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

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.

Authors:

• Yuekun Chen
• Yousef Sardahi
• Salam Hajjar
• Christopher Greer

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.