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 & Continuous Dynamical Systems - A, 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, (1999).   Google Scholar

[2]

R. H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics,, Revised Edition, (1999).  doi: 10.2514/4.861543.  Google Scholar

[3]

P. Billingsley, Probability and Measure,, 2nd edition, (1986).   Google Scholar

[4]

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

[5]

A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control,, American Institute of Mathematical Sciences, (2007).   Google Scholar

[6]

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

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

[8]

A. Désilles, H. Zidani and E. Crück, Collision analysis for an UAV,, Proceedings of the AIAA Guidance, (2012).   Google Scholar

[9]

W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions,, 2nd edition, (2006).   Google Scholar

[10]

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

[11]

R. Isaacs, Differential Games, A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization,, Dover, (1999).   Google Scholar

[12]

I. Ya. Kats and A. A. Martynyuk, Stability and Stabilization of Nonlinear Systems with Random Structure,, Taylor and Francis, (2002).  doi: 10.4324/9780203218891.  Google Scholar

[13]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus,, 2nd edition, (1991).  doi: 10.1007/978-1-4612-0949-2.  Google Scholar

[14]

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

[15]

N. V. Krylov, Introduction to the Theory of Diffusion Processes,, Translations of Mathematical Monographs, (1995).   Google Scholar

[16]

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

[17]

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

[18]

N. N. Krasovskii and A. I. Subbotin, Game-theoretical Control Problems,, Springer-Verlag, (1988).  doi: 10.1007/978-1-4612-3716-7.  Google Scholar

[19]

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

[20]

A. A. Melikyan, On minimal observations in a game of encounter,, Prikladnaya Matematika i Mekhanika, 37 (1973), 407.   Google Scholar

[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, 2 (1996), 169.   Google Scholar

[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, 3 (1995), 227.  doi: 10.1007/978-1-4612-4274-1_11.  Google Scholar

[23]

L. A. Petrosjan, Differential Games of Pursuit,, Series on Optimization, (1993).  doi: 10.1142/1670.  Google Scholar

[24]

H. L. Royden, Real Analysis,, 3rd edition, (1988).   Google Scholar

[25]

M. W. Spong, S. Hutchinson and M. Vidyasagar, Robot Modeling and Control,, John Wiley & Sons, (2005).   Google Scholar

[26]

D. M. Stipanović, A Survey and Some New Results in Avoidance Control,, in 15th International Workshop on Dynamics and Control IWDC 2009, (2009).   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1142/S0219198910002489.  Google Scholar

[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.  doi: 10.1007/s00245-012-9179-8.  Google Scholar

[30]

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

[31]

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

[32]

A.Zatezalo, D. Stipanović, S. Yu and P. McLaughlin, Game-Theoretic Approach to Peer-to-Peer Confrontations,, Technical Paper, (2014).   Google Scholar

[33]

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

[34]

A. Zatezalo, D. Stipanović, R. K. Mehra and K. Pham, Space Collision Threat Mitigation,, Proceedings of SPIE, (2014), 9091.   Google Scholar

show all references

References:
[1]

T. Başar and G. J. Olsder, Dynamic Noncooperative Game Theory,, Revised and updated 2nd edition, (1999).   Google Scholar

[2]

R. H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics,, Revised Edition, (1999).  doi: 10.2514/4.861543.  Google Scholar

[3]

P. Billingsley, Probability and Measure,, 2nd edition, (1986).   Google Scholar

[4]

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

[5]

A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control,, American Institute of Mathematical Sciences, (2007).   Google Scholar

[6]

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

[7]

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

[8]

A. Désilles, H. Zidani and E. Crück, Collision analysis for an UAV,, Proceedings of the AIAA Guidance, (2012).   Google Scholar

[9]

W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions,, 2nd edition, (2006).   Google Scholar

[10]

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

[11]

R. Isaacs, Differential Games, A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization,, Dover, (1999).   Google Scholar

[12]

I. Ya. Kats and A. A. Martynyuk, Stability and Stabilization of Nonlinear Systems with Random Structure,, Taylor and Francis, (2002).  doi: 10.4324/9780203218891.  Google Scholar

[13]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus,, 2nd edition, (1991).  doi: 10.1007/978-1-4612-0949-2.  Google Scholar

[14]

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

[15]

N. V. Krylov, Introduction to the Theory of Diffusion Processes,, Translations of Mathematical Monographs, (1995).   Google Scholar

[16]

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

[17]

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

[18]

N. N. Krasovskii and A. I. Subbotin, Game-theoretical Control Problems,, Springer-Verlag, (1988).  doi: 10.1007/978-1-4612-3716-7.  Google Scholar

[19]

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

[20]

A. A. Melikyan, On minimal observations in a game of encounter,, Prikladnaya Matematika i Mekhanika, 37 (1973), 407.   Google Scholar

[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, 2 (1996), 169.   Google Scholar

[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, 3 (1995), 227.  doi: 10.1007/978-1-4612-4274-1_11.  Google Scholar

[23]

L. A. Petrosjan, Differential Games of Pursuit,, Series on Optimization, (1993).  doi: 10.1142/1670.  Google Scholar

[24]

H. L. Royden, Real Analysis,, 3rd edition, (1988).   Google Scholar

[25]

M. W. Spong, S. Hutchinson and M. Vidyasagar, Robot Modeling and Control,, John Wiley & Sons, (2005).   Google Scholar

[26]

D. M. Stipanović, A Survey and Some New Results in Avoidance Control,, in 15th International Workshop on Dynamics and Control IWDC 2009, (2009).   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1142/S0219198910002489.  Google Scholar

[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.  doi: 10.1007/s00245-012-9179-8.  Google Scholar

[30]

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

[31]

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

[32]

A.Zatezalo, D. Stipanović, S. Yu and P. McLaughlin, Game-Theoretic Approach to Peer-to-Peer Confrontations,, Technical Paper, (2014).   Google Scholar

[33]

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

[34]

A. Zatezalo, D. Stipanović, R. K. Mehra and K. Pham, Space Collision Threat Mitigation,, Proceedings of SPIE, (2014), 9091.   Google Scholar

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