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Direct optimal control for time-delay systems via a lifted multiple shooting algorithm

  • * Corresponding author: Canghua Jiang

    * Corresponding author: Canghua Jiang 

The first author is supported by the Natural Science Foundation of Anhui Province, China, under Grant 2008085MF216

Abstract Full Text(HTML) Figure(5) / Table(3) Related Papers Cited by
  • Aiming at efficient solution of optimal control problems for continuous nonlinear time-delay systems, a multiple shooting algorithm with a lifted continuous Runge-Kutta integrator is proposed. This integrator is in implicit form to remove the restriction of smaller integration step sizes compared with delays. A tangential predictor is applied in the integrator such that Newton iterations required can be reduced considerably. If one Newton iteration is applied, the algorithm has the same structure as direct collocation algorithms whereas derives a condensed nonlinear programming problem. Then, the solution of variational sensitivity equation is decoupled from forward simulation by utilizing the implicit function theorem. Under certain conditions, this function evaluation and derivative computation procedure is proved to be convergent with a global order. Complexity analysis shows that the computational cost can be largely reduced by this lifted multiple shooting algorithm. Then, parallelizable optimal control solver can be constructed by embedding this algorithm in a general-purpose nonlinear programming solver. Simulations on a numerical example demonstrate that the computational speed of multi-threading implementation of this algorithm is increased by $ 36\% $ compared with non-lifted one, and increased by a factor of $ 6.64 $ compared with traditional sequential algorithm; meanwhile, the accuracy loss is negligible.

    Mathematics Subject Classification: Primary: 49M37, 93C23; Secondary: 37M15.


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  • Figure 1.  Forward simulation by multi-threading

    Figure 2.  Connection of state trajectories in two shooting intervals

    Figure 3.  Control trajectories

    Figure 4.  State trajectories

    Figure 5.  Computational times with $ s_\mathrm{pc} = \mathrm{false} $ or $ \mathrm{true} $

    Table 1.  Time complexity per shooting interval

    Newton iter. $ pqL[\frac{2}{3}(rn_x)^3] $
    Sensitivity propagation (11) $ pq[p n_u n_x^2](n_\tau r+1)^2 $
    (12) $ pq[n_p n_x^2](n_\tau r+1)^2 $
    Continuity constraints (15) $ (n_\tau+1)(n_c+1)[(pn_u+n_p)n_x^2] $
    (16) $ (n_\tau+1)(n_c+1)[(pn_u+n_p)n_x^2](n_\tau r+1) $
    Tangential prediction (18) $ pq[n_x(n_x+n_u)(r+1)](n_\tau r+1) $
    (19) $ pq[n_x(n_x+n_u)(r+1)]r $
     | Show Table
    DownLoad: CSV

    Table 2.  Space complexity per shooting interval

    Memory usage
    Forward simulation (7e) $ pqrn_\tau n_x $
    (8b) $ (n_\tau+1)(n_c+1)n_x $
    Sensitivity propagation (10b) $ pqrn_\tau(1+rn_\tau)(n_x+n_u)n_x $
    (10c) $ (n_\tau+1)(n_c+1)(1+rn_\tau)(n_x+n_u)n_x $
    (11b) $ p^2qrn_\tau n_u n_x $
    (12b) $ pqrn_\tau n_p n_x $
    Tangential prediction $ (\sigma,\theta) $ $ pn_u+n_p $
    (7b) $ pqrn_x $
    (9) $ pqr(1+rn_\tau)(n_x+n_u)n_x $
    (10a) $ pq(1+rn_\tau)(n_x+n_u)n_x $
     | Show Table
    DownLoad: CSV

    Table 3.  Computational time (CPU time for $ s = 1 $ and clock time for $ s>1 $) and cost for different $ (s,p,q) $

    $ s $ $ p $ $ q $ total time (ms) cost
    1 64 1 1337 0.891539
    1 8 8 1065 0.899012
    2 4 8 405 0.906770
    4 2 8 263 0.899053
    8 1 8 362 0.899096
     | Show Table
    DownLoad: CSV
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