September  2015, 35(9): 4665-4681. doi: 10.3934/dcds.2015.35.4665

Control of dynamical systems with discrete and uncertain observations

1. 

Scientific Systems Company, Inc, 500 West Cummings Park, Suite 3000, Woburn, MA 01801, United States

2. 

Coordinated Science Laboratory and the Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, United States

Received  April 2014 Revised  October 2014 Published  April 2015

In this paper we provide a design methodology to compute control strategies for a primary dynamical system which is operating in a domain where other dynamical systems are present and the interactions between these systems and the primary one are of interest or are being pursued in some sense. The information from the other systems is available to the primary dynamical system only at discrete time instances and is assumed to be corrupted by noise. Having available only this limited and somewhat corrupted information which depends on the noise, the primary system has to make a decision based on the estimated behavior of other systems which may range from cooperative to noncooperative. This decision is reflected in a design of the most appropriate action, that is, control strategy of the primary system. The design is illustrated by considering some particular collision avoidance problem scenarios.
Citation: Aleksandar Zatezalo, Dušan M. Stipanović. Control of dynamical systems with discrete and uncertain observations. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4665-4681. doi: 10.3934/dcds.2015.35.4665
References:
[1]

T. Başar and G. J. Olsder, Dynamic Noncooperative Game Theory, Revised and updated 2nd edition, SIAM, Philadelphia, PA, 1999.

[2]

R. H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition, AIAA Education Series, 1999. doi: 10.2514/4.861543.

[3]

P. Billingsley, Probability and Measure, 2nd edition, John Willey & Sons, New York, 1986.

[4]

A. M. Bloch (with the collaboration of J. Baillieul, P. Crouch and J. Marsden), Nonholonomic Mechanics and Control, Springer-Verlag, New York, 2003. doi: 10.1007/b97376.

[5]

A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences, Springfield, MO, 2007.

[6]

L. Buonanno, A new Gronwall-Bellman inequality for discontinuous functions, Journal of Interdisciplinary Mathematics, 9 (2006), 543-550.

[7]

F. L. Chernousko and A. A. Melikyan, Some differential games with incomplete information, Lecture Notes in Computer Science, Springer-Verlag, Berlin, 27 (1975), 445-450. doi: 10.1007/3-540-07165-2_62.

[8]

A. Désilles, H. Zidani and E. Crück, Collision analysis for an UAV, Proceedings of the AIAA Guidance, Navigation, and Control Conference, Minneapolis, MN, 2012.

[9]

W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, 2nd edition, Springer, New York, 2006.

[10]

R. A. Freeman and P. V. Kokotović, Robust Nonlinear Control Design: State Space and Lyapunov Techniques, Birkhäuser, Boston, MA, 1996. doi: 10.1007/978-0-8176-4759-9.

[11]

R. Isaacs, Differential Games, A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization, Dover, Mineola, 1999.

[12]

I. Ya. Kats and A. A. Martynyuk, Stability and Stabilization of Nonlinear Systems with Random Structure, Taylor and Francis, London and New York, 2002. doi: 10.4324/9780203218891.

[13]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2nd edition, Springer, New York, 1991. doi: 10.1007/978-1-4612-0949-2.

[14]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.

[15]

N. V. Krylov, Introduction to the Theory of Diffusion Processes, Translations of Mathematical Monographs, Vol. 142, American Mathematical Society, 1995.

[16]

N. V. Krylov and A. Zatezalo, Filtering of finite-state time-nonhomogeneous Markov Processes, a direct approach, Applied Mathematics & Optimization, 42 (2000), 229-258. doi: 10.1007/s002450010012.

[17]

N. V. Krylov and A. Zatezalo, A direct approach to deriving filtering equations for diffusion processes, Applied Mathematics & Optimization, 42 (2000), 315-332. doi: 10.1007/s002450010015.

[18]

N. N. Krasovskii and A. I. Subbotin, Game-theoretical Control Problems, Springer-Verlag, New York, NY, 1988. doi: 10.1007/978-1-4612-3716-7.

[19]

G. Leitmann and J. Skowronski, Avoidance Control, Journal of Optimization Theory and Applications, 23 (1977), 581-591. doi: 10.1007/BF00933298.

[20]

A. A. Melikyan, On minimal observations in a game of encounter, Prikladnaya Matematika i Mekhanika, 37 (1973), 407-414, (In Russian).

[21]

A. A. Melikyan and A. Pourtallier, Games with several pursuers and one evader with discrete observations, Game Theory and Applications (L. A. Petrosjan and V. V. Mazalov, eds.), Nova Science Publishers, New York, NY, 2 (1996), 169-184.

[22]

G. J. Olsder and O. Pourtallier, Optimal selection of observation times in a costly information game, New Trends in Dynamic Games and Applications (G. J. Olsder, ed.), Annals of the International Society of Dynamic Games, Birkhäuser, Boston, MA, 3 (1995), 227-246. doi: 10.1007/978-1-4612-4274-1_11.

[23]

L. A. Petrosjan, Differential Games of Pursuit, Series on Optimization, Vol. 2, World Scientific, Singapore, 1993. doi: 10.1142/1670.

[24]

H. L. Royden, Real Analysis, 3rd edition, Macmillan Publishing Company, New York, 1988.

[25]

M. W. Spong, S. Hutchinson and M. Vidyasagar, Robot Modeling and Control, John Wiley & Sons, Hoboken, NJ, 2005.

[26]

D. M. Stipanović, A Survey and Some New Results in Avoidance Control, in 15th International Workshop on Dynamics and Control IWDC 2009, J. Rodellar and E. Reithmeier (Eds.), Barcelona, 2009.

[27]

D. M. Stipanović, A. Melikyan and N. Hovakimyan, Some sufficient conditions for multi-player pursuit-evasion games with continuous and discrete observations, Annals of the International Society of Dynamic Games, 10 (2009), 133-145.

[28]

D. M. Stipanović, A. Melikyan and N. Hovakimyan, Guaranteed strategies for nonlinear multi-player pursuit-evasion games, International Game Theory Review, 12 (2010), 1-17. doi: 10.1142/S0219198910002489.

[29]

D. M. Stipanović, C. J. Tomlin and G. Leitmann, Monotone approximations of minimum and maximum functions and multi-objective problems, Applied Mathematics & Optimization, 66 (2012), 455-473. doi: 10.1007/s00245-012-9179-8.

[30]

D. M. Stipanović, C. Valicka, C. J. Tomlin and T. R. Bewley, Safe and reliable coverage control, Numerical Algebra, Control and Optimization (NACO), 3 (2013), 31-48. doi: 10.3934/naco.2013.3.31.

[31]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer, New York, 1999. doi: 10.1007/978-1-4612-1466-3.

[32]

A.Zatezalo, D. Stipanović, S. Yu and P. McLaughlin, Game-Theoretic Approach to Peer-to-Peer Confrontations, Technical Paper, AUVSI's Unmanned Systems, 2014.

[33]

A. Zatezalo, D. Stipanović, R. K. Mehra and K. Pham, Constrained Orbital Intercept-Evasion, Proceedings of SPIE, 9085-14, 2014.

[34]

A. Zatezalo, D. Stipanović, R. K. Mehra and K. Pham, Space Collision Threat Mitigation, Proceedings of SPIE, 9091-17, 2014.

show all references

References:
[1]

T. Başar and G. J. Olsder, Dynamic Noncooperative Game Theory, Revised and updated 2nd edition, SIAM, Philadelphia, PA, 1999.

[2]

R. H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition, AIAA Education Series, 1999. doi: 10.2514/4.861543.

[3]

P. Billingsley, Probability and Measure, 2nd edition, John Willey & Sons, New York, 1986.

[4]

A. M. Bloch (with the collaboration of J. Baillieul, P. Crouch and J. Marsden), Nonholonomic Mechanics and Control, Springer-Verlag, New York, 2003. doi: 10.1007/b97376.

[5]

A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences, Springfield, MO, 2007.

[6]

L. Buonanno, A new Gronwall-Bellman inequality for discontinuous functions, Journal of Interdisciplinary Mathematics, 9 (2006), 543-550.

[7]

F. L. Chernousko and A. A. Melikyan, Some differential games with incomplete information, Lecture Notes in Computer Science, Springer-Verlag, Berlin, 27 (1975), 445-450. doi: 10.1007/3-540-07165-2_62.

[8]

A. Désilles, H. Zidani and E. Crück, Collision analysis for an UAV, Proceedings of the AIAA Guidance, Navigation, and Control Conference, Minneapolis, MN, 2012.

[9]

W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, 2nd edition, Springer, New York, 2006.

[10]

R. A. Freeman and P. V. Kokotović, Robust Nonlinear Control Design: State Space and Lyapunov Techniques, Birkhäuser, Boston, MA, 1996. doi: 10.1007/978-0-8176-4759-9.

[11]

R. Isaacs, Differential Games, A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization, Dover, Mineola, 1999.

[12]

I. Ya. Kats and A. A. Martynyuk, Stability and Stabilization of Nonlinear Systems with Random Structure, Taylor and Francis, London and New York, 2002. doi: 10.4324/9780203218891.

[13]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2nd edition, Springer, New York, 1991. doi: 10.1007/978-1-4612-0949-2.

[14]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.

[15]

N. V. Krylov, Introduction to the Theory of Diffusion Processes, Translations of Mathematical Monographs, Vol. 142, American Mathematical Society, 1995.

[16]

N. V. Krylov and A. Zatezalo, Filtering of finite-state time-nonhomogeneous Markov Processes, a direct approach, Applied Mathematics & Optimization, 42 (2000), 229-258. doi: 10.1007/s002450010012.

[17]

N. V. Krylov and A. Zatezalo, A direct approach to deriving filtering equations for diffusion processes, Applied Mathematics & Optimization, 42 (2000), 315-332. doi: 10.1007/s002450010015.

[18]

N. N. Krasovskii and A. I. Subbotin, Game-theoretical Control Problems, Springer-Verlag, New York, NY, 1988. doi: 10.1007/978-1-4612-3716-7.

[19]

G. Leitmann and J. Skowronski, Avoidance Control, Journal of Optimization Theory and Applications, 23 (1977), 581-591. doi: 10.1007/BF00933298.

[20]

A. A. Melikyan, On minimal observations in a game of encounter, Prikladnaya Matematika i Mekhanika, 37 (1973), 407-414, (In Russian).

[21]

A. A. Melikyan and A. Pourtallier, Games with several pursuers and one evader with discrete observations, Game Theory and Applications (L. A. Petrosjan and V. V. Mazalov, eds.), Nova Science Publishers, New York, NY, 2 (1996), 169-184.

[22]

G. J. Olsder and O. Pourtallier, Optimal selection of observation times in a costly information game, New Trends in Dynamic Games and Applications (G. J. Olsder, ed.), Annals of the International Society of Dynamic Games, Birkhäuser, Boston, MA, 3 (1995), 227-246. doi: 10.1007/978-1-4612-4274-1_11.

[23]

L. A. Petrosjan, Differential Games of Pursuit, Series on Optimization, Vol. 2, World Scientific, Singapore, 1993. doi: 10.1142/1670.

[24]

H. L. Royden, Real Analysis, 3rd edition, Macmillan Publishing Company, New York, 1988.

[25]

M. W. Spong, S. Hutchinson and M. Vidyasagar, Robot Modeling and Control, John Wiley & Sons, Hoboken, NJ, 2005.

[26]

D. M. Stipanović, A Survey and Some New Results in Avoidance Control, in 15th International Workshop on Dynamics and Control IWDC 2009, J. Rodellar and E. Reithmeier (Eds.), Barcelona, 2009.

[27]

D. M. Stipanović, A. Melikyan and N. Hovakimyan, Some sufficient conditions for multi-player pursuit-evasion games with continuous and discrete observations, Annals of the International Society of Dynamic Games, 10 (2009), 133-145.

[28]

D. M. Stipanović, A. Melikyan and N. Hovakimyan, Guaranteed strategies for nonlinear multi-player pursuit-evasion games, International Game Theory Review, 12 (2010), 1-17. doi: 10.1142/S0219198910002489.

[29]

D. M. Stipanović, C. J. Tomlin and G. Leitmann, Monotone approximations of minimum and maximum functions and multi-objective problems, Applied Mathematics & Optimization, 66 (2012), 455-473. doi: 10.1007/s00245-012-9179-8.

[30]

D. M. Stipanović, C. Valicka, C. J. Tomlin and T. R. Bewley, Safe and reliable coverage control, Numerical Algebra, Control and Optimization (NACO), 3 (2013), 31-48. doi: 10.3934/naco.2013.3.31.

[31]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer, New York, 1999. doi: 10.1007/978-1-4612-1466-3.

[32]

A.Zatezalo, D. Stipanović, S. Yu and P. McLaughlin, Game-Theoretic Approach to Peer-to-Peer Confrontations, Technical Paper, AUVSI's Unmanned Systems, 2014.

[33]

A. Zatezalo, D. Stipanović, R. K. Mehra and K. Pham, Constrained Orbital Intercept-Evasion, Proceedings of SPIE, 9085-14, 2014.

[34]

A. Zatezalo, D. Stipanović, R. K. Mehra and K. Pham, Space Collision Threat Mitigation, Proceedings of SPIE, 9091-17, 2014.

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