June  2020, 13(6): 1683-1695. doi: 10.3934/dcdss.2020098

Time-scaling transformation for optimal control problem with time-varying delay

1. 

Department of Mathematics, Shanghai University, Shanghai 200444, China

2. 

School of Economics, Shanghai University of Finance and Economics, Shanghai 200433, China

* Corresponding author: Yanqin Bai

Received  February 2018 Revised  May 2018 Published  September 2019

Fund Project: This research was supported by grants from the National Natural Science Foundation of China (No.11771275, No.11871039)

This paper focuses on the solution to nonlinear time-delay optimal control problems with time-varying delay subject to canonical equality and inequality constraints. Traditional control parameterization in conjunction with time-scaling transformation could optimize control parameters and switching times at the same time when the time-delays in the dynamic system are taken as constants. The purpose of this paper is to extend this method to solve the dynamic systems with time-varying delay. We introduce a hybrid time-scaling transformation that converts the given time-delay system into an equivalent system defined on a new time horizon with fixed switching times. Meanwhile, we obtain the value of time-delay state utilizing the relationship between the new time scale and the original one. After computing the gradients of the cost and constraints with respect to the control heights and its durations, we could solve the equivalent optimal control problem using gradient based optimization method.

Citation: Di Wu, Yanqin Bai, Fusheng Xie. Time-scaling transformation for optimal control problem with time-varying delay. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1683-1695. doi: 10.3934/dcdss.2020098
References:
[1]

E. B. M. Bashier and K. C. Patidar, Optimal control of an epidemiological model with multiple time delays, Applied Mathematics and Computation, 292 (2017), 47-56.  doi: 10.1016/j.amc.2016.07.009.

[2]

J. T. Betts, S. L. Campbell and K. C. Thompson, Optimal control software for constrained nonlinear systems with delays. in Proceedings, IEEE Multi Conference on Systems and Control, (2011), 444-449. doi: 10.1109/CACSD.2011.6044560.

[3]

Q. Q. Chai and W. Wang, A computational method for free terminal time optimal control problem governed by nonlinear time delayed systems, Applied Mathematical Modelling, 53 (2018), 242-250.  doi: 10.1016/j.apm.2017.08.023.

[4]

M. DadkhahM. H. Farahi and A. Heydari, Optimal control of a class of non-linear time-delay systems via hybrid functions, IMA Journal of Mathematical Control and Information, 34 (2017), 255-270.  doi: 10.1093/imamci/dnv044.

[5]

R. Dehghan and M. Keyanpour, A numerical approximation for delay fractional optimal control problems based on the method of moments, IMA Journal of Mathematical Control and Information, 34 (2017), 77-92.  doi: 10.1093/imamci/dnv032.

[6]

L. Denis-VidalC. Jauberthie and G. Joly-Blanchard, Identifiability of a nonlinear delayed-differential aerospace model, IEEE Trains. Autom. Control, 51 (2006), 154-158.  doi: 10.1109/TAC.2005.861700.

[7]

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.

[8]

Z. GongC. 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.

[9]

A. Jajarmi and M. Hajipour, An Efficient finite differencr method for the time-delay optimal control problems with time-varying delay, Asian Journal of Control, 19 (2017), 554-563.  doi: 10.1002/asjc.1371.

[10]

H. W. J. LeeK. L. TeoV. Rehbock and L. S. Jennings, Control parametrization enhancing technique for time optimal control problems, Dynamic Systems and Applications, 6 (1997), 243-261. 

[11]

H. W. J. LeeK. L. TeoV. Rehbock and L. S. Jennings, Control parameterization enhancing technique for optimal discrete-valued control problems, Automatica, 35 (1999), 1401-1407.  doi: 10.1016/S0005-1098(99)00050-3.

[12]

J. Lei, Optimal vibration control of nonlinear systems with multiple time-delays: An application to vehicle suspension, Integrated Ferroelectrics, 170 (2016), 10-32.  doi: 10.1080/10584587.2016.1165574.

[13]

G. N. LiH. L. Xu and Y. Lin, Applicationof bat algorithm based time optimal control in multi-robots formation reconfiguration, Journal of Bionic Engneering, 15 (2018), 126-138.  doi: 10.1007/s42235-017-0010-8.

[14]

Q. LinR. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survery, Journal of Industrial and Management Optimization, 10 (2014), 275-309.  doi: 10.3934/jimo.2014.10.275.

[15]

C. LiuZ. 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.

[16]

C. LiuR. Loxton and K. L. Teo, A computational method for solving time-delay optimal control problems with free terminal time, Systems & Control Letters, 72 (2014), 53-60.  doi: 10.1016/j.sysconle.2014.07.001.

[17]

C. Y. LiuR. 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.

[18]

P. LiuG. D. LiX. G. LiuL. XiaoY. L. WangC. H. Yang and W. H. Gui, A novel non-uniform control vector parameterization approach with time grid refinement for flight level tracking optimal control problems, ISA transactions, 73 (2018), 66-78.  doi: 10.1016/j.isatra.2017.12.008.

[19]

G. R. Marzban and S. M. Hoseini, An efficient discretization scheme for solving nonlinear optimal control problems with multiple time delays, Optimal Control Applications and Methods, 37 (2016), 682-707.  doi: 10.1002/oca.2187.

[20]

K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, Essex, 1991.

[21]

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.

[22]

C. Z. WuK. L. TeoR. Li and Y. Zhao, Optimal control of switched systems with time delay, Applied Mathematics Letters, 19 (2006), 1062-1067. 

[23]

Y. WuZ. Yuan and Y. Wu, Optimal tracking control for networked control systems with random time delays and packet dropouts, Journal of Industrial and Management Optimization, 11 (2015), 1343-1354.  doi: 10.3934/jimo.2015.11.1343.

[24]

F. YangK. L. TeoR. LoxtonV. RehbockB. LiC. Yu and L. Jennings, VISUAL MISER: An efficient user-friendly visual program for solving optimal control problems, Journal of Industrial and Management Optimization, 12 (2016), 781-810.  doi: 10.3934/jimo.2016.12.781.

[25]

C. YuB. LiR. 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.

[26]

C. YuQ. LinR. LoxtonK. L. Teo and G. Wang, A hybrid time-scaling transformation for time-delay optimal control problems, Journal of Optimization Theory and Application, 169 (2016), 876-901.  doi: 10.1007/s10957-015-0783-z.

show all references

References:
[1]

E. B. M. Bashier and K. C. Patidar, Optimal control of an epidemiological model with multiple time delays, Applied Mathematics and Computation, 292 (2017), 47-56.  doi: 10.1016/j.amc.2016.07.009.

[2]

J. T. Betts, S. L. Campbell and K. C. Thompson, Optimal control software for constrained nonlinear systems with delays. in Proceedings, IEEE Multi Conference on Systems and Control, (2011), 444-449. doi: 10.1109/CACSD.2011.6044560.

[3]

Q. Q. Chai and W. Wang, A computational method for free terminal time optimal control problem governed by nonlinear time delayed systems, Applied Mathematical Modelling, 53 (2018), 242-250.  doi: 10.1016/j.apm.2017.08.023.

[4]

M. DadkhahM. H. Farahi and A. Heydari, Optimal control of a class of non-linear time-delay systems via hybrid functions, IMA Journal of Mathematical Control and Information, 34 (2017), 255-270.  doi: 10.1093/imamci/dnv044.

[5]

R. Dehghan and M. Keyanpour, A numerical approximation for delay fractional optimal control problems based on the method of moments, IMA Journal of Mathematical Control and Information, 34 (2017), 77-92.  doi: 10.1093/imamci/dnv032.

[6]

L. Denis-VidalC. Jauberthie and G. Joly-Blanchard, Identifiability of a nonlinear delayed-differential aerospace model, IEEE Trains. Autom. Control, 51 (2006), 154-158.  doi: 10.1109/TAC.2005.861700.

[7]

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.

[8]

Z. GongC. 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.

[9]

A. Jajarmi and M. Hajipour, An Efficient finite differencr method for the time-delay optimal control problems with time-varying delay, Asian Journal of Control, 19 (2017), 554-563.  doi: 10.1002/asjc.1371.

[10]

H. W. J. LeeK. L. TeoV. Rehbock and L. S. Jennings, Control parametrization enhancing technique for time optimal control problems, Dynamic Systems and Applications, 6 (1997), 243-261. 

[11]

H. W. J. LeeK. L. TeoV. Rehbock and L. S. Jennings, Control parameterization enhancing technique for optimal discrete-valued control problems, Automatica, 35 (1999), 1401-1407.  doi: 10.1016/S0005-1098(99)00050-3.

[12]

J. Lei, Optimal vibration control of nonlinear systems with multiple time-delays: An application to vehicle suspension, Integrated Ferroelectrics, 170 (2016), 10-32.  doi: 10.1080/10584587.2016.1165574.

[13]

G. N. LiH. L. Xu and Y. Lin, Applicationof bat algorithm based time optimal control in multi-robots formation reconfiguration, Journal of Bionic Engneering, 15 (2018), 126-138.  doi: 10.1007/s42235-017-0010-8.

[14]

Q. LinR. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survery, Journal of Industrial and Management Optimization, 10 (2014), 275-309.  doi: 10.3934/jimo.2014.10.275.

[15]

C. LiuZ. 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.

[16]

C. LiuR. Loxton and K. L. Teo, A computational method for solving time-delay optimal control problems with free terminal time, Systems & Control Letters, 72 (2014), 53-60.  doi: 10.1016/j.sysconle.2014.07.001.

[17]

C. Y. LiuR. 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.

[18]

P. LiuG. D. LiX. G. LiuL. XiaoY. L. WangC. H. Yang and W. H. Gui, A novel non-uniform control vector parameterization approach with time grid refinement for flight level tracking optimal control problems, ISA transactions, 73 (2018), 66-78.  doi: 10.1016/j.isatra.2017.12.008.

[19]

G. R. Marzban and S. M. Hoseini, An efficient discretization scheme for solving nonlinear optimal control problems with multiple time delays, Optimal Control Applications and Methods, 37 (2016), 682-707.  doi: 10.1002/oca.2187.

[20]

K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, Essex, 1991.

[21]

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.

[22]

C. Z. WuK. L. TeoR. Li and Y. Zhao, Optimal control of switched systems with time delay, Applied Mathematics Letters, 19 (2006), 1062-1067. 

[23]

Y. WuZ. Yuan and Y. Wu, Optimal tracking control for networked control systems with random time delays and packet dropouts, Journal of Industrial and Management Optimization, 11 (2015), 1343-1354.  doi: 10.3934/jimo.2015.11.1343.

[24]

F. YangK. L. TeoR. LoxtonV. RehbockB. LiC. Yu and L. Jennings, VISUAL MISER: An efficient user-friendly visual program for solving optimal control problems, Journal of Industrial and Management Optimization, 12 (2016), 781-810.  doi: 10.3934/jimo.2016.12.781.

[25]

C. YuB. LiR. 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.

[26]

C. YuQ. LinR. LoxtonK. L. Teo and G. Wang, A hybrid time-scaling transformation for time-delay optimal control problems, Journal of Optimization Theory and Application, 169 (2016), 876-901.  doi: 10.1007/s10957-015-0783-z.

Figure 1.  Optimal control for Problem 1 for the case of $ q = 7 $
Figure 2.  Optimal state trajectory for Problem 1 using hybrid time-scaling transformation for the case of $ q = 7 $
Figure 3.  Optimal state trajectory for Problem 1 using traditional control parameterization with fixed switching times ($ q = 7 $)
Figure 4.  Optimal control for Problem 2 for the case of $ q = 8 $
Figure 5.  Optimal state trajectory for Problem 2 using hybrid time-scaling transformation for the case of $ q = 8 $
Figure 6.  Optimal state trajectory for Problem 2 using traditional control parameterization with fixed switching times ($ q = 8 $)
Table 1.  Optimal cost of Problem 1
(a) using hybrid time-scaling transformation (b) using traditional control parameterization
partition number $ g_{0}(\mathit{\boldsymbol{u}}^{q,*}) $ partition number $ g_{0}(\mathit{\boldsymbol{u}}^{q,*}) $
$ q=7 $ 2.1275 $ q=7 $ 2.1347
$ q=5 $ 2.1333 $ q=5 $ 2.1446
$ q=3 $ 2.1340 $ q=3 $ 2.1812
(a) using hybrid time-scaling transformation (b) using traditional control parameterization
partition number $ g_{0}(\mathit{\boldsymbol{u}}^{q,*}) $ partition number $ g_{0}(\mathit{\boldsymbol{u}}^{q,*}) $
$ q=7 $ 2.1275 $ q=7 $ 2.1347
$ q=5 $ 2.1333 $ q=5 $ 2.1446
$ q=3 $ 2.1340 $ q=3 $ 2.1812
Table 2.  Parameters in Problem 2
a b c $ t_{f} $ Q R S
0.2 0.5 0.2 1.5 $ I_{2\times2} $ $ I_{2\times2} $ $ 10^{4}I_{2\times2} $
a b c $ t_{f} $ Q R S
0.2 0.5 0.2 1.5 $ I_{2\times2} $ $ I_{2\times2} $ $ 10^{4}I_{2\times2} $
Table 3.  Optimal cost of Problem 2
(a) Using hybrid time-scaling transformation (b) Using traditional control parameterization
partition number $ g_{0}(\mathit{\boldsymbol{u}}^{q,*}) $ partition number $ g_{0}(\mathit{\boldsymbol{u}}^{q,*}) $
$ q=8 $ 3.8750 $ q=8 $ 5.3252
$ q=6 $ 4.2319 $ q=6 $ 6.1942
$ q=4 $ 6.9578 $ q=4 $ 7.9119
(a) Using hybrid time-scaling transformation (b) Using traditional control parameterization
partition number $ g_{0}(\mathit{\boldsymbol{u}}^{q,*}) $ partition number $ g_{0}(\mathit{\boldsymbol{u}}^{q,*}) $
$ q=8 $ 3.8750 $ q=8 $ 5.3252
$ q=6 $ 4.2319 $ q=6 $ 6.1942
$ q=4 $ 6.9578 $ q=4 $ 7.9119
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