Multi-Objective Optimal Design of a Cascade Control System for a Class of Underactuated Mechanical Systems

Authors: Yuekun Chen, Yousef Sardahi, Salam Hajjar, Christopher Greer

Abstract:

This paper presents a multi-objective optimal design of a cascade control system for an underactuated mechanical system. Cascade control structures usually include two control algorithms (inner and outer). To design such a control system properly, the following conflicting objectives should be considered at the same time: 1) the inner closed-loop control must be faster than the outer one, 2) the inner loop should fast reject any disturbance and prevent it from propagating to the outer loop, 3) the controlled system should be insensitive to measurement noise, and 4) the controlled system should be driven by optimal energy. Such a control problem can be formulated as a multi-objective optimization problem such that the optimal trade-offs among these design goals are found. To authors best knowledge, such a problem has not been studied in multi-objective settings so far. In this work, an underactuated mechanical system consisting of a rotary servo motor and a ball and beam is used for the computer simulations, the setup parameters of the inner and outer control systems are tuned by NSGA-II (Non-dominated Sorting Genetic Algorithm), and the dominancy concept is used to find the optimal design points. The solution of this problem is not a single optimal cascade control, but rather a set of optimal cascade controllers (called Pareto set) which represent the optimal trade-offs among the selected design criteria. The function evaluation of the Pareto set is called the Pareto front. The solution set is introduced to the decision-maker who can choose any point to implement. The simulation results in terms of Pareto front and time responses to external signals show the competing nature among the design objectives. The presented study may become the basis for multi-objective optimal design of multi-loop control systems.

Keywords: Cascade control, multi-loop control systems, multi-objective optimization, optimal control.

References:

[1] C. A. Smith and A. B. Corripio, Principles and practice of automatic process control. Wiley New York, 1985, vol. 2.
[2] Y. Lee, S. Park, and M. Lee, “Pid controller tuning to obtain desired closed loop responses for cascade control systems,” Industrial & engineering chemistry research, vol. 37, no. 5, pp. 1859–1865, 1998.
[3] V. M. Alfaro, R. Vilanova, and O. Arrieta, “Two-degree-of-freedom pi/pid tuning approach for smooth control on cascade control systems,” in 2008 47th IEEE Conference on Decision and Control. IEEE, 2008, pp. 5680–5685.
[4] N. B. Almutairi and M. Zribi, “On the sliding mode control of a ball on a beam system,” Nonlinear dynamics, vol. 59, no. 1-2, p. 221, 2010.
[5] V. Pareto et al., “Manual of political economy,” 1971.
[6] Y. H. Sardahi, “Multi-objective optimal design of control systems,” Ph.D. dissertation, UC Merced, 2016.
[7] C. Hern´andez, Y. Naranjani, Y. Sardahi, W. Liang, O. Sch¨utze, and J.-Q. Sun, “Simple cell mapping method for multi-objective optimal feedback control design,” International Journal of Dynamics and Control, vol. 1, no. 3, pp. 231–238, 2013.
[8] D. F. Jones, S. K. Mirrazavi, and M. Tamiz, “Multi-objective meta-heuristics: An overview of the current state-of-the-art,” European journal of operational research, vol. 137, no. 1, pp. 1–9, 2002.
[9] R. T. Marler and J. S. Arora, “Survey of multi-objective optimization methods for engineering,” Structural and multidisciplinary optimization, vol. 26, no. 6, pp. 369–395, 2004.
[10] Y. Tian, R. Cheng, X. Zhang, and Y. Jin, “Platemo: A matlab platform for evolutionary multi-objective optimization
[educational forum],” IEEE Computational Intelligence Magazine, vol. 12, no. 4, pp. 73–87, 2017.
[11] P. Wo´zniak, “Multi-objective control systems design with criteria reduction,” in Asia-Pacific Conference on Simulated Evolution and Learning. Springer, 2010, pp. 583–587.
[12] X. Hu, Z. Huang, and Z. Wang, “Hybridization of the multi-objective evolutionary algorithms and the gradient-based algorithms,” in The 2003 Congress on Evolutionary Computation, 2003. CEC’03., vol. 2. IEEE, 2003, pp. 870–877.
[13] Y. Sardahi and A. Boker, “Multi-objective optimal design of four-parameter pid controls,” in ASME 2018 Dynamic Systems and Control Conference. American Society of Mechanical Engineers, 2018, pp. V001T01A001–V001T01A001.
[14] X. Xu, Y. Sardahi, and C. Zheng, “Multi-objective optimal design of passive suspension system with inerter damper,” in ASME 2018 Dynamic Systems and Control Conference. American Society of Mechanical Engineers, 2018, pp. V003T40A006–V003T40A006.
[15] B. Gadhvi, V. Savsani, and V. Patel, “Multi-objective optimization of vehicle passive suspension system using NSGA-II, SPEA2 and PESA-II,” Procedia Technology, vol. 23, pp. 361–368, 2016.
[16] K. Deb, Multi-Objective Optimization Using Evolutionary Algorithms. New York: Wiley, 2001.
[17] R. C. Dorf and R. H. Bishop, Modern control systems. Pearson, 2011.
[18] K. Ogata, Modern control engineering. Prentice Hall Upper Saddle River, NJ, 2009.
[19] G. F. Franklin, J. D. Powell, and A. Emami-Naeini, Feedback control of dynamic systems. Prentice Hall Press, 2014.
[20] Y. Sardahi and J.-Q. Sun, “Many-objective optimal design of sliding mode controls,” Journal of Dynamic Systems, Measurement, and Control, vol. 139, no. 1, p. 014501, 2017.