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Volumn 96, Issue 3, 1998, Pages 589-626

Approximate solutions to the time-invariant Hamilton-Jacobi-Bellman equation

Author keywords

Feedback synthesis; Galerkin approximation; Hamilton Jacobi Bellman equation; Nonlinear control; Optimal control; Successive approximation

Indexed keywords

APPROXIMATION THEORY; FEEDBACK; GALERKIN METHODS; NONLINEAR CONTROL SYSTEMS; NONLINEAR EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS;

EID: 0032387028     PISSN: 00223239     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1022664528457     Document Type: Article
Times cited : (168)

References (72)
  • 3
    • 0003448648 scopus 로고
    • Prentice-Hall, Englewood Cliffs, New Jersey
    • KIRK, D. E., Optimal Control Theory, Prentice-Hall, Englewood Cliffs, New Jersey, 1970.
    • (1970) Optimal Control Theory
    • Kirk, D.E.1
  • 4
    • 0004163205 scopus 로고
    • John Wiley and Sons, New York, New York
    • LEWIS, F. L., Optimal Control, John Wiley and Sons, New York, New York, 1986.
    • (1986) Optimal Control
    • Lewis, F.L.1
  • 7
    • 0024018006 scopus 로고
    • An iterative method for the finite-time bilinear-quadratic control problem
    • HOFER, E. P., and TIBKEN, B., An Iterative Method for the Finite-Time Bilinear-Quadratic Control Problem, Journal of Optimization Theory and Applications, Vol. 57, pp. 411-427, 1988.
    • (1988) Journal of Optimization Theory and Applications , vol.57 , pp. 411-427
    • Hofer, E.P.1    Tibken, B.2
  • 8
    • 0028498817 scopus 로고
    • The successive approximation procedure for finite-time optimal control of bilinear systems
    • AGANOVIC, Z., and GAJIC, Z., The Successive Approximation Procedure for Finite-Time Optimal Control of Bilinear Systems, IEEE Transactions on Automatic Control, Vol. 39, pp. 1932-1935, 1994.
    • (1994) IEEE Transactions on Automatic Control , vol.39 , pp. 1932-1935
    • Aganovic, Z.1    Gajic, Z.2
  • 9
    • 0021470322 scopus 로고
    • Approximation procedures for the optimal control of bilinear and nonlinear systems
    • CEBUHAR, W. A., and COSTANZA, V., Approximation Procedures for the Optimal Control of Bilinear and Nonlinear Systems, Journal of Optimization Theory and Applications, Vol. 43, pp. 615-627, 1984.
    • (1984) Journal of Optimization Theory and Applications , vol.43 , pp. 615-627
    • Cebuhar, W.A.1    Costanza, V.2
  • 10
    • 0026883116 scopus 로고
    • Global optimization approach to nonlinear optimal control
    • ROSEN, O., and LUUS, R., Global Optimization Approach to Nonlinear Optimal Control, Journal of Optimization Theory and Applications, Vol. 73, pp. 547-562, 1992.
    • (1992) Journal of Optimization Theory and Applications , vol.73 , pp. 547-562
    • Rosen, O.1    Luus, R.2
  • 12
    • 0014469655 scopus 로고
    • Optimal regulation of nonlinear dynamical systems
    • LUKES, D. L., Optimal Regulation of Nonlinear Dynamical Systems, SIAM Journal on Control and Optimization, Vol. 7, pp. 75-100, 1969.
    • (1969) SIAM Journal on Control and Optimization , vol.7 , pp. 75-100
    • Lukes, D.L.1
  • 13
    • 0017537095 scopus 로고
    • Design of nonlinear automatic flight control systems
    • GARRARD, W. L., and JORDAN, J. M., Design of Nonlinear Automatic Flight Control Systems, Automatica, Vol. 13, pp. 497-505, 1977.
    • (1977) Automatica , vol.13 , pp. 497-505
    • Garrard, W.L.1    Jordan, J.M.2
  • 14
    • 0015160843 scopus 로고
    • A method for suboptimal design of nonlinear feedback systems
    • NISHIKAWA, Y., SANNOMIYA, N., and ITAKURA, H., A Method for Suboptimal Design of Nonlinear Feedback Systems, Automatica, Vol. 7, pp. 703-712, 1971.
    • (1971) Automatica , vol.7 , pp. 703-712
    • Nishikawa, Y.1    Sannomiya, N.2    Itakura, H.3
  • 15
    • 0002519776 scopus 로고
    • Feedback control which preserves optimality for systems with unknown parameters
    • WERNER, R. A., and CRUZ, J. B., Feedback Control Which Preserves Optimality for Systems with Unknown Parameters, IEEE Transactions on Automatic Control, Vol. 13, pp. 621-629, 1968.
    • (1968) IEEE Transactions on Automatic Control , vol.13 , pp. 621-629
    • Werner, R.A.1    Cruz, J.B.2
  • 19
    • 0027807138 scopus 로고
    • A new nonlinear optimal feedback control law
    • LU, P., A New Nonlinear Optimal Feedback Control Law, Control Theory and Advanced Technology, Vol. 9, pp. 947-954, 1993.
    • (1993) Control Theory and Advanced Technology , vol.9 , pp. 947-954
    • Lu, P.1
  • 20
    • 0029178462 scopus 로고
    • Optimal nonlinear controllers for feedback linearizable systems
    • Seattle, Washington
    • FREEMAN, R. A., and KOKOTOVIC, P. V., Optimal Nonlinear Controllers for Feedback Linearizable Systems, Proceedings of the American Control Conference, Seattle, Washington, pp. 2722-2726, 1995.
    • (1995) Proceedings of the American Control Conference , pp. 2722-2726
    • Freeman, R.A.1    Kokotovic, P.V.2
  • 21
    • 84967708673 scopus 로고
    • User's guide to viscosity solutions of second-order partial differential equations
    • CRANDALL, M. G., ISHII, H., and LIONS, P. L., User's Guide to Viscosity Solutions of Second-Order Partial Differential Equations, Bulletin of the American Mathematical Society, Vol. 27, pp. 1-67, 1992.
    • (1992) Bulletin of the American Mathematical Society , vol.27 , pp. 1-67
    • Crandall, M.G.1    Ishii, H.2    Lions, P.L.3
  • 22
    • 34250147880 scopus 로고
    • On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming
    • CAPUZZO DOLCETTA, I., On a Discrete Approximation of the Hamilton-Jacobi Equation of Dynamic Programming, Applied Mathematics and Optimization, Vol. 10, pp. 367-377, 1983.
    • (1983) Applied Mathematics and Optimization , vol.10 , pp. 367-377
    • Capuzzo Dolcetta, I.1
  • 23
    • 34250115257 scopus 로고
    • Approximate solutions of the Bellman equation of deterministic control theory
    • CAPPUZZO DOLCETTA, I., and ISHII, H., Approximate Solutions of the Bellman Equation of Deterministic Control Theory, Applied Mathematics and Optimization, Vol. 11, pp. 161-181, 1984.
    • (1984) Applied Mathematics and Optimization , vol.11 , pp. 161-181
    • Cappuzzo Dolcetta, I.1    Ishii, H.2
  • 24
    • 21344488058 scopus 로고
    • Discrete time high-order schemes for viscosity solutions of Hamilton-Jacobi-Bellman equations
    • FALCONE, M., and FERRETTI, R., Discrete Time High-Order Schemes for Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Numerische Mathematik, Vol. 67, pp. 315-344, 1994.
    • (1994) Numerische Mathematik , vol.67 , pp. 315-344
    • Falcone, M.1    Ferretti, R.2
  • 26
    • 0022027092 scopus 로고
    • On deterministic control problems: An approximation procedure for the optimal cost, I: The stationary problem
    • GONZALEZ, R., and ROFMAN, E., On Deterministic Control Problems: An Approximation Procedure for the Optimal Cost, I: The Stationary Problem, SIAM Journal on Control and Optimization, Vol. 23, pp. 242-266, 1985.
    • (1985) SIAM Journal on Control and Optimization , vol.23 , pp. 242-266
    • Gonzalez, R.1    Rofman, E.2
  • 27
    • 0022027390 scopus 로고
    • On deterministic control problems: An approximation procedure for the optimal cost, II: The nonstationary problem
    • GONZALEZ, R., and ROFMAN, E., On Deterministic Control Problems: An Approximation Procedure for the Optimal Cost, II: The Nonstationary Problem, SIAM Journal on Control and Optimization, Vol. 23, pp. 267-285, 1985.
    • (1985) SIAM Journal on Control and Optimization , vol.23 , pp. 267-285
    • Gonzalez, R.1    Rofman, E.2
  • 28
    • 0023123237 scopus 로고
    • A numerical approach to the infinite-horizon problem of deterministic control theory
    • FALCONE, M., A Numerical Approach to the Infinite-Horizon Problem of Deterministic Control Theory, Applied Mathematics and Optimization, Vol. 15, pp. 1-13, 1987.
    • (1987) Applied Mathematics and Optimization , vol.15 , pp. 1-13
    • Falcone, M.1
  • 29
    • 0025484857 scopus 로고
    • Numerical methods for stochastic control problems in continuous time
    • KUSHNER, H. J., Numerical Methods for Stochastic Control Problems in Continuous Time, SIAM Journal on Control and Optimization, Vol. 28, pp. 999-1048, 1990.
    • (1990) SIAM Journal on Control and Optimization , vol.28 , pp. 999-1048
    • Kushner, H.J.1
  • 31
    • 0022440454 scopus 로고
    • Feedback control of nonlinear systems by extended linearization
    • BAUMANN, W. T., and RUGH, W. J., Feedback Control of Nonlinear Systems by Extended Linearization, IEEE Transactions on Automatic Control, Vol. 31, pp. 40-46, 1986.
    • (1986) IEEE Transactions on Automatic Control , vol.31 , pp. 40-46
    • Baumann, W.T.1    Rugh, W.J.2
  • 32
    • 0040619558 scopus 로고    scopus 로고
    • Nonlinear regulation and nonlinear H x-control via the state-dependent Riccati equation technique
    • San Francisco, California
    • CLOUTIER, J. R., D'SOUZA, C. N., and MRACEK, C. P., Nonlinear Regulation and Nonlinear H x-Control Via the State-Dependent Riccati Equation Technique, IFAC World Congress, San Francisco, California, 1996.
    • (1996) IFAC World Congress
    • Cloutier, J.R.1    D'Souza, C.N.2    Mracek, C.P.3
  • 33
    • 0027594098 scopus 로고
    • On the nonlinear optimal regulator problem
    • GOH, C. J., On the Nonlinear Optimal Regulator Problem, Automatica, Vol. 29, pp. 751-756, 1993.
    • (1993) Automatica , vol.29 , pp. 751-756
    • Goh, C.J.1
  • 34
    • 0025517271 scopus 로고
    • Quadratic optimization of motion coordination and control
    • JOHANSSON, R., Quadratic Optimization of Motion Coordination and Control, IEEE Transactions on Automatic Control, Vol. 35, pp. 1197-1208, 1990.
    • (1990) IEEE Transactions on Automatic Control , vol.35 , pp. 1197-1208
    • Johansson, R.1
  • 35
    • 0004178386 scopus 로고
    • Macmillan Publishing Company, New York, New York
    • KHALIL, H. K., Nonlinear Systems, Macmillan Publishing Company, New York, New York, 1992.
    • (1992) Nonlinear Systems
    • Khalil, H.K.1
  • 36
    • 0003787146 scopus 로고
    • Princeton University Press, Princeton, New Jersey
    • BELLMAN, R. E., Dynamic Programming, Princeton University Press, Princeton, New Jersey, 1957.
    • (1957) Dynamic Programming
    • Bellman, R.E.1
  • 37
    • 0039319294 scopus 로고
    • Suboptimal design of intentionally nonlinear controllers
    • REKASIUS, Z. V., Suboptimal Design of Intentionally Nonlinear Controllers, IEEE Transactions on Automatic Control, Vol. 9, pp. 380-386, 1964.
    • (1964) IEEE Transactions on Automatic Control , vol.9 , pp. 380-386
    • Rekasius, Z.V.1
  • 41
    • 0039434283 scopus 로고
    • Suboptimal control of nonlinear stochastic systems
    • SARIDIS, G. N., and WANG, F., Suboptimal Control of Nonlinear Stochastic Systems, Control Theory and Advanced Technology, Vol. 10, pp. 847-871, 1994.
    • (1994) Control Theory and Advanced Technology , vol.10 , pp. 847-871
    • Saridis, G.N.1    Wang, F.2
  • 42
    • 0040026430 scopus 로고
    • An approximate method for the synthesis of optimal control
    • VAISBORD, E. M., An Approximate Method for the Synthesis of Optimal Control, Automation and Remote Control, Vol. 24, pp. 1626-1632, 1963.
    • (1963) Automation and Remote Control , vol.24 , pp. 1626-1632
    • Vaisbord, E.M.1
  • 43
    • 0038841599 scopus 로고
    • Successive approximations for solution of one optimal problem
    • MILSHTEIN, G. N., Successive Approximations for Solution of One Optimal Problem, Automation and Remote Control, Vol. 25, pp. 298-306, 1964.
    • (1964) Automation and Remote Control , vol.25 , pp. 298-306
    • Milshtein, G.N.1
  • 45
    • 0041098692 scopus 로고
    • On error bounds for successive approximation methods
    • BERTSEKAS, D. P., On Error Bounds for Successive Approximation Methods, IEEE Transactions on Automatic Control, Vol. 21, pp. 394-396, 1976.
    • (1976) IEEE Transactions on Automatic Control , vol.21 , pp. 394-396
    • Bertsekas, D.P.1
  • 46
    • 84914965022 scopus 로고
    • On an iterative technique for Riccati equation computations
    • KLEINMAN, D. L., On an Iterative Technique for Riccati Equation Computations, IEEE Transactions on Automatic Control, Vol. 13, pp. 114-115, 1968.
    • (1968) IEEE Transactions on Automatic Control , vol.13 , pp. 114-115
    • Kleinman, D.L.1
  • 48
    • 0016070513 scopus 로고
    • On Newton's method for Riccati equation solution
    • SANDELL, N. R., On Newton's Method for Riccati Equation Solution, IEEE Transactions on Automatic Control, Vol. 19, pp. 254-255, 1974.
    • (1974) IEEE Transactions on Automatic Control , vol.19 , pp. 254-255
    • Sandell, N.R.1
  • 50
    • 0002756084 scopus 로고
    • Invariant subspace methods for the numerical solution of Riccati equations
    • Edited by S. Bittanti and W. Laub, Springer Verlag, New York, New York
    • LAUB, A. J., Invariant Subspace Methods for the Numerical Solution of Riccati Equations, The Riccati Equation, Edited by S. Bittanti and W. Laub, Springer Verlag, New York, New York, pp. 163-196, 1991.
    • (1991) The Riccati Equation , pp. 163-196
    • Laub, A.J.1
  • 53
    • 0022234584 scopus 로고
    • Robust nonlinear regulators based on Hamilton-Jacobi theory and Lyapunov functions
    • Cambridge, Massachusetts
    • GLAD, T., Robust Nonlinear Regulators Based on Hamilton-Jacobi Theory and Lyapunov Functions, IEEE Control Conference, Cambridge, Massachusetts, pp. 276-280, 1985.
    • (1985) IEEE Control Conference , pp. 276-280
    • Glad, T.1
  • 54
    • 0023382820 scopus 로고
    • Robustness of nonlinear state feedback: A survey
    • GLAD, S. T., Robustness of Nonlinear State Feedback: A Survey, Automatica, Vol. 23, pp. 425-435, 1987.
    • (1987) Automatica , vol.23 , pp. 425-435
    • Glad, S.T.1
  • 55
    • 0021482297 scopus 로고
    • Guaranteed robustness properties of multivariable nonlinear stochastic optimal regulators
    • TSITSIKLIS, J. N., and ATHANS, M., Guaranteed Robustness Properties of Multivariable Nonlinear Stochastic Optimal Regulators, IEEE Transactions on Automatic Control, Vol. 29, pp. 690-696, 1984.
    • (1984) IEEE Transactions on Automatic Control , vol.29 , pp. 690-696
    • Tsitsiklis, J.N.1    Athans, M.2
  • 59
    • 0010712771 scopus 로고
    • On a class of K-p.d. and non-K-p.d. operators and operator equations
    • PETRYSHYN, W. V., On a Class of K-p.d. and Non-K-p.d. Operators and Operator Equations, Journal of Mathematical Analysis and Applications, Vol. 10, pp. 1-24, 1965.
    • (1965) Journal of Mathematical Analysis and Applications , vol.10 , pp. 1-24
    • Petryshyn, W.V.1
  • 64
    • 0031332446 scopus 로고    scopus 로고
    • Galerkin approximation of the generalized Hamilton-Jacobi-Bellman equation
    • BEARD, R. W., SARIDIS, G. N., and WEN, J. T., Galerkin Approximation of the Generalized Hamilton-Jacobi-Bellman Equation, Automatica, Vol. 33, 1997.
    • (1997) Automatica , vol.33
    • Beard, R.W.1    Saridis, G.N.2    Wen, J.T.3
  • 65
    • 0004147916 scopus 로고
    • Addison-Wesley, Reading, Massachusetts
    • APOSTOL, T. M., Mathematical Analysis, Addison-Wesley, Reading, Massachusetts, 1974.
    • (1974) Mathematical Analysis
    • Apostol, T.M.1
  • 67
    • 0027589869 scopus 로고
    • Transient stability enhancement and voltage regulation of power systems
    • WANG, Y., HILL, D. J., MIDDLETON, R. H., and GAO, L., Transient Stability Enhancement and Voltage Regulation of Power Systems, IEEE Transactions on Power Systems, Vol. 8, pp. 620-627, 1993.
    • (1993) IEEE Transactions on Power Systems , vol.8 , pp. 620-627
    • Wang, Y.1    Hill, D.J.2    Middleton, R.H.3    Gao, L.4
  • 68
    • 0028429517 scopus 로고
    • Feedback linearizing excitation control on a full-scale power system model
    • KING, C. A., CHAPMAN, J. W., and ILIC, M. D., Feedback Linearizing Excitation Control on a Full-Scale Power System Model, IEEE Transactions on Power Systems, Vol. 9, pp. 1102-1109, 1994.
    • (1994) IEEE Transactions on Power Systems , vol.9 , pp. 1102-1109
    • King, C.A.1    Chapman, J.W.2    Ilic, M.D.3
  • 69
    • 0026925218 scopus 로고
    • A nonlinear control design for power systems
    • GAO, L., CHEN, L., FAN, Y., and MA, H., A Nonlinear Control Design for Power Systems, Automatica, Vol. 28, pp. 975-979, 1992.
    • (1992) Automatica , vol.28 , pp. 975-979
    • Gao, L.1    Chen, L.2    Fan, Y.3    Ma, H.4
  • 71
    • 0028494596 scopus 로고
    • Transient stabilization of power systems with an adaptive control law
    • WANG, Y., HILL, D. J., MIDDLETON, R. H., and GAO, L., Transient Stabilization of Power Systems with an Adaptive Control Law, Automatica, Vol. 30, pp. 1409-1413, 1994.
    • (1994) Automatica , vol.30 , pp. 1409-1413
    • Wang, Y.1    Hill, D.J.2    Middleton, R.H.3    Gao, L.4
  • 72
    • 0027643671 scopus 로고
    • Stabilizing a multimachine power system via decentralized feedback linearizing excitation control
    • CHAPMAN, J. W., ILIC, M. D., KING, C. A., ENG, L., and KAUFMAN, H., Stabilizing a Multimachine Power System via Decentralized Feedback Linearizing Excitation Control, IEEE Transactions on Power Systems, Vol. 8, pp. 830-839, 1993.
    • (1993) IEEE Transactions on Power Systems , vol.8 , pp. 830-839
    • Chapman, J.W.1    Ilic, M.D.2    King, C.A.3    Eng, L.4    Kaufman, H.5


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