# American Institute of Mathematical Sciences

June  2020, 13(6): 1845-1865. doi: 10.3934/dcdss.2020109

## An efficient adjoint computational method based on lifted IRK integrator and exact penalty function for optimal control problems involving continuous inequality constraints

 1 School of Electrical Engineering and Automation, Hefei University of Technology, Hefei 230009, China 2 College of Science, Health, Engineering & Education, Murdoch University, WA 6150, Australia

* Corresponding author: Canghua Jiang

Received  January 2018 Revised  August 2018 Published  September 2019

Adjoint methods applied to solve optimal control problems (OCPs) have a restriction that the number of constraints shall be less than that of optimization variables. Otherwise, they are less efficient than the forward methods. This paper proposes an efficient adjoint method to solve OCPs for index-$1$ differential algebraic systems with continuous-time inequality constraints. The continuous-time inequality constraints are not discretized on time grid but transformed into integrals and penalized in the cost through an exact penalty function. Thus, all the constraints except for box constraints on optimization variables can be removed. Furthermore, a lifted implicit Runge-Kutta (IRK) integrator with adjoint sensitivity propagation is employed to accelerate the function and gradient evaluation procedure. Based on a sensitivity update technique, the number of Newton iterations involved in forward simulation can be reduced to one. Besides this, Lagrange interpolation is applied to approximate the states not on collocation points such that integrals in the penalty function can be evaluated on the same grid for forward simulation. Complexity analysis shows that, for the proposed algorithm, computation involved in the sensitivity propagation is comparable to that of forward one. Numerical simulations on the optimal maneuvering a Delta robot demonstrate that the computational speed of the proposed adjoint algorithm is comparable to that of our previous one, which is based on the lifted IRK integrator and forward sensitivity propagation.

Citation: Canghua Jiang, Zhiqiang Guo, Xin Li, Hai Wang, Ming Yu. An efficient adjoint computational method based on lifted IRK integrator and exact penalty function for optimal control problems involving continuous inequality constraints. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1845-1865. doi: 10.3934/dcdss.2020109
##### References:
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L. Lee, R. Serban, D. E. Shumaker and C. S. Woodward, SUNDIALS, suite of nonlinear and differential/algebraic equation solvers, ACM Transactions on Mathematical Software, 31 (2005), 363-396.  doi: 10.1145/1089014.1089020. [7] W. Huyer and A. Neumaier, A new exact penalty function, SIAM Journal on Optimization, 13 (2003), 1141–1158. doi: 10.1137/S1052623401390537. [8] C. Jiang, Q. Lin, C. Yu, K. L. Teo and G.-R. Duan, An exact penalty method for free terminal time optimal control problem with continuous inequality constraints, Journal of Optimization Theory and Applications, 154 (2012), 30-53.  doi: 10.1007/s10957-012-0006-9. [9] C. Jiang, K. Xie, Z. Guo and K. L. Teo, Implicit integration with adjoint sensitivity propagation for optimal control problems involving differential-algebraic equations, in Proceedings of the 36th Chinese Control Conference, IEEE Computer Society, Dalian, 2017, 2489–2494. doi: 10.23919/ChiCC.2017.8027734. [10] C. Jiang, K. Xie, C. Yu, M. Yu, H. Wang, Y. He and K. L. Teo, A sequential computational approach to optimal control problems for differential-algebraic systems based on efficient implicit Runge-Kutta integration, Applied Mathematical Modelling, 58 (2018), 313-330.  doi: 10.1016/j.apm.2017.05.015. [11] B. Li, C. Xu, K. L. Teo and J. Chu, Time optimal Zermelo's navigation problem with moving and fixed obstacles, Applied Mathematics and Computation, 224 (2013), 866-875.  doi: 10.1016/j.amc.2013.08.092. [12] B. Li, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty function method for continuous inequality constrained optimal control problem, Journal of Optimization Theory and Applications, 151 (2011), 260-291.  doi: 10.1007/s10957-011-9904-5. [13] Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, Journal of Industrial and management optimization, 10 (2014), 275-309.  doi: 10.3934/jimo.2014.10.275. [14] Q. Lin, R. Loxton, K. L. Teo, Y. H. Wu and C. Yu, A new exact penalty method for semi-infinite programming problems, Journal of Computational and Applied Mathematics, 261 (2014), 271-286.  doi: 10.1016/j.cam.2013.11.010. [15] R. Loxton, Q. Lin and K. L. Teo, Switching time optimization for nonlinear switched systems: Direct optimization and the time-scaling transformation, Pacific Journal of Optimization, 10 (2014), 537-560. [16] C. C. Pantelides, The consistent initialization of differential-algebraic systems, SIAM Journal on Scientific and Statistical Computing, 9 (1988), 213-231.  doi: 10.1137/0909014. [17] R. Pytlak, Runge-Kutta based procedure for the optimal control of differential-algebraic equations, Journal of Optimization Theory and Applications, 97 (1998), 675-705.  doi: 10.1023/A:1022698311155. [18] R. Pytlak and T. Zawadzki, On solving optimal control problems with higher index DAEs, Optimization Methods & Software, 29 (2014), 1139-1162.  doi: 10.1080/10556788.2014.892597. [19] R. Quirynen, S. Gros and M. Diehl, Fast auto generated ACADO integrators and application to MHE with multi-rate measurements, in Proceedings of the 2013 European Control Conference, Zurich, Switzerland, 2013, 3077–3082. doi: 10.23919/ECC.2013.6669636. [20] R. Quirynen, S. Gros and M. Diehl, Inexact Newton based lifted implicit integrators for fast nonlinear MPC, in Proceedings of the 5th IFAC Nonlinear Model Predictive Control Conference, 48 (2015), 32–38. [21] R. Quirynen, S. Gros and M. Diehl, Lifted implicit integrators for direct optimal control, in Proceedings of 2015 IEEE 54th Annual Conference on Decision and Control, 2015, 3212–3217. doi: 10.1109/CDC.2015.7402701. [22] A. Rao, A survey of numerical methods for optimal control, in Proceedings of the AAS/AIAA Astrodynamics Specialist Conference, Pittsburgh, PA, USA, 2009, 1–32. [23] I. M. Ross and F. Fahroo, Issues in the real-time computation of optimal control, Mathematical and Computer Modelling, 43 (2006), 1172-1188.  doi: 10.1016/j.mcm.2005.05.021. [24] J. M. Sanz-Serna, Symplectic Runge-Kutta schemes for adjoint equations, automatic differentiation, optimal control, and more, SIAM Review, 58 (2016), 3-33.  doi: 10.1137/151002769. [25] A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, Ser. A, 106 (2006), 25-57.  doi: 10.1007/s10107-004-0559-y. [26] A. Walther and A. Griewank, Combinatorial Scientific Computing, chapter Getting started with ADOL-C, 181–202, Chapman-Hall CRC Computational Science, 2012. [27] C. Yu, K. L. Teo, L. Zhang and Y. Bai, A new exact penalty function method for continuous inequality constrained optimization problems, Journal of Industrial and Management Optimization, 6 (2010), 895-910.  doi: 10.3934/jimo.2010.6.895. [28] C. Yu, K. L. Teo, L. Zhang and Y. Bai, On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem, Journal of Industrial and Management Optimization, 8 (2012), 485-491.  doi: 10.3934/jimo.2012.8.485.

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##### References:
 [1] L. T. Biegler, Nonlinear Programming: Concepts, Algorithms, and Applications to Chemical Processes, SIAM, Philadelphia, 2010. doi: 10.1137/1.9780898719383. [2] A. Codourey, Dynamic modeling of parallel robots for computed-torque control implementation, The International Journal of Robotics Research, 17 (1998), 1325-1336.  doi: 10.1177/027836499801701205. [3] Z. Gong, C. Liu and Y. Wang, Optimal control of switched systems with multiple time-delays and a cost on changing control, Journal of Industrial and Management Optimization, 14 (2018), 183-198.  doi: 10.3934/jimo.2017042. [4] A. Griewank, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, no. 19 in Frontiers in Applied Mathematics., SIAM, Philadelphia, 2000. [5] E. Hairer and G. Wanner, Solving Ordinary Differential Equations: II Stiff and Differential-Algebraic Problems, Springer-Verlag, Berlin, Heidelberg, 1991. doi: 10.1007/978-3-662-09947-6. [6] A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D. E. Shumaker and C. S. Woodward, SUNDIALS, suite of nonlinear and differential/algebraic equation solvers, ACM Transactions on Mathematical Software, 31 (2005), 363-396.  doi: 10.1145/1089014.1089020. [7] W. Huyer and A. Neumaier, A new exact penalty function, SIAM Journal on Optimization, 13 (2003), 1141–1158. doi: 10.1137/S1052623401390537. [8] C. Jiang, Q. Lin, C. Yu, K. L. Teo and G.-R. Duan, An exact penalty method for free terminal time optimal control problem with continuous inequality constraints, Journal of Optimization Theory and Applications, 154 (2012), 30-53.  doi: 10.1007/s10957-012-0006-9. [9] C. Jiang, K. Xie, Z. Guo and K. L. Teo, Implicit integration with adjoint sensitivity propagation for optimal control problems involving differential-algebraic equations, in Proceedings of the 36th Chinese Control Conference, IEEE Computer Society, Dalian, 2017, 2489–2494. doi: 10.23919/ChiCC.2017.8027734. [10] C. Jiang, K. Xie, C. Yu, M. Yu, H. Wang, Y. He and K. L. Teo, A sequential computational approach to optimal control problems for differential-algebraic systems based on efficient implicit Runge-Kutta integration, Applied Mathematical Modelling, 58 (2018), 313-330.  doi: 10.1016/j.apm.2017.05.015. [11] B. Li, C. Xu, K. L. Teo and J. Chu, Time optimal Zermelo's navigation problem with moving and fixed obstacles, Applied Mathematics and Computation, 224 (2013), 866-875.  doi: 10.1016/j.amc.2013.08.092. [12] B. Li, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty function method for continuous inequality constrained optimal control problem, Journal of Optimization Theory and Applications, 151 (2011), 260-291.  doi: 10.1007/s10957-011-9904-5. [13] Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, Journal of Industrial and management optimization, 10 (2014), 275-309.  doi: 10.3934/jimo.2014.10.275. [14] Q. Lin, R. Loxton, K. L. Teo, Y. H. Wu and C. Yu, A new exact penalty method for semi-infinite programming problems, Journal of Computational and Applied Mathematics, 261 (2014), 271-286.  doi: 10.1016/j.cam.2013.11.010. [15] R. Loxton, Q. Lin and K. L. Teo, Switching time optimization for nonlinear switched systems: Direct optimization and the time-scaling transformation, Pacific Journal of Optimization, 10 (2014), 537-560. [16] C. C. Pantelides, The consistent initialization of differential-algebraic systems, SIAM Journal on Scientific and Statistical Computing, 9 (1988), 213-231.  doi: 10.1137/0909014. [17] R. Pytlak, Runge-Kutta based procedure for the optimal control of differential-algebraic equations, Journal of Optimization Theory and Applications, 97 (1998), 675-705.  doi: 10.1023/A:1022698311155. [18] R. Pytlak and T. Zawadzki, On solving optimal control problems with higher index DAEs, Optimization Methods & Software, 29 (2014), 1139-1162.  doi: 10.1080/10556788.2014.892597. [19] R. Quirynen, S. Gros and M. Diehl, Fast auto generated ACADO integrators and application to MHE with multi-rate measurements, in Proceedings of the 2013 European Control Conference, Zurich, Switzerland, 2013, 3077–3082. doi: 10.23919/ECC.2013.6669636. [20] R. Quirynen, S. Gros and M. Diehl, Inexact Newton based lifted implicit integrators for fast nonlinear MPC, in Proceedings of the 5th IFAC Nonlinear Model Predictive Control Conference, 48 (2015), 32–38. [21] R. Quirynen, S. Gros and M. Diehl, Lifted implicit integrators for direct optimal control, in Proceedings of 2015 IEEE 54th Annual Conference on Decision and Control, 2015, 3212–3217. doi: 10.1109/CDC.2015.7402701. [22] A. Rao, A survey of numerical methods for optimal control, in Proceedings of the AAS/AIAA Astrodynamics Specialist Conference, Pittsburgh, PA, USA, 2009, 1–32. [23] I. M. Ross and F. Fahroo, Issues in the real-time computation of optimal control, Mathematical and Computer Modelling, 43 (2006), 1172-1188.  doi: 10.1016/j.mcm.2005.05.021. [24] J. M. Sanz-Serna, Symplectic Runge-Kutta schemes for adjoint equations, automatic differentiation, optimal control, and more, SIAM Review, 58 (2016), 3-33.  doi: 10.1137/151002769. [25] A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, Ser. A, 106 (2006), 25-57.  doi: 10.1007/s10107-004-0559-y. [26] A. Walther and A. Griewank, Combinatorial Scientific Computing, chapter Getting started with ADOL-C, 181–202, Chapman-Hall CRC Computational Science, 2012. [27] C. Yu, K. L. Teo, L. Zhang and Y. Bai, A new exact penalty function method for continuous inequality constrained optimization problems, Journal of Industrial and Management Optimization, 6 (2010), 895-910.  doi: 10.3934/jimo.2010.6.895. [28] C. Yu, K. L. Teo, L. Zhang and Y. Bai, On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem, Journal of Industrial and Management Optimization, 8 (2012), 485-491.  doi: 10.3934/jimo.2012.8.485.
Trajectories of angle $\boldsymbol{\vartheta}$ and position $\mathbf{p}$
Trajectories of motor torque $\mathbf{T}$ and its time derivative $\dot{\mathbf{T}}$
Comparison of computational complexity (flops) and memory usage (floating-point numbers) for function evaluation
 Algorithms 1 and 2 Scheme Ⅱ in [9] (13)(18) $2pqn_k^2[\frac{n_k}{3}+1+n_\xi]$ $2pqn_k^2[\frac{n_k}{3}+1+n_\xi]$ (19)(20) $4pqrn_xn_\xi$ $4pqrn_xn_\xi$ (21) $(n_\mathrm{cp}+1)n_\Psi$ $n_\Psi$ (24) $4pq(n_x+n_\mathrm{cp}+1)(n_x+1)$ $pq[(4+2n_c)n_x+4](n_x+1)$ (25) $2(n_\mathrm{cp}+1)n_p+4pqn_x(n_u+1+n_v)$ $2n_p+(4+2n_c)pqn_x(n_u+1+n_v)$ TP $2pqn_kn_\xi$ $2pqn_kn_\xi$ Memory usage $n_M+(n_\mathrm{cp}+1)[2pq(n_x+1)+n_\omega+1]$ $n_M+n_c(n_x+1)+[2pq(n_x+1)+n_\omega+1]$
 Algorithms 1 and 2 Scheme Ⅱ in [9] (13)(18) $2pqn_k^2[\frac{n_k}{3}+1+n_\xi]$ $2pqn_k^2[\frac{n_k}{3}+1+n_\xi]$ (19)(20) $4pqrn_xn_\xi$ $4pqrn_xn_\xi$ (21) $(n_\mathrm{cp}+1)n_\Psi$ $n_\Psi$ (24) $4pq(n_x+n_\mathrm{cp}+1)(n_x+1)$ $pq[(4+2n_c)n_x+4](n_x+1)$ (25) $2(n_\mathrm{cp}+1)n_p+4pqn_x(n_u+1+n_v)$ $2n_p+(4+2n_c)pqn_x(n_u+1+n_v)$ TP $2pqn_kn_\xi$ $2pqn_kn_\xi$ Memory usage $n_M+(n_\mathrm{cp}+1)[2pq(n_x+1)+n_\omega+1]$ $n_M+n_c(n_x+1)+[2pq(n_x+1)+n_\omega+1]$
States in model (26) for the Delta robot
 State Description $\boldsymbol{\vartheta}\triangleq[\vartheta_1, \vartheta_2, \vartheta_3]^\top$ mounting angles of arms $\mathbf{p}\triangleq[p_1, p_2, p_3]^\top$ position of the nacelle $\mathbf{z}\triangleq[z_1, z_2, z_3]^\top$ Lagrange multiplier $\mathbf{T}\triangleq[T_1, T_2, T_3]^\top$ motor torque $\mathbf{q}\triangleq[\boldsymbol{\vartheta}^\top, \mathbf{p}^\top]^\top$ position of the robot
 State Description $\boldsymbol{\vartheta}\triangleq[\vartheta_1, \vartheta_2, \vartheta_3]^\top$ mounting angles of arms $\mathbf{p}\triangleq[p_1, p_2, p_3]^\top$ position of the nacelle $\mathbf{z}\triangleq[z_1, z_2, z_3]^\top$ Lagrange multiplier $\mathbf{T}\triangleq[T_1, T_2, T_3]^\top$ motor torque $\mathbf{q}\triangleq[\boldsymbol{\vartheta}^\top, \mathbf{p}^\top]^\top$ position of the robot
Parameters in model (26) for the Delta robot
 Parameter Value Parameter Value $l_a$ $0.2$m $\alpha_1$ $0$rad $l_f$ $0.6$m $\alpha_2$ $\frac{2\pi}{3}$rad $m_r$ $0.05$kg $\alpha_3$ $\frac{4\pi}{3}$rad $J_r$ $0.1\mathrm{kgm}^2$ $g$ $9.8\mathrm{ms}^{-2}$
 Parameter Value Parameter Value $l_a$ $0.2$m $\alpha_1$ $0$rad $l_f$ $0.6$m $\alpha_2$ $\frac{2\pi}{3}$rad $m_r$ $0.05$kg $\alpha_3$ $\frac{4\pi}{3}$rad $J_r$ $0.1\mathrm{kgm}^2$ $g$ $9.8\mathrm{ms}^{-2}$
Bounds in Problem (27)
 Parameter Value Parameter Value $\vartheta_1^l, \vartheta_2^l, \vartheta_3^l$ $-\frac{\pi}{2}$rad $\vartheta_1^u, \vartheta_2^u, \vartheta_3^u$ $0$rad $T_1^l, T_2^l, T_3^l$ $-5$Nm $T_1^u, T_2^u, T_3^u$ $5$Nm $\dot{T}_1^l$ $-30\mathrm{Nms}^{-1}$ $\dot{T}_1^u$ $30\mathrm{Nms}^{-1}$ $\dot{T}_2^l$ $-50\mathrm{Nms}^{-1}$ $\dot{T}_2^u$ $50\mathrm{Nms}^{-1}$ $\dot{T}_3^l$ $-50\mathrm{Nms}^{-1}$ $\dot{T}_3^u$ $50\mathrm{Nms}^{-1}$
 Parameter Value Parameter Value $\vartheta_1^l, \vartheta_2^l, \vartheta_3^l$ $-\frac{\pi}{2}$rad $\vartheta_1^u, \vartheta_2^u, \vartheta_3^u$ $0$rad $T_1^l, T_2^l, T_3^l$ $-5$Nm $T_1^u, T_2^u, T_3^u$ $5$Nm $\dot{T}_1^l$ $-30\mathrm{Nms}^{-1}$ $\dot{T}_1^u$ $30\mathrm{Nms}^{-1}$ $\dot{T}_2^l$ $-50\mathrm{Nms}^{-1}$ $\dot{T}_2^u$ $50\mathrm{Nms}^{-1}$ $\dot{T}_3^l$ $-50\mathrm{Nms}^{-1}$ $\dot{T}_3^u$ $50\mathrm{Nms}^{-1}$
Comparison of computational complexity and accuracy
 $p$ $\theta$ tunable $\theta$ fixed Algorithm 1 Scheme Ⅱ in [10] Algorithm 1 Scheme Ⅱ in [10] 5 $n_{\mathrm{iter}}$ $200$ $750$ $100$ $750$ $t_{\mathrm{CPU}_1}$ $0.72$ $0.997$ $0.56$ $1.30$ $t_{\mathrm{CPU}_2}$ $24.3$ $27.5$ $22.2$ $4.19$ $t_{\mathrm{CPU}}$ $25.0$ $28.5$ $22.8$ $5.49$ $e_{\mathrm{cp}}$ $0.0470$ $0$ $0.0368$ $0$ $e_{\mathrm{ct}}$ $0.2556$ $1.7135$ $0.3126$ $2.0710$ $\epsilon$ $6.7679\times10^{-7}$ — $7.2565\times10^{-7}$ — 2 $n_{\mathrm{iter}}$ $230$ $750$ — — $t_{\mathrm{CPU}_1}$ $0.57$ $0.405$ — — $t_{\mathrm{CPU}_2}$ $16.2$ $8.69$ — — $t_{\mathrm{CPU}}$ $16.7$ $9.09$ — — $e_{\mathrm{cp}}$ $0.0452$ $0$ — — $e_{\mathrm{ct}}$ $0.4839$ $1.8563$ — — $\epsilon$ $7.4485\times10^{-7}$ — — —
 $p$ $\theta$ tunable $\theta$ fixed Algorithm 1 Scheme Ⅱ in [10] Algorithm 1 Scheme Ⅱ in [10] 5 $n_{\mathrm{iter}}$ $200$ $750$ $100$ $750$ $t_{\mathrm{CPU}_1}$ $0.72$ $0.997$ $0.56$ $1.30$ $t_{\mathrm{CPU}_2}$ $24.3$ $27.5$ $22.2$ $4.19$ $t_{\mathrm{CPU}}$ $25.0$ $28.5$ $22.8$ $5.49$ $e_{\mathrm{cp}}$ $0.0470$ $0$ $0.0368$ $0$ $e_{\mathrm{ct}}$ $0.2556$ $1.7135$ $0.3126$ $2.0710$ $\epsilon$ $6.7679\times10^{-7}$ — $7.2565\times10^{-7}$ — 2 $n_{\mathrm{iter}}$ $230$ $750$ — — $t_{\mathrm{CPU}_1}$ $0.57$ $0.405$ — — $t_{\mathrm{CPU}_2}$ $16.2$ $8.69$ — — $t_{\mathrm{CPU}}$ $16.7$ $9.09$ — — $e_{\mathrm{cp}}$ $0.0452$ $0$ — — $e_{\mathrm{ct}}$ $0.4839$ $1.8563$ — — $\epsilon$ $7.4485\times10^{-7}$ — — —
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