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doi: 10.3934/jimo.2022108
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Control parameterization approach to time-delay optimal control problems: A survey

1. 

Department of Mathematics, Shanghai University, Shanghai 200444, China

2. 

School of Mathematical Sciences, Sunway University, Malaysia

*Corresponding author: Changjun Yu (yuchangjun@126.com)

Received  March 2022 Revised  April 2022 Early access June 2022

Fund Project: This work is supported by National Natural Science Foundation of China(NSFC), Grant No.11871039, No. 12171307 and Science and Technology Commission of Shanghai Municipality(STCSM), Grant No. 20JC1413900

Control parameterization technique is an effective method to solve optimal control problems. It works by approximating the control function by piece-wise constant (or linear) function. In this way, the optimal control problems are approximated by optimal parameter selection problems, which can be regarded as finite-dimensional optimization problems, where the control heights and switching times of the piece-wise constant function are taken as the decision variables. They can be solved by using gradient-based optimization methods. For this, it requires the gradient formulas of the objective and constraint functions with respect to the decision variables. There are two methods for deriving the required gradient formulas. The aim of this paper is to survey various numerical methods developed based on control parameterization for solving optimal control problems governed by time-delay systems. Particular focus is on those numerical methods for optimizing switching time points. This is because the process of optimizing the switching time points is much more complicated for optimal control problems with time delays. In addition, we shall also review the computational methods developed based on control parameterization technique for solving two optimal control problems involving more complicated time delays.

Citation: Di Wu, Yin Chen, Changjun Yu, Yanqin Bai, Kok Lay Teo. Control parameterization approach to time-delay optimal control problems: A survey. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022108
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show all references

References:
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[2]

C. Abdallah and J. Chiasson, Stability of communication networks in the presence of delays, IFAC Proceedings Volumes, 34 (2001), 171-174.  doi: 10.1016/S1474-6670(17)32885-9.

[3]

M. AlipourM. A. Vali and A. H. Borzabadi, A hybrid parameterization approach for a class of nonlinear optimal control problems, Numerical Algebra, 9 (2019), 493-506.  doi: 10.3934/naco.2019037.

[4]

H. Banks, Necessary conditions for control problems with variable time lags, SIAM Journal on Control, 6 (1968), 9-47.  doi: 10.1137/0306002.

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J. C. Butcher, Numerical Methods for Ordinary Differential Equations, 2$^nd$ edition, John Wiley & Sons, New York, 2008. doi: 10.1002/9780470753767.

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L. Bushnell, Networks and control [guest editorial], IEEE Control System Magazine, 21 (2001), 22-99.  doi: 10.1109/MCS.2001.898789.

[7]

J. A. E. Bryson and Y. C. Ho, Applied optimal control: Optimization, estimation and control, Taylor & Francis, 1975.

[8]

E. Blanchard, R. Loxton and V. Rehbock, A computational algorithm for a class of non-smooth optimal control problems arising in aquaculture operations, Applied Mathematics and Computation, 219 (2013) 8738–8746. doi: 10.1016/j. amc. 2013.02.070.

[9]

C. Bukens and H. Maurer, Nonlinear programming methods for real-time control of an industrial robot, Journal of Optimization Theory and Applications, 107 (2000), 505-527.  doi: 10.1023/A:1026491014283.

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P. T. Boggs and J. W. Tolle, Sequential quadratic programming, Acta Numerica, 4 (1995), 1-51.  doi: 10.1017/S0962492900002518.

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J. Banas and A. Vacroux, Optimal piecewise constant control of continuous time systems with time-varying delay, Automatica, 6 (1970), 809-811.  doi: 10.1016/0005-1098(70)90029-4.

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E. Biberovic, A. Iftar and H. Ozbay, A solution to the robust flow control problem for networks with multiple bottlenecks, in 40th IEEE CDC'01 (Conference on Decision and Control), Orlando, FL, December, 2001, 2303–2308. doi: 10.1109/CDC. 2001.980603.

[13]

F. Ceragioli, Discontinuous ordinary differential equations and stabilization, Universita di Firenze, 1999.

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Q. Q. Chai, R. Loxton, K. L. Teo and C. H. Yang, A class of optimal state-delay control problems, Nonlinear Anal. Real World Appl., 14 (2013) 1536–1550. doi: 10.1016/j. nonrwa. 2012.10.017.

[15]

Q. 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, 219 (2013) 9543–9560. doi: 10.1016/j. amc. 2013.03.015.

[16]

Q. Q. ChaiC. H. YangK. L. Teo and W. H. Gui, Optimal control of an industrial-scale evaporation process: Sodium aluminate solution, Control Engineering Practice, 20 (2012), 618-628.  doi: 10.1016/j.conengprac.2012.03.001.

[17]

P. J. Davis, Iterpolation and approximation, Dover Publications, 1975.

[18]

A. L. Dontchev and W. W. Hager, The Euler approximation in state constrained optimal control, Mathematics of Computation, 70 (2001), 173-203.  doi: 10.1090/S0025-5718-00-01184-4.

[19]

S. Dadebo and R. Luus, Optimal control of time-delay systems by dynamic programming, Optimal Control Applications & Methods, 13 (1992), 29-41.  doi: 10.1002/oca.4660130103.

[20]

D. GiangY. Lenbury and Y. Seidman, Delay effect in models of population growth, Journal of Mathematical Analysis and Applications, 305 (2005), 631-643.  doi: 10.1016/j.jmaa.2004.12.018.

[21]

Z. H. GongC. Y. LiuK. L. Teo and X. P. Yi, Optimal control of nonlinear fractional systems with multiple pantograph-delays, Applied Mathematics and Computation, 425 (2022), 127094.  doi: 10.1016/j.amc.2022.127094.

[22]

D. GargM. PattersonW. W. HagerA. V. RaoD. A. Benson and G. T. Huntington, A unified framework for the numerical solution of optimal control problems using pseudospectral methods, Automatica, 46 (2010), 1843-1851.  doi: 10.1016/j.automatica.2010.06.048.

[23]

C. J. Goh and K. L. Teo, Control parameterization: A unified approach to optimal control problems with general constraints, Automatica, 24 (1988), 3-18.  doi: 10.1016/0005-1098(88)90003-9.

[24]

L. Hasdorff, Gradient optimization and nonlinear control, John Wiley, New York, 1976.

[25]

W. W. Hager, Runge-Kutta methods in optimal control and the transformed adjoint system, Numerische Mathematik, 87 (2000), 247-282.  doi: 10.1007/s002110000178.

[26]

I. Hussein and A. Bloch, Optimal control of underactuated nonholonomic mechanical systems, IEEE Transactions on Automatic Control, 53 (2008), 668-682.  doi: 10.1109/TAC.2008.919853.

[27]

R. F. HartlS. P. Serhi and R. G. Vickson, A survey of the maximum principles for optimal control problems with state constraints, SIAM Review, 37 (1995), 181-218.  doi: 10.1137/1037043.

[28]

C. H. JiangK. XieC. J. YuM. YuH. Wang and Y. G. He, A sequential computational approach to optimal control problems for differential-algebraic systems based on efficient implicit Runge-Kutta integration, Applied Mathematical Modeling, 58 (2018), 313-330.  doi: 10.1016/j.apm.2017.05.015.

[29]

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.

[30]

C. Y. LiuZ. H. GongC. J. YuS. Wang and K. L. Teo, Optimal control computation for nonlinear fractional time-delay systems with inequality constraints, Journal of Optimization Theory and Applications, 191 (2021), 83-117.  doi: 10.1007/s10957-021-01926-8.

[31]

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.

[32]

C. Y. Liu, R. Loxton and K. L. Teo, Optimal parameter selection for nonlinear multistage systems with time-delays, Comput. Optim. Appl., 59 (2014) 285–306. doi: 10.1007/s10589-013-9632-x.

[33]

C. 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.

[34]

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Figure 1.  Piece-wise constant function approximation with equidistant switching times over the planning horizon $ (p=5) $
Figure 2.  The illustration of the time-scaling function $ (p=5) $
Figure 3.  The relationship between $ t $, $ t' $, $ s $ and $ s' $
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