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April  2014, 10(2): 443-460. doi: 10.3934/jimo.2014.10.443

Computation of bang-bang and singular controls in collision avoidance

Helmut Maurer 1, , Tanya Tarnopolskaya 2, and  Neale Fulton 3,

1. 

Institute of Computational and Applied Mathematics, University of Muenster, Einsteinstr. 62, D-48149 Muenster, Germany
2. 

CSIRO Computational Informatics, Locked Bag 17, North Ryde NSW 1670, Australia
3. 

CSIRO Computational Informatics, GPO Box 664, Canberra ACT 2601, Australia

Received  November 2012 Revised  August 2013 Published  October 2013

We study optimal cooperative collision avoidance strategies for two participants in a planar close proximity encounter. Previous research focused on special cases of this problem and showed that bang-bang strategies without switching are optimal in most situations, while singular controls only appear for the case of participants with unequal linear speeds under certain conditions. This paper extends the earlier analyses to a general case of a coplanar close proximity encounter, for which both parameters of the problem may take arbitrary admissible values. For such a case, we present a theoretical and numerical study of the structure of optimal controls. We prove that both controls can not be singular simultaneously and that the only possible singular control is a zero control. We derive formulas for the singular surfaces and verify that sufficient conditions hold for the computed extremal solutions. We identify different types of structural changes of the control strategies and show how the control structure changes with the change in the model parameters and initial conditions.
Keywords: collision avoidance, singular control, Optimal control, bang-bang conrol, switching time optimization..
Mathematics Subject Classification: Primary: 49N90, 49K15, 49K10; Secondary: 49M2.
Citation: Helmut Maurer, Tanya Tarnopolskaya, Neale Fulton. Computation of bang-bang and singular controls in collision avoidance. Journal of Industrial and Management Optimization, 2014, 10 (2) : 443-460. doi: 10.3934/jimo.2014.10.443
References:
[1]

M. S. Aronna, J. F. Bonnans, A. V. Dmitruk and P. A. Lotito, Quadratic conditions for bang-singular extremals, Numer. Algebra Control Optim., 2 (2012), 511-546. doi: 10.3934/naco.2012.2.511.

[2]

M. S. Aronna, Second Order Analysis of Optimal Control Problems with Singular Arcs. Optimality Conditions and Shooting Algorithm, Ph.D thesis, ÉEcole Polytechnique Palaiseau, France, 2011.

[3]

C. Büskens, Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer- und Zustands-Beschränkungen, Ph.D thesis, Institut für Numerische Mathematik, Universität Münster, Münster, Germany, 1998.

[4]

C. Büskens and H. Maurer, SQP-methods for solving optimal control problems with control and state constraints: Adjoint variables, sensitivity analysis and real-time control. SQP-based direct discretization methods for practical optimal control problems, J. Comput. Appl. Math., 120 (2000), 85-108. doi: 10.1016/S0377-0427(00)00305-8.

[5]

C. Büskens and H. Maurer, Sensitivity analysis and real-time optimization of parametric nonlinear programming problems, in Online Optimization of Large Scale Systems (eds. M. Gr\"otschel, S. O. Krumke and J. Rambau), Springer, Berlin, 2001, 3-16.

[6]

R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Duxbury Press, Brooks-Cole Publishing, 1993.

[7]

M. Hestens, Calculus of Variations and Optimal Control Theory, John Wiley & Sons, Inc., New York-London-Sydney, 1966.

[8]

A. J. Krener, The high order maximum principle and its application to singular extremals, SIAM J. Control and Optimization, 15 (1977), 256-293. doi: 10.1137/0315019.

[9]

H. Maurer, Numerical solution of singular control problems using multiple shooting methods, J. Optimization Theory and Applications, 18 (1976), 235-257. doi: 10.1007/BF00935706.

[10]

H. Maurer, C. Büskens, J.-H. R. Kim and Y. Kaya, Optimization methods for the verification of second-order sufficient conditions for bang-bang controls, Optimal Control Appl. Methods, 26 (2005), 129-156. doi: 10.1002/oca.756.

[11]

H. Maurer and H. J. Oberle, Second order sufficient conditions for optimal control problems with free final time: The Riccati approach, SIAM J. Control and Optim., 41 (2002), 380-403. doi: 10.1137/S0363012900377419.

[12]

H. Maurer, T. Tarnopolskaya and N. L. Fulton, Singular controls in optimal collision avoidance for participants with unequal linear speeds, ANZIAM J., 53 (2012), C1-C19.

[13]

H. Maurer, T. Tarnopolskaya and N. L. Fulton, Optimal bang-bang and singular controls in collision avoidance for participants with unequal linear speeds, in 51st IEEE Conference on Decision and Control (CDC), Maui, USA, 2012, 7697-7702. doi: 10.1109/CDC.2012.6426792.

[14]

A. W. Merz, Optimal aircraft collision avoidance, in Proceedings of the Joint Automatic Control Conference, Paper 15-3, 1973, 449-454.

[15]

A. W. Merz, Optimal evasive manoeuvres in maritime collision avoidance, Navigation, 20 (1973), 144-152.

[16]

H. J. Oberle, Numerical computation of singular control functions in trajectory optimization problems, J. of Guidance, Control and Dynam., 13 (1990), 153-159. doi: 10.2514/3.20529.

[17]

N. P. Osmolovskii and H. Maurer, Applications to Regular and Bang-Bang Control. Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control, Advances in Control and Design, Vol. 24, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2012. doi: 10.1137/1.9781611972368.

[18]

N. P. Osmolovskii and H. Maurer, Equivalence of second order optimality conditions for bang-bang control problems. I. Main results, Control and Cybernetics, 34 (2005), 927-950; II. Proofs, variational derivatives and representations, Control and Cybernetics, 36 (2007), 5-45.

[19]

L. S. Pontryagin, W. G. Boltyanski, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Wiley, New York, 1965.

[20]

T. Tarnopolskaya and N. L. Fulton, Optimal cooperative collision avoidance strategy for coplanar encounter: Merz's solution revisited, J. Optim. Theory Appl., 140 (2009), 355-375. doi: 10.1007/s10957-008-9452-9.

[21]

T. Tarnopolskaya and N. L. Fulton, Parametric behavior of the optimal control solution for collision avoidance in a close proximity encounter, in 18th World IMACS Congress and MODSIM09 International Congress on Modelling and Simulation (eds. R. S. Andersson et al.), 2009, 425-431.

[22]

T. Tarnopolskaya and N. L. Fulton, Synthesis of optimal control for cooperative collision avoidance for aircraft (ships) with unequal turn capabilities, J. Optim. Theory Appl., 144 (2010), 367-390. doi: 10.1007/s10957-009-9597-1.

[23]

T. Tarnopolskaya and N. L. Fulton, Dispersal curves for optimal collision avoidance in a close proximity encounter: A case of participants with unequal turn rates, in Lecture Notes in Engineering and Computer Science: Proceedings of The World Congress on Engineering 2010, WCE 2010, IAENG, 2010, 1789-1794.

[24]

T. Tarnopolskaya and N. L. Fulton, Non-unique optimal collision avoidance strategies for coplanar encounter of participants with unequal turn capabilities, IAENG Int. J. Appl. Math., 40 (2010), 289-296.

[25]

T. Tarnopolskaya and N. L. Fulton, Synthesis of optimal control for cooperative collision avoidance in a close proximity encounter: Special cases, in Proceedings of the 18th World Congress of the International Federation of Automatic Control (IFAC), Vol. 18, Milano, Italy, 2011, 9775-9781.

[26]

T. Tarnopolskaya, N. L. Fulton and H. Maurer, Synthesis of optimal bang-bang control for cooperative collision avoidance for aircraft (ships) with unequal linear speeds, J. Optim. Theory Appl., 155 (2012), 115-144. doi: 10.1007/s10957-012-0049-y.

[27]

G. Vossen, Numerische Lösungsmethoden, Hinreichende Optimalitätsbedingungen und Sensitivitätsanalyse für Optimale Bang-Bang und Singuläre Steuerungen, Ph.D thesis, Institut für Numerische und Angewandte Mathematik, Universität Münster, Münster, Germany, 2005.

[28]

G. Vossen, Switching time optimization for bang-bang and singular controls, J. Optim. Theory Appl., 144 (2010), 409-429. doi: 10.1007/s10957-009-9594-4.

[29]

A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), 25-57. doi: 10.1007/s10107-004-0559-y.

show all references

References:
[1]

M. S. Aronna, J. F. Bonnans, A. V. Dmitruk and P. A. Lotito, Quadratic conditions for bang-singular extremals, Numer. Algebra Control Optim., 2 (2012), 511-546. doi: 10.3934/naco.2012.2.511.

[2]

M. S. Aronna, Second Order Analysis of Optimal Control Problems with Singular Arcs. Optimality Conditions and Shooting Algorithm, Ph.D thesis, ÉEcole Polytechnique Palaiseau, France, 2011.

[3]

C. Büskens, Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer- und Zustands-Beschränkungen, Ph.D thesis, Institut für Numerische Mathematik, Universität Münster, Münster, Germany, 1998.

[4]

C. Büskens and H. Maurer, SQP-methods for solving optimal control problems with control and state constraints: Adjoint variables, sensitivity analysis and real-time control. SQP-based direct discretization methods for practical optimal control problems, J. Comput. Appl. Math., 120 (2000), 85-108. doi: 10.1016/S0377-0427(00)00305-8.

[5]

C. Büskens and H. Maurer, Sensitivity analysis and real-time optimization of parametric nonlinear programming problems, in Online Optimization of Large Scale Systems (eds. M. Gr\"otschel, S. O. Krumke and J. Rambau), Springer, Berlin, 2001, 3-16.

[6]

R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Duxbury Press, Brooks-Cole Publishing, 1993.

[7]

M. Hestens, Calculus of Variations and Optimal Control Theory, John Wiley & Sons, Inc., New York-London-Sydney, 1966.

[8]

A. J. Krener, The high order maximum principle and its application to singular extremals, SIAM J. Control and Optimization, 15 (1977), 256-293. doi: 10.1137/0315019.

[9]

H. Maurer, Numerical solution of singular control problems using multiple shooting methods, J. Optimization Theory and Applications, 18 (1976), 235-257. doi: 10.1007/BF00935706.

[10]

H. Maurer, C. Büskens, J.-H. R. Kim and Y. Kaya, Optimization methods for the verification of second-order sufficient conditions for bang-bang controls, Optimal Control Appl. Methods, 26 (2005), 129-156. doi: 10.1002/oca.756.

[11]

H. Maurer and H. J. Oberle, Second order sufficient conditions for optimal control problems with free final time: The Riccati approach, SIAM J. Control and Optim., 41 (2002), 380-403. doi: 10.1137/S0363012900377419.

[12]

H. Maurer, T. Tarnopolskaya and N. L. Fulton, Singular controls in optimal collision avoidance for participants with unequal linear speeds, ANZIAM J., 53 (2012), C1-C19.

[13]

H. Maurer, T. Tarnopolskaya and N. L. Fulton, Optimal bang-bang and singular controls in collision avoidance for participants with unequal linear speeds, in 51st IEEE Conference on Decision and Control (CDC), Maui, USA, 2012, 7697-7702. doi: 10.1109/CDC.2012.6426792.

[14]

A. W. Merz, Optimal aircraft collision avoidance, in Proceedings of the Joint Automatic Control Conference, Paper 15-3, 1973, 449-454.

[15]

A. W. Merz, Optimal evasive manoeuvres in maritime collision avoidance, Navigation, 20 (1973), 144-152.

[16]

H. J. Oberle, Numerical computation of singular control functions in trajectory optimization problems, J. of Guidance, Control and Dynam., 13 (1990), 153-159. doi: 10.2514/3.20529.

[17]

N. P. Osmolovskii and H. Maurer, Applications to Regular and Bang-Bang Control. Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control, Advances in Control and Design, Vol. 24, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2012. doi: 10.1137/1.9781611972368.

[18]

N. P. Osmolovskii and H. Maurer, Equivalence of second order optimality conditions for bang-bang control problems. I. Main results, Control and Cybernetics, 34 (2005), 927-950; II. Proofs, variational derivatives and representations, Control and Cybernetics, 36 (2007), 5-45.

[19]

L. S. Pontryagin, W. G. Boltyanski, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Wiley, New York, 1965.

[20]

T. Tarnopolskaya and N. L. Fulton, Optimal cooperative collision avoidance strategy for coplanar encounter: Merz's solution revisited, J. Optim. Theory Appl., 140 (2009), 355-375. doi: 10.1007/s10957-008-9452-9.

[21]

T. Tarnopolskaya and N. L. Fulton, Parametric behavior of the optimal control solution for collision avoidance in a close proximity encounter, in 18th World IMACS Congress and MODSIM09 International Congress on Modelling and Simulation (eds. R. S. Andersson et al.), 2009, 425-431.

[22]

T. Tarnopolskaya and N. L. Fulton, Synthesis of optimal control for cooperative collision avoidance for aircraft (ships) with unequal turn capabilities, J. Optim. Theory Appl., 144 (2010), 367-390. doi: 10.1007/s10957-009-9597-1.

[23]

T. Tarnopolskaya and N. L. Fulton, Dispersal curves for optimal collision avoidance in a close proximity encounter: A case of participants with unequal turn rates, in Lecture Notes in Engineering and Computer Science: Proceedings of The World Congress on Engineering 2010, WCE 2010, IAENG, 2010, 1789-1794.

[24]

T. Tarnopolskaya and N. L. Fulton, Non-unique optimal collision avoidance strategies for coplanar encounter of participants with unequal turn capabilities, IAENG Int. J. Appl. Math., 40 (2010), 289-296.

[25]

T. Tarnopolskaya and N. L. Fulton, Synthesis of optimal control for cooperative collision avoidance in a close proximity encounter: Special cases, in Proceedings of the 18th World Congress of the International Federation of Automatic Control (IFAC), Vol. 18, Milano, Italy, 2011, 9775-9781.

[26]

T. Tarnopolskaya, N. L. Fulton and H. Maurer, Synthesis of optimal bang-bang control for cooperative collision avoidance for aircraft (ships) with unequal linear speeds, J. Optim. Theory Appl., 155 (2012), 115-144. doi: 10.1007/s10957-012-0049-y.

[27]

G. Vossen, Numerische Lösungsmethoden, Hinreichende Optimalitätsbedingungen und Sensitivitätsanalyse für Optimale Bang-Bang und Singuläre Steuerungen, Ph.D thesis, Institut für Numerische und Angewandte Mathematik, Universität Münster, Münster, Germany, 2005.

[28]

G. Vossen, Switching time optimization for bang-bang and singular controls, J. Optim. Theory Appl., 144 (2010), 409-429. doi: 10.1007/s10957-009-9594-4.

[29]

A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), 25-57. doi: 10.1007/s10107-004-0559-y.

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