of Cambridge, Cambridge, 2003.

15. Van der Shaft A.J., Schumacher H. An Introduction to Hybrid Dynamical Systems. Springer-Verlag, London Ltd, 2000.

Full text (russian)