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Galerkin approximations of the generalized Hamilton-Jacobi-Bellman equation. (English) Zbl 0949.93022
Summary: We study tbe convergence of the Galerkin approximation method applied to the generalized Hamilton-Jacobi-Bellman (GHJB) equation over a compact set containing the origin. The GHJB equation gives the cost of an arbitrary control law and can be used to improve the performance of this control. The GHJB equation can also be used to successively approximate the Hamilton-Jacobi-Bellman equation. We state sufficient conditions that guarantee that the Galerkin approximation converges to the solution of the GHJB equation and that the resulting approximate control is stabilizing on the same region as the initial control. The method is demonstrated on a simple nonlinear system and is compared to a result obtained by using exact feedback linearization in conjunction with the LQR design method.

93B40Computational methods in systems theory
49M15Newton-type methods in calculus of variations
49L20Dynamic programming method (infinite-dimensional problems)
Full Text: DOI
[1] Aeyels, D.: Stabilization of a class of nonlinear systems by a smooth feedback control. System control lett. 5, 289-294 (1985) · Zbl 0569.93056
[2] Aeyels, D.: Local and global stabilizability for nonlinear systems. Theory and applications of nonlinear control systems, 93-105 (1986) · Zbl 0604.93042
[3] Aganovic, Z.; Gajic, Z.: The successive approximation procedure for finite-time optimal control of bilinear systems. IEEE trans. Automat. control 39, No. 9, 1932-1935 (1994) · Zbl 0819.93030
[4] Al’brekht, E. G.: On the optimal stabilization of nonlinear systems. J. appl. Math. mech. 25, No. 5, 836-844 (1961)
[5] Anderson, B. D. O.; Moore, J. B.: Linear optimal control. (1971) · Zbl 0321.49001
[6] Apostol, T. M.: Mathematical analysis. (1974) · Zbl 0309.26002
[7] Battilotti, S.: Global output regulation and disturbance attenuation with global stability via measurement feedback for a class of nonlinear systems. IEEE trans. Automat. control 41, No. 3, 315-327 (1996) · Zbl 0866.93041
[8] Baumann, W. T.; Rugh, W. J.: Feedback control of nonlinear systems by extended linearization. IEEE trans. Automat. control 31, No. 1, 40-46 (1986) · Zbl 0582.93031
[9] Beard, R.: Improving the closed-loop performance of nonlinear systems. Phd thesis (1995)
[10] Beard, R.; Saridis, G.; Wen, J.: Improving the performance of stabilizing control for nonlinear systems. Control systems mag. 16, No. 5, 27-35 (1996)
[11] Beard, R.; Saridis, G.; Wen, J.: Approximate solutions to the time-invariant Hamilton-Jacobi-Bellman equation. J. optim. Theory appl. (1998) · Zbl 0916.49021
[12] Bellman, R.; Dreyfus, S.: Applied dynamic programming. (1962) · Zbl 0106.34901
[13] Bittanti, S.; Laub, A.; Willems, J. C.: The Riccati equation. (1991) · Zbl 0734.34004
[14] Bosarge, W. E.; Johnson, O. G.; Mcknight, R. S.; Timlake, W. P.: The Ritz-Galerkin procedure for nonlinear control problems. SIAM J. Numer. anal. 10, No. 1, 94-110 (1973) · Zbl 0259.49019
[15] Bryson, A. E.; Ho, Y. C.: Applied optimal control. (1975)
[16] Dolcetta, I. Capuzzo: On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming. Appl. math. Optim. 10, 367-377 (1983) · Zbl 0582.49019
[17] Dolcetta, I. Capuzzo; Ishii, H.: Approximate solutions of the Bellman equation of deterministic control theory. Appl math. Optim. 11, 161-181 (1984)
[18] Dolcetta, I. Capuzzo; Falcone, M.: Discrete dynamic programming and viscosity solutions of the Bellman equation. Annales institut henri Poincarè anal nonlinear 6, 161-184 (1989) · Zbl 0674.49028
[19] Cebuhar, W. A.; Costanza, V.: Approximation procedures for the optimal control of bilinear and nonlinear systems. J. optim. Theory appl. 43, No. 4, 615-627 (1984) · Zbl 0518.93030
[20] Chapman, J. W.; Ilic, M. D.; King, C. A.; Eng, L.; Kaufman, H.: Stabilizing a multimachine power system via decentralized feedback linearizing excitation control. IEEE trans. Power systems 8, No. 3, 830-839 (1993)
[21] Cloutier, J. R.; D’souza, C. N.; Mracek, C. P.: Nonlinear regulation and nonlinear H$\infty $control via the state-dependent Riccati equation technique. IFAC world congress (1996)
[22] Crandall, M. G.; Ishii, H.; Lions, P. -L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. am. Math. soc. 27, No. 1, 1-67 (1992) · Zbl 0755.35015
[23] Falcone, M.: A numerical approach to the infinite horizon problem of deterministic control theory. Appl. math. Optim. 15, 1-13 (1987) · Zbl 0715.49023
[24] Falcone, M.; Ferretti, R.: Discrete time high-order schemes for viscosity solutions of Hamilton-Jacobi-Bellman equations. Numer. math. 67, 315-344 (1994) · Zbl 0791.65046
[25] Finlayson, B. A.: The method of weighted residuals and variational principles. (1972) · Zbl 0319.49020
[26] Fleming, W. H.; Soner, H. Mete: Controlled Markov processes and viscosity solutions. (1993) · Zbl 0773.60070
[27] Freeman, R. A.; Kokotovic, P. V.: Optimal nonlinear controllers for feedback linearizable systems. Proc. American control conf., 2722-2726 (1995)
[28] Gao, L.; Chen, Lin; Fan, Yushun; Ma, Haiwu: A nonlinear control design for power systems. Automatica 28, 975-979 (1992) · Zbl 0766.93019
[29] Garrard, W. L.; Jordan, J. M.: Design of nonlinear automatic flight control systems. Automatica 13, 497-505 (1977) · Zbl 0382.93047
[30] Genesio, R.; Tartaglia, M.; Vicino, A.: On the estimation of asymptotic stability regions: state of the art and new proposals. IEEE trans. Automat. control 30, No. 8, 747-755 (1985) · Zbl 0568.93054
[31] Glad, S. T.: Robustness of nonlinear state feedback--a survey. Automatica 23, No. 4, 425-435 (1987) · Zbl 0633.93051
[32] Glad, T.: Robust nonlinear regulators based on Hamilton-Jacobi theory and Lyapunov functions. IEE control conf., 276-280 (1985)
[33] Goh, C. J.: On the nonlinear optimal regulator problem. Automatica 29, 751-756 (1993) · Zbl 0771.93027
[34] Gonzalez, R.; Rofman, E.: On deterministic control problems: an approximation procedure for the optimal cost i: The stationary problem. SIAM J. Control optim. 23, No. 2, 242-266 (1985) · Zbl 0563.49024
[35] Gonzalez, R.; Rofman, E.: On deterministic control problems: an approximation procedure for the optimal cost ii: The nonstationary problem. SIAM J. Control optim. 23, No. 2, 267-285 (1985) · Zbl 0563.49025
[36] Halme, A.; Hamalainen, R. P.: On the nonlinear regulator problem. J. optim. Theory appl. 16, 255-275 (1975) · Zbl 0283.49004
[37] Hofer, E. P.; Tibken, B.: An iterative method for the finite-time bilinear-quadratic control problem. J. optim. Theory appl. 57, No. 3, 411-427 (1988) · Zbl 0622.49011
[38] Hunt, L. R.; Su, R.; Meyer, G.: Design for multi-input nonlinear systems. Differential geometric control theory, 268-298 (1983)
[39] Hunt, L. R.; Su, R.; Meyer, G.: Global transformations of nonlinear systems. IEEE trans. Automat. control 28, No. 1, 24-31 (1983) · Zbl 0502.93036
[40] Isidori, A.: 2nd ed.nonlinear control systems. Communication and control engineering. Nonlinear control systems. Communication and control engineering (1989) · Zbl 0693.93046
[41] Johansson, R.: Quadratic optimization of motion coordination and control. IEEE trans. Automat. control 35, No. 11, 1197-1208 (1990) · Zbl 0721.49032
[42] Kantorovich, L. V.; Krylov, V. I.: Approximate methods of higher analysis. (1958) · Zbl 0083.35301
[43] Khalil, H. K.: Nonlinear systems. (1992) · Zbl 0969.34001
[44] Kirk, D. E.: Optimal control theory. (1970)
[45] Kushner, H. J.: Numerical methods for stochastic control problems in continuous time. SIAM J. Control optim. 28, No. 5, 999-1048 (1990) · Zbl 0721.93087
[46] Leake, R. J.; Liu, R. -W.: Construction of suboptimal control sequences. SIAM J. Control optim. 5, No. 1, 54-63 (1967) · Zbl 0153.13103
[47] Lee, C. S. G.; Chen, M. H.: A suboptimal control design for mechanical manipulators. American control conf., 1056-1061 (1983)
[48] Lewis, F. L.: Optimal control. (1986) · Zbl 0665.93065
[49] Loparo, K. A.; Blankenship, G. L.: Estimating the domain of attraction of nonlinear feedback systems. IEEE trans. Automat. control 23, No. 4, 602-608 (1978) · Zbl 0385.93023
[50] Lu, P.: A new nonlinear optimal feedback control law. Control theory adv. Technol. 9, No. 4, 947-954 (1993)
[51] Lukes, D. L.: Optimal regulation of nonlinear dynamical systems. SIAM J. Control optim. 7, No. 1, 75-100 (1969) · Zbl 0184.18802
[52] Marino, R.: An example of a nonlinear regulator. IEEE trans. Automat. control 29, No. 3, 276-279 (1984) · Zbl 0529.93033
[53] Mikhlin, S. G.: Variational methods in mathematical physics. (1964) · Zbl 0119.19002
[54] Mikhlin, S. G.; Smolitskiy, K. L.: Approximate methods for solution of differential and integral equations. (1967)
[55] Mil’shtein, G. N.: Successive approiximations for solution of one optimal problem. Automation remote control 25, 298-306 (1964)
[56] Nijmeijer, H.; Van Der Schaft, A. J.: Nonlinear dynamical control systems. (1990) · Zbl 0701.93001
[57] Nishikawa, Y.; Sannomiya, N.; Itakura, H.: A method for suboptimal design of nonlinear feedback systems. Automatica 7, 703-712 (1971) · Zbl 0225.49027
[58] Petryshyn, W. V.: On a class of k-pdand non-k-pdoperators and operator equations. J. math. Anal. appl. 10, 1-24 (1965) · Zbl 0135.36503
[59] Rosen, O.; Luus, R.: Global optimization approach to nonlinear optimal control. J. optim. Theory appl. 73, No. 3, 547-562 (1992) · Zbl 0794.49031
[60] Ryan, E. P.: Optimal feedback control of bilinear systems. J. optim. Theory appl. 44, No. 2, 333-362 (1984) · Zbl 0537.93042
[61] Sage, A. P.; Iii, C. C. White: 2nd ed.optimum systems control. Optimum systems control (1977)
[62] Saridis, G. N.; Lee, C. -S.G.: An approximation theory of optimal control for trainable manipulators. IEEE trans. Systems man cybernet. 9, No. 3, 152-159 (1979) · Zbl 0398.49001
[63] Saridis, G. N.; Wang, F. Y.: Suboptimal control of nonlinear stochastic systems. Control theory adv. Technol. 10, No. 4, 847-871 (1994)
[64] Saridis, G. N.; Balaram, J.: Suboptimal control for nonlinear systems. Control theory adv. Technol. 2, No. 3, 547-562 (1986)
[65] Tsitsiklis, J. N.; Athans, M.: Guaranteed robustness properties of multivariable nonlinear stochastic optimal regulators. IEEE trans. Automat. control 29, No. 8, 690-696 (1984) · Zbl 0544.93079
[66] Tzafestas, S. G.; Anagnostou, K. E.; Pimenides, T. G.: Stabilizing optimal control of bilinear systems with a generalized cost. Opt. control appl. Methods 5, 111-117 (1984) · Zbl 0552.93050
[67] Vaisbord, E. M.: An approximate method for the synthesis of optimal control. Automat. remote control 24, 1626-1632 (1963)
[68] Wang, Y.; Hill, D. J.; Middleton, R. H.; Gao, L.: Transient stability enhancement and voltage regulation of power systems. IEEE trans. Power systems 8, No. 2, 620-627 (1993)
[69] Wang, Y.; Hill, D. J.; Middleton, R. H.; Gao, L.: Transient stabilization of power systems with an adaptive control law. Automatica 30, No. 9, 1409-1413 (1994) · Zbl 0925.93447
[70] Werner, R. A.; Cruz, J. B.: Feedback control which preserves optimality for systems with unknown parameters. IEEE trans. Automat. control 13, No. 6, 621-629 (1968)
[71] Zeidler, E.: Nonlinear functional analysis and its applications, 11/A: linear monotone operators. (1990) · Zbl 0684.47028
[72] Zeidler, E.: Nonlinear functional analysis and its applications, II/B: nonlinear monotone operators. (1990) · Zbl 0684.47029