AN ALGORITHM FOR COMPUTING BOUNDARY POINTS OF REACHABLE SETS OF CONTROL SYSTEMS UNDER INTEGRAL CONSTRAINTS

Mikhail I. Gusev     (N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of Russian Academy of Sciences, Ekaterinburg, Russian Federation)

Abstract


In this paper we consider a reachability problem for a nonlinear affine-control system  with  integral constraints, which assumed to be quadratic in the control variables.  Under  controllability assumptions it was  proved [8] that any admissible control, that steers the control system to the boundary of its reachable set, is a local solution to an optimal control problem with an integral cost functional and terminal constraints. This results in the Pontriagyn maximum principle for boundary trajectories. We propose here an numerical algorithm for computing the reachable set boundary  based on the maximum principle and provide some numerical examples.

Keywords


Optimal control, Reachable set, Integral constraints, Boundary points, Pontriagyn maximum principle

Full Text:

PDF

References


Anan'ev B.I. Motion correction of a statistically uncertain system under communication constraints // Automation and Remote Control, 2010. Vol. 71, no. 3. P. 367–378. DOI: 10.1134/S0081543810060039

Baier R., Gerdts M., Xausa I. Approximation of reachable sets using optimal control algorithms // Numerical Algebra, Control and Optimization, 2013. Vol. 3, no. 3. P. 519–548. DOI:10.3934/naco.2013.3.519

Donchev A. The Graves theorem revisited // Journal of Convex Analysis, 1996. Vol. 3, no. 1, P. 45–53. PDF

Filippova T.F., Matviichuk O.G. Algorithms to estimate the reachability sets of the pulse controlled systems with ellipsoidal phase constraints // Automation and Remote Control, 2011. Vol. 72, no. 9. P. 1911–1924. DOI: 10.1134/S000511791109013X

Guseinov K.G., Ozer O., Akyar E., Ushakov V.N. The approximation of reachable sets of control systems with integral constraint on controls // Nonlinear Differential Equations and Applications, 2007. Vol. 14, no. 1–2. P. 57–73. DOI: 10.1007/s00030-006-4036-6

Guseinov Kh.G., Nazlipinar A.S. Attainable sets of the control system with limited resources // Trudy Inst. Mat. i Mekh. Uro RAN, 2010. Vol. 16, no. 5. P. 261–268.

Gusev M.I. Internal approximations of reachable sets of control systems with state constraints // Proc. Steklov Inst. Math., 2014. Vol. 287, Suppl. 1. P. 77–92. DOI: 10.1134/S0081543814090089

Gusev M.I., Zykov I.V. On extremal properties for boundary points of reachable sets under integral constraints on the control // Trudy Inst. Mat. Mekh. UrO RAN, 2017. Vol. 23, no. 1. P. 103–115. (in Russian) DOI: 10.21538/0134-4889-2017-23-1-103-115

Kostousova E.K. On the boundedness of outer polyhedral estimates for reachable sets of linear systems // Comput. Math. And Math. Phys., 2008. Vol. 48. P. 918–932. DOI: 10.1134/S0965542508060043

Kurzhanski A.B., Varaiya P. Dynamic optimization for reachability problems // J. Optim. Theory Appl., 2001. Vol. 108, no. 2. P. 227–251. DOI: 10.1023/A:1026497115405

Kurzhanski A.B., Varaiya P. On ellipsoidal techniques for reachability analysis. Part I. External approximations // Optim. Methods Software, 2002. Vol. 17, no. 2. P. 177–206. DOI: 10.1080/1055678021000012426

Kurzhanski A.B., Varaiya P. Dynamics and control of trajectory tubes. Theory and computation. Basel, 2014. DOI: 10.1007/978-3-319-10277-1

Lee E.B., Marcus L. Foundations of optimal control theory. NY–London–Sydney: John Willey and Sons, Inc., 1967.

Lempio F., Veliov V.M. Discrete approximations of differential inclusions // GAMM Mitt. Ges. Angew. Math. Mech., 1998. Vol. 21. P. 103–135.

Patsko V.S., Pyatko S.G., Fedotov A.A. Three-dimensional reachability set for a nonlinear control system // J. Comput. Syst. Sci. Int., 2003. Vol. 42, no. 3. P. 320–328.

Polyak B.T. Convexity of the reachable set of nonlinear systems under \(L_2\) bounded controls // Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis, 2004. Vol. 11. P. 255–267. PDF

Sinyakov V.V. Method for computing exterior and interior approximations to the reachability sets of bilinear differential systems // Differential Equations, 2015. Vol. 51, no. 8. P. 1097–1111. DOI: 10.1134/S0012266115080145




DOI: http://dx.doi.org/10.15826/umj.2017.1.003

Article Metrics

Metrics Loading ...

Refbacks

  • There are currently no refbacks.