Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-19T03:14:14.468Z Has data issue: false hasContentIssue false
  • English
  • Français

Indirect solution of optimal control problems with state variable inequality constraints: finite difference approximation

Published online by Cambridge University Press:  02 July 2015

A. Nikoobin  and
M. Moradi
A. Nikoobin*
Affiliation:
Faculty of Mechanical Engineering, Semnan University, Semnan, Iran
M. Moradi
Affiliation:
Faculty of Electrical and Computer Engineering, Semnan University, Semnan, Iran Email: m_moradi@aut.ac.ir
*
*Corresponding author. E-mail: anikoobin@semnan.ac.ir
Get access Rights & Permissions [Opens in a new window]

Summary

In this paper, a method for the indirect solution of the optimal control problem (OCP) in the presence of pure state variable inequality constraints (SVICs) and mixed state-control inequality constraints (SCIC), without a need for a close initial guess is presented. In the proposed method, using the finite difference approximation (FDA), the pure SVICs are converted to SCIC. Here, the distance of the constraint function to the feasibility bounds of the constraint is computed in every situation and the control signal is chosen appropriately to facilitate the constraint stays safe. In this method, prior knowledge of the numbers and sequences of activation times is not required. So, it can be simply implemented in continuous boundary value problem (BVP) solvers. The proposed method simply applies the SVICs and since the constraint is directly applied on the control signal, it improves the convergence. On the other hand, because of the convergence problem in the indirect solution of OCP, the simple homotopy continuation method (HCM) is used to overcome the initial guess problem by deploying a secondary OCP for which the initial guess can be zero. The proposed approach is applied on a few comprehensive problems in the presence of different constraints. Simulations are compared with the direct solution of the OCP to confirm the accuracy and with the penalty function method and the sequential constraint-free OCP to confirm the convergence. The results indicate that the FDA method for handling the constraints along with the HCM is easy to apply with acceptable accuracy and convergence, even for highly nonlinear problems in robotic systems such as the constrained time optimal control of a two-link manipulator (TLM) and a three-link common industrial robot.

Keywords

Optimal controlIndirect solutionPontryagin maximum principleState variable inequality constraintHomotopy methodFinite difference approximation
Type
Articles
Information
Robotica , Volume 35 , Issue 1 , January 2017 , pp. 50 - 72
Copyright
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Elbert, P., Ebbesen, S. and Guzzella, L., “Implementation of dynamic programming for n-dimensional optimal control problems with final state constraints,” IEEE Trans. Control Syst. Technol., 21 (3), 924–931 (2013).Google Scholar
2. Betts, J. T., Practical Methods for Optimal Control and Estimation using Nonlinear Programming. vol. 19 (SIAM Press, Philadelphia, 2001).Google Scholar
3. Sargent, R., “Optimal control,” J. Comput. Appl. Math. 124, 361–371 (2000).CrossRefGoogle Scholar
4. Coleman, T. F. and Li, Y., “An interior trust region approach for nonlinear minimization subject to bounds,” SIAM J. Optim. 6, 418–445 (1996).Google Scholar
5. Gill, P. E., Murray, W. and Saunders, M. A., “SNOPT: An SQP algorithm for large-scale constrained optimization,” SIAM J. Optim. 12, 979–1006 (2002).CrossRefGoogle Scholar
6. Waltz, R. A., Morales, J. L., Nocedal, J. and Orban, D., “An interior algorithm for nonlinear optimization that combines line search and trust region steps,” Math. Program. 107, 391–408 (2006).Google Scholar
7. Garg, D., “Advances in Global Pseudospectral Methods for Optimal Control,” (Department of Mechanical and Aerospace Engineering, University of Florida, 2011).Google Scholar
8. Büskens, C. and Maurer, H., “SQP-methods for solving optimal control problems with control and state constraints: Adjoint variables, sensitivity analysis and real-time control,” J. Comput. Appl. Math. 120, 85–108 (2000).Google Scholar
9. Anderson, E. P., Beard, R. W. and McLain, T. W., “Real-time dynamic trajectory smoothing for unmanned air vehicles,” IEEE Trans. Control Syst. Technol. 13, 471–477 (2005).CrossRefGoogle Scholar
10. Nikoobin, A. and Moradi, M., “Optimal balancing of robot manipulators in point-to-point motion,” Robotica, 29, 233–244 (2011).Google Scholar
11. Korayem, M. and Nikoobin, A., “Maximum payload for flexible joint manipulators in point-to-point task using optimal control approach,” Int. J. Adv. Manuf. Technol. 38, 1045–1060 (2008).CrossRefGoogle Scholar
12. Ranieri, C. L. and Ocampo, C. A., “Indirect optimization of three-dimensional finite-burning interplanetary transfers including spiral dynamics,” J. Guid. Control Dyn. 32, 445–455 (2009).Google Scholar
13. Gregory, J., Olivares, A. and Staffetti, E., “Energy-optimal trajectory planning for robot manipulators with holonomic constraints,” Syst. Control Lett. 61, 279–291 (2012).Google Scholar
14. Nikoobin, A., Moradi, M. and Esmaili, A., “Optimal spring balancing of robot manipulators in point-to-point motion,” Robotica, 1, 1–11 (2013).Google Scholar
15. Hannemann, R. and Marquardt, W., “Continuous and discrete composite adjoints for the Hessian of the Lagrangian in shooting algorithms for dynamic optimization,” SIAM J. Sci. Comput. 31, 4675–4695 (2010).Google Scholar
16. Zhulin, S., “Homotopy method for finding extremals in optimal control problems,” Differ. Equ. 43, 1495–1504 (2007).Google Scholar
17. Steinboeck, A., Graichen, K. and Kugi, A., “Dynamic optimization of a slab reheating furnace with consistent approximation of control variables,” IEEE Trans. Control Syst. Technol. 19, 1444–1456 (2011).Google Scholar
18. Hermant, A., “Optimal control of the atmospheric reentry of a space shuttle by an homotopy method,” Optim. Control Appl. Methods, 32, 627–646 (2011).CrossRefGoogle Scholar
19. Liao, S., “Notes on the homotopy analysis method: Some definitions and theorems,” Commun. Nonlinear Sci. Numer. Simul. 14, 983–997 (2009).Google Scholar
20. Graichen, K. and Petit, N., “Constructive methods for initialization and handling mixed state-input constraints in optimal control,” J. Guid. Control Dyn. 31, 1334–1343 (2008).Google Scholar
21. Arutyunov, A. V., Karamzin, D. Y. and Pereira, F. L., “The maximum principle for optimal control problems with state constraints by R. V. Gamkrelidze: Revisited,” J. Optim. Theory Appl. 149, 474–493 (2011).Google Scholar
22. Kelley, H. J., “Method of gradients,” Acad. Press, 2, 1578–1580 (1962).Google Scholar
23. Lasdon, L. S., Waren, A. and Rice, R., “An interior penalty method for inequality constrained optimal control problems,” IEEE Trans. Autom. Control, 12, 388–395 (1967).CrossRefGoogle Scholar
24. Moradi, M., Naraghi, M. and Nikoobin, A., “Indirect Optimal Trajectory Planning of Robotic Manipulators with the Homotopy Continuation Technique,” Second RSI/ISM International Conference on, Robotics and Mechatronics (ICRoM), (2014) pp. 286–291.Google Scholar
25. Malisani, P., Chaplais, F. and Petit, N., “An interior penalty method for optimal control problems with state and input constraints of nonlinear systems,” Optim. Control Appl. Methods (2014 In Press), DOI: 10.1002/oca.2134.Google Scholar
26. Graichen, K. and Petit, N., “Incorporating a class of constraints into the dynamics of optimal control problems,” Optim. Control Appl. Methods, 30, 537–561 (2009).Google Scholar
27. Jacobson, D. and Lele, M., “A transformation technique for optimal control problems with a state variable inequality constraint,” IEEE Trans. Autom. Control, 14, 457–464 (1969).Google Scholar
28. Jacobson, D. H., Lele, M. and Speyer, J. L., “New necessary conditions of optimality for control problems with state-variable inequality constraints,” J. Math. Anal. Appl. 35, 255–284 (1971).Google Scholar
29. Fabien, B. C., “Indirect solution of inequality constrained and singular optimal control problems via a simple continuation method,” J. Dyn. Syst. Meas. Control, 136, 021003 (2014).Google Scholar
30. Gerdts, M. and Hüpping, B., “Virtual control regularization of state constrained linear quadratic optimal control problems,” Comput. Optim. Appl. 51, 867–882 (2012).CrossRefGoogle Scholar
31. Augustin, D. and Maurer, H., “Computational sensitivity analysis for state constrained optimal control problems,” Ann. Oper. Res. 101, 75–99 (2001).Google Scholar
32. Diehl, M., Bock, H. G., Diedam, H. and Wieber, P.-B., “Fast Direct Multiple Shooting Algorithms for Optimal Robot Control,” In: Fast Motions in Biomechanics and Robotics, (Springer, 2006), pp. 65–93.Google Scholar
33. Gerdts, M. and Kunkel, M., “A nonsmooth Newton's method for discretized optimal control problems with state and control constraints,” J. Ind. Manage. Optim. 4, 247 (2008).CrossRefGoogle Scholar
34. Chen, T. W. and Vassiliadis, V. S., “Inequality path constraints in optimal control: A finite iteration ϵ-convergent scheme based on pointwise discretization,” J. Process Control, 15, 353–362 (2005).Google Scholar
35. Rousseau, G., Tran, Q. and Sinoquet, D., “Scop: A sequential Constraint-free Optimal Control Problem Algorithm,” In Control and Decision Conference, 2008. CCDC 2008. (Chinese, 2008) pp. 273–278.Google Scholar
36. Pytlak, R. and Vinter, R., “A feasible directions algorithm for optimal control problems with state and control constraints: Convergence analysis,” SIAM J. Control Optim. 36, 1999–2019 (1998).Google Scholar
37. Teo, K., Jennings, L., Lee, H. and Rehbock, V., “The control parameterization enhancing transform for constrained optimal control problems,” J. Aust. Math. Soc. Ser. B. Appl. Math. 40, 314–335 (1999).Google Scholar
38. Loxton, R., Teo, K. L., Rehbock, V. and Yiu, K. F. C., “Optimal control problems with a continuous inequality constraint on the state and the control,” Automatica, 45, 2250–2257 (2009).CrossRefGoogle Scholar
39. Van Keulen, T., Gillot, J., De Jager, B. and Steinbuch, M., “Solution for state constrained optimal control problems applied to power split control for hybrid vehicles,” Automatica, 50, 187–192 (2014).Google Scholar
40. Murphy, F. H., “A class of exponential penalty functions,” SIAM J. Control, 12, 679–687 (1974).Google Scholar
41. Cominetti, R. and Dussault, J.-P., “Stable exponential-penalty algorithm with superlinear convergence,” J. Optim. Theory Appl. 83, 285–309 (1994).Google Scholar
42. Antczak, T., “A new exact exponential penalty function method and nonconvex mathematical programming,” Appl. Math. Comput. 217, 6652–6662 (2011).Google Scholar
43. Strodiot, J. and Nguyen, V., “An exponential penalty method for nondifferentiable minimax problems with general constraints,” J. Optim. Theory Appl. 27, 205–219 (1979).Google Scholar
44. Becerra, V. M., “Solving optimal control problems with state constraints using nonlinear programming and simulation tools,” IEEE Trans. Educ. 47, 377–384 (2004).CrossRefGoogle Scholar
45. Moradi, M., Nikoobin, A. and Azadi, S., “Adaptive decoupling for open chain planar robots,” Sci. Iranica: Trans. B, Mech. Eng. 17, 376–386 (2010).Google Scholar
46. Fotouhi-c, R. and Szyszkowski, W., “An algorithm for time-optimal control problems,” Trans.-Am. Soc. Mech. Eng. J. Dyn. Syst. Meas. Control, 120, 414–418 (1998).Google Scholar