access icon free Closed-form solution of discrete-time optimal control and its convergence

The convergence of a new closed-form solution for the discrete-time optimal control is presented. First, a new time optimal control law with simple structure is constructed in the form of the state feedback for a discrete-time double-integral system by using the state backstepping approach. The control signal sequence in this approach is determined by the linearised criterion according to the position of the initial state point on the phase plane. This closed-form non-linear state feedback control law clearly shows that time optimal control in discrete time is not necessarily the bang-bang control. Second, the convergence of the time optimal control law is proved by demonstrating the convergence path of the state point sequence driven by the corresponding control signal sequence. Finally, numerical simulation results demonstrate the effectiveness of this new discrete-time optimal control law.

Inspec keywords: nonlinear control systems; time optimal control; convergence of numerical methods; discrete time systems; state feedback; linear systems; control nonlinearities

Other keywords: bang-bang control; discrete-time optimal control; phase plane; double-integral system; convergence path; linearised criterion; numerical simulation; state point sequence; initial state point; nonlinear state feedback control law; state backstepping approach; closed-form solution; control signal sequence

Subjects: Optimal control; Nonlinear control systems; Other numerical methods; Discrete control systems; Linear control systems

References

    1. 1)
      • 11. Maurer, H., Büskens, C., Kim, J.H., et al: ‘Optimization methods for the verification of second order sufficient conditions for bang-bang controls’, Optim. Control Appl. Methods, 2005, 26, (3), pp. 129156.
    2. 2)
      • 27. Poonawala, H.A., Spong, M.W.: ‘Time-optimal velocity tracking control for differential drive robots’, Automatica, 2017, 85, pp. 153157.
    3. 3)
      • 9. Maurer, H., Osmolovskii, N.P.: ‘Second order sufficient conditions for time-optimal bang-bang control’, SIAM J. Control Optim., 2004, 42, (6), pp. 22392263.
    4. 4)
      • 3. Bryson, A.E.: ‘Optimal control-1950 to 1985’, IEEE Control Syst., 1996, 16, (3), pp. 2633.
    5. 5)
      • 15. Huber, O., Acary, V., Brogliato, B., et al: ‘Discrete-time twisting controller without numerical chattering: analysis and experimental results with an implicit method’. Proc. of the 53rd IEEE Conf. on Decision and Control, December 2014, pp. 43734378.
    6. 6)
      • 6. Luo, B., Wu, H.N., Huang, T., et al: ‘Data-based approximate policy iteration for affine nonlinear continuous-time optimal control design’, Automatica, 2014, 50, (12), pp. 32813290.
    7. 7)
      • 18. Johnson, C., Gibson, J.: ‘Singular solutions in problems of optimal control’, IEEE Trans. Autom. Control, 1963, 8, (1), pp. 415.
    8. 8)
      • 4. Adhyaru, D.M., Kar, I.N., Gopal, M.: ‘Fixed final time optimal control approach for bounded robust controller design using Hamilton-Jacobi-Bellman solution’, IET Control Theory Appl., 2009, 3, (9), pp. 11831195.
    9. 9)
      • 1. Hopkin, A.M.: ‘A phase-plane approach to the compensation of saturating servomechanisms’, AIEE Trans., 1951, 70, (1), pp. 631639.
    10. 10)
      • 26. Laschov, D., Margaliot, M.: ‘Minimum-time control of Boolean networks’, SIAM J. Control Optim., 2013, 51, (4), pp. 28692892.
    11. 11)
      • 19. Lastman, G.J.: ‘A shooting method for solving two-point boundary-value problems arising from non-singular bang-bang optimal control problems’, Int. J. Control., 1978, 27, (4), pp. 513524.
    12. 12)
      • 7. Albertini, F., D'Alessandro, D.: ‘Time optimal simultaneous control of two level quantum systems’, Automatica, 2016, 74, pp. 5562.
    13. 13)
      • 24. Gao, Z.: ‘On discrete time optimal control: a closed-form solution’, Proc. Am Control Conf. 2004, 2004, 1, pp. 5258.
    14. 14)
      • 8. Bartolini, G., Ferrara, A., Usai, E.: ‘Chattering avoidance by second-order sliding mode control’, IEEE Trans. Autom. Control, 1998, 43, (2), pp. 241246.
    15. 15)
      • 13. Zhou, K., Doyle, J.C., Glover, K.: ‘Robust and optimal control’ (Prentice Hall, New Jersey, 1996).
    16. 16)
      • 10. LEWIS, F.: ‘Optimal control((Book))’ (Wiley-Interscience, New York, 1986).
    17. 17)
      • 22. Han, J., Wang, W.: ‘Nonlinear tracking-differentiator’, J. Syst. Sci. Math. Sci., 1994, 14, (2), pp. 177183.
    18. 18)
      • 5. Zhang, D.Q., Guo, G.X: ‘Discrete-time sliding mode proximate time optimal seek control of hard disk drives’, Proc. IEE Control Theory Appl., 2000, 147, (4), pp. 440446.
    19. 19)
      • 17. Alt, W., Kaya, C.Y., Schneider, C.: ‘Dualization and discretization of linear-quadratic control problems with bang-bang solutions’, EURO J. Comput. Optim., 2016, 4, (1), pp. 4777.
    20. 20)
      • 20. Bertrand, R., Epenoy, R.: ‘New smoothing techniques for solving bang-bang optimal control problems-numerical results and statistical interpretation’, Optim. Control Appl. Methods., 2002, 23, (4), pp. 171197.
    21. 21)
      • 14. Bellman, R., Glicksberg, I., Gross, O.: ‘On the bang-bang control problem’, Q. Appl. Math., 1956, 14, (1), pp. 1118.
    22. 22)
      • 21. Tsien, H.S.: ‘Engineering cybernetics’, 1954.
    23. 23)
      • 23. Han, J.: ‘From PID to active disturbance rejection control’, IEEE Trans. Ind. Electron., 2009, 56, (3), pp. 900906.
    24. 24)
      • 12. Khaneja, N.: ‘Time optimal control in coupled spin systems: a second order analysis’, arXiv preprint arXiv:1607.02692, 2016.
    25. 25)
      • 2. Desoer, C.A.: ‘The bang bang servo problem treated by variational techniques’, Inf. Control, 1959, 2, (4), pp. 333348.
    26. 26)
      • 25. Zhang, X., Fang, Y., Sun, N.: ‘Minimum-time trajectory planning for underactuated overhead crane systems with state and control constraints’, IEEE Trans. Ind. Electron., 2014, 61, (12), pp. 69156925.
    27. 27)
      • 16. Sussmann, H.J.: ‘The bang-bang problem for certain control systems in GL(n,R)’, SIAM J. Control, 1972, 10, (3), pp. 470476.
http://iet.metastore.ingenta.com/content/journals/10.1049/iet-cta.2017.0749
Loading

Related content

content/journals/10.1049/iet-cta.2017.0749
pub_keyword,iet_inspecKeyword,pub_concept
6
6
Loading