June  2021, 11(2): 283-305. doi: 10.3934/naco.2020026

Examination of solving optimal control problems with delays using GPOPS-Ⅱ

1. 

Applied Mathematical Analysis, 2478 SE Mirromont Pl., Issaquah, WA, 98027, USA

2. 

Department of Mathematics, North Carolina State University, Raleigh, NC, 27695-8205, USA

* Corresponding author: Stephen Campbell

Received  April 2019 Revised  February 2020 Published  June 2021 Early access  May 2020

There are a limited number of user-friendly, publicly available optimal control software packages that are designed to accommodate problems with delays. GPOPS-Ⅱ is a well developed MATLAB based optimal control code that was not originally designed to accommodate problems with delays. The use of GPOPS-Ⅱ on optimal control problems with delays is examined for the first time. The use of various formulations of delayed optimal control problems is also discussed. It is seen that GPOPS-Ⅱ finds a suboptimal solution when used as a direct transcription delayed optimal control problem solver but that it is often able to produce a good solution of the optimal control problem when used as a delayed boundary value solver of the necessary conditions.

Citation: John T. Betts, Stephen Campbell, Claire Digirolamo. Examination of solving optimal control problems with delays using GPOPS-Ⅱ. Numerical Algebra, Control and Optimization, 2021, 11 (2) : 283-305. doi: 10.3934/naco.2020026
References:
[1]

Z. Bartoszewski and M. Kwapisz, On the convergence of waveform relaxation methods for differential-functional systems of equations, J. Math. Anal. Appl., 225 (1999), 478-496.  doi: 10.1006/jmaa.1999.6380.

[2]

Z. Bartoszewski and M. Kwapisz, On error estimates for waveform relaxation methods for delay-differential equations, SIAM J. Numerical Analysis, 38 (2011), 639-659.  doi: 10.1137/S003614299935591X.

[3]

J. T. Betts, Methods for Optimal Control and Estimation using Nonlinear Programming, SIAM, Philadelphia, 2010.

[4]

J. T. BettsN. BiehnS. L. Campbell and W. Huffman, Compensating for order variation in mesh refinement for direct transcription methods Ⅱ: computational experience, J. Comp. Appl. Math., 143 (2002), 237-261.  doi: 10.1016/S0377-0427(01)00509-X.

[5]

J. T. Betts, S. L. Campbell and K. Thompson, Optimal control of delay partial differential equations, , in Control and Optimization with Differential-Algebraic Constraints, SIAM, (2012), 213–231.

[6]

J. T. Betts, S. L. Campbell and K. Thompson, Optimal control software for constrained nonlinear systems with delays, , Proc. IEEE Multi-Conference on Systems and Control (2011 MSC), Denver, (2011), 444–449.

[7]

J. T. BettsS. L. Campbell and K. Thompson, Solving optimal control problems with control delays using direct transcription, Applied Numerical Mathematics, 108 (2016), 185-203.  doi: 10.1016/j.apnum.2015.12.008.

[8]

N. BiehnJ. T. BettsS. L. Campbell and W. Huffman, Compensating for order variation in mesh refinement for direct transcription methods, J. Comp. Appl. Math., 125 (2000), 147-158.  doi: 10.1016/S0377-0427(00)00465-9.

[9]

G. V. Bokov, Pontryagin's maximum principle of optimal control problems with time-delay, J. Mathematical Sciences, 172 (2011), 623–634. (Russian version: Fundam. Prikl. Mat., 15 (2009), Issue 5, 3–19.) doi: 10.1007/s10958-011-0208-y.

[10]

S. L. CampbellJ. T. Betts and C. Digirolamo, Comments on initial guess sensitivity when solving optimal control problems using interior point methods, Numerical Algebra, Control, and Optimization, 10 (2020), 39-41. 

[11]

C. L. DarbyW. W. Hager and A. V. Rao, An hp-adaptive pseudospectral method for solving optimal control problems, Optimal Control Applications and Methods, 32 (2011), 476-502.  doi: 10.1002/oca.957.

[12]

J. F. Frankena, Optimal control problems with delay, the maximum principle and necessary conditions, J. Engineering Mathematics, 9 (1975), 53-64.  doi: 10.1007/BF01535497.

[13]

L. GöllmannD. Kern and H. Maurer, Optimal control problems with delays in state and control subject to mixed state control-state constraints, Optimal Control Applications and Methods, 30 (2009), 341-365.  doi: 10.1002/oca.843.

[14]

L. Göllmann and H. Maurer, Theory and application of optimal control problems with multiple delays, J. Industrial and Management Optimization, 10 (2014), 413-441.  doi: 10.3934/jimo.2014.10.413.

[15]

Z. H. GongC. 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.

[16]

T. Koto, Method of lines approximation of delay differential equations, Computers & Mathematics with Applications, 48 (2004), 45-59.  doi: 10.1016/j.camwa.2004.01.003.

[17]

C. Y. LiuR. LoxtonQ. Lin and K. L. Teo, Dynamic optimization for switched time-delay systems with state-dependent switching conditions, SIAM Journal on Control and Optimization, 56 (2018), 3499-3523.  doi: 10.1137/16M1070530.

[18]

C. Y. LiuZ. GongH. W. Lee and K. L. Teo, Robust bi-objective optimal control of 1, 3-propanediol microbial batch production process, Journal of Process Control, 78 (2019), 170-182. 

[19]

C. LiuZ. GongK. L. TeoR. Loxton and E. Feng, Bi-objective dynamic optimization of a nonlinear time-delay system in microbial batch process, Optimization Letters, 12 (2018), 1249-1264.  doi: 10.1007/s11590-016-1105-6.

[20]

C. Y. LiuR. Loxton and 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.

[21]

C. Y. LiuR. 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. 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, 63 (2014), 957-988.  doi: 10.1007/s10957-014-0533-7.

[23]

M. Maleki and I Hashim, Adaptive pseudospectral methods for solving constrained linear and nonlinear time-delay optimal control problems, J. Franklin Institute, 351 (2014), 811-839.  doi: 10.1016/j.jfranklin.2013.09.027.

[24]

J. Mead and B. Zubik-Kowal, An iterated pseudospectral method for delay partial differential equations, Applied Numerical Mathematics, 55 (2005), 227-250.  doi: 10.1016/j.apnum.2005.02.010.

[25]

M. A. Patterson and A. V. Rao, GPOPS Ⅱ: A MATLAB software for solving multiple-phase optimal control problems using hp-adaptive Gaussian quadrature collocation methods and sparse nonlinear programming, ACM Trans. Math. Software, 41 (2014), 1-37.  doi: 10.1145/2558904.

[26]

H. PengX. WangS. Zhang and B. Chen, An iterative symplectic pseudospectral method to solve nonlinear state-delayed optimal control problems, Commun. Nonlinear. Sci. Numer. Simulat., 48 (2017), 95-114.  doi: 10.1016/j.cnsns.2016.12.016.

[27]

A. V. RaoD. A. BensonC. DarbyM. A. PattersonC. FrancolinI. Sanders and G. T. Huntington, Algorithm 902: Gpops, a MATLAB software for solving multiple-phase optimal control problems using the Gauss pseudospectral method, ACM Transactions Mathematical Software, 37 (2010), 1-39.  doi: 10.1145/2558904.

[28]

L. F. Shampine and S. Thompson, Solving DDEs in Matlab, Applied Numerical Mathematics, 37 (2001), 441-458.  doi: 10.1016/S0168-9274(00)00055-6.

[29]

A. Wäechter and L.T. Biegler, On the implementation of interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Prog., 106 (2006), 25-57.  doi: 10.1007/s10107-004-0559-y.

[30]

Z. Wang and L. Wang, A Legendre-Gauss collocation method for nonlinear delay differential equations, Discrete and Continuous Dynamical Systems, Series B, 13 (2010), 685-708.  doi: 10.3934/dcdsb.2010.13.685.

[31]

D. WuY. Bai and C. Yu, A new computational approach for optimal control problems with multiple time-delay, Automatica, 101 (2019), 388-395.  doi: 10.1016/j.automatica.2018.12.036.

[32]

Z. Wu and W. Michiels, Reliably computing all characteristic roots of delay equations in a given right half plane using a spectral method, J. Comp. and Appl. Math., 236 (2012), 2499-2514.  doi: 10.1016/j.cam.2011.12.009.

[33]

Z. Wu and W. Michiels, Reliably computing all characteristic roots of delay differential equations in a given right half plane using a spectral method, , Internal report TW 596, Department of Computer Science, K. U. Leuven, May, 2011. Available by download from http://twr.cs.kuleuven.be/research/software/delay-control/roots/+. doi: 10.1016/j.cam.2011.12.009.

show all references

References:
[1]

Z. Bartoszewski and M. Kwapisz, On the convergence of waveform relaxation methods for differential-functional systems of equations, J. Math. Anal. Appl., 225 (1999), 478-496.  doi: 10.1006/jmaa.1999.6380.

[2]

Z. Bartoszewski and M. Kwapisz, On error estimates for waveform relaxation methods for delay-differential equations, SIAM J. Numerical Analysis, 38 (2011), 639-659.  doi: 10.1137/S003614299935591X.

[3]

J. T. Betts, Methods for Optimal Control and Estimation using Nonlinear Programming, SIAM, Philadelphia, 2010.

[4]

J. T. BettsN. BiehnS. L. Campbell and W. Huffman, Compensating for order variation in mesh refinement for direct transcription methods Ⅱ: computational experience, J. Comp. Appl. Math., 143 (2002), 237-261.  doi: 10.1016/S0377-0427(01)00509-X.

[5]

J. T. Betts, S. L. Campbell and K. Thompson, Optimal control of delay partial differential equations, , in Control and Optimization with Differential-Algebraic Constraints, SIAM, (2012), 213–231.

[6]

J. T. Betts, S. L. Campbell and K. Thompson, Optimal control software for constrained nonlinear systems with delays, , Proc. IEEE Multi-Conference on Systems and Control (2011 MSC), Denver, (2011), 444–449.

[7]

J. T. BettsS. L. Campbell and K. Thompson, Solving optimal control problems with control delays using direct transcription, Applied Numerical Mathematics, 108 (2016), 185-203.  doi: 10.1016/j.apnum.2015.12.008.

[8]

N. BiehnJ. T. BettsS. L. Campbell and W. Huffman, Compensating for order variation in mesh refinement for direct transcription methods, J. Comp. Appl. Math., 125 (2000), 147-158.  doi: 10.1016/S0377-0427(00)00465-9.

[9]

G. V. Bokov, Pontryagin's maximum principle of optimal control problems with time-delay, J. Mathematical Sciences, 172 (2011), 623–634. (Russian version: Fundam. Prikl. Mat., 15 (2009), Issue 5, 3–19.) doi: 10.1007/s10958-011-0208-y.

[10]

S. L. CampbellJ. T. Betts and C. Digirolamo, Comments on initial guess sensitivity when solving optimal control problems using interior point methods, Numerical Algebra, Control, and Optimization, 10 (2020), 39-41. 

[11]

C. L. DarbyW. W. Hager and A. V. Rao, An hp-adaptive pseudospectral method for solving optimal control problems, Optimal Control Applications and Methods, 32 (2011), 476-502.  doi: 10.1002/oca.957.

[12]

J. F. Frankena, Optimal control problems with delay, the maximum principle and necessary conditions, J. Engineering Mathematics, 9 (1975), 53-64.  doi: 10.1007/BF01535497.

[13]

L. GöllmannD. Kern and H. Maurer, Optimal control problems with delays in state and control subject to mixed state control-state constraints, Optimal Control Applications and Methods, 30 (2009), 341-365.  doi: 10.1002/oca.843.

[14]

L. Göllmann and H. Maurer, Theory and application of optimal control problems with multiple delays, J. Industrial and Management Optimization, 10 (2014), 413-441.  doi: 10.3934/jimo.2014.10.413.

[15]

Z. H. GongC. 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.

[16]

T. Koto, Method of lines approximation of delay differential equations, Computers & Mathematics with Applications, 48 (2004), 45-59.  doi: 10.1016/j.camwa.2004.01.003.

[17]

C. Y. LiuR. LoxtonQ. Lin and K. L. Teo, Dynamic optimization for switched time-delay systems with state-dependent switching conditions, SIAM Journal on Control and Optimization, 56 (2018), 3499-3523.  doi: 10.1137/16M1070530.

[18]

C. Y. LiuZ. GongH. W. Lee and K. L. Teo, Robust bi-objective optimal control of 1, 3-propanediol microbial batch production process, Journal of Process Control, 78 (2019), 170-182. 

[19]

C. LiuZ. GongK. L. TeoR. Loxton and E. Feng, Bi-objective dynamic optimization of a nonlinear time-delay system in microbial batch process, Optimization Letters, 12 (2018), 1249-1264.  doi: 10.1007/s11590-016-1105-6.

[20]

C. Y. LiuR. Loxton and 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.

[21]

C. Y. LiuR. 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. 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, 63 (2014), 957-988.  doi: 10.1007/s10957-014-0533-7.

[23]

M. Maleki and I Hashim, Adaptive pseudospectral methods for solving constrained linear and nonlinear time-delay optimal control problems, J. Franklin Institute, 351 (2014), 811-839.  doi: 10.1016/j.jfranklin.2013.09.027.

[24]

J. Mead and B. Zubik-Kowal, An iterated pseudospectral method for delay partial differential equations, Applied Numerical Mathematics, 55 (2005), 227-250.  doi: 10.1016/j.apnum.2005.02.010.

[25]

M. A. Patterson and A. V. Rao, GPOPS Ⅱ: A MATLAB software for solving multiple-phase optimal control problems using hp-adaptive Gaussian quadrature collocation methods and sparse nonlinear programming, ACM Trans. Math. Software, 41 (2014), 1-37.  doi: 10.1145/2558904.

[26]

H. PengX. WangS. Zhang and B. Chen, An iterative symplectic pseudospectral method to solve nonlinear state-delayed optimal control problems, Commun. Nonlinear. Sci. Numer. Simulat., 48 (2017), 95-114.  doi: 10.1016/j.cnsns.2016.12.016.

[27]

A. V. RaoD. A. BensonC. DarbyM. A. PattersonC. FrancolinI. Sanders and G. T. Huntington, Algorithm 902: Gpops, a MATLAB software for solving multiple-phase optimal control problems using the Gauss pseudospectral method, ACM Transactions Mathematical Software, 37 (2010), 1-39.  doi: 10.1145/2558904.

[28]

L. F. Shampine and S. Thompson, Solving DDEs in Matlab, Applied Numerical Mathematics, 37 (2001), 441-458.  doi: 10.1016/S0168-9274(00)00055-6.

[29]

A. Wäechter and L.T. Biegler, On the implementation of interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Prog., 106 (2006), 25-57.  doi: 10.1007/s10107-004-0559-y.

[30]

Z. Wang and L. Wang, A Legendre-Gauss collocation method for nonlinear delay differential equations, Discrete and Continuous Dynamical Systems, Series B, 13 (2010), 685-708.  doi: 10.3934/dcdsb.2010.13.685.

[31]

D. WuY. Bai and C. Yu, A new computational approach for optimal control problems with multiple time-delay, Automatica, 101 (2019), 388-395.  doi: 10.1016/j.automatica.2018.12.036.

[32]

Z. Wu and W. Michiels, Reliably computing all characteristic roots of delay equations in a given right half plane using a spectral method, J. Comp. and Appl. Math., 236 (2012), 2499-2514.  doi: 10.1016/j.cam.2011.12.009.

[33]

Z. Wu and W. Michiels, Reliably computing all characteristic roots of delay differential equations in a given right half plane using a spectral method, , Internal report TW 596, Department of Computer Science, K. U. Leuven, May, 2011. Available by download from http://twr.cs.kuleuven.be/research/software/delay-control/roots/+. doi: 10.1016/j.cam.2011.12.009.

Figure 1.  Left: Solutions to (8) obtained by GPOPS-Ⅱw and dde23 and Right: solutions to (8) obtained by ode45w and dde23 for $ \sigma = -1.2 $ and $ \tau = 1.0. $
Figure 2.  State (left) and control (right) for (2) with $ \sigma = 1.2 $ using GPOPS-Ⅱm. Computed cost was 44.6641
Figure 3.  State (left) and control (right) for (2) with $ \sigma = 1.2 $ using GPOPS-Ⅱ on the MOS formulation. Computed cost was 43.4214. SOSD gave a similar appearing control and computed cost
Figure 4.  Left: Iterative states of GPOPS-Ⅱow for (9) and Right: states obtained by SOSD, GPOPS-Ⅱm, and control parameterization with $ \sigma = -1.2 $ and $ \tau = 1.0. $
Figure 5.  Left: Iterative controls of GPOPS-Ⅱow for (9) and Right: controls obtained by SOSD, GPOPS-Ⅱm, and control parameterization with $ \sigma = -1.2 $ and $ \tau = 1.0. $
Figure 6.  State (left) and control (right) obtained for (25) using GPOPS-Ⅱ and MOL
Figure 7.  State (left) and control (right) for (31) solving (26) using GPOPS-Ⅱm
Figure 8.  State (left) and control (right) from solving (26), Figure 8 is from [23]
Figure 9.  State (left) and control (right) for (28) using GPOPS-Ⅱm. Computed cost was 52.8417171
Figure 10.  State (left) and control (right) for (28) using SOSD. Computed cost was 53.27103
Figure 11.  State (left) and control (right) for (28) using the modified cost (29) with $ \alpha = 0.01 $ and also with SOSD on the original problem
Figure 12.  State (left) and control (right) for (30) using GPOPS-Ⅱm. Computed cost was 52.8417171
Figure 13.  State (left) and control (right) for (30) using SOSD. Computed cost was 56.187
Figure 14.  State (left) and control (right) for (1) using GPOPS-Ⅱm with prehistory a control variable. Computed cost was 52.8417171
Figure 15.  State (left) and control (right) for (9) with $ \sigma = -1.2, \tau = 1, $ found by solving the necessary conditions using GPOPS-Ⅱm
Figure 16.  State (left) and control (right) for (31) solving the necessary conditions (32) with GPOPS-Ⅱm
Figure 17.  State (left) and control (right) for (28) solving the necessary conditions with GPOPS-Ⅱm and also using SOSD on the original problem, $ \tau = 1, a = -1.14 $
Figure 18.  State (left) and control (right) for (30) solving the necessary conditions with GPOPS-Ⅱm
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