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A Cost Reduction Procedure for Control-Restricted Nonlinear Systems

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This paper describes a numerical scheme to approximate the solution of the optimal control problem for nonlinear systems with restrictions on the manipulated variables. The method proposed here systematically reduces the cost associated with successively updated control strategies after proposing an initial seed trajectory. It follows two main lines of reasoning, the first one relying on linearizations around a seed state/control trajectory and exploiting a theoretical expression for the increment of the cost. This setup is valid in regular situations, and it can be used when saturations occur after some adaptations. One of its advantages is that the decreasing of the cost can be assessed without integrating numerically the nonlinear dynamics. However, and because of the constraints, eventually this method fails, and an alternative approach must be activated to continue decreasing the cost. The alternative approach is based on specific control variations of the current control strategy, and it is activated depending on two theoretical criteria (failure alert) developed here. The first control variation proposed is derived from the differential Riccati equation for the linearized system and appropriate quadratic cost functions. Other variations, similar to those used in Pontryagin theorem for generating the final cone of states, are proposed by modifying the locations of ‘switching times’, producing oscillations in the interior of regular periods. The performance of the numerical proposed method and the related mathematical objects are illustrated by optimizing two-dimensional nonlinear systems with a scalar bounded control.
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Optimal Control; Restricted Controls; Nonlinear Systems; Control Variations; LQR Problems

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A. A. Agrachev, Yu. L. Sachkov, Control Theory from the Geometric Viewpoint (Springer-Verlag, 2004).

M. Athans, P. L. Falb, Optimal Control: An Introduction to the Theory and Its Applications (Dover, 2004).

V. Costanza, P. S. Rivadeneira, Enfoque Hamiltoniano al Control Óptimo de Sistemas Dinámicos (OmniScriptum, 2014).

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, The Mathematical Theory of Optimal Process (Macmillan, 1964).

J. L. Troutman, Variational Calculus and Optimal Control (Springer, 1996).

R. E. Bellman, Dynamic Programming (Dover, 1957).

P. Bernhard, Introducción a la Teória de Control Óptimo, Technical Report Cuaderno Nro 4, Instituto de Matemática “Beppo Levi”, Rosario, 1972.

V. Costanza, C. E. Neuman, Optimal Control of Nonlinear Chemical Reactors Via an Initial-Value Hamiltonian Problem, Optimal Control Applications & Methods, Vol. 27:41-60, 2006.

V. Costanza, P. S. Rivadeneira, Finite-Horizon Dynamic Optimization of Nonlinear Systems in Real Time, Automatica, Vol. 44:2427-2434, 2008.

V. Costanza, P. S. Rivadeneira, R. D. Spies, Equations for the Missing Boundary Values in the Hamiltonian Formulation of Optimal Control Problems, Journal of Optimizations Theory and Applications, Vol. 149:26-46, 2009.

A. V. Rao, D. A. Benson, G. T. Huntington, C. Francolin, C. L. Darby, M. A. Patterson, User’s Manual for Gpops: A Matlab Package for Dynamic Optimization Using the Gauss Pseudospectral Method, Technical Report, University of Florida, August 1972.

S. J. Qin, T. A. Badgwell, A Survey of Industrial Model Predictive Control Technology, Control Engineering Practice, Vol. 11:733-764, 2003.

E. D. Sontag, Mathematical Control Theory (Springer, 1988).

J. L. Speyer, D. H. Jacobson, Primer on Optimal Control Theory (SIAM Books, 2010).

V. Jurdjievic, Geometric Control Theory (Cambridge, 2006).

M. Itik, Optimal Control of Nonlinear Systems with Input Constraints Using Linear Time Varying Approximations, Nonlinear Analysis Modelling and Control, Vol. 21 (Issue 3):400-412, 2016.

X. Wu, K. Zhang, Constrained Optimal Control Problems of Nonlinear Systems Based on Improved Newton Algorithms, In 3rd International Conference on Informative and Cybernetics for Computational Social, 2016.

R. E. Kalman, P. L. Falb, M. A. Arbib, Topics in Mathematical System Theory (McGraw-Hill, 1969).

V. Costanza, P. S. Rivadeneira, J. A. Gómez Múnera, An Efficient Cost Reduction Procedure for Bounded-Control LQR Problems, Computational and Applied Mathematics, 2016.

L. C. Evans, An Introduction to Mathematical Optimal Control Theory Version 0.2.

W. H. Fleming, R. W. Rishel, Deterministic and Sthocastic Optimal Control (Dover, 1975).

V. Costanza, P. S. Rivadeneira, Optimal Satured Feedback Laws for LQR Problems with Bounded Controls, Computational and Applied Mathematics, Vol. 32:355-371, 2013.

V. Costanza, P. S. Rivadeneira, A. H. González, Minimizing Control Energy in a Class Bounded-Control LQR Problems. Optimal Control Applications & Methods, Vol. 35(Issue 3):361-382, 2013.

P. Howlett, The Optimal Control of Train, Annals of Operation Research, 98(1-4):65-87, 2000.

P. Howlett, P. J. Pudney, X. Vu, Local Energy Minimization in Optimal Train Control, Automatica, 45(11):2692-2698, 2009.


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