Marshall University Marshall Digital Scholar Theses, Dissertations and Capstones 2020 Multi-objective Optimization of Multi-loop Control Systems Yuekun Chen 1102306990@qq.com Follow this and additional works at: https://mds.edu/etd Part of the Acoustics, Dynamics, and Controls Commons Recommended Citation Chen, Yuekun, "Multi-objective Optimization of Multi-loop Control Systems" (2020). Theses, Dissertations and Capstones.edu/etd/1270 This Thesis is brought to you for free and open access by Marshall Digital Scholar. It has been accepted for inclusion in Theses, Dissertations and Capstones by an authorized administrator of Marshall Digital Scholar. For more information, please contact zhangj@marshall.edu, beachgr@marshall.
MULTI-OBJECTIVE OPTIMIZATION OF MULTI-LOOP CONTROL SYSTEMS A thesis submitted to the Graduate College of Marshall University In partial fulfillment of the requirements for the degree of Master of Science In Mechanical Engineering by Yuekun Chen Approved by Dr. Yousef Sardahi, Committee Chairperson Dr. Gang Chen Dr. Mehdi Esmaeilpour Marshall University May 2020 ii ACKNOWLEDGMENTS I would like to express my gratitude to all those who helped me during the writing of this thesis.
I gratefully acknowledge the help of my supervisor, Dr. Yousef Sardahi, who has offered me valuable suggestions in the academic studies. Without his consistent and illuminating instruction, this thesis could not have reached its present form. Second, I would like to express my heartfelt gratitude to my thesis committee: Dr.
Gang Chen and Dr. Mehdi Esmaeilpour, for their instruction and assistance. Finally, I would like to thank my beloved family and my friends for their continuous support and encouragement. Without their trust and help, I couldn’t have the strong motivations to urge me working hard on this thesis.
Thank you all. iii TABLE OF CONTENTS List of Tables. vi List of Figures. xii Chapter 1: Introduction .2 Multi-Objective Optimization .4 Outline of the Thesis.
11 Chapter 2: Multi-Objective Optimal Design of a Cascade Control System for a Class of Underactuated Mechanical Systems .1 Cascade control systems .2 Underactuated Ball and Beam System .3 Multi-Objective Optimal Design .4 Results and discussion. 19 Chapter 3: Multi-Objective Optimal Design of an Active Aeroelastic Cascade Control System for an Aircraft Wing With a Leading and Trailing Control Surface .2 Airfoil wing model with two control surfaces .3 LQR-based Outer Control Loop .5 PV-based Inner Control Loop .6 Multi-objective and Multidisciplinary Optimal Design .7 Results and Discussion .1 Pareto Frontier and Set.2 Closed-Loop Eigenvalues .3 Gust Loading Impact. 44 Chapter 4: Summary and future directions. 54 Appendix A: INSITITUTIONAL REVIEW BOARD LETTER.1 Aircraft Flexible Wing .3 Slider-Crank Mechanism.
62 v LIST OF TABLES Table 1: The model parameters (Singh et al.60 Table 2: Motor parameters (Habibi et al.62 vi LIST OF FIGURES Figure 1: NSGA-II algorithm flowchart. 10 Figure 2: Block diagram of two-level cascade control system. 13 Figure 3: Ball and beam system. 16 Figure 4: Projections of the Pareto set: (a) 𝐾𝑑𝑖 versus 𝐾𝑝𝑖 , (b) 𝐾𝑑𝑜 versus 𝐾𝑝𝑜.
The color code indicates the level of ||𝑘||𝐹 , where red denotes the highest value, and dark blue denotes the smallest. 22 Figure 5: Projections of the Pareto front: (a) 𝐹1 versus ||𝑘||𝐹 , (b) 𝐹2 versus ||𝑘||𝐹. The color code indicates the level of ||𝑘||𝐹 , where red denotes the highest value, and dark blue denotes the smallest. 23 Figure 6: Projections of the Pareto front: (a) r versus ||𝑘||𝐹 , (b) 𝐹2 versus 𝐹1.
The color code indicates the level of ||𝑘||𝐹 , where red denotes the highest value, and dark blue denotes the smallest. 23 Figure 7: Pole maps, on the y-axis is the imaginary part of the pole, Im(s), and the x-axis is the real part of the pole, Re(s): (a) Pole map of the inner closed-loop system, (b) Pole map of the outer closed-loop system. The color code indicates the level of ||𝑘||𝐹 , where red denotes the highest. 24 Figure 8: Outer and inner controlled systems’ responses when r = 0.
Red solid line: reference signal, Black solid line: actual system, response with 𝑑𝑖 (t) = 𝑑𝑜 (t) = 0. 24 vii Figure 9: Outer and inner controlled systems’ responses when r = 0. Red solid line: reference signal, Black solid line: actual system response with 𝑑𝑖 (t) = 𝑑𝑜 (t) = 0. 25 Figure 10: Ball position versus time.
Red solid line: reference signal 𝑥𝑑 (𝑡), black solid line: system response with 𝑑𝑖 (t) = 𝑑𝑜 (t) = 0, blue dotted line: system response with 𝑑𝑖 (t) = 𝑑𝑜 (t) = 0. 25 Figure 11: Ball position versus time. Red solid line: reference signal 𝑥𝑑 (𝑡), black solid line: system response with 𝑑𝑖 (t) = 𝑑𝑜 (t)= 0, blue dotted line: system response with 𝑑𝑖 (t) = 𝑑𝑜 (t)= 0. 26 Figure 12: Ball position versus time.
Red solid line: reference signal 𝑥𝑑 (𝑡), black solid line: system response with 𝑛𝑖 (t) = 𝑛𝑜 (t) = 0, blue dotted line: system response with 𝑛𝑖 (t) = 𝑛𝑜 (t) = WN. 26 Figure 13: Cascade control system of aeroelastic structure and actuators. 29 Figure 14: Airfoil wing model with two control surfaces (Singh et al. 30 Figure 15: A generic EMA system (Habibi et al.
36 Figure 16: Control surface driven by slider-crank mechanism. 37 Figure 17: Projections of the Pareto front: (a) Eav versus Dav , (b) Eav versus r. The color code indicates the level of Eav , where red denotes the highest value, and dark blue denotes the smallest. 45 viii Figure 18: Projections of the Pareto set: (a) k pT versus k dT (b) k pL versus k dL.
The color code indicates the level of Eav , where red denotes the highest value, and dark blue denotes the smallest. 45 Figure 19: Projections of the Pareto set: (a) Q1 versus Q3 (b) Q2 versus Q4. The color code indicates the level of Eav , where red denotes the highest value, and dark blue denotes the smallest. 46 Figure 20: A Projection of the Pareto set: R1 versus R 2.
The color code indicates the level of Eav , where red denotes the highest value, and dark blue denotes the smallest. 46 Figure 21: Pole maps, on the y-axis is the imaginary part of the pole, imag(λ), and the x-axis is the real part of the pole, real(λ): (a) Pole map of the outer controlled system: outer control loop and aeroelastic structure, (b) Pole map of the inner controller applied to the trailing actuator, and (c) Pole map of the inner controller applied to the leading actuator. 47 Figure 22: Dominant pole maps, the x-axis is the location of pole closer to the imaginary axis, max(real(λ)) the y-axis is unlabeled, and: (a) Dominant pole map of the outer controlled system: outer control loop and aeroelastic structure, (b) Dominant pole map of the trailing and leading inner controllers, (c) Dominant pole map of the inner controller applied to the trailing actuator, and (d) Dominant pole map of the inner controller applied to the leading actuator. 47 Figure 23:Gust load wg (𝑡) profile versus time.
48 Figure 24: Controlled systems’ responses when the disturbance rejection is the best min (𝐷𝑎𝑣 ). Top left: time versus the plunging displacement (h). Top right: time versus the plunging the pitching angle α. Bottom left: time versus the actual X T and desired XdT ball-screw mechanism displacement of the actuator at the trailing aileron.
Bottom Right: time versus the actual XL and desired XdL ball-screw mechanism displacement of the actuator at the leading aileron. 48 ix Figure 25: Controlled systems’ responses when the disturbance rejection is the worst max (Dav ). Top left: time versus the plunging displacement (h). Top right: time versus the plunging the pitching angle α.
Bottom left: time versus the actual XT and desired XdT ball-screw mechanism displacement of the actuator at the trailing aileron. Bottom Right: time versus the actual XL and desired XdL ball-screw mechanism displacement of the actuator at the leading aileron. 49 Figure 26: Controlled systems’ responses when the control energy is the maximum max (Eav ). Top left: time versus the plunging displacement (h).
Top right: time versus the plunging the pitching angle α. Bottom left: time versus the actual XT and desired XdT ball-screw mechanism displacement of the actuator at the trailing aileron. Bottom Right: time versus the actual XL and desired XdL ball-screw mechanism displacement of the actuator at the leading aileron. 49 Figure 27: Controlled systems’ responses when the control energy is the minimum min(Eav ).
Top left: time versus the plunging displacement (h). Top right: time versus the plunging the pitching angle α. Bottom left: time versus the actual XT and desired XdT ball-screw mechanism displacement of the actuator at the trailing aileron. Bottom Right: time versus the actual XL and desired XdL ball-screw mechanism displacement of the actuator at the leading aileron.
50 Figure 28: Controlled systems’ responses when the inner closed-loop algorithms are way faster than outer control loop max (r). Top left: time versus the plunging displacement (h). Top right: time versus the plunging the pitching angle α. Bottom left: time versus the actual XT and desired XdT ball-screw mechanism displacement of the actuator at the trailing aileron.
Bottom Right: time versus the actual XL and desired XdL ball-screw mechanism displacement of the actuator at the leading aileron. 51 Figure 29: Controlled systems’ responses when the inner closed-loop algorithms are way slower than outer control loop max (r). Top left: time versus the plunging displacement (h). Top right: x time versus the plunging the pitching angle α.
Bottom left: time versus the actual XT and desired XdT ball-screw mechanism displacement of the actuator at the trailing aileron. Bottom Right: time versus the actual XL and desired XdL ball-screw mechanism displacement of the actuator at the leading aileron. 51 xi ABSTRACT Cascade Control systems are composed of inner and outer control loops. Compared to the traditional single feedback controls, the structure of cascade controls is more complex.
As a result, the implementation of these control methods is costly because extra sensors are needed to measure the inner process states. On the other side, cascade control algorithms can significantly improve the controlled system performance if they are designed properly. For instance, cascade control strategies can act faster than single feedback methods to prevent undesired disturbances, which can drive the controlled system’s output away from its target value, from spreading through the process. As a result, cascade control techniques have received much attention recently.
In this thesis, we present a multi-objective optimal design of linear cascade control systems using a multi-objective algorithm called the non-dominated sorting genetic algorithm (NSGA-II), which is one of the widely used algorithms in solving multi-objective optimization problems (MOPs). Two case studies have been considered. In the first case, a multi-objective optimal design of a cascade control system for an underactuated mechanical system consisting of a rotary servo motor, and a ball and beam is introduced. The setup parameters of the inner and outer control loops are tuned by the NSGA-II to achieve four objectives: 1) the closed-loop system should be robust against inevitable internal and outer disturbances, 2) the controlled system is insensitive to inescapable measurement noise affecting the feedback sensors, 3) the control signal driving the mechanical system is optimum, and 4) the dynamics of the inner closed-loop system has to be faster than that of the outer feedback system.
By using the NSGA- II algorithm, four design parameters and four conflicting objective functions are obtained. The second case study investigates a multi-objective optimal design of an aeroelastic cascade controller applied to an aircraft wing with a leading and trailing control surface. The dynamics of xii the actuators driving the control surfaces are considered in the design. Similarly, the NSGA-II is used to optimally adjust the parameters of the control algorithm.