Correctness of a constrained control Mayer's problem for a class of singularly perturbed functional-differential systems

• Autorzy: Glizer V. Y.
• Języki publikacji: EN

Abstrakty

EN

A Mayer's problem for a singularly perturbed controlled system with the general type of a small state delay is considered. The control is subject to geometrical constraints. The cost functional is a function of the terminal value of the slow state variable. A simpler parameter-free optimal control problem (the reduced problem) is associated with the original problem. A convergence of the optimal value of the cost functional in the original problem to the optimal value of the cost functional in the reduced problem, as a parameter of singular perturbation tends to zero, is established. An asymptotic suboptimality of the optimal control of the reduced problem in the original problem is shown. These results are extended to some more general optimal control problems. An illustrative example is presented.

Słowa kluczowe

EN

functional differential system; time delay; Mayer's problem; bounded control; singular perturbation; correctness; suboptimal control

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2008
• Tom: Vol. 37, no 2
• Strony: 329--351
• Opis fizyczny: Bibliogr. 38 poz.

Twórcy

autor

Glizer V. Y.

• Department of Mathematics, Ort Braude College P.O. Box 78, Karmiel 21982, Israel

Bibliografia

• ARTSTEIN, Z. (2005) Bang-bang controls in the singular perturbations limit. Control Cybernet. 34 (3), 645-663.
• ARTSTEIN, Z. and SLEMROD,M. (2001) On singularly perturbed retarded functional differential equations. J. Differential Equations 171 (1), 88-109.
• BERNHARD, P. (1979) Linear-quadratic, two-person, zero-sum differential games: necessary and sufficient conditions. J. Optim. Theory Appl. 27 (1), 51-69.
• COOKE, K.L. and MEYER, K.R. (1966) The condition of regular degeneration for singularly perturbed systems of linear differential-difference equations. J. Math. Anal. Appl. 14 (1), 83-106.
• DMITRIEV, M.G. and KURINA, G.A. (2006) Singular perturbations in control problems. Autom. Remote Control 67 (1), 1-43.
• DONCHEV, T. and SLAVOV, I. (1995) Singularly perturbed functional-differential inclusions. Set-Valued Anal. 3 (2), 113-128.
• DONCHEV, T. and SLAVOV, I.I. (1997) Singularly perturbed delayed differential inclusions. Annuaire Univ. Sofia Fac. Math. Inform. 91 (1-2), 61-71.
• DONTCHEV, A.L. (1983) Perturbations, Approximations and Sensitivity Analysis of Optimal Control Systems. Springer-Verlag, Berlin.
• DONTCHEV, A.L. and ZOLEZZI, T. (1993) Well-Posed Optimization Problems. Springer-Verlag, Berlin.
• FRIDMAN, E. (1990) Decomposition of linear optimal singularly perturbed systems with aftereffect. Autom. Remote Control 51 (11) part 1, 1518-1527.
• FRIDMAN, E. (1996) Decoupling transformation of singularly perturbed systems with small delays. Z. Angew. Math. Mech. 76 (2), 201-204.
• FRIDMAN, E. (2006) Robust sampled-data H∞ control of linear singularly perturbed systems. IEEE Trans. Automat. Control 51 (3), 470-475.
• GAJIC, Z. and LIM, M.-T. (2001) Optimal Control of Singularly Perturbed Linear Systems and Applications. High-Accuracy Techniques. Marcel Dekker, New York.
• GLIZER, V.Y. (1998) Asymptotic solution of a singularly perturbed set of functional-differential equations of Riccati type encountered in the optimal control theory. NoDEA Nonlinear Differential Equations Appl. 5 (4), 491-515.
• GLIZER, V.Y. (2000) Asymptotic solution of a boundary-value problem for linear singularly-perturbed functional differential equations arising in optimal control theory. J. Optim. Theory Appl. 106 (2), 309-335.
• GLIZER, V.Y. (2003) Blockwise estimate of the fundamental matrix of linear singularly perturbed differential systems with small delay and its application to uniform asymptotic solution. J. Math. Anal. Appl. 278 (2), 409-433.
• GLIZER, V.Y. (2004) On stabilization of nonstandard singularly perturbed systems with small delays in state and control. IEEE Trans. Automat. Control 49 (6), 1012-1016.
• GLIZER, V.Y. (2005) Suboptimal solution of a cheap control problem for linear systems with multiple state delays. J. Dyn. Control Syst. 11 (4) 527-574.
• GLIZER, V.Y. (2006) Cheap control problem of linear systems with delays: a singular perturbation approach. In: F. Ceragioli, A. Dontchev, H. Furuta, K. Marti and L. Pandolfi, eds., Systems, Control, Modeling and Optimization. IFIP Series 202, Springer, New York, 183-193.
• GLIZER, V.Y. (2007) Infinite horizon quadratic control of linear singularly perturbed systems with small state delays: an asymptotic solution of Riccati-type equations. IMA J. Math. Control Inform. 24 (4), 435-459.
• HAGENAARS, H.L., IMURA, J. and NIJMEIJER, H. (2004) Approximate continuous-time optimal control in obstacle by time/space discretization of non-convex constraints. Proc. 2004 IEEE Intern. Conf. Control Applications, Taipei, Taiwan, 2, 878-883.
• HALANAY, A. (1966) Differential Equations: Stability, Oscillations, Time Lags. Academic Press, New York.
• HALE, J.K. and LUNEL, S.M.V. (1993) Introduction to Functional Differential Equations. Springer, New York.
• HALE, J.K. and TANAKA, S.M. (2000) Square and pulse waves with two delays. J. Dynam. Differential Equations 12 (1), 1-30.
• KOKOTOVIC, P.V., KHALIL, H.K. and O’REILLY, J. (1999) Singular Perturbation Methods in Control: Analysis and Design. SIAM, Philadelphia.
• KOPEIKINA, T.B. (1989) Controllability of singularly perturbed linear systems with time-lag. Differential Equations 25 (9), 1055-1064.
• LANGE, C.G. and MIURA, R.M. (1994) Singular perturbation analysis of boundary-value problems for differential-difference equations. VI. Small shifts with rapid oscillations. SIAM J. Appl. Math. 54 (1), 273-283.
• LIN-CHEN, Y.Y. and GOODALL, D.P. (2004) Stabilizing feedbacks for imperfectly known, singularly perturbed nonlinear systems with discrete and distributed delays. Internat. J. Systems Sci. 35 (15), 869-887.
• LIZANA, M. (1999) Global analysis of the sunflower equation with small delay. Nonlinear Anal. Ser. B: Real World Appl. 36 (6), 697-706.
• LUKOYANOV, N.Yu. and RESHETOVA, T.N. (1998) Problems of conflict control of high dimensionality functional systems. J. Appl. Math. Mech. 62 (4), 545-554.
• MAGALHÃES, L.T. (1984) Convergence and boundary layers in singularly perturbed linear functional-differential equations. J. Differential Equations 54 (3), 295-309.
• MALLET-PARET, J. and NUSSBAUM, R.D. (1989) A differential-delay equation arising in optics and physiology. SIAM J. Math. Anal. 20 (2), 249-292.
• MITROPOL’SKII, Yu.A., FODCHUK V.I. and KLEVCHUK, I.I. (1986) Integral manifolds, stability and bifurcation of solutions of singularly perturbed functional-differential equations. Ukrainian Math. J. 38 (3), 290-294.
• NAIDU, D.S. (2002) Singular perturbations and time scales in control theory and applications: an overview. Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 9 (2), 233-278.
• REDDY, P.B. and SANNUTI, P. (1974) Optimal control of singularly perturbed time delay systems with an application to a coupled core nuclear reactor. Proc. 1974 IEEE Conf. Decision Control, Tucson, Arizona, USA, 793-803.
• REDDY, P.B. and SANNUTI, P. (1975) Optimal control of a coupled-core nuclear reactor by a singular perturbation method. IEEE Trans. Automat. Control 20 (6), 766-769.
• SLAVOV, I. (1995) Continuity properties of the attainable set of singularly perturbed control system with time delay. Proc. Tech. Univ. Sofia 48, 29-36.
• TURETSKY, V. (1999) Differential game solubility condition in H∞-optimization. In: V.D. Batukhtin, P.M. Kirillova, and V.I. Ukhobotov, eds., Non-smooth and Discontinuous Problems of Control and Optimization. Pergamon Press, New York, 209-214.