Optimal-tuning of proportional-integral-derivative-like controller for constrained nonlinear systems and application to ship steering control. (English) Zbl 1451.93161
Summary: In this paper, we consider an optimal-tuning problem of proportional-integral-derivative-like (PID-like) controller for constrained nonlinear systems with continuous-time inequality constraints and a terminal state constraint. Due to the complexity of such constraints, it is difficult to solve this problem by using conventional optimization methods. To overcome this difficulty, a constraint transcription approach and a smoothing function are applied to these continuous-time inequality constraints. An augmented cost functional is obtained by appending these continuous-time inequality constraints to the cost functional. Then, this optimal-tuning problem can be converted into an approximate optimal parameter selection problem. The gradient formulae of the augmented cost function and the terminal constraint function are derived, and a gradient-based numerical algorithm is developed for solving this approximate problem. Convergence results show that there only exist a finite number of iterates such that the sequence obtained by using the algorithm is infeasible, and any optimal solution of the approximate problem is also an optimal solution of the original problem. Finally, a ship steering control problem is solved to illustrate the effectiveness of the approach proposed by us.
MSC:
| 93C10 | Nonlinear systems in control theory |
| 93C95 | Application models in control theory |
| 49J40 | Variational inequalities |
Keywords:
constrained nonlinear systems; ship steering control; optimal-tuning problem of PID-like controllerSoftware:
PID_tuningReferences:
| [1] | Liu, G. P.; Daley, S., Optimal-tuning PID controller design in the frequency domain with application to a rotary hydraulic system, Control Eng. Pract., 7, 7, 821-830, (1999) |
| [2] | Liu, G. P.; Daley, S., Optimal-tuning nonlinear PID control of hydraulic systems, Control Eng. Pract., 8, 9, 1045-1053, (2000) |
| [3] | Visioli, A., Optimal tuning of PID controllers for integral and unstable processes, IEE Proc. Control Theory Appl., 148, 2, 180-184, (2001) |
| [4] | Ang, K. H.; Chong, G.; Li, Y., PID control system. analysis design, and technology, IEEE Trans. Control Syst. Technol., 13, 4, 559-576, (2005) |
| [5] | Fang, H.; Chen, L.; Shen, Z., Application of an improved PSO algorithm to optimal tuning of PID gains for water turbine governor, Energy Convers. Manag., 52, 4, 1763-1770, (2011) |
| [6] | Kumar, D. B.S.; Sree, R. P., Tuning of IMC based PID controllers for integrating systems with time delay, ISA Trans., 63, 242-255, (2016) |
| [7] | Ziegler, J. G.; Nichols, N. B., Optimum settings for automatic controllers, Trans. ASME, 64, 11, 759-768, (1942) |
| [8] | Xu, J. X.; Hang, C. C.; Liu, C., Parallel structure and tuning of a fuzzy PID controller, Automatica, 36, 5, 673-684, (2000) · Zbl 0959.93036 |
| [9] | Chang, W. D.; Hwang, R. C.; Hsieh, J. G., A self-tuning PID control for a class of nonlinear systems based on the Lyapunov approach, J. Process. Control, 12, 2, 233-242, (2002) |
| [10] | Dey, C.; Mudi, R. K.; Simhachalam, D., A simple nonlinear PD controller for integrating processes, ISA Trans., 53, 1, 162-172, (2014) |
| [11] | Anil, C.; Sree, R. P., Tuning of PID controllers for integrating systems using direct synthesis method, ISA Trans, 57, 211-219, (2015) |
| [12] | Vanavil, B.; Chaitanya, K. K.; Rao, A. S., Improved PID controller design for unstable time delay processes based on direct synthesis method and maximum sensitivity, Int. J. Syst. Sci., 46, 8, 1349-1366, (2015) · Zbl 1312.93049 |
| [13] | Sabir, M. M.; Ali, T., Optimal PID controller design through swarm intelligence algorithms for Sun tracking system, Appl. Math. Comput., 274, 690-699, (2016) · Zbl 1410.93085 |
| [14] | Kumar, A.; Kumar, V., A novel interval type-2 fractional order fuzzy PID controller: design, performance, evaluation and its optimal time domain tuning, ISA Trans., (2017) |
| [15] | Srivastava, S.; Misra, A.; Thakur, S. K.; Pandit, V. S., An optimal PID controller via LQR for standard second order plus time delay systems, ISA Trans., 60, 244-253, (2016) |
| [16] | Padula, F.; Visioli, A., Tuning rules for optimal PID and fractional-order PID controllers, J. Process. Control, 21, 1, 69-81, (2011) |
| [17] | Wang, Y. J., Determination of all feasible robust PID controllers for open-loop unstable plus time delay processes with gain margin and phase margin specifications, ISA Trans., 53, 2, 628-646, (2014) |
| [18] | Anwar, M. N.; Pan, S., A frequency response model matching method for PID controller design for processes with dead-time, ISA Trans., 55, 175-187, (2015) |
| [19] | Villarreal-Cervantes, M. G.; Alvarez-Gallegos, J., Off-line PID control tuning for a planar parallel robot using DE, variants, Expert Syst. Appl., 64, 444-454, (2016) |
| [20] | Sanchez, H. S.; Visioli, A.; Vilanova, R., Optimal Nash tuning rules for robust PID controllers, (2017), J. Frankl. Inst. · Zbl 1367.93169 |
| [21] | Mahmoodabadi, M. J.; Maafi, R. A.; Taherkhorsandi, M., An optimal adaptive robust PID controller subject to fuzzy rules and sliding modes for MIMO uncertain chaotic systems, Appl. Soft. Comput., 52, 1191-1199, (2017) |
| [22] | Chen, W. H.; Ballance, D. J.; Gawthrop, P. J., Optimal control of nonlinear systems: a predictive control approach, Automatica, 39, 4, 633-641, (2003) · Zbl 1017.93044 |
| [23] | Kirk, D. E., Optimal control theory: an introduction, (2012), New York: Courier, Corporation |
| [24] | Jiang, Y.; Jiang, Z. P., Global adaptive dynamic programming for continuous-time nonlinear systems, IEEE Trans Automat Control, 60, 11, 2917-2929, (2015) · Zbl 1360.49017 |
| [25] | Bian, T.; Jiang, Y.; Jiang, Z. P., Adaptive dynamic programming and optimal control of nonlinear nonaffine systems, Automatica, 50, 10, 2624-2632, (2014) · Zbl 1301.49081 |
| [26] | Wei, Q.; Liu, D.; Lin, H., Value iteration adaptive dynamic programming for optimal control of discrete-time nonlinear systems, IEEE Trans. Cybern., 46, 3, 840-853, (2016) |
| [27] | Kiumarsi, B.; Lewis, F. L.; Levine, D. S., Optimal control of nonlinear discrete time-varying systems using a new neural network approximation structure, Neurocomputing, 156, 157-165, (2015) |
| [28] | Sakawa, Y.; Shindo, Y., Optimal control of container cranes, Automatica, 18, 3, 257-266, (1982) · Zbl 0488.93021 |
| [29] | Wu, X.; Lei, B.; Zhang, K.; Cheng, M., Hybrid stochastic optimization method for optimal control problems of chemical processes, Chem. Eng. Res. Des., 126, 297-310, (2017) |
| [30] | Dong, R.; Pedrycz, W., Approximation grid evaluation-based PID control in cascade with nonlinear gain, J. Frankl. Inst., 352, 4279-4296, (2015) · Zbl 1395.93225 |
| [31] | Wu, C. J.; Huang, C. H., A hybrid method for parameter tuning of PID controllers, J. Frankl. Inst., 334, 547-562, (1997) · Zbl 0875.93253 |
| [32] | Eriksson, L.; Oksanen, T.; Mikkola, K., PID controller tuning rules for integrating processes with varying time-delays, J. Frankl. Inst., 346, 470-487, (2009) · Zbl 1167.93342 |
| [33] | Bagis, A., Tabu search algorithm based PID controller tuning for desired system specifications, J. Frankl. Inst., 348, 2795-2812, (2011) · Zbl 1254.93070 |
| [34] | Teng, T. K.; Shieh, J. S.; Chen, C. S., Genetic algorithms applied in online autotuning PID parameters of a liquid-level control system, Trans. Inst. Meas Control, 25, 433-450, (2003) |
| [35] | Wang, W.; Zhang, J.; Chai, T., A survey of advanced PID parameter tuning methods, Acta Autom. Sinica, 26, 3, 347-355, (2000) |
| [36] | Wu, X.; Zhang, K.; Sun, C., Parameter tuning of multi-proportional-integral-derivative controllers based on optimal switching algorithms, J. Optim. Theory Appl., 159, 454-472, (2013) · Zbl 1281.49031 |
| [37] | Bryson, A. E.; Ho, Y. C., Applied optimal control: optimization, estimation, and control, (1975), New York: Taylor & Francis |
| [38] | Fiacco, A. V.; McCormick, G. P., Nonlinear programming: sequential unconstrained minimization techniques, (1990), Philadelphia: Siam · Zbl 0713.90043 |
| [39] | Unar, M. A., Ship Steering Control using Feedforward Neural Networks, (1999), Glasgow: University of Glasgow, Doctor’s thesis · Zbl 0979.93536 |
| [40] | Tang, Y.; Zhao, L.; Han, Z.; Bi, X.; Guan, X., Optimal gray PID controller design for automatic voltage regulator system via imperialist competitive algorithm, Int. J. Mach. Learn. Cybern., 7, 2, 229-240, (2016) |
| [41] | Aghababa, M. P., Optimal design of fractional-order PID controller for five bar linkage robot using a new particle swarm optimization algorithm, Soft. Comput., 20, 10, 4055-4067, (2016) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
