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Optimal control for stochastic nonlinear singular system using neural networks. (English) Zbl 1165.93343
Summary: Optimal control for stochastic nonlinear singular system with quadratic performance is obtained using neural networks. The goal is to provide optimal control with reduced calculus effort by comparing the solutions of the matrix Riccati differential equation (MRDE) obtained from the well-known traditional Runge-Kutta (RK) method and nontraditional neural network method. To obtain the optimal control, the solution of MRDE is computed by feedforward neural network (FFNN). The accuracy of the solution of the neural network approach to the problem is qualitatively better. The advantage of the proposed approach is that, once the network is trained, it allows instantaneous evaluation of solution at any desired number of points spending negligible computing time and memory. The computation time of the proposed method is shorter than the traditional RK method. An illustrative numerical example is presented for the proposed method.
93E20Optimal stochastic control (systems)
60H35Computational methods for stochastic equations
62M45Neural nets and related approaches (inference from stochastic processes)
49N10Linear-quadratic optimal control problems
65L06Multistep, Runge-Kutta, and extrapolation methods
[1]Athens, M.: Special issues on linear quadratic Gaussian problem, IEEE automat. Control 16, 527-869 (1971)
[2]Bensoussan, A.: Lecture on stochastic control part I, Lecture notes in math 972, 1-39 (1983)
[3]Bucci, F.; Pandolfi, L.: The regulator problem with indefinite quadratic cost for boundary control systems: the finite horizon case, Systems control lett. 39, 79-86 (2000) · Zbl 0943.49024 · doi:10.1016/S0167-6911(99)00091-2
[4]Davis, M. H. A.: Linear estimation and stochastic control, (1997)
[5]Wonham, W. M.: On a matrix Riccati equation of stochastic control, SIAM J. Control optim. 6, 681-697 (1968) · Zbl 0182.20803
[6]Chen, S. P.; Li, X. J.; Zho, X. Y.: Stochastic linear quadratic regulators with indefinite control weight costs, SIAM J. Control optim. 36, No. 5, 1685-1702 (1998) · Zbl 0916.93084 · doi:10.1137/S0363012996310478
[7]Zhu, J.; Li, K.: An iterative method for solving stochastic Riccati differential equations for the stochastic LQR problem, Optim. methods. Softw. 18, 721-732 (2003) · Zbl 1067.93063 · doi:10.1080/10556780310001636657
[8]Balasubramaniam, P.; Samath, J. Abdul; Kumaresan, N.; Kumar, A. Vincent Antony: Solution of matrix Riccati differential equation for the linear quadratic singular system using neural networks, Appl. math. Comput. 182, 1832-1839 (2006) · Zbl 1107.65057 · doi:10.1016/j.amc.2006.06.020
[9]Balasubramaniam, P.; Samath, J. Abdul; Kumaresan, N.: Optimal control for nonlinear singular systems with quadratic performance using neural networks, Appl. math. Comput. 187, 1535-1543 (2007) · Zbl 1114.65336 · doi:10.1016/j.amc.2006.09.072
[10]De Wilde, P.: Neural network models, (1997)
[11]Karakasoglu, A.; Sudharsanan, S. L.; Sundareshan, M. K.: Identification and decentralized adaptive control using neural networks with application to robotic manipulators, IEEE trans. Neural netw. 4, 919-930 (1993)
[12]Narendra, K. S.; Parathasarathy, K.: Identification and control of dynamical systems using neural networks, IEEE trans. Neural netw. 1, 4-27 (1990)
[13]Ellacott, S. W.: Aspects of the numerical analysis of neural networks, Acta numer. 5, 145-202 (1994) · Zbl 0807.65007
[14]F.M. Ham, E.G. Collins, A neurocomputing approach for solving the algebraic matrix Riccati equation, in: Proceedings IEEE International Conference on Neural Networks 1, 1996, pp. 617–622
[15]Wang, J.; Wu, G.: A multilayer recurrent neural network for solving continuous time algebraic Riccati equations, Neural netw. 11, 939-950 (1998)
[16]Campbell, S. L.: Singular systems of differential equations, (1980) · Zbl 0444.34062 · doi:10.1080/00036818008839326
[17]Campbell, S. L.: Singular systems of differential equations II, (1982)
[18]Lewis, F. L.: A survey of linear singular systems, Circuits systems signal process. 5, No. 1, 3-36 (1986) · Zbl 0613.93029 · doi:10.1007/BF01600184
[19]Brenan, K. E.; Campbell, S. L.; Petzold, L. R.: Numerical solution of initial value problems in differential-algebraic equations, (1989)
[20]Campbell, S. L.; Griepentrog, E.: Solvability of general differential-algebraic equations, SIAM J. Sci. comput. 16, 257-270 (1995) · Zbl 0821.34005 · doi:10.1137/0916017
[21]Mcclamroch, N. H.: Feedback stabilization of control systems described by a class of nonlinear differential algebraic equations, Systems control lett. 15, 53-60 (1990) · Zbl 0704.93060 · doi:10.1016/0167-6911(90)90044-U
[22]Dai, L.: Singular control systems, Lecture notes in control and information sciences (1989)
[23]Banks, S. P.; Mhana, K. J.: Optimal control and stabilization for nonlinear systems, IMA J. Math. control inform. 9, 179-196 (1992) · Zbl 0773.49018 · doi:10.1093/imamci/9.2.179
[24]Mccaffrey, D.; Banks, S. P.: Lagrangian manifolds and asymptotically optimal stabilizing feedback control, Systems control lett. 43, 219-224 (2001) · Zbl 1032.49036 · doi:10.1016/S0167-6911(01)00101-3
[25]Da Prato, G.; Ichikawa, A.: Quadratic control for linear periodic systems, Appl. math. Optim. 18, 39-66 (1988) · Zbl 0647.93057 · doi:10.1007/BF01443614
[26]Zhou, K.; Khargonekar, P.: An algebraic Riccati equation approach to H optimization, Systems control lett. 11, 85-91 (1998) · Zbl 0666.93025 · doi:10.1016/0167-6911(88)90080-1
[27]Jamshidi, M.: An overview on the solutions of the algebraic matrix Riccati equation and related problems, Large scale syst. 1, 167-192 (1980) · Zbl 0453.93025
[28]Vaughu, D. R.: A negative exponential solution for the matrix Riccati equation, IEEE trans. Automat. control 14, 72-75 (1969)
[29]Lovren, N.; Tomic, M.: Analytic solution of the Riccati equation for the homing missile linear quadratic control problem, J. guid. Control. dyn. 17, 619-621 (1994)
[30]Razzaghi, M.: Solution of the matrix Riccati equation in optimal control, Inform. sci. 16, 61-73 (1978) · Zbl 0443.93020 · doi:10.1016/0020-0255(78)90011-7
[31]Razzaghi, M.: A computational solution for a matrix Riccati differential equation, Numer. math. 32, 271-279 (1979) · Zbl 0386.34018 · doi:10.1007/BF01397001
[32]Razzaghi, M.: A Schur method for the solution of the matrix Riccati equation, Int. J. Math. math. Sci. 20, 335-338 (1997) · Zbl 1047.34502 · doi:10.1155/S0161171297000446
[33]Jodar, L.; Navarro, E.: Closed analytical solution of Riccati type matrix differential equations, Indian J. Pure appl. Math. 23, 185-187 (1992) · Zbl 0765.34001
[34]Lagaris, I. E.; Likas, A.; Fotiadis, D. I.: Artificial neural networks for solving ordinary and partial differential equations, IEEE trans. Neural netw. 9, 987-1000 (1998)
[35]D. Nguyen, B. Widrow, Improving the learning speed of 2-layer neural networks by choosing initial values of the adaptive weights, in: Proceedings of International Joint Conference on Neural Networks, vol. III, 1990, pp. 21–26
[36]Paplinski, A. P.: Lecture notes on feedforward multilayer neural networks, Lecture notes on feedforward multilayer neural networks (L.5) (2004)
[37]Hagan, M. T.; Menhaj, M.: Training feedforward networks with the Marquardt algorithm, IEEE trans. Neural netw. 5, No. 6, 989-993 (1994)