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Solution of generalized matrix Riccati differential equation for indefinite stochastic linear quadratic singular system using neural networks. (English) Zbl 1157.65397
Summary: The solution of a generalized matrix Riccati differential equation (GMRDE) for an indefinite stochastic linear quadratic singular system is obtained using neural networks. The goal is to provide an optimal control with reduced calculus effort by comparing the solutions of GMRDE obtained from the well known traditional Runge Kutta (RK) method and a nontraditional neural network method.
To obtain the optimal control, the solution of the GMRDE is computed by a feed forward neural network (FFNN). The accuracy of the solution of the neural network approach to the problem is qualitatively better. The neural network solution is also compared with the solution of ode45, a standard solver available in MATLAB which implements the RK method for a variable step size.
The advantage of the proposed approach is that, once the network is trained, it allows an instantaneous evaluation of the 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.

65K10 Numerical optimization and variational techniques
49N10 Linear-quadratic optimal control problems
49J55 Existence of optimal solutions to problems involving randomness
65C30 Numerical solutions to stochastic differential and integral equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
Full Text: DOI
[1] Ait Ram, M.; Moore, J.B.; Zhou, Xun Yu, Indefinite stochastic linear quadratic control and generalized differential Riccati equation, SIAM J. control optim., 40, 4, 1296-1311, (2001) · Zbl 1009.93082
[2] Athens, M., Special issues on linear quadratic Gaussian problem, IEEE automat. control, AC-16, 527-869, (1971)
[3] Balasubramaniam, P.; Abdul Samath, J.; Kumaresan, N.; Vincent Antony Kumar, A., 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
[4] Balasubramaniam, P.; Abdul Samath, J.; Kumaresan, N., Optimal control for nonlinear singular systems with quadratic performance using neural networks, Appl. math. comput., 187, 1535-1543, (2007) · Zbl 1114.65336
[5] Balasubramaniam, P.; Abdul Samath, J.; Kumaresan, N.; Vincent Antony Kumar, A., Neuro approach for solving matrix Riccati differential equation, Neural parallel sci. comput., 15, 125-135, (2007) · Zbl 1192.93050
[6] Bensoussan, A., Lecture on stochastic control part I, (), 1-39
[7] Bucci, F.; Pandolfi, L., The regulator problem with indefinite quadratic cost for boundary control systems: the finite horizon case, Syst. control lett., 39, 79-86, (2000) · Zbl 0943.49024
[8] Campbell, S.L., Singular systems of differential equations, (1980), Pitman Marshfield · Zbl 0419.34007
[9] Campbell, S.L., Singular systems of differential equations II, (1982), Pitman Marshfield · Zbl 0482.34008
[10] Chen, S.P.; Li, X.J.; Zho, X.Y., Stochastic linear quadratic regulators with indefinite control weight costs, SIAM J. control optim., 36, 5, 1685-1702, (1998) · Zbl 0916.93084
[11] Choi, Chiu H., A survey of numerical methods for solving matrix Riccati differential equation, IEEE proc. southeastcon, 696-700, (1990)
[12] Da Prato, G.; Ichikawa, A., Quadratic control for linear periodic systems, Appl. math. optim., 18, 39-66, (1988) · Zbl 0647.93057
[13] Davis, M.H.A., Linear estimation and stochastic control, (1977), Chapman and Hall London · Zbl 0437.60001
[14] Ellacott, S.W., Aspects of the numerical analysis of neural networks, Acta numer., 5, 145-202, (1994) · Zbl 0807.65007
[15] Hagan, M.T.; Menhaj, M., Training feedforward networks with the Marquardt algorithm, IEEE trans. neural networ., 5, 6, 989-993, (1994)
[16] Ham, F.M.; Collins, E.G., A neurocomputing approach for solving the algebraic matrix Riccati equation, Proc. IEEE int. conf. neural networ., 1, 617-622, (1996)
[17] Karakasoglu, A.; Sudharsanan, S.L.; Sundareshan, M.K., Identification and decentralized adaptive control using neural networks with application to robotic manipulators, IEEE trans. neural networ., 4, 919-930, (1993)
[18] Lagaris, I.E.; Likas, A.; Fotiadis, D.I., Artificial neural networks for solving ordinary and partial differential equations, IEEE trans. neural networ., 9, 987-1000, (1998)
[19] Lewis, F.L., A survey of linear singular systems, Circ. syst. sig. proc., 5, 1, 3-36, (1986) · Zbl 0613.93029
[20] Narendra, K.S.; Parathasarathy, K., Identification and control of dynamical systems using neural networks, IEEE trans. neural networ., 1, 4-27, (1990)
[21] Nguyen, D.; Widrow, B., Improving the learning speed of 2-layer neural networks by choosing initial values of the adaptive weights, Proc. int. joint conf. neural networ., III, 21-26, (1990)
[22] A.P. Paplinski, Lecture Notes on Feedforward Multilayer Neural Networks, NNet (L.5) 2004.
[23] Wang, J.; Wu, G., A multilayer recurrent neural network for solving continuous time algebraic Riccati equations, Neural networ., 11, 939-950, (1998)
[24] De Wilde, P., Neural network models, (1997), Springer-Verlag London · Zbl 0869.68070
[25] Wonham, W.M., On a matrix Riccati equation of stochastic control, SIAM J. control optim., 6, 681-697, (1968) · Zbl 0182.20803
[26] 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
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