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Vibration isolation active control techniques and testing in a micro-gravity environment Boulet, Jason 1997

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V I B R A T I O N ISOLATION A C T I V E C O N T R O L T E C H N I Q U E S A N D T E S T I N G IN A M I C R O - G R A V I T Y E N V I R O N M E N T By Jason Boulet B.Sc.E.E. University of Manitoba, 1994 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S E L E C T R I C A L E N G I N E E R I N G We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A Apri l 1997 © Jason Boulet, 1997. In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Electrical Engineering The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: /) Abstract Since orbitting space stations do not provide a perfect micro-gravity environment, vi-bration isolation techniques must be employed for sensitive experiments, such as crystal growing. A non-contact approach using magnetic levitation for active vibration isolation has been developed previously that can be used in all micro-gravity situations. Using advanced controller techniques, it should be possible to lower the stiffness of the magnetic coupling below levels achievable by PID control. This thesis examines controlling magnetic levitation (MAGLEV) for improved vibra-tion isolation on Earth and in a micro-gravity environment. Lorentz forces are used to levitate a large platform holding the application in an experimental system using this MAGLEV technology. This technology is described and two designs of it's ap-plication are presented. Vibration control techniques are investigated, with and Q-parameterization algorithms explained and employed in active control in a normal gravity environment. A smaller magnetically levitated wrist device is then described and employed in a coarse-fine approach to isolating acceleration disturbances in the micro-gravity environment provided by NASA's DC-9 performing parabolic flights. Results of the testing done on the flight is given and analyzed. Table of Contents Abstract ii List of Figures vii Acknowledgments x 1 Introduction 1 1.1 Vibration Isolation ". .' . 1 1.2 Contact vs. Non-Contact 2 1.3 Control . . • • • 4 1.4 Magnetic Levitation Large Motion Isolation Mount . . . . . . . . . . . . . 6 1.5 Micro-Gravity Testing 6 1.5.1 Two Stage System , 8 1.6 Thesis Overview. 8 2 Motion Isolation Mount Modeling 10 2.1 MAGLEV LMIM 10 2.1.1 Actuation . 12 2.1.2 Sensing 13 2.2 Model 14 2.3 Effect of Filters . . 17 2.4 Destabilizing Feedback 19 2.5 Plant Identification Y(s) 23 i n 2.5.1 Position Plant 24 2.5.2 Acceleration Plant 29 3 Controller 33 3.1 Design Objectives 33 3.2 Theory : 34 3.2.1 Plant and Design • 34 3.2.2 Weights 37 3.2.3 Restrictions 37 3.2.4 7-Optimization 38 3.2.5 SISO Simplification 39 3.3 Position Controller 40 3.3.1 Weight Selection • . 40 3.3.2 Simple Plant . .". 42 3.3.3 Destabilizing Feedback 44 3.3.4 MATLAB Code 47 3.3.5 Implementation • • • ^ 3.3.6 Impulse Response •. - 49 3.3.7 The Sensitivity Problem 49 3.4 Acceleration Feedback 51 3.4.1 Controller Design Considerations . 52 3.4.2 Acceleration Controller 57 3.4.3 Force Rejection 61 3.4.4 Acceleration Controller Implementation 61 3.5 Conclusions 61 iv 4 The Coarse-Fine System 64 4.1 Hardware and Software 64 4.1.1 Hardware . . . . 64. 4.1.2 Software 66 4.2 Model Development 67 4.3 Open Loop System Analysis 71 4.4 Closed Loop System Analysis 72 4.5 Simulink Simulations 74 5 Micro-Gravity Testing 79 5.1 Installation and Testing 79 5.2 Results ; 81 5.3 Analysis • 83 5.4 Recommendations for Future Study • • • 86 6 Conclusions 89 Bibliography 91 A Theory Notes 95 A. l Augmented Plant 95 A.2 Stabilizable and Detectable 96 A.3 Plant Transformations . . . . 96 A.3.1 Axis Shifting 96 A.3.2 Bilinear Transform 97 B Nomenclature 99 v C Hoc Controller Design Code 102 C.l Main design file: design.m 102 C.2 Position weight generation: weightsjc.m . . . . . . . . . . . 104 C.3 Acceleration weight generation: weights_a.m . . . " 105 C.4 Bilinear transform: hinLbilin.m 105 C.5 State space system: msys.m . 106 C. 6 Plotting summary: pltsum.m 106 D Coarse-Fine Tracker Software Particulars 109 D. l Coarse-Fine Software Algorithm . . . . : 109 D.2 Coarse Stage Control Signal . . . . . 112 D.3 Coarse Stage Code Variables . ; 112 E Data Plots 114 VI List of Figures 2.1 Simple schematic of the MAGLEV LMIM system 11 2.2 Freebody diagram of one DOF of the LMIM 12 2.3 A simple representation of the LMIM platform 15 2.4 Bode plot of double integrator plant G(s) = 18 2.5. Simple LMIM model 18 2.6 Effect of filters 20 2.7 Effect of filters 21 2.8 Model of plant with unstable feedback 22 2.9 Bode plot of position plant with unstable feedback of 500 N/m 22 2.10 Closed loop system used for plant identification 23 2.11 Spectral analysis of position plant 25 2.12 Position plant identification. . . 28 2.13 Bode plot of Yp(s). . . . - 29 2.14 Spectral analysis of acceleration plant. . , 30 2.15 Acceleration plant identification 32 3.16 The classic feedback loop for control systems 34 3.17 The unique model representation for control theory 35 3.18 Standard form for the HQQ control problem. 35 3.19 The position of the weights in the classic feedback model 37 3.20 Augmented Plant . 41 3.21 Model for Hco design - simple double integrator plant 43 vii 3.22 #oo controller 45 3.23 Characteristics of the designed system with unstable force feedback. . . . 46 3.24 Theory and data comparison 48 3.25 Plot to evaluate the area of the sensitivity function 50 3.26 The acceleration plant with an inner closed-loop position feedback. . . . 53 3.27 Bode plot for acceleration plant Gax(s) 54 3.28 General shape of the weights for acceleration controller design 55 3.29 The complete position plant with acceleration feedback. . . . 56 3.30 A position transfer function with acceleration feedback 59 3.31 Position transfer function with acceleration feedback . . 60 3.32 Force disturbance rejection . . 62 4.33 Diagram of the 1 DOF coarse-fine tracker. . 65 4.34 Freebody diagram of the coarse fine coupled system. . 67 4.35 Nyquist plots showing open loop stability of the system to airplane vibra-tions 73 4.36 Bode plots showing closed loop response of the system to airplane vibrations. 74 4.37 Model of the coarse-fine control system 75 4.38 Plots showing the simulation response to a sample vibration signal with Kp = 1200, Kv = 15. 77 4.39 Plots showing the simulation response to a sample vibration signal with Kp = 1200, Kv = 0 78 5.40 Typical parabola cycle 80 5.41 Simulation compared to actual data 82 5.42 Data plots for a parabola . . . . . . . . . 84 vm 5.43 Comparison of data and simulation with low frequency mode for the timing belt and static friction for the sliding car 87 A. 44 Special form of the bilinear transform that maps the jio-axis to a circle. . 98 B. 45 Closed loop position system with acceleration feedback 99 D. 46 State switching algorithm - I l l E. 47 Parabolic data plot 1 115 E.48 Parabolic data plot 2 116 E.49 Parabolic data plot 3 . • • 117 E.50 Parabolic data plot 4 118 E.51 Parabolic data plot 5. . . 119 E.52 Parabolic data plot 6 120 E.53 Parabolic data plot 7. . . . . . . . . . 121 E.54 Parabolic data plot 8 122 ix Acknowledgments I wish to extend my gratitude to many individuals who assisted me over the course of my research here at the University of British Columbia. First I would like to thank Tim Salcudean for his guidance during my project and for introducing me to this exciting field and interesting technology he has developed. I am greatly appreciative to him for the opportunity to ride the DC-9 in Cleveland, which was the experience of a lifetime. I would also like to thank Niall Parker for all his help in setting up and coding the system and making my work so much easier. The people in Cleveland were very helpful as well, especially B i l l Stewart and Bjarni Tryggvason of the Canadian Space Agency who gave up a good chunk of their time to assist me in overcoming some bugs with the hardware and software. I arrived in Vancouver knowing no one, and I thank all the peers and colleagues I met who gave me positive experiences and memories. And finally I thank my family, my Mom and Dad for being always supportive and always believing in my abilities, and my daughter Ashley for always bringing a smile to my face. x Chapter 1 Introduction 1.1 Vibration Isolation Things vibrate and cause vibrations - it is a fact. Whether sitting in a car at the rail crossing as the freight train rumbles by or riding in an airplane while the engines cause the seat to "tingle", vibrations are felt. Or to take an extreme example, doing a delicate experiment when suddenly the whole table moves as a mild earthquake shakes the area and ruins the work. However vibrations affect life, there are certainly instances when it is prefered not to be concerned with them. That plane ride would be much better if the ride was smooth, and that experiment will have to be started all over again. Enter vibration isolation techniques. These are aimed at reducing the effects of the source (that which causes the vibrations) on the application (that which is sensitive to the vibrations). In a normal gravity environment, vibration isolation is used in many applications. Scanning tunnel microscopy depends on reliable control of tip-to-sample distance and is degraded if noise is present. Other precise measuring applications require isolation from noise. In delicate fabrication and assembly procedures, reliable position and orientation is required. Vibration noise in normal environment can be quite large, with high power machines bolted to floors or large vehicles driving around. Vibration isolation is important in micro-gravity experiments, such as crystal growth research and other chemical experiments. Vibrations cause disturbance motions, and 1 Chapter 1. Introduction 2 therefore disturbance accelerations. The purpose of micro-gravity is to have net acceler-ations on the experiment to be zero, and vibrations degrade this result. This has become a large concern for orbitting labratories and needs to be addressed. Two places to achieve a micro-gravity environment are in orbit and in a plane performing parabolic flights. In both cases, high frequency vibration is caused by the engines and other disturbances are caused by the movement of personnel in the craft. There are two types of control available: passive and active. Passive techniques have been used extensively in many applications utilizing fluidics and dampers and springs. Active techniques are newer and include vibration cancellation and feed forward designs. Cancelation is the method by which the vibrations from the source are countered before getting to the ambient surroundings. This method works well for a large gener-ator to be isolated from the factory floor. The success of cancellation depends on the knowledge of the vibrations caused by the source, whether by design or measurement, and tends to be less successful for random vibrations (noise). [6] makes use of the idea that the best way to reduce ground vibrations is to not let it get to ground in the first place. The authors mount vibration causing equipment (motors, circular saws) on rigid supports with electro-hydraulic servo mechanisms. The vibrations of the isolated system are measured and fed-forward to a controller that has been determined in a parameter optimization-like procedure. 1.2 Contact vs. Non-Contact The term contact is used to express the idea that the application is attached in some way to. the ambient surroundings by some sort of rigid support - table legs, shock absorbers, and the like. It is reasonable to expect that the application be mechanically linked to the ground in this manner and many people have investigated isolation using this set up. Chapter 1. Introduction 3 The use of a Stewart Platform is investigated in [9]. The Stewart Platform used is an innovative design and the authors look specifically at the 6 degree of freedom case. By controlling the hydraulics of the system in a specific manner, a spring-damper connection to the ground is achieved to attenuate vibrations. A method to rigidly mount the experimental platform to a space structure is presented in [5]. The legs of the platform are connected to the hull of the lab through piezo-ceramic mounts, which contract and expand with electric currents. The amount of movement is very limited and the piezo-ceramic materials might be subject to breakage. Of course the biggest problem with any contact system is exactly that - it is actually touching the ambient surroundings and therefore some vibrations are bound to be trans-mitted to the application. The best solution would be a non-contact system - one that does not physically touch the surroundings at all, except perhaps through an umbilical cord that supplies power and measurements to a controller. Since the application is not in contact with the surroundings, only its own vibrations affect it. A simple analogy would be flying in a plane during an earthquake. [4] takes this idea to the extreme and proposes that the application (an experiment platform) be free floating. This assumes that the isolation system be located in space of course. An umbilical cord must be attached between the platform and hull for power and measurements and such, but otherwise the system is completely isolated. However, due to trajectory changes and corrections, the platform is bound to not stay centered in the research space. The authors address this with the use of air jets to keep the platform centered. They are careful to find the range of linear force for the air jets. This system has limited application on the ground and also limited centering control. An option that works for earth based systems as well as micro-gravity is magnetic levitation, and much has been reported on its use in various applications, mostly in transportation. Levitated car seats and levitated high velocity trains are some of the Chapter 1. Introduction 4 ideas for levitation. It can also be used in vibration isolation to gain some of the benefits of a non-contact system while allowing better control than the complete non-contact system; [11] uses magnetic bearing levitation for fine vibration isolation. There is still no direct contact,.but better control can be achieved. The platform is levitated above a Stewart platform which controls the coarse motion of the platform. The platform controller tends to return to the center of the Stewart platform, which in turn tends to the center of the research space. This idea is expanded in [15], which looks at optimizing the hardware for ease of controller design. Given the properties of the plate levitated by magnetic bearings, the paper looks into determining the best position and number of sensors and actuators required. Theorems supporting their results plus the instability of magnetic bearings are presented, and addresses very briefly the magnetic non-linearity. A system closer in design to what is presently being worked with is given in [8] - the magnetic levitation is different than in previous papers. The platform is levitated on a coarse motion stage (a robot arm) attached to the wall of the craft which tends to the center of the space. The actuator and sensor locations are fixed so hardware optimization is not possible. The platform remains unconnected to the surroundings and can be moved large displacements (slowly) with the coarse stage while the fine stage levitation isolates the platform from the entire system. 1.3 Control After hardware has been chosen, a controller must be designed. Some papers are hardware non-specific and show the effect popular control algorithms on the vibration isolation problem. These designs assume linear plant's and knowledge of system dynamics for the most part. PD or PID control can be used, but other techniques can be used to improve Chapter 1. Introduction 5 the system response. [7] compares two types of active control - the usual cooperative optimization technique vs. the authors' non-cooperative technique. A numerical example is provided for a simple plant and indicates that the non-cooperative technique has better isolation while remaining just as robust as the cooperative technique. Another paper of note is [10] which actively controls vibrations by cancelling the vi-bration nodes of the system. This of course requires knowledge of the vibration nodes before design. Another controller examined is one designed by using LQG theory. Sim-ulations of these are also positive. A general control scheme for linear forces is presented in [12]. Using a quadratic performance index, a robust controller is designed based on one of five possible repre-sentations of the system. The controller has an LQR feedback loop and a constant gain feedforward loop. An LQG controller which is designed based on the predicted disturbances of the system and also depends on knowledge of the response of the flexible, structure to the vibrations is employed in [5] to control the electric currents for the piezo-ceramic mounts. The system to be used may be non-linear and attempts can be made to linearize it. [19] directly addresses the magnetic non-linearity by putting a feedback PD loop around the actuator. [18] uses this tool to remove.the negative stiffness of their novel actuator for their isolation technique. With the accelerometers on the base of the system, the authors use PID as their control algorithm. The gains are adjustable, and are changed until the desired response is obtained. A lot of the controllers depend on knowledge of the system dynamics and vibration nodes that exist. One option is as in [9] where the authors use adaptive control since the specific dynamics of the system are unknown, plus the controller then becomes much more flexible in possible applications. Chapter 1. Introduction 6 1.4 Magnetic Levitation Large Motion Isolation Mount The Large Motion Isolation Mount uses magnetic levitation technology (MAGLEV) [38] and is developed in [17], [14]. The hardware was designed and built under contract [29] [30] [31] at the University of British Columbia for the Canadian Space Agency., The CSA was looking to use long magnetic bearing rods for vibration isolation in space and approached Tim Salcudean. Dr. Salcudean suggested a coarse-fine approach to give the same workspace while using Lorentz forces to control the fine stage. In this approach, force is not proportional to the position. A fine stage only design was sent up on the Space Station MIR and has provided vibration isolation reliably for over six months. The control for this MIM was provided by Tim Salcudean and Niall Parker after several control techniques were studied [16]. A PID controller yielded acceptable response controllers but it was thought other tech-niques would be better. Slotine and Li methods had slow convergence and a recursive least squares method did not converge satisfactorily at all. Q-parameter methods were investigated and found to have good responses and were easy to design. 1.5 Micro-Gravity Testing There are two popular methods of simulating micro-gravity conditions in normal grav-ity. The first method of micro-gravity simulation is the use of a drop tower - a very tall structure with a padded vertical shaft that usually extends into the ground. The experiment is released from a drop tower where the experiment then accelerates toward a soft landing area. This method has excellent micro-gravity characteristics but has the disadvantage of the sudden stop at the end, which might not be desirable for some of the more delicate experiments. The second and cheaper way to conduct micro-gravity experiments is to use an aircraft Chapter 1. Introduction 7 performing parabolic flights, such as NASA Lewis Research Center's DC-9. The airplane follows a parabolic trajectory that gives a period of micro-gravity conditions. First the plane pulls up steeply, increasing the gravity to almost twice normal. Then the plane begins to decrease in pitch lowering the effect of the acceleration due to gravity on the plane. At the top of the parabola, as the pitch angle changes to negative, the occupants of the plane experience free fall conditions and continue to do so until the plane pulls up level with the horizon again. Further information on the flight can be obtained from NASA's DC-9 World Wide Web site at http://zeta.lerc.nasa.gov/jpw/cover.htm or [13]. Parabolic flights provide approximately 20 seconds of noisy micro-gravity. The noise comes in the form of high frequency vibrations caused by the aircraft engines and air turbulence and low frequency motion from trajectory errors and corrections. Due to this noise, to obtain a clean micro-gravity environment the experiment is usually conducted as a free float. A free float experiment is released by the experimenter during the zero gravity phase of the parabola and is allowed to drift around the aircraft until it. hits the wall. This is not optimal because the motion of the free float is unconstrained and could be a hazard to other experiments, plus when releasing the experiment it is impossible not to impart initial conditions. Free floats are used because any experiment bolted to the aircraft frame experiences all the vibrations and noise that the frame does. The noise problems associated with the parabolic flights can be rectified by using a magnetic levitation approach. A magnetically levitated platform will isolate the ex-periment from the high frequency vibrations and will slowly move the platform to the center of the rattle space to correct the low frequency errors. The platform is attached to the aircraft, so there will be some noise transmission to the experiment, but it will be considerably less than if no magnetic levitation was used. One problem with free floats is uncontrollable initial conditions. With a magnetically levitated platform, initial conditions can be set to desired values. Chapter 1. Introduction 8 1.5.1 Two Stage System On the parabolic flights the low frequency errors are large in magnitude and will cause the platform to reach the limits of its workspace quickly if the control bandwidth is small. What is required, if the bandwidth of the platform is to be kept low, is for the stator to move as well. This means there are two systems working together to isolate the system: a coarse stage and a fine stage. The anticipated problem with this setup would be having two systems trying to work together while at the same time reacting against each other. The fine stage is a version of the magnetically levitated platform and reduces high frequency noise. The coarse stage follows low frequency errors and would be a robot arm of some type, such as a six degree of freedom PUMA. Thus as the fine stage is left to drift freely, the coarse stage tracks the flotor, keeping it in the rattle space. See [38], [37], [36]. The motion of the coarse stage is limited by the workspace of the airplane, but the combined workspace will be much larger than the fine stage alone. Another benefit of the coarse-fine approach is the bandwidth of the fine stage can be lowered further since less centering force is required. 1.6 Thesis Overview Vibrations are a problem that need to be considered. The goal is to have an isolation system that is effective against all disturbance vibrations, both high frequency and low frequency. The system used in this thesis consists of two stages. A fine stage isolates, the system from small high frequency disturbances and a coarse stage cancels large low frequency disturbances. This increases the workspace during isolation procedures. The two stage approach gives better isolation characteristics than a one stage system. The fine stage uses magnetic levitation technology [17], which isolates the system from Chapter 1. Introduction 9 high frequency disturbances, instead of cancelling disturbances. With the MAGLEV technology, a lower stiffness should be achievable. Other benefits of using Lorentz forces are the force is not proportional to position so the actuators are more linear, and there is no positioning backlash with the actuators. Actuator control is developed using HQO theory. This should provide better robustness in the presence of model uncertainties and provide good closed-loop shaping. This thesis provides some background information on the MAGLEV system and de-velops Hoe control theory for use on an actual MAGLEV apparatus. The effectiveness of the coarse-fine approach in a micro-gravity environment is evaluated using a one degree of freedom set-up tested on a DC-9 performing parabolic flights. A one degree of freedom system has less of the dynamic considerations necessary for a full six degree of freedom system and should prove the concept works. The first chapter looks at the system to be used and explains how it works. Next H^o control theory is explained and used to attempt design of a robust controller that uses the available position data. Experimental data was studied to determine the success or failure of the system after the controller is tested in simulation. The next chapter explains the smaller version of the magnetically levitated platform and its coarse stage tracker. The system is- modeled and simulated in the next chapter. The subsequent chapter describes the testing on the DC-9 and analyzes the data collected. The final chapter draws conclusions and presents ideas for future work. Chapter 2 Motion Isolation Mount Modeling This chapter looks at the Large Motion Isolation Mount used in normal gravity. A description of the system is presented, including a brief overview of the hardware. The system is modeled as a rigid body and the development of this model is presented. Some plant identification is performed and the benefits of using different plants is discussed. 2.1 M A G L E V LMIM The Large Motion Isolation Mount (LMIM) is a hardware intensive system, and a simple schematic is shown in Figure 2.1. A magnetically levitated (MAGLEV) system has two distinct parts: the flotor and the stator. The flotor is called flotor because it is levitated, or floats, above the base. In the case of the LMIM, the flotor weighs 22 kg. The stator is the base of the system. The only connection between the flotor and stator is an umbilical cord that provides power for the flotor sensing devices and tranmits the signal data from these devices. A one degree of freedom freebody diagram of the system is presented in Figure 2.2. The actuator couples the flotor and stator, with the actuator applying fact to the flotor. The umbilical cord physically couples the flotor to the stator, and its force effect is included in fdist- The vibrations that act on the stator are those from which the flotor is to be isololated. The main actuation and sensing components are shown in Figure 2.1. 10 Chapter 2. Motion Isolation Mount Modeling 11 Front view at cutt ing plane A - A Figure 2.1: Simple schematic of the MAGLEV LMIM system. Chapter 2. Motion Isolation Mount Modeling 12 F l o t o r c a c t i s t A c t u a t o r l ± l U m b i l i c a l c o r d S t a t o r v i b r a t i o n s Figure 2.2: Freebody diagram of one DOF of the LMIM. . 2.1.1 Actuation The flotor has eight pairs of permanent magnets in a square configuration that create the magnetic field necessary for operation. The stator has eight wire coils, each one fitting between a pair of the permanent magnets on the flotor. Lorentz forces are used to levitate the flotor above the stator by controlling a current through the wire coils in the stator in the magnetic field produced by the magnets in the flotor. The force is , F = I / (dl x B) (2.1) where I is the current through the coil, B is the magnetic field produced by the magnets and dl is a differential length of wire in the direction of the current. Lorentz levitation has the benefit of frictionless motion with no backlash, so it is ideal for fine positioning applications. The advantage for vibration isolation is instead of actively cancelling high frequency noise, as is the case for contact technology, magnetic levitation cannot follow the high frequency noise above the system bandwidth. This is an advantage because it Chapter 2. Motion Isolation Mount Modeling 13 should allow the bandwidth to be decreased significantly resulting in a lower stiffness coupling which will isolate better against noise. Using Lorentz forces also creates problems. The magnetic field strength is not con-stant over the entire space between the permanent magnets. Non-linearities grow as the gap between the magnets increases - the flux fringes to a greater extent - and the mag-netic field weakens. This then becomes a tradeoff between flotor motion range (rattle space) and controllability. The flotor can only move as much as the gap between its magnets so a larger gap means a greater rattle space, but also means less linearity for control. The LMIM has a gap of ±1.5 mm in the x direction and ± 2 mm in the y direction. Design specifications were for a ± 4 mm air gap, but sloppy tolerances on the flotor construction prevents it from achieving the full workspace. 2.1.2 Sensing Position and orientation of the flotor is determined by Position Sensing Diodes (PSD). Three narrow beam LED's on the flotor project onto a detector attached to the stator. The hardware, calibration, and signal conditioning for the PSD's is given in [39], and the matrix manipulations required to give flotor position and orientation are supplied in [33]. The acceleration of the flotor and stator in each of the axes x, y, and z are measured by accelerometers. The signals are conditioned by Butterworth filters located in the stator. The filters are low pass with cut off frequency at 100 Hz. The acceleration filters are second order and the position filters are fourth order. • Chapter 2. Motion Isolation Mount Modeling 14 2.2 Model The LMIM is obviously a very complicated system with many different components. To obtain a model that will be easy to work with some assumptions are made. First, it is assumed the flotor behaves as a rigid body and any flexible modes are either small or high enough in frequency to be negligible. This allows the flotor to be represented as a simple rigid body. The actuators, as mentioned earlier, are non-linear. The transformation from currents to wrenches is location-dependent and there will be some eddy current coupling. For the actuator model, linearity and negligible eddy, currents are assumed. The sensors have a very high resolution and are subject to small deformities. The slightest concave curvature of the sensor could push the flotor to the side and limit the lowest frequency available in normal gravity. All sensors are assumed to be linear. The filters are assumed to be all identical and precise. Using these assumptions, a model of the LMIM platform is shown in Figure 2.3. The sensed signals are position (x) and acceleration (a). The position vector has both trans-lation and rotation components while the acceleration vector has only three translation components, as angular accelerations are not measured. In Figure 2.3, the wrench vector is applied to the flotor. I maps the six element wrench vector to three translational and three rotational acceleration elements and then G(s) converts the accelerations to position elements. The force-torque transformation for a rigid body of a point not at the center of mass is where / and r are the force and torque applied to the flotor, m and J are the mass and / = mo + mw x (c — o) + m(wx)2(c — b) (2.2) r 0 = (c — o) x mo + JQW + w x JQW (2-3) Chapter 2. Motion Isolation Mount Modeling 15 Flotor wrench M I 9j l umbilical force acce,e ra,,„„ sensor noise G(s) 4* 6 + • « = — disturbance force Anti-aliasing Filter o + ««=— position sensor noise Accelerometer Output AAF Reconstruction Filter u RCF K(z) PSD Output AAF Anti-aliasing Filter Digital Controller Figure 2.3: A simple representation of the LMIM platform. inertia matrix, o is the linear acceleration of the point, w is the angular acceleration, and c is the center of mass point. If we assume the second order terms are small, the quadratics can be neglected, giving / = mo + mw x (c — 5) f0 = (c — o) x mo + JQW , (2.4) (2.5) When the point is the center of mass, Tc — JrW . (2.6) Applying the wrench fc + (c-f)xf f (2.7) Chapter<2. Motion Isolation Mount Modeling 16 JcW + rcf x (mo + mw x rcf) mo + mw x fcf JcW + fcf x mf mf + mw x fc/ (2.8) (2.9) (2.10) we get the force-torque transformation ml mrcf x ' } ' mfcf x J-c w (2.11) where r~cf is the vector from c to the center of the flotor / . When the center of the flotor coincides with the sensor center and the center of mass, the diagonals of the force-torque transformation reduce to zero, and the matrix / is the inverse of this transformation I = hh 0 0 (2.12) where 73 is the 3x3 identity matrix. Continuing with Figure 2.3, G(s) transforms the accelerations to position and is a 6x6 matrix of transfer functions. K(z) models the digital controller plus the digital-to-analog and analog-to-digital converters. The response of the digital controller K(z) approaches the response of the continuous controller K(s) at sufficiently high sampling rate. The inputs to the computer are the three translational accelerations of the flotor with added accelerometer noise and the six element position and orientation of the flotor with added PSD noise. [1] identifies this noise to be white with magnitude of 2.5 fan. AAF is the anti-aliasing filter for the input signals. The signals are filtered by low pass Butterworth filters of cut off frequency 100 Hz and order four for position and order Chapter 2. Motion Isolation Mount Modeling. 17 two for acceleration. RCF is the reconstruction filter and is the filter for the output signal. The filter is a low pass fourth order Butterworth with cut off frequency 100 Hz. The AAF will also be refered to as the A/D filter, and the RCF will also be refered to as the DAC filter. The output of the computer is conditioned by the DAC filter and is the six element wrench command. This wrench command is affected by any disturbance forces such as changing acceleration levels and the umbilical force. The umbilical cord acts as a spring-damper connection with positive spring constant between the flotor and stator and the umbilical force is dependent on the position of the flotor. This six degree of freedom model is very cumbersome with large matrices and six element vectors. For ease of controller design, the model will be reduced to a one degree of freedom approximation using the afore mentioned assumptions and assuming all degrees of freedom behave identically. The rotational elements are ignored and the design will be done for an axis in the horizontal plane so the force due to gravity does not need to be considered. In one degree of motion, the mass transform / is ^ and G(s) is jj. This model is shown in Figure 2.5 and the Bode plot of ^ is shown in Figure 2.4. K(s) is made up of two controllers, position (Cp(s)) and.acceleration (Ca(s)). 2.3 Effect of Filters Figure 2.5 shows a model with the conversion filters and the plant as simply ^ . The model would be of much lower order if the filters could be disregarded, since the filters are up to fourth order. The effect of the filters near the bandwidth must be investigated. A simple test can be done to determine if the filters need to be included in the model. Chapter 2. Motion Isolation Mount Modeling 18 Figure 2.5: Simple LMIM model. Chapter 2. Motion Isolation Mount Modeling 19 Using a simple PD controller to force the closed loop response bandwidth to be around 10 Hz (very high), the difference in response with different filters can be determined. Figure 2.6 shows the effect on the system for a second order reconstruction filter at 30 Hz and a fourth order anti-aliasing filter at 250 Hz. The effect of the reconstruction filter is the most pronounced and alters the system by creating a 6 dB resonant peak. Figure 2.7 changes the filters slightly to fourth order at 100 Hz for the RCF and fourth order at 200 Hz for the AAF. The effect of these filters are insubstantial at this bandwidth, so as long as the filter bandwidth is large enough, at least 100 Hz, and the bandwidth of the closed-loop system is less than 10 Hz, the filters can be ignored in the model. 2.4 Destabilizing Feedback There is a destabilizing force on the flotor that varies as a function of position, similar to an inverted pendulum [28]. An increasing force causes the flotor to reach the edge of the workspace, and the controller is not able to counter this force. There are two probable causes of this - actuator non-linearity and PSD detector curvature. Since the flotor tends to go to one side more often, it is likely PSD detector curvature causes the position measurement center to not coincide with the center of the magnetic gap. And once off-center, the inverted pendulum effect occurs. Figure 2.8 is a model of this actuator non-linearity. K(s) is the controller, and the new plant Gp(s) is G ' W = 5 ? T f f <2:i3> where m is the mass of the flotor and H is the positive linear force feedback. The Bode plot of this transfer function is shown in Figure 2.9. A value for H was determined indirectly. Initial measurements were done with a force meter. A more accurate method involved changing the steady state force acting on the Chapter 2. Motion Isolation Mount Modeling 20 Position Closed Loop Response ~ i 1 1 — i — | - "T 1 1 r-0 CD T3 "c CO -20 h -40 h -601 - No filters - - DAC filter - A/D filter both: filters m CD T3 13 'c CO 0 -20 -40 -60' 10" 10' 10^  Frequency [Hz] Position Closed Loop with Acceleration Feedback Response No filters - - DAC filter - - A/D filter both: filters 10" 10' Frequency [Hz] 10' Figure 2.6: Effect of filters on closed loop system under PD control... Second order reconstruction filter at 30 Hz and fourth order anti-aliasing filter at 250 Hz. Chapter 2. Motion Isolation Mount Modeling 21 Position Closed Loop Response Frequency [Hz] Position Closed Loop with Acceleration Feedback Response T I I I I I I I | r n 1 i i 1 1 — i — i — | 1 1 r 10"1 10° 101 Frequency [Hz] Figure 2.7: Effect of filters on closed loop system under PD control. Fourth order recon-struction filter at 100 Hz and fourth order anti-aliasing filter at 200 Hz. Chapter 2. Motion Isolation Mount Modeling x r e f o K(s! G(s) f x H Figure 2.8: Model of plant with unstable feedback. 0 CD 73 C '(0 CD £ -50 -100 10 -2 10u -179 -179.5 cn d) to $ -180 CO -180.5 -181 2 -2 10 10 10u 10' Figure 2.9: Bode plot of position plant with unstable feedback of 500 N/m. Chapter 2. Motion Isolation Mount Modeling 23 Cqmputer Figure 2.10: Closed loop system used for plant identification. flotor when the controller stiffness was too low to center the flotor. The steady state force was increased from zero until the flotor moved off the edge of the work space and hit the other side. The strength of the force required to move the flotor away from the edge of the workspace was approximately 0.5 N. With workspace limit of about 1.5 mm, ^becomes 333 N/m. The expression "falls to the side" may be used in discussion of the LMIM. The in-stability comes from the position and force non-linearity of the system and amounts to an inverted pendulum mode [28]. When the flotor is centered any displacement will add increasing force and the flotor will hit the edge of the workspace, as an inverted pendulum that is displaced will fall due to the force of gravity. Thus the term "falls to the side". 2.5 Plant Identification Y(s) In an attempt to get a more accurate model some simple identification was performed. Figure 2.10 is the closed loop system used and defines the unknown plant from the RCF output u to position and acceleration to be G(s). The controller is simple proportional-plus-integral-plus-derivative and stabilizes the flotor in the workspace with no exceptional isolation characteristics. The output of the PID controller is / and is more readily Chapter 2. Motion Isolation Mount Modeling 24 available than u. As a consequence, the plant that will be identified is Ct(S) plus the reconstruction Butterworth filters. This plant will be called Y(s) and is = ' Y(s) f ' ' (2.14) a x Ideally, to get a good measure of the output response of a system, the input signal is white noise. It is not possible to set / to this because the system would be open loop and unstable. Instead, xref is driven by an approximation of white noise which results in a noisy signal / as the PID controller attempts to follow the reference position. By performing spectral analyses on / to x and a, the plant can be approximated. Data signals / , x, and a were collected for 30 minutes as the flotor position was driven by the noise. More data points give better representation of all frequencies and a better spectral analysis. 2.5.1 Position Plant The analysis.of the force and position data is shown in Figure 2.11. The first plot shows the power spectral density of the Control Input to the plant and the second shows the power spectral density of the Flotor Position. Note the units are inconsistent. Based on these power spectral densities the transfer function from / to x is represented graphically in the Tf fen mag plot of Figure 2.11. The coherence of this transfer function is shown in the last plot and shows confidence between 0.2 Hz and 10 Hz. Coherence is a measure of the correlation between the input signal and the output signal, and has a value between one and zero. A coherence of one indictates that the signals are a linear transformation of each other, and a low coherence indicates non-linear a x YP(s) f (2.15) Chapter 2. Motion Isolation Mount Modeling 25 10 u 3 i o - 2 10" Control Input [N] Flotor Position [mm] 10" o o o CO -20 -40 O M . ' — / 1 " - J l 1 1 ' 1 1 f i l m 1 • 1 1 1 i 1 10"° 10° Frequency (Hz) Tf fen mag. 10 test hinf 10 0.8 0.6 0.4 0.2 0 10" File: h10 Coherence 10"' 10" 10' Fri Aug 2 02:08:12 1996 Figure 2.11: Spectral analysis of controller output to flotor position. The Control Input is / , the Flotor Position is x, and the Tf fen mag is j. Chapter 2. Motion Isolation Mount Modeling 26 terms and noise in the output signal. The next step in this simple identification attempt is to fit a transfer function in the Laplace domain to the shape shown in the Tf fen mag plot of Figure 2.11. The low frequency gain of Tf fen mag appears to be constant near 9 dB, keeping in mind position units are mm. The DC response is approximately | F p (27 r / ) | » 2.6,/ < 0.1. (2.16) After the roll off the data plot has a slope of -40 dB/decade for a time. As the frequency increases past 9 or 10 Hz, the plot rolls off at a greater rate. The new rate cannot be determined due to the bad coherence of the data past "10 Hz. The first part of the roll off can be expressed as l ^ ( 2 7 r / ) l ~ ( 2 ^ ' 2 < / < 9 ( 2 ' 1 7 ) and then the denominator order increases. Although the phase of the transfer function is not plotted separately, it can be seen in Figure 2.12. In this instance, the phase starts at —180° and increases slightly to —170° before decreasing at 1 Hz. The low frequency phase being negative with a constant mag-nitude gain implies a right half plane pole, and the slight increase in phase is indicative of a small positive damping in a second order function. Using these points, the transfer function Yp(s) is at least third order in the denomi-nator and zero order in the numerator. For this simple identification, the denominator will be chosen as third order with a.first order and a second order factor. The second order factor is chosen because of the damping required to fit the phase data. The general form for the transfer function between / and x is defined as . Chapter 2. Motion Isolation Mount Modeling 27 where a is the DC gain, u>i is the first corner frequency, w2 is the second corner frequency, and ( is the damping ratio of the plant. Both frequencies W{ are in rad/s.. From (2.16) Yp(0) = a = 2.6 . (2.19) and from (2.17) aw\ = 4 5 -»• ™i ~ 4 - 1 6 - ( 2 - 2 0 ) After iterating the values of ( from 0.05 to 0.5 and w2 from 50 rad/s to 300 rad/s, the best fit was obtained when C = 0.15 (2.21) w2 = 140. (2.22) Putting all these values into (2.18), Y(s)- ^ (2 23) P [ ) " (lio + l)(*2 + 2(0.15)(4.16)5 - 4.16») where the numerator has been divided by 1000 to convert units from mm to m. Figure 2.12 compares the shape of this transfer function to the data in Figure 2.11. The closed loop response of the system is also presented for comparison. In Figure 2.12, the open loop plant magnitude is very close and the phase is off by 5° by 5 Hz. The closed loop matching is very good, much better than the simple double integrator plant provides. The effect of a second order model in the form —TTT—z is better understood, and a good approximation of Yp(s) is Y'(s) = — •— . (2.24) p W 22.2s2 + 27.7355 - 384.615 \ -This approximates the mass of the flotor as 22.2 kg, very close to the actual mass. The Bode plot of this function is shown in Figure 2.13. Chapter 2. Motion Isolation Mount Modeling 28 Figure 2.12: Position plant identification. Note position is measured in mm. Chapter 2. Motion Isolation Mount Modeling 29 -50 m T3 c 'CO -100 -150 Figure 2.13: Bode plot of Yp(s). 2.5.2 Acceleration Plant The acceleration plant can be estimated in the same fashion by collecting controller out-put and flotor acceleration data and using spectral analysis to find the transfer function j = Ya(s). Figure 2.14 shows the power spectral densities of the input to the system and of the flotor acceleration. These units are consistent. The transfer function and coher-ence functions are also given. The coherence becomes jumpy below 1 Hz and the transfer function plot below this frequency becomes suspect. Possible reasons for this are low magnitude low frequency components from the input signal and the output magnitude being at the same scale as the noise of the system. Examining the transfer function in Figure 2.14 it is difficult to get an accurate low frequency response of the plant since the coherence is bad and the plant rolls off at low frequency. The flat magnitude band is about -27 dB, which is approximately 0.045. The bandwidth is quite high, almost 50 Hz, so it can be expected that the filters will play a role. Chapter 2. Motion Isolation Mount Modeling 30 Control Input [N] Flotor Acceleration [m/s/s] 10v 10 Frequency (Hz) 10u File: h10 Tf fen mag. Coherence S-40 10 test hint 0.8 0.6 0.4 0.2 0 \ ^ 1 1 . 1 • 10"' 10" 10' FriAug 2 02:08:12 1996 Figure 2.14: Spectral analysis of controller output to flotor acceleration. The Control Input is / , the Flotor Acceleration is a, and the Tf fen mag is j. Chapter 2. Motion Isolation Mount Modeling 31 The starting guess for the transfer function Ya(s) will be s2Yp(s). This transfer func-tion fits the low frequency data well but does not roll off at 60 Hz. Since this is getting close to the filter corner frequency, a fourth order low pass Butterworth with corner fre-quency 30 Hz was added. The transfer function had the correct shape but rolled off much too early. By increasing the filter frequency to 325 rad/s a better fit was obtained. The acceleration plant is defined as • ' 0 045s2 y - . M = (ft + + 2(0.15)(4.16), - 4.16*) * B " t o < 4 - 3 2 5 > <2'25> where Butter(4,325) is a fourth order Butterworth filter with corner frequency 325 rad/s. This transfer function is compared to data in Figure 2.15 which shows the open loop plant has good phase agreement and magnitude agreement up to 30 Hz. The closed loop system matching is very good. Since the interesting bandwidth of the system is' much less than 30 Hz, it is reasonable to disregard the Butterworth filter in this model. Chapter 2. Motion Isolation Mount Modeling 32 Magnitude [dB] Magnitude [dB] , ro -•• -i. o c n . f i . c o r o o o o - ± 0 0 0 o o ro Figure 2.15: Acceleration plant identification. Chapter 3 Hoo Controller This chapter investigates control theory for use in vibration isolation. First, the design objectives of the controller are investigated. Then the theory of Hoc design is presented and design parameters are explained. A position controller using this theory is obtained, and more theory on its design is presented. Acceleration feedback is used to improve the system response and the theory for this is presented. 3.1 Design Objectives The objective of controller design for vibration isolation is to keep the acceleration levels of the flotor as low as possible. The acceleration of the flotor should have low frequency tracking to keep the flotor returning to the center of the workspace. The controller should also attenuate high frequency disturbance accelerations on the flotor. It has been shown [16] that isolation effectiveness of the controller can be found by examining position'excitation. Thus the objective is to design a low bandwidth position controller. Acceleration feedback can then be used to further reduce the bandwidth, creating a better isolation controller. Acceleration control is also used to improve the force disturbance rejection response. 33 Chapter 3. Controller 34 o e K(s) u G(s) Figure 3.16: The classic feedback loop for control systems. 3.2 Theory 3.2.1 Plant and Design The simple feedback control system is usually drawn as in Figure 3.16, with K(s) being the controller, G(s) the plant, r the input signal, c the output signal, e the tracking error, and u the control input. For control theory, the plant changes somewhat. The loop is drawn in a unique way, such that the system has two inputs and two outputs. This allows for better understanding of how the system is affected by different changes. Figure 3.17 shows this modified block diagram for Hoc theory. The input w is the external input and includes all inputs to the system such as disturbances and tracking signals. The other input signal is u and is the control signal from the controller. The two outputs are z and y, which are the control output and measured output respectively. The signal z is not necessarily available for measurement (it doesn't even have to exist) but contains the states of interest for the control design. The various Wi(s) are weights that are applied to the control output to shape the controller and dictate the response of the states in question. The entire system without the controller K(s) is known as the augmented plant, P(s). The block diagram is often simplified to Figure 3.18 which is known as standard form. The augmented plant, P(s), is the transfer matrix from w and u to z and y . (3.26) z w = P(s) . y. u Chapter 3. Controller Augmented Plant_ K(s) Figure 3.17: The unique model representation for HQO control theory. •>• z Figure 3.18: Standard form for the control problem. Chapter 3. Controller 36 Section A. l has more augmented plant details. The basic idea for control theory is to minimize the oo-norm for the closed-loop transfer function between z and and w [25], [23]. More specifically, design the system so that |r^(a)||oo < i (3.27) where the transfer function is W2R{s) W3T(s) (3.28) and S(s) = (I + GKis))-1 R(s) = K(s)(I + GK(s)y1 T(s) = GK(s)(I + G/iT(s)) _ 1 . (3.29) (3.30) (3.31) S(s) is known as the Sensitivity function, T(s) is the Complementary Sensitivity function and / is the identity matrix of appropriate dimensions. Using this, a successful Hoo controller has \W^(s)\>a[S{j<o)] (3.32) \W^(a)\>&[R{jw)] (3.33) \W^(s)\ > a[T(jw)} (3.34) where iWf^jiw)! is.the desired disturbance attenuation factor. The definitions Wi(a) = Ws(s) (3.35) W2{s) = WR{s) (3.36) W3{s) •-= WT{s) . (3.37) Chapter 3. Controller 37 w K(s) u G(s) Figure 3.19: The position of the weights in the classic feedback model, are used interchangably. 3.2.2 Weights How the choice of weights affects the performance of the system can be examined two ways: shaping the singular values of the various transfer functions defining the system using (3.32) to (3.34); or, by looking at Figure 3.19, Wi(s) shapes the error, W2(s) shapes the control signal, and W^s) shapes the measured output. 3.2.3 Restrictions There are certain restrictions in design that must be accounted for else the theory will not work [25], [2]. First the plant must be stabilizable and detectable. This is further explained in Section A.2. The next restriction on design is that -D\2 and D2\ must be full rank. Examining the rank requirements, the relationships between D\2 and D2\ and the system can be seen by writing the equations z = C\x + D\\w + Di2u, (3.38) Chapter 3. Controller 38 y = C2x + D2lw + D22u. (3.39) In words, there must be some direct transmission from u to z, and from w to y. The rank of Di2 can be made full by making the control input a new output at z, but weighted so heavily by a constant it becomes insignificant. Another option is to make the polynomial WsG(s) strictly proper. To make the rank of D2i full, w must have direct transmission to y. Another requirement deals specifically with the choice of weights. W\(jw) and Ws{jw). must be chosen in such a way that '^[W^iJw^ + aiW-^jw)] > 1, yW (3.40) so (3.32) and (3.34) can be achieved [25, page 1-40], [24]. The final restriction is the existence of poles or zeroes of the plant on the jw-axis: this is not allowed. As a result, in order to develop a controller for a plant with illegal poles and zeroes, it is required that the plant be transformed in such a way that the infringing poles and zeroes positions are changed. Two methods of doing this are axis shifting and bilinear transform (see Section A.3). 3.2.4 7-Optimization The solution with theory is not necessarily optimal - it may be possible to get a better solution using the same shape weights. Or, it is possible, and really quite likely, that the initial choice of weights will not yield a solution to the design problem. In these cases, a 7-iteration technique can be employed [25]. This is the equivalent of doing multiple designs with varying weights. 7 has the initial value of 1, and the weights are adjusted such that the new design is run with 7^1(5), 7^2(5) and 7^3(5) , For each successful solution, 7 can be increased. Each unsuccessful Chapter 3. Controller 39 attempt would require 7 to be decreased. In many cases, a binary search is the best vehicle to determine the optimal value of 7. Of course, it is not necessary to multiply each weight by 7 - perhaps due to some tough closed-loop specifications certain weights cannot be altered. Then only the weights that are more flexible should be changed. 3.2.5 SISO Simplification In Single Input Single Output systems, such as a one degree of freedom model, certain simplifications in set up and theory can be done. Given the standard feedback transfer function, the closed-loop responses of the system are s^=TTGTiT) <3-41> ' TU\ _GK[s)_ • T { S ) = 1 + GK{s) • ( 3 - 4 2 ) Interpretation of these functions is easy for a single-input single-output system, and design therefore becomes easy as well. Getting right back, to basics (and leaving out the Laplace operator for simplicity), S - 1 b ~ 1+GK~' GK T 1 + GK G K (3.43) 1 GK < 1 1 GK > 1 (3.44) GK GK < 1 which means to get a flat response of unity gain, \GK\ should be larger than 1, and the larger the value, the closer \T\ will be to unity. Also, the sensitivity of the system is for large \GK\. Chapter 3. Controller 40 Of perhaps of more import in the SISO case are the following relations: when \GK\ > 1, GK «. (1 + GK) = Wx when \GK\ < 1, GK w GK(l + GK)-1 = W3 - l (3.45) The implications of this are apparent when deciding on weights for Hoo design. • Unity gain implies \GK\ > 1. It can be calculated that if \GK\ = 25 dB, \T\ « - \ • When \GK\ ^> 1, as is desired in the bandwidth, GK = W\, and T '« 1. • The bandwidth of W 3 _ 1 dictates the bandwidth of T. • The slope of W^1 dictates the slope of T after the roll off, since when \GK\ <C 1, as is the case out of the bandwidth, T = GK = W 3 X . Of course, a design that looks only at magnitudes (the train of thought followed here) could cause resonant peaks in the closed-loop transfer function. 3.3 Position Controller 3.3.1 Weight Selection The augmented plant is shown as Figure 3.20 and indicates the signals to which each weight applies. W/(s), as explained previously, is small and exists only to make the rank of D12 full and is not necessary if the polynomial WTG(S) is strictly proper. In this model, it is possible to achieve this with a non-proper WT{S) with order 2. Otherwise Wj(s) would be necessary. As noted in a previous section, the weights affect the open loop system GK(s) in a direct fashion. W~T(S) controls the bandwidth of the open loop plant, and as such , dB. Chapter 3. H^, Controller Au gme n t e d _ P1 ant_ P(s) W s(s] Wf (s] o u K(s) Figure 3.20: Augmented Plant. Chapter 3. Controller 42 is to have unity gain at the desired bandwidth of the system and roll off at the desired rate for the system. Ws(s) dictates the response below the cut off frequency, and should be low pass with greater than unity gain for the closed-loop system to approach unity gain at low frequency. One set of possible weights are defined as WT(s) = — (3.47) (3.48) where f3 is the desired DC gain of the open loop system, a is the desired HF gain of the open loop system, wc is the desired bandwidth of the system in rad/s, ( is the damping, and 20m dB/decade is the roll off of the system. These weights are only two of a set. Other possibilities include changing Ws to fourth order, or increasing the roll off rate of the system. It is necessary to note, however, that if m > 2 is desired, = ( f ) ' ( r z ^ j - r ) " " < 3 - « ) \ wc/ \ 1 0 0 0 w c ' / so that the polynomial WTG(S) remains proper. 3.3.2 Simple Plant In its most basic form, the LMIM platform is a simple double integrator plant and therefore the model for design will be as shown in Figure 3.21. The augmented plant is required in state space form and can be calculated for this mixed sensitivity approach using the MATLAB function augtf. If WTG(S) is strictly Chapter 3. Controller 43 x r e f O x e r r K ( s ] x ms Figure 3.21: Model for Hoc design - simple double integrator plant, proper, the matrices can be worked out to be A B1 B2 cx A i D12 C2 A i D22 AG 0 0 0 BG —BWSCG Aws 0 BWs —BWSDG 0 0 AwTG 0 BwTG —DWSCG Cws 0 D W S —DGDWS 0 0 CwTG 0 D\vTG - C G .0 0 1 - D G (3.50) where (AG,~BG, CG, DQ) is the state space representation of the simple plant, G(s), such that G(s) = DG + CG{SI - AG)BG (3.51) {AwSi Bws, CwSi Dws) is the state space representation of the sensitivity weight Ws(s), and (AWTG\ BWTG, CWTG, DWTG) is the state space representation of the cascade function WTG(s). The resulting augmented plant' has both jw-axis poles and zeroes so a bilinear trans-form technique (Section A.3) is employed to allow Hoo design. In this case, the dominant pole placement was chosen to be at -0.1 to start. With the weight variables chosen as wc = 2TT1.0 m = 2 Chapter 3. Hrx, Controller 44 P = 15 a = 0 C = 0.7 a controller, «, was designed. Its particulars are shown as Figure 3.22, showing the bandwidth of the closed-loop system to be 0.9 Hz. A second order roll off was desired because the plant has a "natural" slope of -40 dB/decade and this choice will simplify the controller response and design. It is also theoretically possible to remove controller poles and zeroes well past the bandwidth of the system without adversely affecting the system response. This idea may be used for reducing high state controllers for complex plants. 3.3.3 Destabilizing Feedback Using the plant • . G(s) = — * r n n (3.52) v ' • ms2 - 500 K ; and,with the weights as defined previously and parameters wc = 27r3.0 ro = 2 P = 60 a = 0 C = 0.7 a controller can be designed. It will be designated «y. The specifics of the controller are shown in Figure 3.23, where a large resonant peak is evident in the closed-loop transfer function and the bandwidth is 3.1 Hz. The sensitivity weight is the limiting factor in this design, as it can be seen that the complementary sensitivity weight is not approached. Chapter 3. Controller 45 Controller Controller 601 50 S 4 0 30 20 L 10" 1 0 Frequency [Hz] Sensitivity and Weight -60 10" 10u Frequency [Hz] 10"' 10u Frequency [Hz] Complementary Sensitivity and Weight 201 — ~ 0 CD CD 1-20 'c CO ^-40 -60 10 -2 10° Frequency [Hz] Figure 3.22: controller. (a,b) Frequency response of controller, (c) Sensitivity of the system with the sensitivity weight superimposed, (d) Closed-loop response of the system with the complementary sensitivity weight superimposed. Chapter 3. Hr*, Controller Controller Controller 10 10 Frequency [Hz] Sensitivity and Weight 10 10" Frequency [Hz] Complementary Sensitivity and Weight 20, ^ 0 m T3 o "§-20 •*—* 'c CD CO 2 - 4 0 10" Frequency [Hz] -60 10 , 1 I ! 1 \ \ \ ' ' \ A \ \ \ \ \ \ - ITf -n/wt 10" Frequency [Hz] Figure 3.23: Characteristics of the designed system with unstable force feedback. Chapter 3. Controller 47 3.3.4 M A T L A B Code A design tool was developed to aid in producing controllers with different plants.. The available plant options are double integrator, positive feedback, and Y(s). The weights are as defined in this section and need variables / c , m, /3, and a. Summary plots are drawn as well. The code can be found in Appendix C. 3.3.5 Implementation Implementation of controller K on the LMIM reveals that the controller fails by allowing the flotor to fall to the edge of the workspace, which indicates that the bandwidth of the system is not high enough. Simulation with plant Y(s) gives a similar situation which indicates designing a controller with the simple plant is limited in application. There is obviously an increase in the force required as the flotor moves to the limits of the workspace. This corresponds to a negative spring constant for the system which makes it unstable. Implementing controller Ky on the LMIM platform was successful. The flotor centered quickly and a spectral analysis of the motion was done to compare to the theoretical transfer function. This comparison is shown in the first plot of Figure 3.24 and shows good agreement. The difference is a result of improper.modeling, including inaccurate modeling of the feedback and lack of modeling the filters for analog and digital conversion. The size of H is a variable, and its affect on the system was examined. Increasing the value would just make the system stiffer, so the value was lowered as much as possible. As H decreased, it was easier to design controllers - a wider range was possible. At first, the bandwidth of 3.11 Hz was preserved for comparison to the case where H = 500. Good characteristics, defined as smooth and timely convergence to the center plus no drift, were observed in controllers designed down to H = 40. At this value, the centering Chapter 3. HQO Controller 48 20 Comparision of Theory and Data i 1 1 1 1 — i — p ~ m Oh : o "§-20 c D) CO -40 -60 Data Theory 10 -2 10" 10" Frequency [Hz] Frequency [Hz] 10' Figure 3.24: A comparison of closed-loop transfer functions of disturbance input to flotor position in theory and practice, (a) H = 500 and fc = 3.113 Hz, (b) H = 40 and fc — 1.3 Hz. Chapter 3. Hoc Controller 49 motion was initially slightly oscillatory until the center was found. The resonant peak for the closed-loop system also decreased as H decreased, to less than 4 dB. 3.3.6 Impulse Response One measure of performance to a disturbance is the closed-loop systems response to an impulse function. To effect this input in practice, the flotor was suddenly displaced from center and the result studied. Similar results were found when simulated with the plant Y{s). In the case of the controllers, after the displacing force was removed, the flotor returned quickly to the center of the workspace. So quickly, in fact, that overshoot resulted and a short period of oscillations followed. In real terms, this means that while the flotor is safe from falling to the edge of the workspace, the quickness of its zero return causes extra, and unnecessary, accelerations. 3.3.7 The Sensitivity Problem The sensitivity of a system is bounded by the constraint that the area bounded by the graph of liSfjitf)! on a log scale as a function of w on a linear scale be greater or equal to zero [24]: poo / log\S(jw)\dw > 0 (3.53) Jo The sensitivity function of the system with K with these plot axes is shown as Figure 3.25. Obviously the lower the frequency range of attenuation, the less positive area must be made up. But for a higher bandwidth, more positive area is necessary - and hard to get. The solutions are a collection of trade offs. The sensitivity is mathematically linked to the open loop function, and the sensitivity can only be greater than one when |GA"(5)| is near one as well, and with parts of the Chapter 3. Hrx, Controller 50 Figure 3.25: Plot to evaluate the area of the sensitivity function. Chapter 3. Controller 51 complex number negative. One solution would be to keep |G-rY(s)| near one for a longer range and this could be achieved by having a slower roll off at the bandwidth. This increases the chance for positive \S(jw)\, but at the same time, if |GiC(s)| is decreasing slower, then more negative ^(jiw)! accumulates. Another solution would be to have the angle of GK(s) be closer to —180° near the crossover so that the |G/f(s) | counts more to the positive area. This has the negative effect of lowering the robustness and stability of the system. Yet another thought is to have smaller \GK(s)\ at lower frequency. This would be the most direct way to decrease the negative area, and also decreases the closed-loop tracking at low frequency, but reduces disturbance rejection. 3.4 Acceleration Feedback It is not possible to design controllers with arbitrary weight characteristics. As before, trying to design the system for bandwidth much greater than 3 Hz causes problems with the sensitivity. Also, the low frequency seems bounded as well by near 0.3 Hz. Not only does this cause problems when designing but the resulting closed-loop system has a bad transfer function (large DC gain and large resonant peak). If only position feedback was used this large resonant peak could cause a problem as it is not going to disappear. However, additional data is available to use for control: the accelerations of the flotor. It may be possible to remove the resonant peak while decreasing the bandwidth of the closed-loop position system. Chapter 3. Controller 52 3.4.1 Controller Design Considerations Now that a stabilizing position controller has been achieved it is time to see if the response of the system can be improved using the acceleration data that is available. A new plant can be defined that includes the closed-loop position system and will be the plant to be controlled. This new plant, Gax is shown in Figure 3.26 in relation to the other important blocks in the design procedure, and a sample Bode plot is shown in Figure 3.27. Gax(s) can be manipulated so that its transfer function is Gax(s) (3.54) ms2 - H + Kx(s) where Kx(s) is the position controller designed previously and all other variables are as before. Wf •= e for rank requirements, and only if WTG(S) is not strictly proper. If both WT(S) and G(s) are proper (NB. Gax(s) is strictly proper and therefore WT(S) must be proper) and there are no pole or zero cancellations, the simplest augmented plant can be determined to be AGAX 0 0 0 Boax BwTCaax AwT 0 0 BwTDGax A Br B2 ' B W s C G a x 0 Aws Bws BwsDGax Ci D n D12 = 0 0 0 0 e C2 D21 D22 DwTCGax PwT 0 0 DGaxDwT DwsCGax 0 C W s •D-ws D G a x D W s GGOX 0 0 1 Daax (3.55) where the state space matrices are defined similar to as before. The desired acceleration function is one with decreased sensitivity to noise in a fre-quency range with tracking of inputs in this range. This bandwidth cannot extend to DC, but should include the cut off of the closed-loop position plant. The general shape Chapter 3. Controller Figure 3.26: The acceleration plant with an inner closed-loop position feedback. Chapter 3. Controller 54 -50 CD •o c CO O -100 -150 -135 -180 CD T3 -225 CD CO CO . C -270 D_ -315 -360 10" 10u 10' 10" 10" 10' Figure 3.27: Bode plot for acceleration plant Gax(s). of the desired weights is shown in Figure 3.28, where can be seen that the sensitivity weight is large.in the bandwidth to keep.the sensitivity small, and the complementary sensitivity weight shapes the closed-loop response outside of the bandwidth. A possible set of weights with second order slope is Ws{s) WT(s) ((^)2 + 2C^ + l)((^)2 + 2C-^  + l) • ( ^ — + 1)4 (3.56) (3.57) (21^ + 0.01)2(To^ + l ) 2 -where wc\ is the cut in of the acceleration controller, wC2 is the cut off, ( is a damping ratio, and B is a gain that, brings the magnitude of Ws(s) to one at the crossover. Although these weights only dictate second order roll off, they have an order 4. So increasing the desired roll off greatly complicates the weights. The weights are also strictly proper, since the plant is strictly proper. Assuming successful design of an acceleration controller, the new transfer function Chapter 3. Controller 55 Acceleration Weights Figure 3.28: General shape of the weights for acceleration controller design. from disturbance to position becomes x 7 1 + -^KJs) - -±?H - ±KJs) GxKx(s) 1 + GxKx{s) - GJ<a(s) based on Figure 3.29, and where 1 (3.58) (3.59) Gx(s) = Ga(s) = H H (3.60) (3.61) By examining this transfer function, certain things can be inferred about the behavior of the closed-loop system. For low frequencies the system behaves as the position only design since \GxKx(s)\ ^> \GaKa(s)\. The new system continues to follow GxKx(s) until the order of the magnitudes approach. At this time the phases of the functions come into play. If the phase difference is very large while \GxKx(s)\ ~ \GaKa(s)[ then the closed-loop response will show a resonant peak. As |G0isTa(5)| continues to increase, | | | begins Chapter 3. i/oo Controller 56 K a ( s ) k K x ( s ) . + — 1 m x H Figure 3.29: The complete position plant with acceleration feedback. to decrease and is equal to the ratio of \GxKx{s)\l\GaKa{s)\. This is how the system bandwidth is decreased, so the value of this new cut off is dependent on the frequency where \GxKx(s)\ « \GaKa{s)\ > 1, and in theory could be placed anywhere with the proper choice of controller. As \GaKa(s)\ decreases, the system again approaches the simple position only closed-loop function. Note this will also eliminate the resonant peak of the position only transfer function. A better indication for design parameters comes with a little manipulation of (3.58) (see Appendix B for more closed-loop manipulations): --^KJs) ms* V / • Txa(s) = 1 -(- -^KJs) - -^H - -t-KJs) ' ms z x \ I ms* m " \ / Kjf) m s 2 + K X - H - s2Ka{s) KJj) (rns* + Kx(s) - H)(l - m/_K4l(s)) Kx(s) ms2-H 1 1 + J¥ £W \l-GaxKa{s)t ms* —H / x v ' ' GxKx(s) \( 1 \l + GxKx(s)J\l-GaxKa{s)i (3.62) where Txa(s).is the closed-loop position function with acceleration feedback, Tx(s) is the Chapter 3. Controller 57 closed-loop position function without acceleration feedback, and Sax(s) is the sensitivity of the acceleration system with position controller. The last definition also gives Ws(s), since attempts to near equalize S(s) and wl(s) • Ws(s) should be chosen to reflect the desired new response of the system, and WT(S) should be chosen to complement this. Knowing the desired sensitivity, choose Ws(s) such that \Wg 1(s)\ > IS^5)!- The Bode plot of Gax(s) shows that it increases at 40 dB/decade at low frequency and levels off to a gain less than one. In an attempt to keep the controller simple, WT(S) was picked to match the controller roll up at low frequency, and to have a 20 dB/decade slope through the cut off (for robustness and so that the controller doesn't have to deviate so much from Gax(s)). WT(S) also has a leveling term so that it is a proper function. These simple weights have the same shape as (3.56) and (3.57), confirming the previous selection. 3.4.2 Acceleration Controller A controller that should in theory work can be found by choosing the acceleration weights to be as in (3.56) and (3.57) | m/2 Ws(s) [( J_)2 + 2£-*_ + 1][(^)2 + + 1] (3.63) W2(s) = e (3.64) where wc\ and wc2 are the low and high frequency ranges for the bandpass, C i , c 2 are frequencies a little smaller than the cut in and a little larger than the cut off respectively (done so that |Ws(.7w')|_1 + IWTO'W)!-1 > 1), and m is the desired slope' of W\. By setting the bandpass range to be 0.05 rad/s to 20 rad/s, and choosing a simple second order slope for Ws with unity damping, the weights are s4 +40.1s3 + 404s2 +40.15 + 1 , , S [ S ) 1.63s4 + 6.54s3 + 9.81s2 + 6.54s+ 1.63 [ ' ' Chapter 3. Hoo Controller 58 W2(s) = 10~5 (3.67) „ . , A 1.09s4 + 4.25s3 + 6.25s2 + 4.08s + 1 ^ T ( S ) = 3.13s3 + 625s2 + 0.5s + 10~* (3"68) The new position transfer function is shown in Figure 3.30 and it indicates a small resonant peak of less than 1 dB exists at the new cut off of approximately 0.4 rad/s. The behavior of the transfer function is easily explained. The new function can be represented as Xref 1 + GXKX — GaKa where subscript x refers to position design and subscript a refers to acceleration design. The peak is a result of the phase characteristics of GaKa. As | G a . / f a | approaches I G ^ i ^ l , the phases of the functions must not be opposite, or near opposite, else the peak is created. Another thing to notice about Figure 3.30 is that the magnitude only drops to about — 10 dB for the first while. This is because ICTa/fal is not much larger than I G ^ i ^ l , and when the two magnitudes are similar, mathematically fa \ = — 6 dB. To effect a larger decrease in magnitude would either require making |GXKX\ smaller, which would affect the DC and low frequency gain of the system, or by making ICroifol larger, which would require making the order of Wia larger. Making the order larger, however, dramat-ically increases the resonant peak, which can be somewhat reduced by choosing larger damping ratios. Of course, larger order also means a larger controller order. Technically, the roll off can be placed anywhere with the proper choice of bandwidth, but practically there would be limitations. If I C T J ; / ^ ! is around 25 dB, it takes about 1/2 a decade for GaKa to get that large, so that is when the new roll off starts. For example, if a desired new roll off was 0.05 rad/s, we choose the low frequency bandpass for acceleration to be 0.01 rad/s and after some tuning we get Figure 3.31. Chapter 3. HQO Controller 59 Position TF with Acceleration Feedback 10"3 10~2 10"1 10° 101 102 Frequency [rad/s] Figure 3.30: A position transfer function with acceleration feedback. Chapter 3. Controller 60 Figure 3.31: An attempt to get the new corner frequency to 0.05 rad/s. The parameters used in the design: Position - DC = 40, HF = 0.01, slope = -2, bandwidth = 10 rad/s. Acceleration - frequency range 0.01 rad/s to 50 rad/s, damping ratio = 10, order = 4. Note that this controller has 19 states. Chapter 3. Hoo Controller 61 The important design plots are those of GaKa and GXKX. Proper manipulation of these will give the desired design. Looking at the plots can also give a good indication of the resulting system before it is even designed. 3.4.3 Force Rejection The purpose in adding acceleration force feedback was two fold. First the position roll off was to be pushed to a lower frequency, and that was accomplished earlier. Second, a better response to the disturbance force was expected. This has yet to be confirmed. The hope is that a disturbance force, applied just before the plant, has a lesser effect on the plant acceleration with acceleration feedback than without. To test this, the controllers designed leading to Figure 3.30 were used. The effect of the disturbance force can be seen in Figure 3.32 and shows a better rejection function with feedback than without. In fact, disturbance rejection can also be controlled by I G ^ i ^ l , where in a higher magnitude over a wider range decreases the disturbance effect accordingly. 3.4.4 Acceleration Controller Implementation No designed acceleration controllers stabilized the LMIM platform. The LMIM is an inverted pendulum and needs a higher bandwidth controller to be stable. 3.5 Conclusions Hoc design theory was able to successfully design a position controller but not an imple-mentable acceleration controller. Hoc design for the single degree of freedom case is very straight forward to set up since the weights have direct relevance to the open loop, and hence closed-loop system. However, even if the problem is set up properly, a solution does not always exist. Chapter 3. Controller 62 Disturbance Force Effect 10"2 10"1 10° 101 102 103 Frequency [rad/s] Figure 3.32: The effect of a disturbance force added just before the plant on the plant acceleration. Acceleration feedback decreases the effect. Chapter 3. Hoo Controller 63 This chapter provided a lot of insight into selecting design weights specifically for a single input single output system. The sensitivity function desired dictates the shape of Ws for the performance specification. Wj limits the bandwidth of the closed-loop system. Using this information, suitable weights can be formulated for control design. For this case the plant created complications in the controller solution that were difficult to overcome. The j^-axis poles of the simple plant and the right half plane pole [20] of the more accurate plant hinder finding solutions to all problems. Even with this problem, the Hoo controller designed showed slightly better perfor-mance than PID controllers. If the problem can be addressed satisfactorily then the use of Hoo design will provide good vibration isolation controllers. Chapter 4. The Coarse-Fine System This chapter describes the coarse-fine system that will be tested on the DC-9. A de-scription of the hardware and software is followed by the development of a model for simulation. 4.1 Hardware and Software 4.1.1 Hardware The hardware for the one degree of freedom Coarse-Fine Tracking System itself consists of two main parts: the fine stage wrist and the'coarse stage rail. This system was designed and developed by Tim Salcudean and Chia-Tung Chen at the University of British Columbia [29, 30, 31]. A simple representation is shown in Figure 4.33. The wrist is a similar and smaller version of the LMIM, described previously, and operates in the same fashion. The wrist also has a flotor that is suspended above the stator by magnetic field forces, but instead the flotor contains the permanent magnets and the stator holds the wire coils. The position of the flotor with respect to the stator is determined by position sensing diodes, and the acceleration of the flotor is determined by an accelerometer mounted externally. The flotor workspace is limited, and in the vertical direction the maximum displacement from centre is ±4.5 mm. The stator is attached to a plate that is in turn bolted to a linear bearing car. The acceleration level of the stator is determined by an accelerometer attached to the plate. The flotor is the fine stage. The 64 Chapter 4. The Coarse-Fine System M o t o r S l i d i n g C a r T i m i n g B e l t f l o t o r s t a t o r Figure 4.33: Diagram of the 1 DOF coarse-fine tracker. Chapter 4. The Coarse-Fine System 66 plate, stator, accelerometers, and linear bearing car are collectively the coarse stage. The coarse stage is set on a linear bearing rail with low friction bearings. This rail is 1600 mm long and has a small shock damper at each end to soften any impact on the wrist. The rail is attached to a 1915 mm long aluminum I-beam that is clamped vertically to a camera pole in the DC-9. The coarse stage is also attached to a timing belt on one side of the I-beam that is wrapped tightly around two pulleys, one of which is attached to a DC motor. The velocity of the timing belt is reduced by viscous forces of the bearing system and acceleration limits of the motor. The other side of the I-beam has a position transducer and the magnet is attached to and moves with the coarse stage. The approximate motion limit of the coarse stage on the rail is ±630 mm. The wrist gets power from a larger six channel current driver which is in turn powered by a 25 Volt DC supply, as is the motor current driver. All signals are filtered and connected to the computer by cables. The signals are interfaced with a computer through the digital/analog converter on the DSP board. The output signals from the computer are routed to the two current amplifiers, and the input signals are from the wrist PSD's, the coarse transducer, and the accelerometers. 4.1.2 Software The software used to move control code onto the DSP board is one developed at UBC by Nelson Ho called IN. Separate data storing software is necessary to write the variables to the computer hard.drive; this was developed at UBC by Chia-Tung Chen. The wrist is controlled by a simple PID controller, but is certainly not restricted to this. The method . described previously in the thesis could be adapted for the wrist with a few changes in the variables. Appendix D gives some particulars for the software used to control the coarse-fine Chapter 4. The Coarse-Fine System 67 Figure 4.34: Freebody diagram of the coarse fine coupled system. system. The variables that will be used most often for varying the controllers are kv-crse, kp-crs,eJso, / „ , and (. Another variable of interest is ORIENTATION, which can be VERTICAL, HORIZONTAL, or SPACE, depending on the current application. By changing ORIENTATION, variables are initially set to allow the flotor to be levitated. 4.2 Model Development To develop the model, the freebody diagram in Figure 4.34 will be referenced. It indicates the variables and coordinate systems of interest for model development. All units are S.I., and displacements are measured in the inertial frame. The coordinate systems are defined as follows: Oi - inertial reference oc - centre of the I-beam Chapter 4. The Coarse-Fine System 68 op - position of the pulley, linearly os - position of the stator oj - position of the flotor Using these coordinate systems, several variables are defined. Xpsd — Of — os : value of the position sensing diodes Xf = Of — Oi : position of flotor from inertial reference xs = os — oc : value of the position transducer xa = oc — 0{ : error displacement of the airplane rO = op — oc : displacement of the timing belt at the pulley Since Oi is assigned the inertial reference, Oi = 0 OQ — xa Op = r9 + xa Os — T Of = Xf •Kpsd — f Using Figure 4.34, the freebody equations of motion can be obtained to develop a simulation model and system transfer functions. The flotor is a mass suspended above the stator by magnetic forces that effect a spring-damper connection. The force applied to the flotor is Ffi = -[b{6f - 6S) + k(o} - os)} (4.70) Chapter 4. The Coarse-Fine System 69 where b and k are the effective damping and spring constants obtained when the flotor is stabilized. The motion of the flotor is given by mof = Fji = -b(6f - 6S) - k(oj - os) (4-71) mxf = -[b(xj - xs — xa) + k(xf - xs - xa)] (4-72) where m is the mass of the flotor. In the Laplace domain, (k + bs + ms2)x} = (bs + k)(xs + xa) (4-73) •A{s)xf = D(s)(xa + xa). (4.74) The main motion of the stator is produced by the motor through the timing belt. The forces acting on the stator are the reaction force from the flotor, any disturbance forces to the system, the sliding friction of the linear bearing car, and the force applied by the timing belt. The force of the timing belt is given by Ftb = B(6P - 6S) + K(op - os) = B(r0 - £ , ) + -K(r6 - xs) where B and K identify the mode of the timing belt. The sliding friction is Fst = BS(6S — 6C) < y ' . (4.76) — B sx s where Bs is the sliding friction constant. The sum of the forces on the stator is Fst = -Ff, + fd- FsJ + Ftb • (4.77) which yields the equation of motion Mxs = [b(if - xs - xa) + k{xj - xs - xa)] + fd -B3xs + [B(r6 - x.) + K{r0 - xs)] (4.78) Chapter 4. The Coarse-Fine System 70 where M is the mass of the stator. In the Laplace domain, the equation becomes xs[Ms2 + (b + Bs + B)s + {k + K)] = fd + (xf- xa)(bs + k) + r§{Bs + K) (4.79) G(s)xs = fd + D(s)(x}-xa) + N{s)rd.. (4.80) The torques acting on the pulley are the torque of the motor and the reaction torque of the timing belt. The torque of the motor is characterized by the equation Ti = va-bJ (4.81) where va is the input voltage, Im is the motor inertia, and bm is the back emf. The torque of the motor-pulley system is Tm = Ti — rFtb (4.82) and the equation of motion is Ij = V a - bj - r[B{rO - x„) + K{r0 - xs)} . (4.83) The equation becomes ' . • 0[Ims2 + (bm + r2B)s + r2K] = va + xsr(Bs + K) (4.84) P(s)6 = va + rN(s)xs (4.85) in the Laplace domain. The system variables need to be estimated for analysis and simulation. For the flotor, the values that will be used during micro-gravity are m = 0.88 Chapter 4. The Coarse-Fine System 71 wf = 2?r0.1 Pf = 0.7 J k = w2 b = 2pjwj which is a soft coupling to the stator. The values for the stator and timing belt are M = 10.0 ws = 27r60 ps = 0.5 K = w2s B = 2psws Bs = 5 with the frequency of the timing belt quite high to represent that the timing belt does not slip over the pulley. The motor system variables are Im= 0.05 bm = 0.3 r = 0.0025 where the back emf was estimated from horizontal test runs. 4.3 Open Loop System Analysis For stability analysis of the system, the notation defined previously as A(s) = ms2 + bs + k D(s) = bs + k G{s) = Ms2 + (BS + B + b)s + (K + k) N(s) = Bs + K P(s) = ImS2 + \r2B + bm)s + r2K (4.86a) (4.86b) (4.86c) (4.86d) (4.86e) Chapter 4. The Coarse-Fine System 72 can be used to develop open loop transfer functions for the system, these, the transfer functions can be derived.as 1 Manipulating x} APG(s) - r2N2A(s) - PD2{s) 1 \{D - A)PD{s)xa + AP{s)fd + rNA(s)va] (4.87) APG{s) - r2N2A(s) - PD2{s) Letting [(PG-PD-r2N2)D(s)xa + DP(s)fd + rND(s)va]. (4.88) (4.89) E(s) = APG{s) - r2N2A{s) - PD2{s) the open loop equations can be put into matrix form as AP rNA PD{D - A) DP rND D{PG -PD^ r2N2) -X S 1 ~ E XQ, (4.90) The stability of the open loop system will be examined using Nyquist criteria and plots. There are no right half plane poles in the system. The plots from the inputs va and fd show stability over a very large gain margin. The plots from the input xa are shown in Figure 4.35. These plots indicate that the flotor is stable for all gain values and the stator has a gain margin of 1250. Thus, the system is anticipated to be stable for all inputs. 4.4 Closed Loop System Analysis For closed loop analysis, the controller in the computer must be defined. The stator is to follow the flotor's displacement from the centre of the workspace, given by the signal xpsd. To ensure the stator doesn't gain too much speed and overshoot the centre position, velocity damping will be included as well. The output of the computer will be defined as VA — Kp^psd KVXS (4.91) Chapter 4. The Coarse-Fine System 73 .x 10" xs/xa Open Loop xf/xa Open Loop Figure 4.35: Nyquist plots showing open loop stability of the system to airplane vibra-tions. where Kp and Kv are the proportional and velocity constants respectively. Since the value of XpSd is in the millimetre range, the proportional constant will be quite large. Putting the voltage into the open loop equations (4.87) and (4.88), the closed loop system transfer functions can be derived in the Laplace domain as. 1 Xf E + rNKp(A -D) + rNAKvs 1 [APfd + (rNKp + PD)(D - A)xa] (4.92) E + rNKP(A - D)T7NAl5[DPf< + ° { P G ~ P D " ^  + ^ where E is E(s) as defined previously. The frequency response of these equations to airplane motion xa for Kp — 1200 and Kv = 15 is shown in Figure 4.36. The first plot shows that the stator cancels the airplane motion at low frequency and does not respond to high frequency motion. There is a resonant peak in the response of small magnitude that should not affect the system too greatly. The reduction at very low frequency is the Chapter 4. The Coarse-Fine System 74 xs/xa Closed Loop xf/xa Closed Loop Frequency (rad/sec) Frequency (rad/sec) a) b) ^ ' Xa I x a Figure 4.36: Bode plots showing closed loop response of the system to airplane vibrations. effect of the flotor on the system. As shown in the second plot, the flotor follows very low frequency airplane motions. This is consistent with the desired response of the system. 4.5 Simulink Simulations The equations of motion were developed into a Simulink system, which is shown in Figure 4.37. This model can be used to test the system against various expected input airplane vibrations. Note that in this model, the workspace limits of the flotor and stator have not been included. Figure 4.38 shows the results when the model is subjected to a sample of airplane acceleration data. The acceleration levels of the flotor are 100 to 1000 times smaller than the input acceleration and at a frequency of approximately 5 Hz. If this had been an actual test run, the flotor would have hit the workspace limits in under 3 seconds, and the stator exceeded its bounds in under 5 seconds. To stop the flotor from reaching it's limits, the stator has to move faster, though the stator would reach it's bounds sooner Chapter 4. The Coarse-Fine System 75 F tb x_psd x s v a C o m p u t e r Motor > x _ a > x s _ _ ^ Stator *3-: lo tor F J I x _ p s d a) Complete system in Simulink F tb T = F . R v a 3 T m 1/LrrT 1/1 m 1/s w _m| b m R s 1 r the ta r the ta r the ta x s b) Motor block expanded t Bs+K r t h e t a - x s T iming Belt 1 F tb x a x s 1 M 1/s Ma_s a s v s B F sf c) Stator block expanded o s x_psd= x f - o s - ( b s + k ) Restor ing Force x_psd 1/s x_s x_s ms* x f F f d) Flotor block expanded Figure 4.37: Model of the coarse-fine control system. Chapter 4. The Coarse-Fine System 76 as a result. Figure 4.39 shows the response of the system when the velocity damping of the stator, Kv, is reduced to zero. The flotor in this case stays within its workspace as the stator is able to move faster. The stator exceeds its workspace in under 4 seconds. Chapter 4. The Coarse-Fine System 77 System Acceleration Levels CO c g "co 0 8 -o.oi o < -0.02 2.2 2.4 Time [si a) Sample of acceleration levels of the airplane and flotor. System Positions Position Sensing Diodes 4 6 Time [s] b) PSD signal output. i o u o . in -1 1 r ^  _ ' v . ' "S. ' \ ' N. • • — xp - xs - -xf 4 6 Time [s] 10> c) Displacements of system elements Figure 4.38: Plots showing the simulation response to a sample vibration signal with Kp = 1200, Kv = 15. Chapter 4. The Coarse-Fine System 78 System Acceleration Levels 0.05 CO o^ E, c g ed i— 0> CD O O < -0.05 2 2.2 2.4 2.6 "TJ m o TeO a) Sample of acceleration levels of the airplane and flotor. System Positions x10 Position Sensing Diodes 4 6 Time [s] b) PSD signal output a) i o o in ' -1 -2 \ " — x a - xs - -xf 4 6 Time [s] 10 c) Displacements of system elements Figure 4.39: Plots showing the simulation response to a sample vibration signal with Kp = 1200, Kv = 0. Chapter 5 Micro-Gravity Testing This chapter explains briefly about the micro-gravity testing and presents the data from the DC-9 flight. Analysis of data, comparison to simulation predictions, and recommen-dations for future testing is also presented. 5.1 Installation and Testing The system was installed on NASA Lewis Research Center's DC-9 to be tested during micro-gravity flights. The I-beam was clamped to a vertical pole in the plane and long wires were used to connect the hardware to the computer, which was situated in a rack with the voltage and current supplies.' There, were some problems encountered while setting up the system. First, the pole to which the I-beam was attached was very flexible and would add increased vibrations to the system. Another problem was electrical. A ground reference error with the coarse current amplifier had to be fixed. The first flights were used to judge the response of the flotor in low gravity situations to get an intuitive feel for the bandwidth necessary for successful control. With the coarse stage strapped down and the flotor control bandwidth rather large, data from the system was collected. The signals stored were acceleration of plane, acceleration of flotor, and position of flotor (xpad). The next step was to release the coarse stage and observe the entire system. A typical micro-gravity run produced variables with the trends shown in Figure 5.40. The acceleration of the plane starts at -9.8 m/s/s and approaches zero. The acceleration 79 Chapter 5. Micro-Gravity Testing 80 Acceleration of Plane Position of Coarse Stage 10 Time [s] 10 Time [s] 20 Figure 5.40: Typical parabola cycle. remains around zero for a time (approximately 20 seconds) until the plane begins it's pull up phase. Then the acceleration becomes large negative again. The position of the coarse stage is at the bottom of the I-beam until the acceleration level becomes small enough, at which time the coarse stage is centred. This causes large vibrations and motion on the flotor. When the coarse stage is centred and the acceleration of the plane remains small enough, the coarse stage is controlled to follow the flotor. Control continues until the coarse position approaches the edge of the workspace, at which time the coarse stage is centred along the I-beam again, or until the acceleration level of the plane becomes too large, then the coarse stage is returned to the base of the I-beam. Several things were attempted during flights to control the system. The effect of a dead band switch on flotor position was investigated as well as different gains for the system. All controllers could not be tested due to a memory error with the DSP board. Chapter 5. Micro-Gravity Testing 81 The ambient weather conditions were rougher than normal during the testing on the DC-9. This caused larger amplitude vibrations and also increased the low frequency errors. As a result, the coarse stage reached the workspace limits quicker than would be expected under average conditions. 5.2 Results The data for the wrist only experiment (when the coarse stage was strapped down) was plotted and compared to simulation predictions. The comparisons are shown in Figure 5.41 and indicate very good agreement between theory and experiments for the wrist. The difference between actual and simulation data comes as a result of a steady state force on the flotor, which is set when the flotor position is reset to the centre of the workspace. Since each parabola has different acceleration characteristics, this steady state force is not correct for all instances, but provides a good starting force to near centre the wrist. Indeed, the first graph shows the simulation is a constant 0.2 mm off the actual data, which is the effect of an extra steady state force. The second set of data has a better centred wrist and the simulation and data are in good agreement. Some of the data collected when the coarse stage was controlled to follow the flotor is shown in Appendix E. The first three show control for the parameters /„ = 0.2, kp = 80 and kv = 1 with the coarse controller only in effect when the flotor position exceeds ±1 mm, a dead band. The position of the flotor vibrates greatly, due to the large coarse control signal constantly turning on and off as the flotor vibrates around the 1 mm mark. This was verified later on the model in simulation. The dead band was not included in the simulation model since its effect was underestimated. The dead band was then removed from the digital controller and tracking was at-tempted again. The next three graphs in the Appendix show the data collected for the Chapter 5. Micro-Gravity Testing 82 0.4 0.3h Comparison of Actual Data to Simulation E 0-2 E . | 0.1 CO £ 0 -0.1 -0.2 0 4 5 6 Time [s] 8 Actua Data Sinr 10 c o 0.8 0.6 0.4 0.2 0 8-0.2 -0.4 -0.6 -0.8 0 Simulation/ / : \ \ Actual Data 2 3 4 5 6 Time [s] 8 10 Figure 5.41: Comparison of actual data to simulation of wrist. In the top graph, /„ = 5, the lower fn = 2. Chapter 5. Micro-Gravity Testing 83 same values for the variables. Again the flotor position is oscillatory, but not as much as the previous attempt, and not always. At first it was thought that the problem was due to initial conditions the flotor acquired rapidly being centred on the I-beam. The velocity constant was increased tp limit the centering speed, but this did not improve the flotor response. The velocity constant was then set to zero and the response of the system improved greatly. However, modifications to the control code increased the size of the compiled file to beyond that available on the DSP board and memory allocation errors were occuring. As a result, the response of the system was inconsistent and the data was unreliable. Some data can be analyzed for trends but no concrete conclusions can be drawn. 5.3 Analysis Figure 5.42 shows plots of data collected during one parabola that the system behaved relatively as desired. The variables for this parabola were /„ = 0.1 and kp-crseJso = 50. At the start of the micro-gravity phase the acceleration of the flotor is much smaller than the acceleration of the plane. The magnet levitation system has removed the slow oscillations and attenuated by more than a factor of two the high frequency noise. The coarse stage has a slow downward drift, changing 200 mm in just over 7 seconds, and the flotor position is negative. This agrees with the acceleration of the plane which is more negative at the beginning of the parabola. After time approximately 12 seconds, the acceleration of the plane tends to be more positive and the flotor position confirms this by becoming positive as well. The coarse stage begins to move rapidly upwards following the flotor and finally hits the workspace limits. During this part of the parabola the flotor acceleration degrades and shows the action-reaction effect of the stator on the flotor, indicating that a lower bandwidth is still required for the system to have greater Chapter 5. Micro-Gravity Testing 84 Figure 5.42: Data plots for parabola number 50 on day 4. The flotor acceleration is considerably less than the plane acceleration while the flotor position is near centre. Chapter 5. Micro-Gravity Testing 85 effectiveness. The response of the system during micro-gravity parabola is in disagreement with the system model. There are several things that may contribute to this, both linear and non-linear considerations. The acceleration due to gravity, while small, is not zero and will cause the flotor and stator to have disturbance forces. For effective vibration isolation, a small Wf is necessary for the flotor, and the disturbance will have a larger effect on the system. The acceleration changes direction rapidly. This either increases or decreases the power necessary to move the coarse stage by the motor. For example, if the desired coarse motion is negative, and the force due to gravity is also negative, the static friction of the bearings is much easier to overcome. However, if the force due to gravity is positive, then a much larger signal is necessary to overcome both static friction and this positive force. This unmodelled force, which is a function of the acceleration of the plane, can be found and added to the model. The timing belt for the coarse stage also introduces unmodelled effects. The belt is stretched between two pulleys and has a small amount of elasticity and a large amount of flexibility. The belt has several vibration modes at low frequency that cause perturbations on the system, the most significant one at 7 Hz. This effect will cause oscillations in the coarse position when the coarse motion is not steady. There are non-linear effects to consider as well. The MAGLEV wrist, being a smaller version of it, will have similar concerns as the LMIM. This includes acting as an inverted pendulum due to the magnetic non-linearities. This was not included in the model since the effect was expected to be small. Also, the hard contact the flotor experiences as it hits the edge of its workspace has not been modelled, but is not important if xpsa< remains within acceptable limits. While the timing belt has known vibration nodes, these can not simply be modelled Chapter 5. Micro-Gravity Testing 86 as a spring-damper connection with a 7 Hz bandwidth - there is a maximum stretch the timing belt can experience. If no maximum stretch is considered and the timing belt frequency reduced, the system simulation becomes unstable as xs oscillates with increasing magnitude and does not follow rO, which it must. No effective way of modelling the maximum stretch was found. Another non-linear consideration is that of static friction of the linear bearing car on the rail, which is in addition to sliding friction. The static friction causes the linear bear-ing car to remain stationary until sufficient force is applied. When it does start to move, the friction force is reduced and the stator moves faster than necessary, overshooting the desired response. Then the stator stops briefly as the flotor catches up and passes the stator once again. The stator must once again overcome the static friction, and continues this jerky motion. Attempts to model these effects to verify their effect proved difficult, in particular the maximum stretch of the belt. A rough approximation of the non-linear timing belt with a lower vibration frequency was modelled and added to the simulation. As well, an approximation of the static friction was included in the new model. Figure 5.43 shows the comparison between actual data and simulation using the same airplane acceleration signal. While the simulation does not exactly match the observations on the plane, the behaviour of the simulation is similar to actual data; Therefore, using coarse control with a flexible timing belt and a sliding car produces undesired oscillations. 5.4 Recommendations for Future Study Recommendations for future work with the coarse-fine tracker to improve the system: • Use a different coarse motion system. Chapter 5. Micro-Gravity Testing 87 0 2 4 6 8 10 Time [s] Figure 5.43: Comparison of data and simulation with low frequency mode for the timing belt and static friction for the sliding car. Chapter 5. Micro-Gravity Testing 88 • To remove initial conditions of the flotor the stiffness should be quite high when the coarse stage is being centred. This way the flotor will converge to the centre of its workspace quickly instead of being pushed around at the beginning. • Use previous data to predict the initial acceleration levels and provide beneficial initial conditions to the flotor. For example, if the acceleration is drifting positive during the centering mode, a negative velocity could be added to the flotor to extend the length of time the system stays in the tracking mode, thus longer improved micro-gravity time. • The stiffness of the wrist can be increased as the flotor approaches the edge of the rattle space to give the flotor a centering velocity before hitting the edge. This becomes a tradeoff between hitting the edge of the rattle space and allowing greater transmission of vibrations. • The control output to the motor could be an increasing centering force as the coarse stage nears the edge of the I-beam. Again, it depends if a swift movement to the centre of the workspace, as is the case presently, is a better option than transmitting greater vibrations or forcing the flotor.to hit the edge of the rattle space. Chapter 6 Conclusions Magnetic levitation technology using Lorentz forces is a good method for isolating ap-plications from external vibrations and acceleration disturbances. The LMIM platform, designed under contract at UBC, and the smaller version known as the wrist use this theory to create a non-contact system for vibration isolation. This thesis looked at the LMIM and wrist and explained briefly how Lorentz theory has been applied to create the non-contact technology. An accurate model of the LMIM was developed for possible use in designing controllers. The identified plant had an unstable pole and confirmed the mass of the flotor to be approximately 22 kg. The LMIM system acts as an inverted pendulum and it was difficult to adequately control the system with this problem. Using the LMIM for testing, control techniques were employed to actively control the system. Several assumptions were made on the LMIM model to facilitate controller design. The Hoo control theory proved difficult to design with. Solutions to the control problem do not always exist which forces an iterative procedure to finding answers. The poles of the LMIM plant also contributed problems to the design of a successful controller, as Bode's Integral theorem must be satisfied. The effect of the weights on controller design was studied and weight selection insight was provided. Some successful position controllers were developed with bandwidth of the closed-loop system near 3 Hz that stabilized the LMIM. Further reduction of the bandwidth was not possible with 89 Chapter 6. Conclusions 90 the position controller. Adding acceleration feedback was straight-forward in theory, but design solutions were difficult. Acceleration feedback controller testing on the LMIM was unsuccessful. The controllers that worked in practice showed good disturbance attenuation for position control. There is one important point when adding acceleration feedback. With Hoo theory the solution does not always exist, but if one does it is shaped as expected and gives the desired closed-loop response. The wrist was used in a coarse-fine tracking approach to vibration isolation in a micro-gravity environment. A model of the wrist was developed and was in good agreement with actual responses to noise. Simple PID control was employed for the wrist with the coarse stage tracking the flotor position in a DC-9 performing parabolic flights. The data collected showed effects not included in the model of the coarse stage play a significant role in the system. The data also showed disturbance attenuation while tracking the flotor was possible but requires further testing. Bibliography J. Yan, "Design and Control of a Bilateral Motion-Scaling System Using Magnetic Levitation," Master's thesis, University of British Columbia, March 1994. J.C Doyle, K. Glover, P.P. Khargonekar, and B.A. Francis, "State-Space Solutions to Standard H2 and Control Problems," IEEE Trans. Automatic Control, vol. AC-34, pp. 831-847, 1989. J.M. Maciejowski, Multivariable Feedback Design. New York:Addison-Wesley, 1989. R.M. Alexander, C.H. Gerhold, C.B. Atwood, and J.F. Cordera, "Vibration Isola-tion in a Microgravity Environment," J. of Vibration and Acoustics, vol. 115, no. 4, pp. 477-483, Oct. 1993.. L.A. Sievers, and A.H. von Flotow, "Linear Control Design for Active Vibration Isolation of Narrow Band Disturbances," Proc. of IEEE Conf. on Decision and Control, vol. 2, pp. 1032-1037, 1988. N. Tanaka, Y. Kikushima, "Optimal Design of Active Vibration Isolation Systems," Trans, of ASME Journ. of Vibration, Acoustics, Stress, and Reliability in Design, vol. 110, no. l,pp.'42-48, 1988. N.G. Fitz-Coy, and.J.Jang, "Active Vibration Isolation of a Space Platform: A Game Theoretic Approach," Advances in,the Astronautical Sciences, vol. 85, pt. 3, pp. 2199-2110, 1993. 5. E. Salcudean; H. Davis, C T . Chen, D.E. Goertz, and B.V. Tryggvason, "Coarse-Fine Residual Gravity Cancellation System with Magnetic Levitation," Proc. of IEEE Conf. on Robotics and Automation, pp. 1641-1647, 1992. Z.J. Geng and L.S. Flaynes, "Six Degree-of-Freedom Active Vibration Control Using the Stewart Platforms," IEEE Trans, on Control Systems Tech., vol. 2, no. 1, pp. 45-53, 1994. S. Li and R. 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Appendix A HQO Theory Notes A . l Augmented Plant The augmented plant, -P(s), is the transfer matrix from w and u to z and y z y w u Pu(s) P12\s) Puis) Puis) w u z and Knowing u = K(s)y the transfer function between w is Tzw(s) - Pn(a) + Pi2K(s){I - P22K{s))-1 P21(s). The augmented plant is most often represented in state-space form X A Bi B2 X z — Ci Du D12 r C2 D21 D22 u which can be found by converting the transfer matrix to state space, i.e. P(s) => (A, B\ B2 ' Ci Dn D12 D2i D22 ) (A.94) (A.95) (A.96) (A.97) where the notation (A, B, C, D) is the state space representation of the transfer function T{s) = D + C{sI-A)-1B. (A.98) 95 Appendix A. Theory Notes The B{, d, and Dij matrices must be partitioned to reflect the number of external inputs, control inputs, control outputs and measured outputs. A.2 Stabilizable and Detectable These are defined as [3], [25] • (A, B) is stabilizable if the uncontrollable modes are stable. • (A, C) is detectable if the unobservable modes are stable. The uncontrollable and unobservable modes can be determined using the theorems • (A, B) is controllable if the rank of [(si — A) B] is equal to the size of A for all s. The values of s where this is not true are the uncontrollable modes. is equal to the size of A for all s. (si-A) • (A, C) is observable if the rank of C The values of s where this is not true are the unobservable modes A.3 Plant Transformations A.3.1 Axis Shifting With axis shifting the idea is to shift all the poles of the plant a small amount before the design is run, and then after a successful design is achieved, to shift the poles back. This can be thought of as shifting the jw-axis back and forth. The pole shifting can be done with the transform on the A matrix of the plant state space A = A + eI (A.99) where / is the identity matrix of same dimension as A. This is the same as shifting the jw-a,xis by e units to the left. Shifting the axis to the left puts the poles in the RHP, i.e. Appendix A. Hoo Theory Notes 97 makes the plant unstable. This is required as a consequence of Hoo theory, which will place the closed loop poles of the successful design in the LHP. By shifting the axis left, one insures that the poles remain in the LHP when the axis is shifted back. If the axis was shifted right, any closed loop poles with value less than e, will be in the RHP after the axis is shifted back. The proper choice of e is essential for determining the closed loop poles of the system. Hoo design theory will not design unstable systems, nor marginally stable ones. Therefore, the smallest pole of the system will be larger than e. This smallest pole also dictates the response of the system. Obviously, e must be less than the desired bandwidth so that the designed controller is able to react fast enough. At the same time, e can not be too small or the plant matrix may get badly conditioned. The choice of e is critical in the system design - it dictates the smallest allowable pole in the closed loop response. A.3.2 Bilinear Transform In bilinear transform [25], as with the axis shifting, the controller will be designed for the transformed plant, and then the solution is transformed back into the original plane. The bilinear transform takes the form s = ^ ± i (A.100) and the inverse transform is I z + . B z = 6 - - ^ . (A.101) 7 5 — a A special form of the transform maps the jioaxis to a circle and a circle to the new jto-axis, as shown in Figure A.44. This operation has the form z + ih - + 1 P2 (A.102) Appendix A. Theory Notes 98 s-plane A z-plane c B B Figure A.44: Special form of the bilinear transform that maps the jw-ax.is to a circle. so that a = 1,6 = Pi, 7 = ^ , and 3 = 1. When p2 = oo (7 = 0), this becomes the simple axis shifting and gives the same results as previous. The choice of locations of pi and p2 are important since they limit the placements of the closed loop poles: p2 affects the high frequency response of the system, pi determines the dominant poles of the closed loop system since the controller designed in the z-plane will have its poles in the LHP, which is inside the circle (region A) when transformed back. All poles of the designed closed loop system are in the circle, so the size and location of it directly determines the possible configuration of the closed loop system. Choosing it too small or in a bad position could result in a solution with totally unacceptable results. Appendix B Nomenclature Based on the closed loop shown in Figure B.45, the following nomenclature is used. Position Plant 1 Gx — ms2 — H Acceleration Plant Ga — rns2 — H Position Plant with Acceleration Feedback Gxa — ms2 — s2Ka — H Acceleration Plant with Position Feedback s2Ka Gnx — ms2-H + Kx D + Q x e r r = > K x ( s ) • a + I K a ( s ) 1 in H (B.103) (B.104) (B.105) (B.106) x Figure B.45: Closed loop position system with acceleration feedback. 99 Appendix B. Nomenclature 100 The open loop plants with cascade controller have the same subscripts, but G is replaced with GK. Sensitivity of Position S* = TTGK  ( A I 0 7 ) Sensitivity of Acceleration &-T=W (B'108) Sensitivity of Position with Acceleration Feedback 1 J x a ~ 1 + GKxa Sensitivity of Acceleration with Position Feedback 1 - GK, Position Closed Loop 1 + GKX Acceleration Closed Loop 1 - GKa Position Closed Loop with Acceleration Feedback . 1 + GK Acceleration Closed Loop with Position Feedback (B.109) Sax = - ^ - — (B.110) GK T* = 7^%~ (B-lll) GK T» = T ^ f e " (B.112) T*a = T^T~ (B.113) Appendix B. Nomenclature Other interesting relationships: GKxa — GKxSa GKax — GKaSx S - 1 * x a 1 + GKxSa S = I a x 1 - GKaSx T - T ? -1- xa — -1- x^a T — T <? Appendix C Hoo Controller Design Code The following is a listing of the design code for H^ control for use with MATLAB. C . l Main design file: design.m dispC P o s i t i o n C o n t r o l l e r • ' ) wrad = l o g s p a c e ( - 2 , 3 , 6 0 0 ) ; m = 2 2 ; ok = ' n ' ; w h i l e ok == ' n ' , c h o i c e = menu( 'Choose p l a n t t y p e ' , ' D o u b l e i n t e g r a t o r ' , . . . ' P o s i t i v e F e e d b a c k ' , ' M o r e A c c u r a t e P l a n t ' , ' P D S t a b i l i z e d ' ) ; i f c h o i c e == 1 , [ a g x . b g x . c g x . d g x ] = t f 2 s s ( l , [ m 0 0 ] ) ; e l s e i f c h o i c e = = 2 , H = - 4 0 ; [ a g x , b g x , c g x , d g x ] = t ! 2 s s ( l , [ m O H ] ) ; e l s e i f ( c h o i c e == 3) I ( c h o i c e = = 4 ) , dnx = c o n v ( [ 1 / 1 4 0 , 1 ] , [ 1 , 2 * . 1 5 * s q r t ( 4 5 / 2 . 6 ) , - 4 5 / 2 . 6 ] ) ; [ a g x . b g x . c g x . d g x ] = t f 2 s s ( 4 5 / 1 0 0 0 , d n x ) ; i f c h o i c e = = 4 , [ a p d . b p d . c p d . d p d ] = t f 2 s s ( [ 6 0 0 2 0 0 0 ] , [ 0 . 0 0 1 1 ] ) ; [ a g x , b g x , c g x , d g x ] = f e e d b a c k ( a g x , b g x , c g x , d g x , a p d , b p d , c p d , d p d , - 1 ) ; end end ss_gx = m k s y s ( a g x , b g x , c g x , d g x ) ; [Wlx,W2x,W3x] = w e i g h t s . x ; [ g a m x , s s _ k x , s s _ T x , n o r m x ] = h i n f _ b i l i n(ss _ g x , W l x , W 2 x , W 3 x ) ; [ a k x , b k x , c k x , d k x ] = b r a n c h ( s s _ k x ) ; [ a g k x , b g k x , c g k x , d g k x ] = s e r i e s ( a k x , b k x , c k x , d k x , a g x , b g x , c g x , d g x ) ; [ a t x . b t x . c t x.dtx] = c l o o p ( a g k x , b g k x , c g k x , d g k x , - l ) ; i f m a x ( b o d e ( a t x , b t x , c t x , d t x ) ) > 1 , f c ( a t x , b t x , c t x . d t x ) end f i g u r e ( l ) j b o d e ( a t x , b t x , c t x , d t x , l , w r a d ) ; s u b p l o t ( 2 1 1 ) t i t l e ( ' C l o s e d Loop R e s p o n s e ' ) a x i s ( [ . 0 1 20 - 4 0 1 0 ] ) s u b p l o t ( 2 1 2 ) v = a x i s ; v ( l : 2 ) = [ . 0 1 2 0 ] ; axis ( v ) 102 Appendix C. Hoo Controller Design Code 103 ok = i n p u t ( ' I s t h i s s a t i s f a c t o r y ? ( [ y ] / n ) : ' , ' s ' ) ; end p l t s u m ( s s _ g x , W l x , W 3 x , s s _ k x , g a m x ) y, _ _ _ _ d i s p ( " ) ; d i s p ( " ) ; d i s p ( " ) ; d i s p ( " ) ; d i s p ( " ) ; d i s p ( " ) ; d i s p ( " ) ; d i s p ( " ) ; d i s p ( ' A c c e l e r a t i o n C o n t r o l l e r — ' ) [ a k a , b k a , c k a , d k a ] = t f 2 s s ( 0 , l ) ; [ a g a . b g a . c g a . d g a ] = t f 2 s s ( 0 , l ) ; a c c _ c o n t r o l = i n p u t ( ' A c c e l e r a t i o n c o n t r o l l e r ? ( y / [ n ] ) : ' , ' s ' ) ; w h i l e a c c _ c o n t r o l == ' y ' , i f c h o i c e == 1 , [ a i , b i , c i , d i ] = t f 2 s s ( l , [ l 0 0 ] ) ; [ a h . b h , c h . d h ] = s e r i e s ( a i , b i , c i , d i , a k x , b k x , c k x , d k x ) ; [ a g a . b g a . c g a . d g a ] = f e e d b a c k ( [ ] , [ ] , • , 1 / m , a h , b h , c h , d h , - l ) ; e l s e i f c h o i c e == 2 , [ a h . b h , c h . d h ] = p a r a l l e l ( [ ] , • , • , H , a k x , b k x , - c k x , - d k x ) ; [ a h . b h . c h . d h ] = s e r i e s ( [ 0 0 ; 1 0 ] , [ 1 ; 0 ] , [ 0 1 ] , [ 0 ] . a h . b h . c h . d h ) ; [ a g a . b g a . c g a . d g a ] = f e e d b a c k ( [ ] , [ ] , [ ] , 1 / m , a h , b h , c h , d h , + l ) ; e l s e i f ( c h o i c e == 3) I ( c h o i c e == 4 ) , [ n m f . d n f ] = b u t t e r ( 4 , 3 2 5 , ' s ' ) ; [nmg.dng] = s e r i e s ( [ 4 5 / 2 . 6 0 0 ] . d n x . n m f , d n f ) ; [ a g a , b g a , c g a , d g a ] = t f 2 s s ( n m g , d n g ) ; end ss_ga = m k s y s ( a g a , b g a , c g a , d g a ) ; [Wla,W2a,W3a] = w e i g h t s _ a ; [ g a m a , s s _ k a , s s _ T a , n o r m a ] = h i n f _ b i l i n ( s s _ g a , W l a , W 2 a , W 3 a ) ; '/, A c c e l e r a t i o n sys tem [ a k a , b k a , c k a , d k a ] = b r a n c h ( s s _ k a ) ; [ a g k a . b g k a , c g k a . d g k a ] = s e r i e s ( a k a , b k a , c k a , d k a , a g a , b g a , c g a , d g a ) ; [ a t a . b t a , c t a . d t a ] = c l o o p ( a g k a , b g k a , c g k a , d g k a , + l ) ; [ a s . b s . c s . d s ] = f e e d b a c k ( [ ] , [ ] , [ ] , 1 , a g k a . b g k a , c g k a , d g k a , + l ) ; '/, Complete sys tem i f c h o i c e == 1 , ag = [ 0 . 0 ; 1 0 ] ; bg = [ l / m ; 0 ] ; eg = [0 1;0 0 ] ; dg = [ 0 ; l / m ] ; e l s e i f c h o i c e = = 2 , [ a g . b g , e g , d g ] = t f 2 s s ( l , [ m 0 - H ] ) ; eg = [ e g ; H / m * c g ] ; dg = [dg ; 1 / m ] ; e l s e i f c h o i c e == 3 , [ a x . b x . c x . d x ] = t f 2 s s ( 4 5 / 1 0 0 0 , [ 1 , 2 * . 1 5 * s q r t ( 4 5 / 2 . 6 ) , - 4 5 / 2 . 6 ] ) ; [ a l . b l . c l . d l ] = t f 2 s s ( l , [ 1 / 1 4 0 1 ] ) ; [ a 2 , b 2 , c 2 , d 2 ] = t f 2 s s ( [ l 0 0 ] * n m f ( 5 ) , d n f ) ; [ a g . b g , e g , d g ] = a p p e n d ( a l , b l , c l , d l , a 2 , b 2 , c 2 , d 2 ) ; bg = [ b l ; b 2 ] ; dg = [ d l ; d 2 ] ; [ a g . b g , e g , d g ] = s e r i e s ( a x , b x . c x , d x . a g . b g , e g , d g ) ; end ss_g = m k s y s ( a g , b g , c g , d g ) ; [ a k . b k ] = a p p e n d ( a k x , b k x , c k x , d k x , a k a , b k a , c k a , d k a ) ; ck = [ ckx c k a ] ; dk = [ dkx d k a ] ; ss_k = m k s y s ( a k , b k , c k . d k ) ; [ a g k , b g k , c g k , d g k ] = s e r i e s ( a k , b k , c k . d k , a g . b g , e g , d g ) ; [ a l . b l . c l . d l ] = f e e d b a c k ( a g k , b g k , c g k , d g k , [ ] , [ ] , [ ] , [ - l 0 ; 0 1 ] , + 1 ) ; f i g u r e ( 2 ) j b o d e ( a l , b l , c l , d l , l , w r a d ) ; s u b p l o t ( 2 1 1 ) Appendix C. Controller Design Code t i t l e ( ' C l o s e d Loop R e s p o n s e ' ) a x i s ( [ . 0 1 20 - 4 0 2 0 ] ) s u b p l o t ( 2 1 2 ) v = a x i s ; v ( l : 2 ) = [ . 0 1 2 0 ] ; a x i s ( v ) a c c _ c o n t r o l = i n p u t ( ' W o u l d you l i k e t o t r y a g a i n ? ( y / [ n ] ) : ' , ' s ' ) ; end C.2 Position weight generation: weights_x.m f u n c t i o n [ W I , W2, W3] = w e i g h t s '/, Se tup f o r r o b u s t c o n t r o l l e r d e s i g n '/, D e f i n e s w e i g h t s o n l y . Se ts i t up f o r p o s i t i o n , o n l y d e s i g n . '/, Robus tness Spec (W3) d i s p ( ' W 3 c o n t r o l s t h e r o b u s t n e s s spec o f t h e s y s t e m ' ) bw = i n p u t ( ' E n t e r b a n d w i d t h (Hz) o f t h e c l o s e d l o o p p o s i t i o n s y s t e m : ' ) ; bw = 2 * p i * b w ; wc = bw; s l o p e = i n p u t ( ' E n t e r s l o p e ( i e : - 1 f o r - 2 0 d B / d e c a d e , - 2 f o r - 4 0 d B / d e c a d e ) : ' ) ; c r o s s o v e r = 1 0 " ( - s l o p e * l o g l 0 ( w c ) ) ; num = [ 1 ] ; den = [ c r o s s o v e r ] ; f o r k = 1 : - s l o p e , num = c o n v ( n u m , [ l , 0 ] ) ; end i f ( s l o p e < - 2 ) , f o r k = 3 : - s l o p e , den = c o n v ( d e n , [ l / w c / 1 0 " ( 2 ) 1 ] ) ; end end [ r , c ] = s i z e ( n u m ) ; [ x , r ] = s i z e ( d e n ) ; den = [ z e r o s ( 1 , c - r ) , den ] ; W3 = [ num ; den ] ; '/, D i s t u r b a n c e r e j e c t i o n (WI) d i s p C • ' ) d i s p ( ' W l a f f e c t s t h e s e n s i t i v i t y f u n c t i o n ' ) d i s p ( ' T h e DC g a i n a f f e c t s t h e low f r e q u e n c y r e s p o n s e o f T ( s ) . A t o o l o w v a l u e ' ) d i s p ( ' w i l l cause e a r l y r o l l o f f . A t o o h i g h v a l u e w i l l c r e a t e r e s o n a n t p e a k s . ' ) d i s p ( ' T h e HF g a i n , i f s m a l l enough ( < . l ) has l i t t l e e f f e c t on t h e s y s t e m . ' ) DC = i n p u t ( ' D C g a i n : ' ) ; HF = i n p u t ( ' HF g a i n : ' ) ; */, H i g h f r e q u e n c y g a i n wc = 0 . 5 * b w ; /, f i l t e r c r o s s o v e r f r e q u e n c y d l = 0 . 7 ; '/, damping r a t i o 1 num = DC*[HF 2 * d l * w c * s q r t ( H F ) w c " 2 ] ; den = [DC 2 * d l * w c * s q r t ( D C ) wc~2] ; i f 0 , '/. F o u r t h o r d e r WI num = c o n v ( n u m , n u m ) ; den = c o n v ( d e n . d e n ) ; end WI = [ num ; den ] ; W2 = [ l . e - 4 ; 1 ] ; '/. S m a l l t o f u l f i l l r a n k r e q u i r e m e n t s on complex p l a n t s Appendix C. Controller Design Code C.3 Acceleration weight generation: weights_a.m f u n c t i o n [ W I , W2, W3] = w e i g h t s _ a */. [ W I , W2, W3] = w e i g h t s _ a '/, We igh ts f o r a c c e l e r a t i o n o n l y c o n t r o l l e r . d i s p ( ' T h e b a n d w i d t h f o r t h e a c c e l e r a t i o n c o n t r o l i s band p a s s . ' ) w c l = i n p u t ( ' P l e a s e e n t e r t h e low f r e q u e n c y c r o s s o v e r ( H z ) : ' ) ; w c l = 2 * p i * w c l ; wc2 = i n p u t ( ' P l e a s e e n t e r t h e h i g h f r e q u e n c y c r o s s o v e r ( H z ) : ' ) ; wc2 = 2 * p i * w c 2 ; d l = i n p u t ( ' D a m p i n g r a t i o 1 : ' ) ; d2 = i n p u t ( ' D a m p i n g r a t i o 2 : ' ) ; o r d = i n p u t ( ' I n p u t s l o p e / o r d e r o f p e r f o r m a n c e spec (see n o t e s ) : ' ) ; c l = 2 / 3 * w c l ; c2 = 3 / 2 * w c 2 ; den = c o n v ( [ l / 4 / w c i 1 ] , [ 1 / 0 . 2 5 / w c 2 1] ) ; den = c o n v ( d e n . d e n ) ; num = [ 1 / w c l 0 ] ; num = c o n v ( n u m , n u m ) ; WI = [ 0 0 num ; den ] ; num = c o n v ( [ l / 2 / c l 1 ] , [ I / O . 2 5 / c 2 1 ] ) ; num = c o n v ( n u m , [ 1 / 2 / c l 1 ] ) ; den = [ 1 / c l 0 ] ; den = c o n v ( d e n . d e n ) ; den = c o n v ( d e n , [ l / 1 0 / c 2 1 ] ) ; W3 = [ num ; deh ] ; W2 = [ l . e - 4 ; 1 ] ; WI = [0 5 0 . 9 ; 0 . 0 5 0 . 6 1 ] ; W3 = [ 0 . 0 5 0 0 1.0500 1 .0000 ; 0 .0100 1 0 . 0 0 0 1 0 . 1 0 0 0 ] ; C.4 Bilinear transform: hinf.bilin.m f u n c t i o n [gam, s s _ k , ss_Tzw, norm_Tzw] = h i n f _ b i l i n ( s s _ g , W l , W 2 , W 3 ) ; '/, [gam, s s _ k , ' ss_Tzw, n o r m h i n f ] = h i n f _ b i l i n ( s s _ g , W l ,W2,W3); 7. '/, B i l i n e a r t r a n s f o r m H - i n f i n i t y d e s i g n */. az + d '/. s = */, cz + b '/. '/, P l a n t i n d e p e n d a n t '/, R e q u i r e s */, P l a n t ~ > ( s s _ g ) */, We igh ts — > W I , W2, W3 as f o r use i n a u g t f .m */, C a l c u l a t e s */, Augmented p l a n t ( w i t h gam*W) — > Tss '/, Gamma — > gam */, C o n t r o l l e r — > ( a k . b k . c k . d k ) o r ss_k '/. Tzw l o o p — > ( a c l , b c l , c c l , d c l ) o r ss_Tzw [ a g . b g , e g , d g ] = b r a n c h ( s s _ g ) ; [ n W l . q ] = s i z e ( W l ) ; nWl = n W l / 2 ; '/, A s s i g n v a l u e s a = 1 ; '/. A x i s s c a l i n g Appendix C. Hoo Controller Design Code 106 b = 1 ; •/. c = O.O; '/. l / p 2 d = . - 0 . 1 ; '/, Dominant p o l e p l a c e m e n t '/, Augment p l a n t , Tss = a u g t f ( s s _ g , W l , W 2 , W 3 ) ; '/. Bunch o f s t u f f t o be done f o r t r a n s f o r m on augmented p l a n t [ a l , b l , b 2 , c l , c 2 , d l 1 , d l 2 , d 2 1 , d 2 2 ] = b r a n c h ( T s s ) ; [ q , n i ] = s i z e ( b 2 ) ; [ n o . q ] = s i z e ( c 2 ) ; [ a l , b l , c l , d l ] = b i l i n ( a l , [ b l , b 2 ] , [ c l ; c 2 ] , [ d l l , d l 2 ; d 2 1 , d 2 2 ] , 1 , ' G _ B i l i ' , [ a . b . c . d ] ) ; Tss = m s y s ( a l , b l , c l , d l . n i . n o ) ; '/, Do r o b u s t d e s i g n * / . [ss_k,ss_Tzw] = h i n f ( T s s ) ; gam = 1 ; [ g a m , s s _ k , s s _ T z w ] = h i n f o p t ( T s s , [ 1 3 ] ) ; ' / , , l : n W l ) ; '/, Compute c o n t r o l l e r and do i n v e r s e t r a n s f o r m ss_k = b i l i n ( s s _ k , - l , ' G _ B i l i ' , [ a . b . c . d ] ) ; '/. Compute Tzw and do i n v e r s e t r a n s f o r m ss_Tzw = b i l i n ( s s _ T z w , - l , ' G _ B i l i ' , [ a , b , c , d ] ) ; norm_Tzw = n o r m h i n f ( s s _ T z w ) ; C.5 State space system: msys.m f u n c t i o n t s s _ p = m s y s ( a p , b p , c p , d p , u , y ) '/, Changes t h e s t a t e space r e p r e s e n t a t i o n o f a p l a n t i n t o one f o r t h e h i n f '/. r o u t i n e i n M a t l a b . ( S t a n d a r d S t a t e - s p a c e t o T w o - p o r t S t a t e - s p a c e ) '/. '/, t s s _ p = m s y s ( a p , b p , c p , d p , n i , n o ) */, where n i = number o f c o n t r o l i n p u t s ( s i z e o f u ) '/, no = number o f c o n t r o l o u t p u t s ( s i z e o f y ) [ n , m ] = s i z e ( d p ) ; a = a p ; b l = b p ( : , 1 : m - u ) ; b2 = b p ( : , m - u + l : m ) ; c l = c p ( l : n - y , : ) ; c2 = c p ( n - y + l : n , : ) ; d l l = d p ( l : n - y , 1 : m - u ) ; d l 2 = d p ( l : n - y , m - u + l : m ) ; d21 = d p ( n - y + l : n , 1 : m - u ) ; d22 = d p ( n - y + l : n , m - u + l : m ) ; t s s _ p = m k s y s ( a , b l , b 2 , c l , c 2 , d l l , d l 2 , d 2 i , d 2 2 , ' t s s ' ) ; C.6 Plotting summary: pltsum.m f u n c t i o n p l t s u m ( s s _ g , W l , W 3 , s s _ k , g a m ) */, p l t s u m ( s s _ g , W l , W 3 ) s s _ k , g a m ) '/. '/, Summary p l o t t i n g r o u t i n e '/. [ a g , b g , c g , d g ] = b r a n c h ( s s _ g ) ; Appendix C. Controller Design Code [ a k , b k , c k , d k ] = b r a n c h ( s s _ k ) ; [ a g k . b g k . c g k . d g k ] = s e r i e s ( a k . b k , c k . d k , a g . b g , e g , d g ) ; [ a s , b s , c s , d s ] = f e e d b k ( a g k , b g k , c g k , d g k , 1 ) ; [ a t , b t , c t , d t ] = c l o o p ( a g k , b g k , c g k , d g k , - l ) ; wrad = l o g s p a c e ( - l . 5 , 2 , 8 0 0 ) ; f i g u r e s u b p l o t ( 2 2 1 ) [ m a g i . p h i l ] = b o d e ( a k , b k , c k . d k , 1 , w r a d ) ; s e m i l o g x ( w r a d / 2 / p i , 2 0 * l o g l 0 ( m a g i ) ) ; x l a b e l ( ' F r e q u e n c y [ H z ] ' ) t i t l e ( ' C o n t r o l l e r ' ) y l a b e l ( ' | K | [ d B ] ' ) > v = a x i s ; v ( l : 2 ) = [ . 0 1 2 0 ] ; a x i s ( v ) ; g r i d s u b p l o t ( 2 2 2 ) s e m i l o g x ( w r a d / 2 / p i , p h i l ) ; t i t l e ( ' C o n t r o l l e r ' ) x l a b e l ( ' F r e q u e n c y [ H z ] ' ) ylabeK ' A n g l e (K) [ d e g ] ' ) v = a x i s ; v ( l ) = 0 . 0 1 ; v ( 2 ) = 2 0 ; a x i s ( v ) y t i c k = g e t ( g c a , ' y t i c k ' ) ; y l i m = g e t ( g c a , ' y l i m ' ) ; y r a n g e = y l i m ( 2 ) - y l i m ( l ) ; n o _ o f _ p t s = l o g ( y r a n g e / ( l e n g t h ( y t i c k ) * 9 0 ) ) / l o g ( 2 ) ; n = r o u n d ( l o g ( y r a n g e / ( l e n g t h ( y t i c k ) * 9 0 ) ) / l o g ( 2 ) ) ; s e t ( g c a , ' ' y l i m m o d e ' , ' m a n u a l ' ) i f n o _ o f _ p t s >= - 1 . 1 5 */, 4 5 , 9 0 , 180, 3 6 0 , . . . d e g r e e i n c r e m e n t s y t i c k = [ - 9 0 * 2 - n : - ( 9 0 * 2 ~ n ) : y l i m ( l ) , 0 : ( 9 0 * 2 ~ n ) : y l i m ( 2 ) ] ; y t i c k = y t i c k ( f i n d ( y t i c k >= y l i m ( l ) & y t i c k <= y l i m ( 2 ) ) ) ; s e t ( g c a , ' y t i c k ' , y t i c k ) ; e l s e i f n >= - 2 '/, S p e c i a l case f o r 30 d e g r e e i n c r e m e n t s r a t h e r t h a n 2 2 . 5 y t i c k = [ - 3 0 : - 3 0 : y l i m ( l ) , 0 : 3 0 : y l i m ( 2 ) ] ; y t i c k = y t i c k ( f i n d ( y t i c k >= y l i m ( l ) & y t i c k <= y l i m ( 2 ) ) ) ; s e t ( g c a , ' y t i c k ' , y t i c k ) ; end g r i d s u b p l o t ( 2 2 3 ) mag2 = b o d e ( a s , b s , c s , d s , l , w r a d ) ; mag3 = b o d e ( W l ( 2 , : ) , g a r a * W i ( l , : ) , w r a d ) ; s e m i l o g x ( w r a d / 2 / p i , 2 0 * l o g l 0 ( m a g 2 ) , w r a d / 2 / p i , 2 0 * l o g l 0 ( m a g 3 ) , ' - . ' ) a x i s ( [ . 0 1 20 - 6 0 2 0 ] ) g r i d t i t l e ( ' S e n s i t i v i t y and W e i g h t ' ) x l a b e l ( ' F r e q u e n c y [ H z ] ' ) y l a b e l ( ' M a g n i t u d e [dB] ' ) l e g e n d C | S | ' , ' 11/Wsl ' ) s u b p l o t ( 2 2 4 ) mag4 = b o d e ( a t , b t , c t , d t , 1 , w r a d ) ; mag5 = b o d e ( W 3 ( 2 , : ) , g a m * W 3 ( l , : ) , w r a d ) ; s e m i l o g x ( w r a d / 2 / p i , 2 0 * l o g l 0 ( m a g 4 ) , w r a d / 2 / p i , 2 0 * l o g l 0 ( m a g 5 ) , ' - . ' ) a x i s ( [ . 0 1 20 - 6 0 2 0 ] ) Appendix C. Hoo Controller Design Code g r i d t i t l e ( ' C o m p l e m e n t a r y S e n s i t i v i t y and W e i g h t ' ) x l a b e l ( ' F r e q u e n c y [ H z ] ' ) y l a b e l ( ' M a g n i t u d e [ d B ] ' ) l e g e n d C | T | ' , ' I i / W t | ' ) s e t ( g c f , ' P a p e r P o s i t i o n ' , [ 1 2 6 7 ] ) Appendix D Coarse-Fine Tracker Software Particulars The software was developed by.Tim Salcudean and Chia-Tung Chen. D. l Coarse-Fine Software Algorithm The controller code and the include files are compiled and linked with a C30. compiler. The control code is executed every time the DSP has a specific interrupt set, causing the following functions to execute: 1. get A I ) 2. psd_position, measure_accel, measure_coarse_pos 3. determine_state 4. coarse_control ' -5. flotor_control 6. force_to_current, current _to_volt 7. put DA ' In short, the code reads in the signal data and calculates relevant position, velocity, and acceleration variables, calculates the control signals required to drive the system, and outputs these signals. The above is explained in greater detail now. 109 Appendix D. Coarse-Fine Tracker Software Particulars 110 1. The raw data available at the A/D port is read in and stored. 2. The raw data is manipulated through algebra and matrix multiplication to get the position and orientation of the flotor, the acceleration levels of the flotor, stator, and plane, and the coarse position. 3. The type of control to be used is dependent on the state of the system. There are five states: READY, ISOLATE, CATCH, LOWER, and REST. The state the system is in depends on the plane acceleration and coarse position. Figure D.46 shows the state switching diagram which can be summarized, as follows: READY Magnitude of the acceleration of plane less than MAXLEV but greater than MAXISO. ISOLATE Magnitude of the acceleration of plane less than MAXISO. C A T C H Coarse position in ISOLATE nears the end of the rail. LOWER Acceleration of plane less than -MAXLEV and coarse position not at minimum. REST Acceleration of plane less than -MAXLEV and coarse position at minimum. MAXLEV is the maximum acceleration allowed to keep the coarse stage centered on the rail, while MAXISO is the maximum acceleration allowed during the vibration isolation and tracking phase. 4. The control of the coarse position is dependent on the State of the system. The coarse control signal, ucrse, is an integer between ±500 that gets converted to a voltage to the current amp. See Appendix D for the control relationships. 5. The flotor is controlled by a simple PID, f — kpXerr -j- h{ f XerT "j" krlXgrr (D.121) Appendix D. Coarse-Fine Tracker Software Particulars 111 A — t-H V (0 A cl) 1 V ' *. CATCH >cl) -CATCH X A Figure D.46: State switching algorithm, x is the coarse position, a is the acceleration of the plane, i is MAXISO, 1 is MAXLEV, p is the lower limit of the I-beam, and cl is the coarse position limits during operation, Appendix D. Coarse-Fine Tracker Software Particulars 112 where X e r r — Xref •Kfine' 6. The force just calculated is transformed first into a current, then a voltage. 7. The voltages for the coarse control and flotor control are put to the D/A converter for use by the system. D.2 Coarse Stage Control Signal READY — kp-crscrdy * (CRSE-DB — X-crse-err) — kvjerse * vjcrse-f This centers the coarse stage on the rail. ISOLATE ucrse = kp-crseJso * x^fine — kvjerse * v_crse_/ The coarse stage is driven to follow the displacement of the flotor with velocity damping. C A T C H ucrse = 0 Stops the motion toward the rail limits. LOWER ucrse = kv-crse * (PARKA/EL - vjcrsej) Lowers, the coarse stage to the bottom of the rail. REST ucrse = -70 Constant negative force to keep the coarse stage at the bottom of the rail. D.3 Coarse Stage Code Variables All position variables have units mm and velocity variables mm/s. kp-crsejrdy Proportional constant for coarse control in State READY. Appendix D. Coarse-Fine Tracker Software Particulars 113 CRSE-DB Minimum displacement of coarse position (dead band) from center before control is started. X-crse Coarse position. crse-cen Set point for centering the coarse stage. xjcrsejerr — xjcrse — crse-cen kv^crse Derivative constant for the coarse position. V-crse-f Velocity of coarse position, filtered. kp-.crse-iso Proportional constant for coarse control in State ISOLATE. X-fine Position of flotor as measured by the PSD's. PARK-VEL Maximum velocity when lowering the coarse stage. kp Proportional constant for flotor control in N.mm. kp = 4 7 T 2 / 2 * massfi0tor/WOO ki Integral constant for flotor control, equal to 0.1 N/s/mm arbitrarily. kd Derivative constant for flotor control in N.s/mm. kd = i7r(fn * mass fi0tor/1000 fn Desired bandwidth of fine motion system in Hz. ( Desired damping ratio of fine motion system. Appendix E Data Plots 114 Appendix E. Data Plots 115 Acceleration of Plane 5 10 15 Time [s] 20 Acceleration of Flotor liiimnnnninmmn 5 10 15 Time [s] 20 Position of Stator Position of Flotor E E c o 600 400 200 0 rx -200 -400 -600 0 5 10 15 Time [s] 20 5 10 15 Time [s] 20 Figure E.47: Data plots for control when the dead band was not removed. It is possible to see the coarse stage vibrations in the Position of Stator plot, kp.crse.iso = 80, kv-crse = 1, /„ = 0.2. Appendix E. Data Plots 116 Acceleration of Plane Acceleration of Flotor Time [s] Time [s] Position of Stator Position of Flotor Time [s] Time [s] Figure E.48: Data plots for control when the dead band was not removed. It is possible to see the coarse stage vibrations in the Position of Stator plot. kp.crseJso = 80, kv-crse = 1, /„ — 0.2. Appendix E. Data Plots 117 Acceleration of Plane Acceleration of Flotor Time [s] Time [s] Figure E.49: Data plots for control when the dead band was not removed. It is possible to see the coarse stage vibrations in the Position of Stator plot. kp-crseJso = 80, kv-crse = ! , / „ = 0.2. Appendix E. Data Plots 118 Acceleration of Plane 0.5 r- 1 ; ; — Time [s] Position of Stator 600 \ • • : 400 -600 M •; ; ; 0 5 10 15 20 Time [s] Acceleration of Flotor Time [s] Figure E.50: Data plots for control with no dead band but with a velocity variable causing vibrations. Less coarse stage noise is seen in Position of Stator plot but acceleration of flotor remains poor. kp-CrseJso = 80, kv-crse = 1, fn — 0.2. Appendix E. Data Plots 119 Acceleration of Plane Acceleration of Flotor 10 15 Time [s] 20 5 10 15 20 Time [s] Position of Stator Position of Flotor E E c o 600 400 200 CO L? -200 -400 -6001 10 Time [s] 20 10 Time [s] 20 Figure E.51: Data plots for control with no dead band but with a velocity variable causing vibrations. Less coarse stage noise is seen in Position of Stator plot but acceleration of flotor remains poor. kp.crseJso = 80, kvjcrse = ! , / „ = 0.2. Appendix E. Data Plots 120 Acceleration of Plane Acceleration of Flotor 0 5 10 15 20 0 5 10 15 20 Time [s] Time [s] Position of Stator Position of Flotor Time [s] Time [s] Figure E.52: Data plots for control with no dead band but with a velocity variable causing vibrations. Less coarse stage noise is seen in Position of Stator plot but acceleration of flotor remains poor. kp-crseJso = 80, kvjcrse = 1, /„ = 0.2. Appendix E. Data Plots 121 Acceleration of Plane Acceleration of Flotor J O CO E E, c o Cfl _g> CD o o < -0.5 10 Time [s] Position of Stator E E c o 600 400 200 0 CO o -200 -400 -600 J 0.5 j o l o E c o 2 0) CD O O < 0 10 Time [s] 0 20 -0.5 "0 10 Time [s] Position of Flotor 20 20 10 Time [s] 20 Figure E.53: Data plots for unreliable data obtained when the system was not working properly. There is no velocity damping in ISOLATE state and the acceleration of. the flotor is considerably improved. Notice the position of the flotor at the start of the parabola while coarse stage centering is attempted. kp-.crseJso = 50, /„ = 0.1 Appendix E. Data Plots 122 Acceleration of Plane 0.51 ; 1 ! — i i i i i 0 5 10 15 20 Time [s] Acceleration of Flotor 0.51 in ; T T ~ CO E E -0.5' 1 : : u 0 5 10 15 20 Time [s] Position of Flotor 0 5 10 15 20 Time [s] Figure E.54: Data plots for unreliable data obtained when the system was not working properly. There is no velocity damping in ISOLATE state and the acceleration of the flotor is considerably improved. kp-crseJso = 50, fn =0.1 

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