# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021135
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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

 1 School of Electrical Engineering and Automation, Hefei University of Technology, Hefei 230009, China 2 CRSC Research & Design Institute Group Co., Ltd., Beijing 100070, China

* Corresponding author: Canghua Jiang

Received  May 2021 Revised  May 2021 Early access August 2021

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

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.

Citation: Canghua Jiang, Cheng Jin, Ming Yu, Zongqi Xu. Direct optimal control for time-delay systems via a lifted multiple shooting algorithm. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021135
##### References:

show all references

##### References:
Connection of state trajectories in two shooting intervals
Control trajectories
State trajectories
Computational times with $s_\mathrm{pc} = \mathrm{false}$ or $\mathrm{true}$
Time complexity per shooting interval
 flops 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) $pq[n_x(n_x+n_u)(r+1)]r$
 flops 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) $pq[n_x(n_x+n_u)(r+1)]r$
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) $pq(1+rn_\tau)(n_x+n_u)n_x$
 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) $pq(1+rn_\tau)(n_x+n_u)n_x$
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
 $s$ $p$ $q$ 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
 [1] Sihong Shao, Huazhong Tang. Higher-order accurate Runge-Kutta discontinuous Galerkin methods for a nonlinear Dirac model. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 623-640. doi: 10.3934/dcdsb.2006.6.623 [2] Antonia Katzouraki, Tania Stathaki. Intelligent traffic control on internet-like topologies - integration of graph principles to the classic Runge--Kutta method. Conference Publications, 2009, 2009 (Special) : 404-415. doi: 10.3934/proc.2009.2009.404 [3] Da Xu. Numerical solutions of viscoelastic bending wave equations with two term time kernels by Runge-Kutta convolution quadrature. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2389-2416. doi: 10.3934/dcdsb.2017122 [4] Akram Kheirabadi, Asadollah Mahmoudzadeh Vaziri, Sohrab Effati. Linear optimal control of time delay systems via Hermite wavelet. Numerical Algebra, Control & Optimization, 2020, 10 (2) : 143-156. doi: 10.3934/naco.2019044 [5] Z. Foroozandeh, Maria do rosário de Pinho, M. Shamsi. On numerical methods for singular optimal control problems: An application to an AUV problem. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2219-2235. doi: 10.3934/dcdsb.2019092 [6] Martin Benning, Elena Celledoni, Matthias J. Ehrhardt, Brynjulf Owren, Carola-Bibiane Schönlieb. Deep learning as optimal control problems: Models and numerical methods. Journal of Computational Dynamics, 2019, 6 (2) : 171-198. doi: 10.3934/jcd.2019009 [7] Xiaowei Pang, Haiming Song, Xiaoshen Wang, Jiachuan Zhang. Efficient numerical methods for elliptic optimal control problems with random coefficient. Electronic Research Archive, 2020, 28 (2) : 1001-1022. doi: 10.3934/era.2020053 [8] Chunjuan Hou, Yanping Chen, Zuliang Lu. Superconvergence property of finite element methods for parabolic optimal control problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 927-945. doi: 10.3934/jimo.2011.7.927 [9] Eugene Kashdan, Dominique Duncan, Andrew Parnell, Heinz Schättler. Mathematical methods in systems biology. Mathematical Biosciences & Engineering, 2016, 13 (6) : i-ii. doi: 10.3934/mbe.201606i [10] Gildas Besançon, Didier Georges, Zohra Benayache. Towards nonlinear delay-based control for convection-like distributed systems: The example of water flow control in open channel systems. Networks & Heterogeneous Media, 2009, 4 (2) : 211-221. doi: 10.3934/nhm.2009.4.211 [11] Tayel Dabbous. Adaptive control of nonlinear systems using fuzzy systems. Journal of Industrial & Management Optimization, 2010, 6 (4) : 861-880. doi: 10.3934/jimo.2010.6.861 [12] Jake Bouvrie, Boumediene Hamzi. Kernel methods for the approximation of some key quantities of nonlinear systems. Journal of Computational Dynamics, 2017, 4 (1&2) : 1-19. doi: 10.3934/jcd.2017001 [13] Ta T.H. Trang, Vu N. Phat, Adly Samir. Finite-time stabilization and $H_\infty$ control of nonlinear delay systems via output feedback. Journal of Industrial & Management Optimization, 2016, 12 (1) : 303-315. doi: 10.3934/jimo.2016.12.303 [14] Liangquan Zhang, Qing Zhou, Juan Yang. Necessary condition for optimal control of doubly stochastic systems. Mathematical Control & Related Fields, 2020, 10 (2) : 379-403. doi: 10.3934/mcrf.2020002 [15] Elena Goncharova, Maxim Staritsyn. Optimal control of dynamical systems with polynomial impulses. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4367-4384. doi: 10.3934/dcds.2015.35.4367 [16] Michael Basin, Pablo Rodriguez-Ramirez. An optimal impulsive control regulator for linear systems. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 275-282. doi: 10.3934/naco.2011.1.275 [17] Rein Luus. Optimal control of oscillatory systems by iterative dynamic programming. Journal of Industrial & Management Optimization, 2008, 4 (1) : 1-15. doi: 10.3934/jimo.2008.4.1 [18] Qiying Hu, Wuyi Yue. Optimal control for resource allocation in discrete event systems. Journal of Industrial & Management Optimization, 2006, 2 (1) : 63-80. doi: 10.3934/jimo.2006.2.63 [19] Simone Göttlich, Patrick Schindler. Optimal inflow control of production systems with finite buffers. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 107-127. doi: 10.3934/dcdsb.2015.20.107 [20] Leonardo Colombo, David Martín de Diego. Optimal control of underactuated mechanical systems with symmetries. Conference Publications, 2013, 2013 (special) : 149-158. doi: 10.3934/proc.2013.2013.149

2020 Impact Factor: 1.801

## Tools

Article outline

Figures and Tables