Approximation of the feasible control set of discrete-time system for the minimum-time problem

Аuthors

Ibragimov D. N.

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

e-mail: rikk.dan@gmail.com

Abstract

The minimum-time problem of the linear discrete-time system with convex set of the feasible controls is considered in this paper, i.e. problem of transferred given system to 0 by means of feasible control in minimum number of steps. Specifity of 0-controllability problem for linear discrete-time systems is related to the difficulty of using conventional methods. These features is characterized by lack of explicit form of the solve of the problem of dynamic programming and lack of analogue of Belman’s equation, incorrectness of maximum principle for minimum time problem in discrete-time case, nonuniqueness of optimal solution.

The algorithm of the construction an optimal solution has been developed in previous works. But this method can be applied only to the systems with linear constraints of the feasible controls set. The main improvement of the new algorithm is its workability for any systems with the convex constraints on the controls. This fact is reached by using polyhedron approximation instead the real set of the feasible controls. The theorem of the convergence of the sequence of polyhedrons to the compact convex set, which is approximated, is formulated and proved in this paper.

Also the method of the estimating of the accuracy of approximation is represented in this paper. It is based on the calculating of Hausdorff distance between two polyhedrons. This problem is reduced to projecting exterior points of the first polyhedron to the faces of the second polyhedron, i.e. this is the quadratic programming problem with the linear constraints.

The designed algorithm is tested on the problem of the minimum time orientation of the aerostat. As a mathematical model, a solid body suspended on a cord and able to perform rotational movements is used. The upper end of the cord is fixedly mounted and operated with two fan motors, generating the opposite control moments about the vertical axis. The body is subjected to the torque is linearly dependent on the angle of rotation caused by an elastic cord and the moment of viscous friction linearly dependent on the angular velocity. It is assumed that the engines instantly gaining momentum, influence on the running engine center of gravity is neglected, and the atmosphere is considered to be stationary and homogeneous. This model is described by linear discrete-time system with convex set of the feasible controls.

Keywords:

linear discrete-time system, minimum-time problem, polyhedron, Hausdorff distance

References

1. Li Je.B., Markus L. Osnovy teorii optimal’nogo upravleniya (Base of the theory of the optimal control), Moscow, Nauka, 1972, 576 p.

2. Alekseev V.M., Tihomirov V.M., Fomin S.V. Optimal’noe upravlenie (Optimal control), Moscow, Nauka, 430 p.

3. Pontrjagin L.S., Boltjanskij V.G., Gamkrelidze R.V., Mishhenko B.F. Matematicheskaya teoriya optimal’nykh protsessov (Mathematical theory of the optimal processes), Moscow, Nauka, 1969, 393 p.

4. Bellman R. Dinamicheskoe programmirovanie (Dynamic programming), Moscow, Inostrannaya Literatura, 1960, 400 p.

5. Boltjanskij V.G. Matematicheskie metody optimal’nogo upravleniya (Mathematical methods of the optimal control), Moscow, Nauka, 1969, 408 p.

6. Ibragimov D.N., Sirotin A.N. Avtomatika i Telemekhanika, 2015, no.9, pp. 3-30.

7. Ibragimov D.N. Zhurnal «Trudy MAI», 2015, no. 83: http://www.mai.ru/science/trudy/published.php?ID=62313

8. Kolmogorov A.N., Fomin S.V. Elementy teorii funktsii i funktsional’nogo analiza (Elements of the theory of functions and functional analysis), Moscoiw, FIZMATLIT, 2012, 572 p.

9. Kronover R.M. Fraktaly i khaos v dinamicheskikh sistemakh (Fractals and chaos in the dynamic systems), Moscow, POSTMARKET, 2000, 352 p.

10. Polovinkin E.S., Balashov M.V. Elementy vypuklogo i sil’no vypuklogo analiza (Elements of the theory of convex and strong convex analysis), Moscow, FIZMATLIT, 2004, 416 p.

11. Ivanov D.S., Ovchinnikov M.Ju., Izvestiya RAN. Teoriya i sistemy upravleniya, 2011, no., pp. 107-119.

Download