A time-scaling technique for time-delay switched systems

Linna Li; Changjun Yu; Ning Zhang; Yanqin Bai; Zhiyuan Gao

doi:10.3934/dcdss.2020108

Previous Article
An efficient adjoint computational method based on lifted IRK integrator and exact penalty function for optimal control problems involving continuous inequality constraints
DCDS-S Home
This Issue
Next Article
An optimal pid tuning method for a single-link manipulator based on the control parametrization technique

June 2020, 13(6): 1825-1843. doi: 10.3934/dcdss.2020108

A time-scaling technique for time-delay switched systems

Linna Li ^1, , Changjun Yu ^1,, , Ning Zhang ^1, , Yanqin Bai ^1, and Zhiyuan Gao ^2,

1.	Department of Mathematics, Shanghai University, Shanghai 200444, China
2.	Mechatronics Engineering and Automation, Shanghai University, Shanghai 200444, China

^* Corresponding author: Changjun Yu

Received May 2018 Revised August 2018 Published September 2019

Fund Project: This work is supported by National Natural Science Foundation of China (NSFC grant number 11871039, 11771275)

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.

Keywords: Switched system, time-delay system, switching time optimization, time-scaling transformation, gradient-based optimization.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation: Linna Li, Changjun Yu, Ning Zhang, Yanqin Bai, Zhiyuan Gao. A time-scaling technique for time-delay switched systems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1825-1843. doi: 10.3934/dcdss.2020108

References:

[1]	N. U. Ahmed, Dynamic Systems and Control with Applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. doi: 10.1142/6262. Google Scholar
[2]	N. U. Ahmed, Elements of Finite-Dimensional Systems and Control Theory, Longman Scientific and Technical, New York, 1988. Google Scholar
[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. Google Scholar
[4]	L. D. Berkovitz and N. G. Medhin, Nonlinear Optimal Control Theory, CRC Press, Boca Raton, FL, 2013. Google Scholar
[5]	F. Ceragioli, Discontinuous Ordinary Differential Equations and Stabilization, Ph.D thesis, Universita di Firenze, 1999. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[33]	J. Nocedal and S. J. Wright, Numerical Optimization, 2$^nd$ edition, Springer Series in Operations Research and Financial Engineering, New York, 2006. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[39]	J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York-London, 1972. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[2]	N. U. Ahmed, Elements of Finite-Dimensional Systems and Control Theory, Longman Scientific and Technical, New York, 1988. Google Scholar
[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. Google Scholar
[4]	L. D. Berkovitz and N. G. Medhin, Nonlinear Optimal Control Theory, CRC Press, Boca Raton, FL, 2013. Google Scholar
[5]	F. Ceragioli, Discontinuous Ordinary Differential Equations and Stabilization, Ph.D thesis, Universita di Firenze, 1999. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[33]	J. Nocedal and S. J. Wright, Numerical Optimization, 2$^nd$ edition, Springer Series in Operations Research and Financial Engineering, New York, 2006. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[39]	J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York-London, 1972. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

Figure 1. The relationship between $ (s, \mu(s|\mathit{\boldsymbol{\sigma}})) $ and $ (\omega(s|\mathit{\boldsymbol{\sigma}}), \mu(s|\mathit{\boldsymbol{\sigma}})-h) $. Note that $ \mu(s|\mathit{\boldsymbol{\sigma}}) $ is a piecewise linear function

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. The relationship between $ (s, t) $ and $ (\omega(s), \mu(\omega(s))) $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3. A demonstration for two scenarios of $ \sigma_{\lfloor s\rfloor +1} $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4. Optimal state trajectory for Problem 1 for $ \sigma_i\in[0, 1) $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5. Optimal state trajectory for Problem 2

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6. Optimal control $ \xi_1, \xi_2, \xi_3, \xi_4 $ for Problem 2

Figure Options

Download full-size image

Download as PowerPoint slide

Table 1. Optimal cost if the interval of $ \sigma_i $ is different

the interval of $ \sigma_i $	$ \tilde{{G}}^*_{0}(s, \mathit{\boldsymbol{\sigma}}) $	the optimal $ \mathit{\boldsymbol{\sigma}} $
$ \sigma_i\in[0, 1] $	$ 3.9855\times 10^{-3} $	$ \mathit{\boldsymbol{\sigma}}^\star=(0.000, 0.46836125, 0.53163875) $
$ \sigma_i\in[0.001, 1) $	$ 4.0432\times 10^{-3} $	$ \mathit{\boldsymbol{\sigma}}^\star=(0.001, 0.46779421, 0.53120579) $
$ \sigma_i\in[0.01, 1) $	$ 4.5411\times 10^{-3} $	$ \mathit{\boldsymbol{\sigma}}^\star=(0.010, 0.46257412, 0.52742588) $

Table Options

Download as excel

the interval of $ \sigma_i $	$ \tilde{{G}}^*_{0}(s, \mathit{\boldsymbol{\sigma}}) $	the optimal $ \mathit{\boldsymbol{\sigma}} $
$ \sigma_i\in[0, 1] $	$ 3.9855\times 10^{-3} $	$ \mathit{\boldsymbol{\sigma}}^\star=(0.000, 0.46836125, 0.53163875) $
$ \sigma_i\in[0.001, 1) $	$ 4.0432\times 10^{-3} $	$ \mathit{\boldsymbol{\sigma}}^\star=(0.001, 0.46779421, 0.53120579) $
$ \sigma_i\in[0.01, 1) $	$ 4.5411\times 10^{-3} $	$ \mathit{\boldsymbol{\sigma}}^\star=(0.010, 0.46257412, 0.52742588) $

[1]	Di Wu, Yanqin Bai, Fusheng Xie. Time-scaling transformation for optimal control problem with time-varying delay. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1683-1695. doi: 10.3934/dcdss.2020098
[2]	Maoli Chen, Xiao Wang, Yicheng Liu. Collision-free flocking for a time-delay system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1223-1241. doi: 10.3934/dcdsb.2020251
[3]	Lei Liu, Shaoying Lu, Cunwu Han, Chao Li, Zejin Feng. Fault estimation and optimization for uncertain disturbed singularly perturbed systems with time-delay. Numerical Algebra, Control & Optimization, 2020, 10 (3) : 367-379. doi: 10.3934/naco.2020008
[4]	Juanjuan Huang, Yan Zhou, Xuerong Shi, Zuolei Wang. A single finite-time synchronization scheme of time-delay chaotic system with external periodic disturbance. Mathematical Foundations of Computing, 2019, 2 (4) : 333-346. doi: 10.3934/mfc.2019021
[5]	Changjun Yu, Lei Yuan, Shuxuan Su. A new gradient computational formula for optimal control problems with time-delay. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021076
[6]	Richard H. Rand, Asok K. Sen. A numerical investigation of the dynamics of a system of two time-delay coupled relaxation oscillators. Communications on Pure & Applied Analysis, 2003, 2 (4) : 567-577. doi: 10.3934/cpaa.2003.2.567
[7]	Zhong-Jie Han, Gen-Qi Xu. Dynamical behavior of networks of non-uniform Timoshenko beams system with boundary time-delay inputs. Networks & Heterogeneous Media, 2011, 6 (2) : 297-327. doi: 10.3934/nhm.2011.6.297
[8]	Hanyu Cao, Meiying Zhang, Huanxi Cai, Wei Gong, Min Su, Bin Li. A zero-forcing beamforming based time switching protocol for wireless powered internet of things system. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2913-2922. doi: 10.3934/jimo.2019086
[9]	Stefano Galatolo, Mark Holland, Tomas Persson, Yiwei Zhang. Anomalous time-scaling of extreme events in infinite systems and Birkhoff sums of infinite observables. Discrete & Continuous Dynamical Systems, 2021, 41 (4) : 1799-1841. doi: 10.3934/dcds.2020341
[10]	Elnaz Delpisheh, Aijun An, Heidar Davoudi, Emad Gohari Boroujerdi. Time aware topic based recommender system. Big Data & Information Analytics, 2016, 1 (2&3) : 261-274. doi: 10.3934/bdia.2016008
[11]	Victor Gergel, Konstantin Barkalov, Alexander Sysoyev. Globalizer: A novel supercomputer software system for solving time-consuming global optimization problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 47-62. doi: 10.3934/naco.2018003
[12]	B. Cantó, C. Coll, A. Herrero, E. Sánchez, N. Thome. Pole-assignment of discrete time-delay systems with symmetries. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 641-649. doi: 10.3934/dcdsb.2006.6.641
[13]	Xiaochen Mao, Weijie Ding, Xiangyu Zhou, Song Wang, Xingyong Li. Complexity in time-delay networks of multiple interacting neural groups. Electronic Research Archive, , () : -. doi: 10.3934/era.2021022
[14]	Chongyang Liu, Meijia Han, Zhaohua Gong, Kok Lay Teo. Robust parameter estimation for constrained time-delay systems with inexact measurements. Journal of Industrial & Management Optimization, 2021, 17 (1) : 317-337. doi: 10.3934/jimo.2019113
[15]	Ming He, Xiaoyun Ma, Weijiang Zhang. Oscillation death in systems of oscillators with transferable coupling and time-delay. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 737-745. doi: 10.3934/dcds.2001.7.737
[16]	Qinqin Chai, Ryan Loxton, Kok Lay Teo, Chunhua Yang. A unified parameter identification method for nonlinear time-delay systems. Journal of Industrial & Management Optimization, 2013, 9 (2) : 471-486. doi: 10.3934/jimo.2013.9.471
[17]	Caibin Zhang, Zhibin Liang, Kam Chuen Yuen. Portfolio optimization for jump-diffusion risky assets with regime switching: A time-consistent approach. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020156
[18]	Xiang Xie, Honglei Xu, Xinming Cheng, Yilun Yu. Improved results on exponential stability of discrete-time switched delay systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 199-208. doi: 10.3934/dcdsb.2017010
[19]	Zhanyou Ma, Pengcheng Wang, Wuyi Yue. Performance analysis and optimization of a pseudo-fault Geo/Geo/1 repairable queueing system with N-policy, setup time and multiple working vacations. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1467-1481. doi: 10.3934/jimo.2017002
[20]	Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003

2019 Impact Factor: 1.233

Tools

Metrics

Tools

Article outline

Figures and Tables

2019 Impact Factor: 1.233

Tools

Figures and Tables

2019 Impact Factor: 1.233

American Institute of Mathematical Sciences

A time-scaling technique for time-delay switched systems

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

A time-scaling technique for time-delay switched systems

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors