Article Contents
Article Contents

# A new gradient computational formula for optimal control problems with time-delay

• * Corresponding author: Lei Yuan

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

• In this paper, we consider a class of time-delay optimal control problem (TDOCP) with canonical equality and inequality constraints. By applying control parameterization method together with time-scaling transformation, a TDOCP can be readily solved by gradient-based optimization methods. The partial derivative of the cost as well as the constraint functions with respect to the decision variables are obtained by variational approach, which is inefficient when the discretization for the control function is relatively dense. For general optimal control problem without time-delay, co-state approach is an effective way to compute the gradients, however, when time-delay is involved in the dynamic system, the co-state system is not known. In this paper, we derive the co-state system for TDOCP to compute the gradients of the cost and constraints. Numerical results show that the computational efficiency is much higher when compared with the traditional variational approach.

Mathematics Subject Classification: Primary: 90C30, 90-08; Secondary: 34H05.

 Citation:

• Table 3.  Experimental results of co-state method and variational method

 Problem numbers of partitions co-state method CPU time(s) variational method CPU time(s) optimal cost Prob 1 p=4 9.62 11.98 1.7500 p=6 18.04 29.49 1.7407 p=8 26.14 31.73 1.7405 p=12 61.41 117.46 1.7403 Prob 2 p=5 10.2 147.23 2.4046 Prob 3 p=10 183 2702 2.0356 Prob 4 p=1 1.180 1.729 0.0218 p=3 4.485 22.39 0.0176 p=6 12.92 329.01 0.0142 Prob 5 p=5 8.72 51.56 2.1502

Table 1.  Parameters in Problem 2

 a b c $t_f$ Q R S 0.2 0.5 0.2 1.5 $I_{2\times2}$ $I_{2\times2}$ $10^4I_{2\times2}$

Table 2.  Parameters in Problemk 5

 a b c h $t_f$ Q R S 0.2 0.5 0.2 1 2 $I_{2\times2}$ $I_{2\times2}$ $10^4I_{2\times2}$
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