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

Computation of bang-bang and singular controls in collision avoidance

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.
Citation: Helmut Maurer, Tanya Tarnopolskaya, Neale Fulton. Computation of bang-bang and singular controls in collision avoidance. Journal of Industrial & 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.  doi: 10.3934/naco.2012.2.511.  Google Scholar

[2]

M. S. Aronna, Second Order Analysis of Optimal Control Problems with Singular Arcs. Optimality Conditions and Shooting Algorithm,, Ph.D thesis, (2011).   Google Scholar

[3]

C. Büskens, Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer- und Zustands-Beschränkungen,, Ph.D thesis, (1998).   Google Scholar

[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.  doi: 10.1016/S0377-0427(00)00305-8.  Google Scholar

[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\, (2001), 3.   Google Scholar

[6]

R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming,, Duxbury Press, (1993).   Google Scholar

[7]

M. Hestens, Calculus of Variations and Optimal Control Theory,, John Wiley & Sons, (1966).   Google Scholar

[8]

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

[9]

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

[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.  doi: 10.1002/oca.756.  Google Scholar

[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.  doi: 10.1137/S0363012900377419.  Google Scholar

[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).   Google Scholar

[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), (2012), 7697.  doi: 10.1109/CDC.2012.6426792.  Google Scholar

[14]

A. W. Merz, Optimal aircraft collision avoidance,, in Proceedings of the Joint Automatic Control Conference, (1973), 15.   Google Scholar

[15]

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

[16]

H. J. Oberle, Numerical computation of singular control functions in trajectory optimization problems,, J. of Guidance, 13 (1990), 153.  doi: 10.2514/3.20529.  Google Scholar

[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, (2012).  doi: 10.1137/1.9781611972368.  Google Scholar

[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.   Google Scholar

[19]

L. S. Pontryagin, W. G. Boltyanski, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, Wiley, (1965).   Google Scholar

[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.  doi: 10.1007/s10957-008-9452-9.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s10957-009-9597-1.  Google Scholar

[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, (2010), 1789.   Google Scholar

[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.   Google Scholar

[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), (2011), 9775.   Google Scholar

[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.  doi: 10.1007/s10957-012-0049-y.  Google Scholar

[27]

G. Vossen, Numerische Lösungsmethoden, Hinreichende Optimalitätsbedingungen und Sensitivitätsanalyse für Optimale Bang-Bang und Singuläre Steuerungen,, Ph.D thesis, (2005).   Google Scholar

[28]

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

[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.  doi: 10.1007/s10107-004-0559-y.  Google Scholar

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.  doi: 10.3934/naco.2012.2.511.  Google Scholar

[2]

M. S. Aronna, Second Order Analysis of Optimal Control Problems with Singular Arcs. Optimality Conditions and Shooting Algorithm,, Ph.D thesis, (2011).   Google Scholar

[3]

C. Büskens, Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer- und Zustands-Beschränkungen,, Ph.D thesis, (1998).   Google Scholar

[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.  doi: 10.1016/S0377-0427(00)00305-8.  Google Scholar

[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\, (2001), 3.   Google Scholar

[6]

R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming,, Duxbury Press, (1993).   Google Scholar

[7]

M. Hestens, Calculus of Variations and Optimal Control Theory,, John Wiley & Sons, (1966).   Google Scholar

[8]

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

[9]

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

[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.  doi: 10.1002/oca.756.  Google Scholar

[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.  doi: 10.1137/S0363012900377419.  Google Scholar

[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).   Google Scholar

[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), (2012), 7697.  doi: 10.1109/CDC.2012.6426792.  Google Scholar

[14]

A. W. Merz, Optimal aircraft collision avoidance,, in Proceedings of the Joint Automatic Control Conference, (1973), 15.   Google Scholar

[15]

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

[16]

H. J. Oberle, Numerical computation of singular control functions in trajectory optimization problems,, J. of Guidance, 13 (1990), 153.  doi: 10.2514/3.20529.  Google Scholar

[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, (2012).  doi: 10.1137/1.9781611972368.  Google Scholar

[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.   Google Scholar

[19]

L. S. Pontryagin, W. G. Boltyanski, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, Wiley, (1965).   Google Scholar

[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.  doi: 10.1007/s10957-008-9452-9.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s10957-009-9597-1.  Google Scholar

[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, (2010), 1789.   Google Scholar

[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.   Google Scholar

[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), (2011), 9775.   Google Scholar

[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.  doi: 10.1007/s10957-012-0049-y.  Google Scholar

[27]

G. Vossen, Numerische Lösungsmethoden, Hinreichende Optimalitätsbedingungen und Sensitivitätsanalyse für Optimale Bang-Bang und Singuläre Steuerungen,, Ph.D thesis, (2005).   Google Scholar

[28]

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

[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.  doi: 10.1007/s10107-004-0559-y.  Google Scholar

[1]

Karl Kunisch, Lijuan Wang. The bang-bang property of time optimal controls for the Burgers equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3611-3637. doi: 10.3934/dcds.2014.34.3611

[2]

Karl Kunisch, Lijuan Wang. Bang-bang property of time optimal controls of semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 279-302. doi: 10.3934/dcds.2016.36.279

[3]

Walter Alt, Robert Baier, Matthias Gerdts, Frank Lempio. Error bounds for Euler approximation of linear-quadratic control problems with bang-bang solutions. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 547-570. doi: 10.3934/naco.2012.2.547

[4]

Gengsheng Wang, Yubiao Zhang. Decompositions and bang-bang properties. Mathematical Control & Related Fields, 2017, 7 (1) : 73-170. doi: 10.3934/mcrf.2017005

[5]

M. Soledad Aronna, J. Frédéric Bonnans, Andrei V. Dmitruk, Pablo A. Lotito. Quadratic order conditions for bang-singular extremals. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 511-546. doi: 10.3934/naco.2012.2.511

[6]

Jiaqin Wei. Time-inconsistent optimal control problems with regime-switching. Mathematical Control & Related Fields, 2017, 7 (4) : 585-622. doi: 10.3934/mcrf.2017022

[7]

Thomas I. Seidman. Optimal control of a diffusion/reaction/switching system. Evolution Equations & Control Theory, 2013, 2 (4) : 723-731. doi: 10.3934/eect.2013.2.723

[8]

Fabio Bagagiolo. Optimal control of finite horizon type for a multidimensional delayed switching system. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 239-264. doi: 10.3934/dcdsb.2005.5.239

[9]

Galina Kurina, Sahlar Meherrem. Decomposition of discrete linear-quadratic optimal control problems for switching systems. Conference Publications, 2015, 2015 (special) : 764-774. doi: 10.3934/proc.2015.0764

[10]

Fabio Bagagiolo. An infinite horizon optimal control problem for some switching systems. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 443-462. doi: 10.3934/dcdsb.2001.1.443

[11]

Matthias Gerdts, René Henrion, Dietmar Hömberg, Chantal Landry. Path planning and collision avoidance for robots. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 437-463. doi: 10.3934/naco.2012.2.437

[12]

M. Delgado-Téllez, Alberto Ibort. On the geometry and topology of singular optimal control problems and their solutions. Conference Publications, 2003, 2003 (Special) : 223-233. doi: 10.3934/proc.2003.2003.223

[13]

Enkhbat Rentsen, J. Zhou, K. L. Teo. A global optimization approach to fractional optimal control. Journal of Industrial & Management Optimization, 2016, 12 (1) : 73-82. doi: 10.3934/jimo.2016.12.73

[14]

Piermarco Cannarsa, Cristina Pignotti, Carlo Sinestrari. Semiconcavity for optimal control problems with exit time. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 975-997. doi: 10.3934/dcds.2000.6.975

[15]

Jérome Lohéac, Jean-François Scheid. Time optimal control for a nonholonomic system with state constraint. Mathematical Control & Related Fields, 2013, 3 (2) : 185-208. doi: 10.3934/mcrf.2013.3.185

[16]

Piermarco Cannarsa, Carlo Sinestrari. On a class of nonlinear time optimal control problems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 285-300. doi: 10.3934/dcds.1995.1.285

[17]

Adriano Festa, Andrea Tosin, Marie-Therese Wolfram. Kinetic description of collision avoidance in pedestrian crowds by sidestepping. Kinetic & Related Models, 2018, 11 (3) : 491-520. doi: 10.3934/krm.2018022

[18]

Shihchung Chiang. Numerical optimal unbounded control with a singular integro-differential equation as a constraint. Conference Publications, 2013, 2013 (special) : 129-137. doi: 10.3934/proc.2013.2013.129

[19]

Térence Bayen, Marc Mazade, Francis Mairet. Analysis of an optimal control problem connected to bioprocesses involving a saturated singular arc. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 39-58. doi: 10.3934/dcdsb.2015.20.39

[20]

Pierluigi Colli, Gianni Gilardi, Gabriela Marinoschi, Elisabetta Rocca. Optimal control for a phase field system with a possibly singular potential. Mathematical Control & Related Fields, 2016, 6 (1) : 95-112. doi: 10.3934/mcrf.2016.6.95

2018 Impact Factor: 1.025

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (2)

[Back to Top]