Symplectic Numerical Approach for Nonlinear Optimal Control of Systems with Inequality Constraints
Abstract: This paper proposes a system representation for unifying control design and numerical calculation in nonlinear optimal control problems with inequality constraints in terms of the symplectic structure. The symplectic structure is derived from Hamiltonian systems that are equivalent to Hamilton-Jacobi equations. In the representation, the constraints can be described as an input-state transformation of the system. Therefore, it can be seamlessly applied to the stable manifold method that is a precise numerical solver of the Hamilton-Jacobi equations. In conventional methods, e.g., the penalty method or the barrier method, it is difficult to systematically assign the weights of penalty functions that are used for realizing the constraints. In the proposed method, we can separate the adjustment of weights with respect to objective functions from that of penalty functions. Furthermore, the proposed method can extend the region of computable solutions in a state space. The validity of the method is shown by a numerical example of the optimal control of a vehicle model with steering limitations.
Cite this paper: Abe, Y. , Nishida, G. , Sakamoto, N. and Yamamoto, Y. (2015) Symplectic Numerical Approach for Nonlinear Optimal Control of Systems with Inequality Constraints. International Journal of Modern Nonlinear Theory and Application, 4, 234-248. doi: 10.4236/ijmnta.2015.44018.
References

[1]   Liberzon, D. (2012) Calculus of Variations and Optimal Control Theory. Princeton University Press, Princeton.

[2]   Sakamoto, N. and van der Schaft, A.J. (2008) Analytical Approximation Methods for the Stabilizing Solution of the Hamilton-Jacobi Equation. IEEE Transactions on Automatic Control, 53, 2335-2350.
http://dx.doi.org/10.1109/TAC.2008.2006113

[3]   Sakamoto, N. (2013) Case Studies on the Application of the Stable Manifold Approach for Nonlinear Optimal Control Design. Automatica, 49, 568-576.
http://dx.doi.org/10.1016/j.automatica.2012.11.032

[4]   Courant, R. (1943) Variational Methods for the Solution of Problems of Equilibrium and Vibrations. Bulletin of the American Mathematical Society, 49, 1-23.
http://dx.doi.org/10.1090/S0002-9904-1943-07818-4

[5]   Karmarkar, N. (1984) A New Polynomial-Time Algorithm for Linear Programming. Combinatorica, 4, 373-395.
http://dx.doi.org/10.1007/BF02579150

[6]   Jacobson, D. and Lele, M. (1969) A Transformation Technique for Optimal Control Problems with a State Variable Inequality Constraint. IEEE Transactions on Automatic Control, 14, 457-464.
http://dx.doi.org/10.1109/TAC.1969.1099283

[7]   Goh, C.J. and Teo, K.L. (19880 Control Parametrization: A Unified Approach to Optimal Control Problems with General Constraints. Automatica, 24, 3-18.
http://dx.doi.org/10.1016/0005-1098(88)90003-9

[8]   Hager, W.W. (1990) Multiplier Methods for Nonlinear Optimal Control. SIAM Journal on Numerical Analysis, 27, 1061-1080.
http://dx.doi.org/10.1137/0727063

[9]   Neittaanmaki, P. and Stachurski, A. (1992) Solving Some Optimal Control Problems Using the Barrier Penalty Function Method. Applied Mathematics and Optimization, 25, 127-149.
http://dx.doi.org/10.1007/BF01182477

[10]   Willsa, A.G. and Heath, W.P. (2004) Barrier Function Based Model Predictive Control. Automatica, 40, 1415-1422.
http://dx.doi.org/10.1016/j.automatica.2004.03.002

[11]   Graichen, K. and Petit, N. (2009) Incorporating a Class of Constraints into the Dynamics of Optimal Control Problems. Optimal Control Applications and Methods, 30, 537-561.
http://dx.doi.org/10.1002/oca.880

[12]   Foroozandeh, Z. and Shamsi, M. (2012) Solution of Nonlinear Optimal Control Problems by the Interpolating Scaling Functions. Acta Astronautica, 72, 21-26.
http://dx.doi.org/10.1016/j.actaastro.2011.10.004

[13]   Isidori, A. (1995) Nonlinear Control Systems. 3rd Edition, Springer, Berlin.
http://dx.doi.org/10.1007/978-1-84628-615-5

[14]   Boyd, S. and Vandenberghe, L. (2004) Convex Optimization. Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/CBO9780511804441

[15]   van der Schaft, A.J. (1991) On a State-Space Approach To Nonlinear -Control. Systems & Control Letters, 16, 1-8.
http://dx.doi.org/10.1016/0167-6911(91)90022-7

[16]   Rajamani, R. (2012) Vehicle Dynamics and Control. 2nd Edition, Springer, Berlin.
http://dx.doi.org/10.1007/978-1-4614-1433-9

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