# American Institute of Mathematical Sciences

## 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 & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020109
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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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