Abstract
We formulate the optimal control problem for a class of nonlinear objects that can be represented as objects with linear structure and state-dependent coefficients. We assume that the system is subjected to uncontrollable bounded disturbances. The linear structure of the transformed nonlinear system and the quadratic quality functional let us, in the optimal control synthesis, to pass from Hamilton-Jacobi-Isaacs equations to a state-dependent Riccati equation. Control of nonlinear uncertain object in the problem of moving along a given trajectory is considered using the theory of differential games. We also give an example that illustrates how theoretical results of this work can be used.
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References
Person, J.D., Approximation Methods in Optimal Control, J. Electron. Control, 1962, no. 12, pp. 453–469.
Mrasek, C.P. and Clouter, J.R., Control Design for the Nonlinear Benchmark Problem via SDRE Method, Int. J. Robust Nonlin. Control, 1998, no. 8, pp. 401–433.
Menon, P.K. and Ohlmeyer, E.J., Integrated Design of Angel Missile Guidance and Control Systems, Proc. 17 Mediterran.Conf. Control Automat. (MED99), Haifa, Israel, June 28–30, 1999, pp. 1470–1494.
Mrasek, C.P., SDRE Autopilot for Dual Controlled Missiles, Proc. 17th IFAC Sympos. Automat. Control Aerospace, Toulouse, France, 2007.
Friedland, B., Quasi Optimal Control and the SDRE Method, Proc. 17th IFAC Sympos. Automat. Control Aerospace, Toulouse, France, 2007.
Salnci, M.U. and Gokbilen, B., SDRE Missile Autopilot Design Using Sliding Mode Control with Sliding Surfaces, Proc. 17th IFAC Sympos. Automat. Control Aerospace, Toulouse, France, 2007.
Cimen, T., On the Existence of Solutions Characterized by Riccati Equations to Infinite-Time Horizon Nonlinear Optimal Control Problems, Proc. 18th World Conf. IFAC, Milano (Italy), 28.08–2.09, 2011, pp. 9620–9626.
Ruderman, M., Weigel, D., Hoffmann, F., and Bertram, T., Extended SDRE Control of 1-DOF Robotic Manipulator with Nonlinearities, Proc. 18th World Conf. IFAC, Milano (Italy), 28.08–2.09, 2011, pp. 10940–10945.
Athans, M. and Falb, P.L., Optimal Controls, New York: McGraw-Hill, 1966. Translated under the title Optimal’noe upravlenie, Moscow: Mashinostroenie, 1968.
Afanas’ev, V.N., Kolmanovskii, V.B., and Nosov, V.R., Matematicheskaya teoriya konstruirovaniya sistem upravleniya (Mathematical Theory for Constructing Control Systems), Moscow: Vysshaya Shkola, 2003.
Sain, M.K., Won, C.-H., Spencer, B.F., Jr., and Liberty, S.R., Cumulates and Risk-Sensitive Control: A Cost Mean and Variance Theory with Application to Seismic Protection of Structures, in Advanced in Dynamic Games and Applications, Ann. Int. Soc. Dynam. Games, 2000, vol. 5, pp. 427–459.
Albert, A., Regression, and the Moor-Penrose Pseudoinverse, New York: Academic, 1972. Translated under the title Regressiya, psevdoinversiya i rekurrentnoe otsenivanie, Moscow: Nauka, 1977.
Lancaster, P. and Tismenetsky, M., Theory of Matrices, New York: Academic, 1985, 2nd ed.
Fedorenko, R.P., Priblizhennoe reshenie zadach optimal’nogo upravleniya (Approximate Solution of Optimal Control Problems), Moscow: Nauka, 1978.
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Original Russian Text © V.N. Afanas’ev, 2015, published in Avtomatika i Telemekhanika, 2015, No. 1, pp. 3–20.
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Afanas’ev, V.N. Control of nonlinear uncertain object in the problem of motion along the given trajectory. Autom Remote Control 76, 1–15 (2015). https://doi.org/10.1134/S0005117915010014
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DOI: https://doi.org/10.1134/S0005117915010014