Response speed optimal air-balloon movement control

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 paper presents the algorithm of response speed optimum position control in the form of an air-balloon movement control problem. 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, which 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 gain momentum, and influence of the running engine center of gravity is neglected, The atmosphere is considered to be stationary and homogeneous. This model is described by linear discrete-time system with polyhedron set of the feasible controls.

The subject of discussion is problem of zero-controllability of linear discrete-time system with the bounded set of feasible controls, i. e. the problem of transfer the given system to zero by means of feasible control in minimum number of steps. Specificity of zero-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 solver of the problem with dynamic programming and lack of analogue of Belman’s equation, incorrectness of maximum principle for minimum time problem in discrete-time case, non-uniqueness of optimal solution.

The solution of the problem is based on the conception of a zero-controllability set in N steps, i. e. the set of points from which the system can be transferred to its origin by no more than in N steps, using a certain number of control actions. It is proven, that in this case the zero-controllability set is a convex compact polyhedron that can be represented as the algebraic sum of other polyhedrons. The algorithm of constructing faces of a zero-controllability set in any N steps is developed. The method of computation of minimum time for reaching the origin from any initial state based on exact description of zero-controllability sets is constructed. While the optimal feedback control has explicit form if description of zero-controllability sets is available for any step.

At last, the considered problem of the construction of the minimum time air-balloon movement control is completely solved by the investigated methods.

Keywords:

linear discrete-time control system, minimum optimal response time control, polyhedron, linear programming

References

1. Ibragimov D.N., Sirotin A.N. Avtomatika i telemekhanika. 2015. no.8. pp. 100-121.

2. Kwakernaak H., Sivan R. Lineynie optimal’nie sistemy upravlenia (Linear optimal control systems), Moscow, Mir, 1977, 653 p.

3. Boltianskiy V.G., Optimal’noe upravlenie diskretnymi sistemami (Optimal control of discrete-time systems), Moscow, Nauka, 1973, 448 p.

4. Tabak D., Kuo B. Optimal’noe upravlenie i matematicheskoe programmirovanie (Optimal control and mathmatical programming), Moscow, Nauka, 1975, 280 p.

5. Propoy A.I. Elementy teorii optimal’nykh diskretnykh processov (Elements of theory of optimal discrete-time processes), Moscow, Nauka, 1973, 256 p.

6. Combastel C., Raka S.A. On Computing Envelopes for Discrete-time Linear Systems with Affine Parametric Uncertainties and Bounded Inputs // Preprints 18 IFAC World Congr. Milano (Italy) August 28 — September 2, 2011. P. 4525-4533.

7. Kurzhanskiy A.F., Varaiya P. Theory and computational techniques for analysis of discrete-time control systems with disturbancens. Optim. Method Software, 2011, V.26. No.4-5. P.719-746.

8. Keerthi S.S., Gilbert E.G. Computatuon of minimum-time feedback control laws for discrete-time cyctems with state-control // IEEE Transaction Automat. Control. 1987. V.32. No.5. P.432-434.

9. Lin W.-S. Time-optimal control strategy for saturating linear discrete systems // Int. J. Control. 1986. V.43. No.5. P.1343-1351.

10. Eldem V., Selbuz H. On the general solytion of the state deadbeat control problem // IEEE Transaction Automat. Control. 1994. V.39. No.5. P.1002-1006.

11. Benvenuti L., Farina L. The geometry of the reachability set for linear Discrete-time systems with positive controls // SIAM J. Matrix Anal. Appl. 2006. V.28. No.2. P.306—325.

12. de Leon-Canion P., Lunze J. Dependable control of uncertain linear systems based on set-theoretic methods. Int. J. Control. 2010. V.83. No.6. P.1248-1264.

13. Fisher M.E., Gayek J.E. Estimating Reachable Sets for Two-Dimensional Linear Discrete Systems. J. Optim. Theory Appl. 1988. V.56. No.1. P.67-88.

14. Kostousova E. K., Vychislitel’nye Tekhnologii, 2004, V.9, no.4, pp.54-72.

15. Kostousova E. K. External Polyhedral Estimates For Reachable Sets Of Linear Discrete-Time Systems with Integral Bounds On Controls. Int. J. Pure Appl Math. 2009. V.50. No.2. P.187-194.

16. Ding M.-F., Liu Y., Gear J. A. A Modified Centered Climbing Algorithm for Linear Programming. Appl. Math. 2012. V.3. P.1423-1429.

17. Telgen J. Minimal representation of convex polyhedral sets. J. Optim. Theory Appl. 1982. V.38. No.1. P.1-24.

18. Ivanov D.S., Ovchinnikov M.U., Tkachev S.S., Izv. RAN. Teoria I sistemy upravlenia, 2011, no.1, pp.107-119.

Download