A NUMERICAL METHOD FOR SOLVING LINEAR–QUADRATIC CONTROL PROBLEMS WITH CONSTRAINTS

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

Abstract


The paper is devoted to the optimal control problem for a linear system with integrally constrained control function. We study the problem of minimization of a linear terminal cost with terminal constraints given by a set of linear inequalities. For the solution of this problem we propose two-stage numerical algorithm, which is based on construction of the reachable set of the system. At the first stage we find a solution to finite–dimensional optimization problem with a linear objective function and linear and quadratic constraints. At the second stage we solve a standard linear–quadratic control problem, which admits a simple and effective solution.


Keywords


Optimal control; Reachable set; Integral constraints; Convex programming; Semi-infinite linear programming.

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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

Antipin A.S., Khoroshilova E.V. Linear programming and dynamics // Trudy Inst. Mat. i Mekh. UrO RAN. 2013. Vol. 19, no. 2. P. 7–25. [in Russian]

Antipin A.S., Khoroshilova E.V. Linear programming and dynamics // Ural Mathematical Journal, 2015. Vol. 1, no. 1. P. 3–19. DOI: 10.15826/umj.2015.1.001

Arutyunov A.V., Magaril-Il’yaev G.G. and Tikhomirov V.M. Pontryagin maximum principle. Proof and applications. Moscow: Factorial press, 2006. 124 p. [in Russian]

Dar’in A.N., Kurzhanski A.B. Control under indeterminacy and double constraints // Differential Equations. 2003. Vol. 39, no. 11. P. 1554–1567.

Goberna M.A.,Lopez M.A.Linear semi-infinite programming theory: An updated survey // European J. of Operational Research. 2002. Vol. 143, Issue 2. P. 390–405.

Gusev M.I. On optimal control problem for the bundle of trajectories of uncertain system // Lecture Notes in Computer Sciences. Springer. LNCS 5910. 2010. P. 286–293. DOI: 10.1007/978-3-642-12535-5_33

Krasovskii N.N. Theory of Control of Motion. Moscow: Nauka, 1968. 476 p. [in Russian]

Kurzhanski A.B. Control and Observation under Conditions of Uncertainty. Moscow: Nauka, 1977. 392 p. [in Russian]

Lee E.B., Marcus L. Foundations of Optimal Control Theory. Jhon Willey and Sons, Inc., 1967. 124 p.

Martein L., Schaible S. On solving a linear program with one quadratic constraint // Rivista di matematica per le scienze economiche e sociali. 1988. P. 75–90.

Ushakov V.N. Extremal strategies in differential games with integral constraints // J. of Applied Mathematics and Mechanics. 1972. Vol. 36, no. 1. P. 15–23.

Ukhobotov V.I. On a class of differential games with an integral constraint // J. of Applied Mathematics and Mechanics. 1977. Vol. 41, no. 5. P. 838–844.




DOI: http://dx.doi.org/10.15826/umj.2016.2.009

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