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An optimal pid tuning method for a single-link manipulator based on the control parametrization technique
A time-scaling technique for time-delay switched systems
1. | Department of Mathematics, Shanghai University, Shanghai 200444, China |
2. | Mechatronics Engineering and Automation, Shanghai University, Shanghai 200444, China |
In this paper, we consider a class of optimal control problems governed by nonlinear time-delay switched systems, in which the system parameters and switching times between different subsystems are decision variables to be optimized. We propose a new computational approach to deal with the computational difficulties caused by variable switching times. The original time-delay switched system is firstly transformed into an equivalent switched system defined on a new time horizon where the switching times are fixed, but each of the subsystems contain a variable time-delay that depends on the durations of each sub-system in the original system. By deriving the analytical form for the variable time-delay in the new time horizon, we can solve the new time-delay switched system. Then, gradient-based optimization algorithm can be applied to solve the equivalent problem efficiently. Numerical results show that this new approach is effective.
References:
[1] |
N. U. Ahmed, Dynamic Systems and Control with Applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006.
doi: 10.1142/6262. |
[2] |
N. U. Ahmed, Elements of Finite-Dimensional Systems and Control Theory, Longman Scientific and Technical, New York, 1988. |
[3] |
I. Area, F. Ndairou, J. J. Nieto, C. J. Silva and D. F. M. Torres,
Ebola model and optimal control with vaccination constraints, Journal of Industrial and Management Optimization, 14 (2017), 427-446.
doi: 10.3934/jimo.2017054. |
[4] |
L. D. Berkovitz and N. G. Medhin, Nonlinear Optimal Control Theory, CRC Press, Boca
Raton, FL, 2013. |
[5] |
F. Ceragioli, Discontinuous Ordinary Differential Equations and Stabilization, Ph.D thesis, Universita di Firenze, 1999. |
[6] |
Q. Chai, R. Loxton, K. L. Teo and and C. Yang,
A max-min control problem arising in gradient elution chromatography, Industrial and Engineering Chemistry Research, 51 (2012), 6137-6144.
doi: 10.1021/ie202475p. |
[7] |
Q. Chai, R. Loxton, K. L. Teo and C. Yang,
A unified parameter identification method for nonlinear time-delay systems, Journal of Industrial and Management Optimization, 9 (2013), 471-486.
doi: 10.3934/jimo.2013.9.471. |
[8] |
Q. Chai, R. Loxton, K. L. Teo and C. H. Yang,
Time-delay estimation for nonlinear systems with piecewise-constant input, Applied Mathematics and Computation, 17 (2013), 9543-9560.
doi: 10.1016/j.amc.2013.03.015. |
[9] |
Y. Chen and Y. Zhu,
Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems, Journal of Industrial and Management Optimization, 14 (2018), 913-930.
doi: 10.3934/jimo.2017082. |
[10] |
W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Applications of Mathematics, No.1. Springer-Verlag, Berlin-New York, 1975.
doi: 10.1007/978-1-4612-6380-7. |
[11] |
G. S. F. Frederico and D. F. M. Torres,
Noether's symmetry theorem for variational and optimal control problems with time delay, Numerical Algebra, Control and Optimization, 2 (2012), 619-630.
doi: 10.3934/naco.2012.2.619. |
[12] |
L. Göllmann and H. Maurer,
Theory and applications of optimal control problems with multiple time-delays, Journal of Industrial and Management Optimization, 10 (2014), 413-441.
doi: 10.3934/jimo.2014.10.413. |
[13] |
L. Göllmann, D. Kern and H. Maurer,
Optimal control problems with delays in state and control variables subject to mixed control-state constraints, Optimal Control Applications and Methods, 30 (2009), 341-365.
doi: 10.1002/oca.843. |
[14] |
Z. H. Gong, C. Y. Liu and Y. J. 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. |
[15] |
K. Kaji and K. H. Wong,
Nonlinearly constrained time-delayed optimal control problems, Journal of Optimization Theory and Applications, 82 (1994), 295-313.
doi: 10.1007/BF02191855. |
[16] |
C. Y. Kaya and J. L. Noakes,
Computational method for time-optimal Switching Control, Journal of Optimization Theory and Applications, 117 (2003), 69-92.
doi: 10.1023/A:1023600422807. |
[17] |
H. W. J. Lee, K. L. Teo, V. Rehbock and L. S. Jennings,
Control parametrization enhancing technique for optimal discrete-valued control problems, Automatica, 35 (1999), 1401-1407.
doi: 10.1016/S0005-1098(99)00050-3. |
[18] |
H. W. J. Lee and K. H. Wong,
Semi-infinite programming approach to nonlinear time-delayed optimal control problems with linear continuous constraints, Optimization Methods and Software, 21 (2006), 679-691.
doi: 10.1080/10556780500142306. |
[19] |
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. |
[20] |
Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu,
A new computational method for optimizing nonlinear impulsive systems, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, 18 (2011), 59-76.
doi: 20.500.11937/48414. |
[21] |
C. Y. Liu, R. Loxton and K. L. Teo,
Optimal parameter selection for nonlinear multistage systems with time-delays, Computational Optimization and Applications, 59 (2014), 285-306.
doi: 10.1007/s10589-013-9632-x. |
[22] |
C. Y. Liu and Z. H. Gong,
Modelling and optimal control of a time-delayed switched system in fed-batch process, Journal of the Franklin Institute, 351 (2014), 840-856.
doi: 10.1016/j.jfranklin.2013.09.014. |
[23] |
C. Y. Liu, Z. H. Gong, K. L. Teo, J. Sun and L. Caccetta,
Robust multi-objective optimal switching control arising in 1, 3-propanediol microbial fed-batch process, Nonlinear Analysis: Hybrid Systems, 25 (2017), 1-20.
doi: 10.1016/j.nahs.2017.01.006. |
[24] |
C. Y. Liu, Z. H. Gong and K. L. Teo,
Robust parameter estimation for nonlinear multistage time-delay systems with noisy measurement data, Applied Mathematical Modelling, 53 (2018), 353-368.
doi: 10.1016/j.apm.2017.09.007. |
[25] |
C. Y. Liu, R. Loxton and K. L. Teo,
Switching time and parameter optimization in nonlinear switched systems with multiple time-delays, Journal of Optimization Theory and Applications, 163 (2014), 957-988.
doi: 10.1007/s10957-014-0533-7. |
[26] |
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.
doi: 20.500.11937/19892. |
[27] |
R. Loxton, K. L. Teo and V. Rehbock,
Optimal control problems with multiple characteristic time points in the objective and constraints, Automatica, 44 (2018), 2923-2929.
doi: 10.1016/j.automatica.2008.04.011. |
[28] |
R. Loxton, K. L. Teo and V. Rehbock,
An optimization approach to state-delay identification, IEEE Transactions on Automatic Control, 55 (2010), 2113-2119.
doi: 10.1109/TAC.2010.2050710. |
[29] |
R. Loxton, K. L. Teo, V. Rehbock and K. F. C. Yiu,
Control problems with a continuous inequality constraint on the state and the control, Automatica, 45 (2009), 2250-2257.
doi: 10.1016/j.automatica.2009.05.029. |
[30] |
R. Loxton, K. L. Teo, V. Rehbock and W. K. Ling,
Optimal switching instants for a switched-capacitor DC/DC power converter, Automatica, 45 (2009), 973-980.
doi: 10.1016/j.automatica.2008.10.031. |
[31] |
R. Luus,
Use of Luus-Jaakola optimization procedure for singular optimal control problems, Nonlinear Analysis, Theory, Methods & Applications, 47 (2001), 5647-5658.
doi: 10.1016/S0362-546X(01)00666-6. |
[32] |
H. Maurer and N. P. Osmolovskii,
Second order sufficient conditions for time-optimal bang-bang control, SIAM Journal on Control Optimization, 42 (2004), 2239-2263.
doi: 10.1137/S0363012902402578. |
[33] |
J. Nocedal and S. J. Wright, Numerical Optimization, 2$^nd$ edition, Springer Series in Operations Research and Financial Engineering, New York, 2006. |
[34] |
M. Schlegel, K. Stockmann, T. Binder and W. Marquardt,
Dynamic optimization using adaptive control vector parameterization, Computers and Chemical Engineering, 29 (2005), 1731-1751.
doi: 10.1016/j.compchemeng.2005.02.036. |
[35] |
A. Siburian and V. Rehbock,
Numerical procedure for solving a class of singular optimal control problems, Optimization Methods and Software, 19 (2004), 413-426.
doi: 10.1080/10556780310001656637. |
[36] |
K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, Essex, 1991. |
[37] |
L. Y. Wang, W. H. Gui, K. L. Teo, R. Loxton and C. H. Yang,
Optimal control problems arising in the zinc sulphate electrolyte purification process, Journal of Global Optimization, 54 (2012), 307-323.
doi: 10.1007/s10898-012-9863-x. |
[38] |
L. Y. Wang, W. H. Gui, K. L. Teo, R. Loxton and C. H. Yang,
Time delayed optimal control problems with multiple characteristic time points: Computation and industrial applications, Journal of Industrial and Management Optimization, 5 (2009), 705-718.
doi: 10.3934/jimo.2009.5.705. |
[39] |
J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New
York-London, 1972. |
[40] |
K. H. Wong, L. S. Jennings and F. Benyah,
The control parametrization enhancing transform for constrained time–delayed optimal control problems, ANZIAM Journal, 43 (2002), 154-185.
doi: 10.21914/anziamj.v43i0.469. |
[41] |
S. F. Woon, V. Rehbock and R. Loxton,
Towards global solutions of optimal discrete-valued control problems, Optimal Control Applications and Methods, 33 (2012), 576-594.
doi: 10.1002/oca.1015. |
[42] |
C. Z. Wu and K. L. Teo,
Optimal impulsive control computation, Journal of Industrial and Management Optimization, 2 (2006), 435-450.
doi: 10.3934/jimo.2006.2.435. |
[43] |
C. Z. Wu, K. L. Teo, R. Li and Y. Zhao,
Optimal control of switched systems with time delay, Applied Mathematics Letters, 19 (2006), 1062-1067.
doi: 10.1016/j.aml.2005.11.018. |
[44] |
X. Xiang and Y. Peng,
Second order nonlinear impulsive time-variant systems with unbounded perturbation and optimal controls, Journal of Industrial and Management Optimization, 4 (2008), 17-32.
doi: 10.3934/jimo.2008.4.17. |
[45] |
C. J. Yu, B. Li, R. Loxton and K. L. Teo,
Optimal discrete-valued control computation, Journal of Global Optimization, 56 (2013), 503-518.
doi: 10.1007/s10898-012-9858-7. |
[46] |
C. J. Yu, Q. Lin, R. Loxton, K. L. Teo and G. Wang,
A hybrid time-scaling transformation for time-delay optimal control problems, Journal of Optimization Theory and Applications, 169 (2016), 876-901.
doi: 10.1007/s10957-015-0783-z. |
[47] |
C. J. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai,
A new exact penalty function method for continuous inequality constrained optimization problems, Journal of Industrial Management and Optimization, 6 (2010), 895-910.
doi: 10.3934/jimo.2010.6.895. |
[48] |
Y. Zhao and M. A. Stadtherr,
Rigorous global optimization for dynamic systems subject to inequality path constraints, Industrial and Engineering Chemistry Research, 50 (2011), 12678-12693.
doi: 10.1021/ie200996f. |
show all references
References:
[1] |
N. U. Ahmed, Dynamic Systems and Control with Applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006.
doi: 10.1142/6262. |
[2] |
N. U. Ahmed, Elements of Finite-Dimensional Systems and Control Theory, Longman Scientific and Technical, New York, 1988. |
[3] |
I. Area, F. Ndairou, J. J. Nieto, C. J. Silva and D. F. M. Torres,
Ebola model and optimal control with vaccination constraints, Journal of Industrial and Management Optimization, 14 (2017), 427-446.
doi: 10.3934/jimo.2017054. |
[4] |
L. D. Berkovitz and N. G. Medhin, Nonlinear Optimal Control Theory, CRC Press, Boca
Raton, FL, 2013. |
[5] |
F. Ceragioli, Discontinuous Ordinary Differential Equations and Stabilization, Ph.D thesis, Universita di Firenze, 1999. |
[6] |
Q. Chai, R. Loxton, K. L. Teo and and C. Yang,
A max-min control problem arising in gradient elution chromatography, Industrial and Engineering Chemistry Research, 51 (2012), 6137-6144.
doi: 10.1021/ie202475p. |
[7] |
Q. Chai, R. Loxton, K. L. Teo and C. Yang,
A unified parameter identification method for nonlinear time-delay systems, Journal of Industrial and Management Optimization, 9 (2013), 471-486.
doi: 10.3934/jimo.2013.9.471. |
[8] |
Q. Chai, R. Loxton, K. L. Teo and C. H. Yang,
Time-delay estimation for nonlinear systems with piecewise-constant input, Applied Mathematics and Computation, 17 (2013), 9543-9560.
doi: 10.1016/j.amc.2013.03.015. |
[9] |
Y. Chen and Y. Zhu,
Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems, Journal of Industrial and Management Optimization, 14 (2018), 913-930.
doi: 10.3934/jimo.2017082. |
[10] |
W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Applications of Mathematics, No.1. Springer-Verlag, Berlin-New York, 1975.
doi: 10.1007/978-1-4612-6380-7. |
[11] |
G. S. F. Frederico and D. F. M. Torres,
Noether's symmetry theorem for variational and optimal control problems with time delay, Numerical Algebra, Control and Optimization, 2 (2012), 619-630.
doi: 10.3934/naco.2012.2.619. |
[12] |
L. Göllmann and H. Maurer,
Theory and applications of optimal control problems with multiple time-delays, Journal of Industrial and Management Optimization, 10 (2014), 413-441.
doi: 10.3934/jimo.2014.10.413. |
[13] |
L. Göllmann, D. Kern and H. Maurer,
Optimal control problems with delays in state and control variables subject to mixed control-state constraints, Optimal Control Applications and Methods, 30 (2009), 341-365.
doi: 10.1002/oca.843. |
[14] |
Z. H. Gong, C. Y. Liu and Y. J. 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. |
[15] |
K. Kaji and K. H. Wong,
Nonlinearly constrained time-delayed optimal control problems, Journal of Optimization Theory and Applications, 82 (1994), 295-313.
doi: 10.1007/BF02191855. |
[16] |
C. Y. Kaya and J. L. Noakes,
Computational method for time-optimal Switching Control, Journal of Optimization Theory and Applications, 117 (2003), 69-92.
doi: 10.1023/A:1023600422807. |
[17] |
H. W. J. Lee, K. L. Teo, V. Rehbock and L. S. Jennings,
Control parametrization enhancing technique for optimal discrete-valued control problems, Automatica, 35 (1999), 1401-1407.
doi: 10.1016/S0005-1098(99)00050-3. |
[18] |
H. W. J. Lee and K. H. Wong,
Semi-infinite programming approach to nonlinear time-delayed optimal control problems with linear continuous constraints, Optimization Methods and Software, 21 (2006), 679-691.
doi: 10.1080/10556780500142306. |
[19] |
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. |
[20] |
Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu,
A new computational method for optimizing nonlinear impulsive systems, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, 18 (2011), 59-76.
doi: 20.500.11937/48414. |
[21] |
C. Y. Liu, R. Loxton and K. L. Teo,
Optimal parameter selection for nonlinear multistage systems with time-delays, Computational Optimization and Applications, 59 (2014), 285-306.
doi: 10.1007/s10589-013-9632-x. |
[22] |
C. Y. Liu and Z. H. Gong,
Modelling and optimal control of a time-delayed switched system in fed-batch process, Journal of the Franklin Institute, 351 (2014), 840-856.
doi: 10.1016/j.jfranklin.2013.09.014. |
[23] |
C. Y. Liu, Z. H. Gong, K. L. Teo, J. Sun and L. Caccetta,
Robust multi-objective optimal switching control arising in 1, 3-propanediol microbial fed-batch process, Nonlinear Analysis: Hybrid Systems, 25 (2017), 1-20.
doi: 10.1016/j.nahs.2017.01.006. |
[24] |
C. Y. Liu, Z. H. Gong and K. L. Teo,
Robust parameter estimation for nonlinear multistage time-delay systems with noisy measurement data, Applied Mathematical Modelling, 53 (2018), 353-368.
doi: 10.1016/j.apm.2017.09.007. |
[25] |
C. Y. Liu, R. Loxton and K. L. Teo,
Switching time and parameter optimization in nonlinear switched systems with multiple time-delays, Journal of Optimization Theory and Applications, 163 (2014), 957-988.
doi: 10.1007/s10957-014-0533-7. |
[26] |
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.
doi: 20.500.11937/19892. |
[27] |
R. Loxton, K. L. Teo and V. Rehbock,
Optimal control problems with multiple characteristic time points in the objective and constraints, Automatica, 44 (2018), 2923-2929.
doi: 10.1016/j.automatica.2008.04.011. |
[28] |
R. Loxton, K. L. Teo and V. Rehbock,
An optimization approach to state-delay identification, IEEE Transactions on Automatic Control, 55 (2010), 2113-2119.
doi: 10.1109/TAC.2010.2050710. |
[29] |
R. Loxton, K. L. Teo, V. Rehbock and K. F. C. Yiu,
Control problems with a continuous inequality constraint on the state and the control, Automatica, 45 (2009), 2250-2257.
doi: 10.1016/j.automatica.2009.05.029. |
[30] |
R. Loxton, K. L. Teo, V. Rehbock and W. K. Ling,
Optimal switching instants for a switched-capacitor DC/DC power converter, Automatica, 45 (2009), 973-980.
doi: 10.1016/j.automatica.2008.10.031. |
[31] |
R. Luus,
Use of Luus-Jaakola optimization procedure for singular optimal control problems, Nonlinear Analysis, Theory, Methods & Applications, 47 (2001), 5647-5658.
doi: 10.1016/S0362-546X(01)00666-6. |
[32] |
H. Maurer and N. P. Osmolovskii,
Second order sufficient conditions for time-optimal bang-bang control, SIAM Journal on Control Optimization, 42 (2004), 2239-2263.
doi: 10.1137/S0363012902402578. |
[33] |
J. Nocedal and S. J. Wright, Numerical Optimization, 2$^nd$ edition, Springer Series in Operations Research and Financial Engineering, New York, 2006. |
[34] |
M. Schlegel, K. Stockmann, T. Binder and W. Marquardt,
Dynamic optimization using adaptive control vector parameterization, Computers and Chemical Engineering, 29 (2005), 1731-1751.
doi: 10.1016/j.compchemeng.2005.02.036. |
[35] |
A. Siburian and V. Rehbock,
Numerical procedure for solving a class of singular optimal control problems, Optimization Methods and Software, 19 (2004), 413-426.
doi: 10.1080/10556780310001656637. |
[36] |
K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, Essex, 1991. |
[37] |
L. Y. Wang, W. H. Gui, K. L. Teo, R. Loxton and C. H. Yang,
Optimal control problems arising in the zinc sulphate electrolyte purification process, Journal of Global Optimization, 54 (2012), 307-323.
doi: 10.1007/s10898-012-9863-x. |
[38] |
L. Y. Wang, W. H. Gui, K. L. Teo, R. Loxton and C. H. Yang,
Time delayed optimal control problems with multiple characteristic time points: Computation and industrial applications, Journal of Industrial and Management Optimization, 5 (2009), 705-718.
doi: 10.3934/jimo.2009.5.705. |
[39] |
J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New
York-London, 1972. |
[40] |
K. H. Wong, L. S. Jennings and F. Benyah,
The control parametrization enhancing transform for constrained time–delayed optimal control problems, ANZIAM Journal, 43 (2002), 154-185.
doi: 10.21914/anziamj.v43i0.469. |
[41] |
S. F. Woon, V. Rehbock and R. Loxton,
Towards global solutions of optimal discrete-valued control problems, Optimal Control Applications and Methods, 33 (2012), 576-594.
doi: 10.1002/oca.1015. |
[42] |
C. Z. Wu and K. L. Teo,
Optimal impulsive control computation, Journal of Industrial and Management Optimization, 2 (2006), 435-450.
doi: 10.3934/jimo.2006.2.435. |
[43] |
C. Z. Wu, K. L. Teo, R. Li and Y. Zhao,
Optimal control of switched systems with time delay, Applied Mathematics Letters, 19 (2006), 1062-1067.
doi: 10.1016/j.aml.2005.11.018. |
[44] |
X. Xiang and Y. Peng,
Second order nonlinear impulsive time-variant systems with unbounded perturbation and optimal controls, Journal of Industrial and Management Optimization, 4 (2008), 17-32.
doi: 10.3934/jimo.2008.4.17. |
[45] |
C. J. Yu, B. Li, R. Loxton and K. L. Teo,
Optimal discrete-valued control computation, Journal of Global Optimization, 56 (2013), 503-518.
doi: 10.1007/s10898-012-9858-7. |
[46] |
C. J. Yu, Q. Lin, R. Loxton, K. L. Teo and G. Wang,
A hybrid time-scaling transformation for time-delay optimal control problems, Journal of Optimization Theory and Applications, 169 (2016), 876-901.
doi: 10.1007/s10957-015-0783-z. |
[47] |
C. J. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai,
A new exact penalty function method for continuous inequality constrained optimization problems, Journal of Industrial Management and Optimization, 6 (2010), 895-910.
doi: 10.3934/jimo.2010.6.895. |
[48] |
Y. Zhao and M. A. Stadtherr,
Rigorous global optimization for dynamic systems subject to inequality path constraints, Industrial and Engineering Chemistry Research, 50 (2011), 12678-12693.
doi: 10.1021/ie200996f. |





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