http://iet.metastore.ingenta.com
1887

Differential games, continuous Lyapunov functions, and stabilisation of non-linear dynamical systems

Differential games, continuous Lyapunov functions, and stabilisation of non-linear dynamical systems

For access to this article, please select a purchase option:

Buy article PDF
£12.50
(plus tax if applicable)
Buy Knowledge Pack
10 articles for £75.00
(plus taxes if applicable)

IET members benefit from discounts to all IET publications and free access to E&T Magazine. If you are an IET member, log in to your account and the discounts will automatically be applied.

Learn more about IET membership 

Recommend Title Publication to library

You must fill out fields marked with: *

Librarian details
Name:*
Email:*
Your details
Name:*
Email:*
Department:*
Why are you recommending this title?
Select reason:
 
 
 
 
 
IET Control Theory & Applications — Recommend this title to your library

Thank you

Your recommendation has been sent to your librarian.

In this study, the authors address the two-player zero-sum differential game problem for non-linear dynamical systems with non-linear-non-quadratic cost functions over the infinite time horizon. The pursuer's goal is to minimise the cost function and guarantee asymptotic stability of the closed-loop system, whereas the evader's goal is to maximise the cost function. Closed-loop asymptotic stability is certified by continuous Lyapunov functions that are viscosity solutions of the steady-state Hamilton–Jacobi–Isaacs equation for the controlled system. Since it is difficult to find viscosity solutions of partial differential equations for numerous problems of practical interest, they extend an inverse optimality framework to provide explicit closed-form solutions of differential game problems, which involve affine in the controls dynamical systems with quadratic cost functions and linear dynamical systems with Lagrangians in polynomial form. The authors' framework allows also to solve optimal robust control problems involving non-linear dynamical systems with non-linear-non-quadratic cost functionals and provides a generalisation of the mixed-norm optimal robust control framework. Two numerical examples illustrate the applicability of theoretical results provided.

References

    1. 1)
      • 1. Oshman, Y., Rad, D.A.: ‘Differential-game-based guidance law using target orientation observations’, IEEE Trans. Aerosp. Electron. Syst., 2006, 42, (1), pp. 316326.
    2. 2)
      • 2. Harini Venkatesan, R., Sinha, N.K.: ‘A new guidance law for the defense missile of nonmaneuverable aircraft’, IEEE Trans. Control Syst. Technol., 2015, 23, (6), pp. 24242431.
    3. 3)
      • 3. Ekneligoda, N.C., Weaver, W.W.: ‘Game-theoretic cold-start transient optimization in DC microgrids’, IEEE Trans. Ind. Electron., 2014, 61, (12), pp. 66816690.
    4. 4)
      • 4. Kumar, P.: ‘Optimal mixed strategies in a dynamic game’, IEEE Trans. Autom. Control, 1980, 25, (4), pp. 743749.
    5. 5)
      • 5. Shen, S., Li, H., Han, R., et al: ‘Differential game-based strategies for preventing malware propagation in wireless sensor networks’, IEEE Trans. Inf. Forensics Sec., 2014, 9, (11), pp. 19621973.
    6. 6)
      • 6. Gibbons, R.: ‘Game theory for applied economists’ (Princeton University Press, Princeton, NJ, 1992).
    7. 7)
      • 7. Isaacs, R.: ‘Games of pursuit’. In RAND Corporation, Research Memorandum, 1951.
    8. 8)
      • 8. Isaacs, R.: ‘Differential games: a mathematical theory with applications to warfare and pursuit, control and optimization’ (Dover, New York, NY, 1999).
    9. 9)
      • 9. Case, J.H.: ‘Toward a theory of many player differential games’, SIAM J. Control, 1969, 7, (2), pp. 179197.
    10. 10)
      • 10. Leitmann, G.: ‘Multicriteria decision making and differential games’ (Springer, New York, NY, 2013).
    11. 11)
      • 11. Bressan, A.: ‘Noncooperative differential games’, Milan J. Math., 2011, 79, (2), pp. 357427.
    12. 12)
      • 12. Petrosjan, L.A.: ‘Cooperative differential games’, in Szajowski, A.S., Nowak, K. (Eds.): ‘Advances in dynamic games’, vol. 7, (Birkhäuser, Boston2005), pp. 183200.
    13. 13)
      • 13. Haddad, W.M., Chellaboina, V.S.: ‘Nonlinear dynamical systems and control: a Lyapunov-based approach’ (Princeton University Press, Princeton, NJ, 2008).
    14. 14)
      • 14. Zhukovskiy, V.I.: ‘Lyapunov functions in differential games’ (Taylor & Francis, New York, NY, 2003).
    15. 15)
      • 15. Adiatulina, R.A., Taras'yev, A.M.: ‘A differential game of unlimited duration’, J. Appl. Math. Mech., 1987, 51, (4), pp. 415420.
    16. 16)
      • 16. Soravia, P.: ‘The concept of value in differential games of survival and viscosity solutions of Hamilton–Jacobi equations’, Diff. Integral Equ., 1992, 5, (5), pp. 10491068.
    17. 17)
      • 17. Doyle, J.C., Glover, K., Khargonekar, P.P., et al: ‘State-space solutions to standard H2 and H control problems’, IEEE Trans. Autom. Control, 1989, 34, (8), pp. 831847.
    18. 18)
      • 18. Jacobson, D.: ‘On values and strategies for infinite-time linear quadratic games’, IEEE Trans. Autom. Control, 1977, 22, (3), pp. 490491.
    19. 19)
      • 19. Limebeer, D.J.N., Anderson, B.D.O., Khargonekar, P.P., et al: ‘A game theoretic approach to H control for time-varying systems’, SIAM J. Control Optim., 1992, 30, (2), pp. 262283.
    20. 20)
      • 20. Mageirou, E.: ‘Values and strategies for infinite time linear quadratic games’, IEEE Trans. Autom. Control, 1976, 21, (4), pp. 547550.
    21. 21)
      • 21. Ball, J.A., Helton, J.W.: ‘H control for nonlinear plants: connections with differential games’. IEEE Conf. Decision and Control, 1989, vol. 2, pp. 956962.
    22. 22)
      • 22. Basar, T., Bernhard, P.: ‘Hoptimal control and related minimax design problems’ (Cambridge University Press, Cambridge, UK, 2000).
    23. 23)
      • 23. Basar, T., Olsder, G.: ‘Dynamic noncooperative game theory’ (Society for Industrial and Applied Mathematics, New York, NY, 1998, 2nd edn.).
    24. 24)
      • 24. McEneaney, W.M.: ‘A uniqueness result for the Isaacs equation corresponding to nonlinear ∞ H control’, Math. Control Signals Syst., 1998, 11, (4), pp. 303334.
    25. 25)
      • 25. Schaft, A.J.: ‘Essays on control: perspectives in the theory and its applications’, in Trentelman, H. L., Willems, J. C. (Eds.): ‘Chapter nonlinear state space H control theory’ (Birkhäuser, Boston, MA, 1993), pp. 153190.
    26. 26)
      • 26. Ball, J.A., Helton, J.W., Walker, M.L.: ‘H control for nonlinear systems with output feedback’, IEEE Trans. Autom. Control, 1993, 38, (4), pp. 546559.
    27. 27)
      • 27. James, M.R.: ‘Asymptotic analysis of nonlinear stochastic risk-sensitive control and differential games’, Math. Control Signals Syst., 1992, 5, (4), pp. 401417.
    28. 28)
      • 28. Lim, A.E.B., Zhou, X.Y.: ‘A new risk-sensitive maximum principle’, IEEE Trans. Autom. Control, 2005, 50, (7), pp. 958966.
    29. 29)
      • 29. Whittle, P.: ‘A risk-sensitive maximum principle’, Syst. Control Lett., 1990, 15, (3), pp. 183192.
    30. 30)
      • 30. Jacobson, D.: ‘Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games’, IEEE Trans. Autom. Control, 1973, 18, (2), pp. 124131.
    31. 31)
      • 31. Whittle, P.: ‘Risk-sensitive linear/quadratic/gaussian control’, Adv. Appl. Probab., 1981, 13, (4), pp. 764777.
    32. 32)
      • 32. Bardi, M.: ‘Viscosity solutions and applications: lectures given at the 2nd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Montecatini Terme, Italy, 12–20 June 1995’, in Capuzzo-Dolcetta, I., Lions, P. L. (Eds.): ‘Chapter Some applications of viscosity solutions to optimal control and differential games’ (Springer, Berlin, Germany, 1997), pp. 4497.
    33. 33)
      • 33. Clarke, F.H., Ledyaev, Y.S., Stern, R.J., et al: ‘Nonsmooth analysis and control theory’ (Springer, Secaucus, NJ, USA, 1998).
    34. 34)
      • 34. Subbotin, A.I.: ‘Generalized solutions of first order PDEs: the dynamical optimization perspective’ (Birkhäuser, Boston, MA, 1994).
    35. 35)
      • 35. Kotz, S., Krasovskii, N.N., Subbotin, A.I.: ‘Game-theoretical control problems’ (Springer, New York, NY, 2011).
    36. 36)
      • 36. Clarke, F.H., Ledyaev, Y.S., Subbotin, A.I.: ‘The synthesis of universal feedback pursuit strategies in differential games’, SIAM J. Control Optim., 1997, 35, (2), pp. 552561.
    37. 37)
      • 37. Barron, E.N., Evans, L.C., Jensen, R.: ‘Viscosity solutions of Isaacs’ equations and differential games with Lipschitz controls', J. Diff. Equ., 1984, 53, (2), pp. 213233.
    38. 38)
      • 38. Crandall, M.G., Lions, P.-L.: ‘Viscosity solutions of Hamilton–Jacobi equations’, Trans. Am. Math. Soc., 1983, 277, (1), pp. 142.
    39. 39)
      • 39. Aubin, J.-P.: ‘Differential games: a viability approach’, SIAM J. Control Optim., 1990, 28, (6), pp. 12941320.
    40. 40)
      • 40. Aubin, J.P., Bayen, A.M., Saint-Pierre, P.: ‘Viability theory: new directions’ (Springer, Berlin, Germany, 2011).
    41. 41)
      • 41. Cardaliaguet, P., Quincampoix, M., Saint-Pierre, P.: ‘Advances in dynamic game theory: numerical methods, algorithms, and applications to ecology and economics’, in Jørgense, S., Quincampoix, M., Vincent, T. L. (Eds.): Chapter differential games through viability theory: old and recent results’ (Birkhäuser, Boston, MA, 2007), pp. 335.
    42. 42)
      • 42. Bernstein, D.S.: ‘Nonquadratic cost and nonlinear feedback control’, Int. J. Robust Nonlinear Control, 1993, 3, (3), pp. 211229.
    43. 43)
      • 43. Agarwal, R.P., Lakshmikantham, V.: ‘Uniqueness and nonuniqueness criteria for ordinary differential equations’ (World Scientific, Singapore, 1993).
    44. 44)
      • 44. Filippov, A.F.: ‘Differential equations with discontinuous right-hand sides’, ‘Mathematics and its Applications (Soviet Series)’ (Kluwer Academic Publishers, Dordrecht, The Netherlands, 1988).
    45. 45)
      • 45. Yoshizawa, T.: ‘Stability theory by Liapunov's second methods’ (Mathematical Society of Japan, Tokyo, Japan, 1966).
    46. 46)
      • 46. Bernstein, D.S., Haddad, W.M.: ‘LQG control with an H performance bound: a Riccati equation approach’, IEEE Trans. Autom. Control, 1989, 34, (3), pp. 293305.
    47. 47)
      • 47. Haddad, W.M., Chellaboina, V.S.: ‘Optimal nonlinear-nonquadratic feedback control for systems with L2 and L disturbances’, Nonlinear Anal., Theory Methods Appl., 1998, 34, (2), pp. 229255.
    48. 48)
      • 48. L'Afflitto, A.: ‘Differential games, asymptotic stabilization, and robust optimal control of nonlinear systems’. IEEE Conf. Decision and Control, 2016, vol. 2, pp. 19331938.
    49. 49)
      • 49. L'Afflitto, A.: ‘Differential games’, partial-state stabilization, and model reference adaptive control', J. Franklin Inst., 2017, 354, (1), pp. 456478.
    50. 50)
      • 50. Bass, R.W., Webber, R.: ‘Optimal nonlinear feedback control derived from quartic and higher-order performance criteria’, IEEE Trans. Autom. Control, 1966, 11, (3), pp. 448454.
    51. 51)
      • 51. Freeman, R., Kokotovic, P.: ‘Inverse optimality in robust stabilization’, SIAM J. Control Optim., 1996, 34, (4), pp. 13651391.
    52. 52)
      • 52. Jacobson, D.H.: ‘Extensions of linear-quadratic control optimization and matrix theory’ (Academic Press, New York, NY, 1977).
    53. 53)
      • 53. Molinari, B.: ‘The stable regulator problem and its inverse’, IEEE Trans. Autom. Control, 1973, 18, (5), pp. 454459.
    54. 54)
      • 54. Moylan, P., Anderson, B.: ‘Nonlinear regulator theory and an inverse optimal control problem’, IEEE Trans. Autom. Control, 1973, 18, (5), pp. 460465.
    55. 55)
      • 55. Sepulchre, R., Jankovic, M., Kokotovic, P.V.: ‘Constructive nonlinear control’ (Springer, London, UK, 1997).
    56. 56)
      • 56. Garcia, E., Casbeer, D.W., Pachter, M.: ‘Cooperative strategies for optimal aircraft defense from an attacking missile’, J. Guid. Control Dyn., 2015, 38, (8), pp. 15101520.
    57. 57)
      • 57. Haddad, W.M., Chellaboina, V.S., Nersesov, S.G.: ‘Impulsive and hybrid dynamical systems: stability, dissipativity, and control’ (Princeton University Press, Princeton, NJ, 2006).
    58. 58)
      • 58. Katzourakis, N.: ‘An introduction to viscosity solutions for fully nonlinear pde with applications to calculus of variations inL’ (Springer, London, UK, 2014).
    59. 59)
      • 59. Stein, E.M., Shakarchi, R.: ‘Real analysis: measure theory, integration, and Hilbert spaces’ (Princeton University Press, Princeton, NJ, 2009).
    60. 60)
      • 60. Sontag, E., Sussmann, H.J.: ‘Nonsmooth control-Lyapunov functions’. IEEE Conf. Decision and Control, 1995, vol. 3, pp. 27992805.
    61. 61)
      • 61. Kalman, R.E., Bertram, J.E.: ‘Control system analysis and design via the ‘second method’ of Lyapunov: I – continuous-time systems’, Trans. ASME, D, J. Basic Eng., 1960, 82, pp. 371393.
    62. 62)
      • 62. Green, M., Limebeer, D.J.N.: ‘Linear robust control’ (Prentice-Hall, Mineola, NY, 1995).
    63. 63)
      • 63. Anderson, B.D.O., Moore, J.B.: ‘Optimal control: linear quadratic methods’ (Prentice-Hall, Englewood Cliffs, NJ, 1990).
    64. 64)
      • 64. Mitchell, I.M.: ‘A toolbox of level set methods’. http://www.cs.ubc.ca/mitchell/ToolboxLS/, 2007, Accessed 3/12/2017.
    65. 65)
      • 65. Sethian, J.A.: ‘Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science’ (Cambridge University Press, Cambridge, UK, 1999).
    66. 66)
      • 66. Falcone, M.: ‘A numerical approach to the infinite horizon problem of deterministic control theory’, Appl. Math. Optim., 1987, 15, (1), pp. 113.
    67. 67)
      • 67. Sethian, J.A.: ‘A fast marching level set method for monotonically advancing fronts’, IEEE Trans. Autom. Control, 1995, 40, (9), pp. 15281538.
http://iet.metastore.ingenta.com/content/journals/10.1049/iet-cta.2017.0271
Loading

Related content

content/journals/10.1049/iet-cta.2017.0271
pub_keyword,iet_inspecKeyword,pub_concept
6
6
Loading
This is a required field
Please enter a valid email address