@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix dc: . @prefix skos: . vivo:departmentOrSchool "Science, Faculty of"@en, "Physics and Astronomy, Department of"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Greenall, Russ"@en ; dcterms:issued "2009-11-24T21:25:23Z"@en, "2004"@en ; vivo:relatedDegree "Master of Science - MSc"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description """This thesis explores techniques to actively control the position of large masses such as focusing magnets with precision on the order of 1 nm against vibrations. The technique applied (labeled as an "optical anchor") is to actively "stiffen" the support structure using an optical interference method to measure distance to a remote reference point. The magnet is modeled as a mass on a spring, with a piezo electric actuator. In this model, proportional and differential control applied to the piezo allows the mass to be critically damped and the spring coefficient to be arbitrarily increased. A digital implementation with finite sampling rate has a finite stable region in control parameter space. If there are more mechanical degrees of freedom, the stable region and the quality of control can be greatly reduced. An interferometric instrument design for remote distance measurement is discussed and measurement results reflecting an accuracy of 0.2nm RMS are demonstrated. The instrument requires only two light detectors in a Michelson interferometer configuration. The algorithm design is implemented at a 5KHz sample rate using a circa 2000 DSP processor with 4-byte floating point operations running at a 40 MHz clock rate. Control tests on a one degree-of-freedom experimental platform are performed using proportional and differential control. These tests demonstrate active control which significantly damps fundamental mode excitations but are insufficient to stiffen the system. More sophisticated models and algorithms will be necessary. Nevertheless, some insight is gained into techniques which will allow control on the nanometer scale against "standard" ground vibrations. In particular, a successful implementation of coherent ground disturbance modeling provides a three-fold reduction in RMS vibration of our test system over our simple PID control."""@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/15660?expand=metadata"@en ; dcterms:extent "16502151 bytes"@en ; dc:format "application/pdf"@en ; skos:note "NANOVIBRATION C O N T R O L by R U S S G R E E N A L L B . S c , Simon Fraser University, 2001 A THESIS SUBMITTED IN PARTIAL FULLFILLMENT OF 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 OF Master of Science In The Faculty of Graduate Studies ^ Department c ' •= Physics^ University of British Columbia We accept this thesjs as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA June 2004 © Russ Greenall, 2004 Library Authorization In presenting this thesis in partial fulfillment 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. Title of Thesis: /\\J ^ n n w f L rr, f S o „ C „ ( D e 9 r e e : nOn . cK , ' , e f P / w . „ , c Year: Z O O 4-Name of Author (please print) Date (dd/mm/yyyy) The University of British Columbia Vancouver, BC Canada Department of This thesis explores techniques to actively control the position of large masses such as focusing magnets with precision on the order of 1 nm against vibrations. The technique applied (labeled as an \"optical anchor\") is to actively \"stiffen\" the support structure using an optical interference method to measure distance to a remote reference point. The magnet is modeled as a mass on a spring, with a piezo electric actuator. In this model, proportional and differential control applied to the piezo allows the mass to be critically damped and the spring coefficient to be arbitrarily increased. A digital implementation with finite sampling rate has a finite stable region in control parameter space. If there are more mechanical degrees of freedom, the stable region and the quality of control can be greatly reduced. An interferometric instrument design for remote distance measurement is discussed and measurement results reflecting an accuracy of 0.2nm RMS are demonstrated. The instrument requires only two light detectors in a Michelson interferometer configuration. The algorithm design is implemented at a 5KHz sample rate using a circa 2000 DSP processor with 4-byte floating point operations running at a 40 M H z clock rate. Control tests on a one degree-of-freedom experimental platform are performed using proportional and differential control. These tests demonstrate active control which significantly damps fundamental mode excitations but are insufficient to stiffen the system. More sophisticated models and algorithms will be necessary. Nevertheless, some insight is gained into techniques which wil l allow control on the nanometer scale against \"standard\" ground vibrations. In particular, a successful implementation of coherent ground disturbance modeling provides a three-fold reduction in RMS vibration of our test system over our simple PID control. i i mMm @ff €®trafteffifts ABSTRACT ii TABLE OF CONTENTS iii TABLE OF FIGURES v 1. INTRODUCTION 1 2. CONTROL THEORY FOR PHYSICISTS 3 2.1. SIMPLE CONTINUOUS MODEL 3 2.2. CONTINUOUS CONTROL 5 2.3. BODE PLOTS AND TRANSFER FUNCTIONS.. 9 2.4. STABILITY IN CONTINUOUS FEEDBACK SYSTEMS 13 2.5. DISCRETE TIME 16 2.6. DISCRETE CONTROL 19 2.7. CONTROL OF TWO-RESONANCE SYSTEM IN DISCRETE TIME 23 2.8. CONCLUSION .- 27 3. APPARATUS 29 3.1. INTERFEROMETER 29 3.2. ELECTRONICS 30 3.3. DATA ACQUISITION 31 3.4. LOCK-IN AMPLIFIER 32 3.5. POSITION CONTROL TEST PLATFORM 33 4. INTERFEROMETER 37 4.1. INTERFEROMETER SIGNAL MODEL ; 37 4.1.1. Physics 37 4.1.2. Direct Parameterization 39 4.1.3. Calibration 40 4.1.4. Additional Algorithm Design Criteria 41 4.2. GEOMETRIC PARAMETERIZATION 43 4.2.1. Geometric Parameterization Validity 44 4.2.2. Parameter Iterations in Geometric Model 47 4.2.3. AI go rithm Tests 56 4.2.4. Non-Correlated Noise Test 58 5. CONTROL TESTS 60 5.1. PLATFORM RESPONSE 60 5.2. GROUND-MOTION CHARACTERISTICS 62 5.3. PID CONTROL MODEL 66 5.4. RESULTS 67 5.5. EXPLORATION OF GAIN LIMITS 70 5.6. PID CONTROL WITH RESONANCE NOTCHING 72 5.7. COHERENT DISTURBANCE CORRECTION 73 5.8. BEST COMPOSITE PERFORMANCE -. 76 6. CONCLUSIONS 77 iii APPENDIX A NLC VIBRATION CONTROL EDITORIAL 79 APPENDIX B DEADBEAT GAIN 82 APPENDIX C NOTCH FILTER ALGORITHM .84 INDEX 88 iv TatoEfe ©f iFigpires Figure 1: Simplified Model of a Single Mass with Position Control .3 Figure 2: Sample Evolution of the Simplified Model 4 Figure 3: Evolution of \"Stiffened\" System Kp=10 6 Figure 4: Damped System 7 Figure 5: Simple Mass on a Spring Viewed as a Transfer Function 9 Figure 6: Bode Plot for Simple Mass on Spring 10 Figure 7: Mass on a Spring Control Schematic 11 Figure 8: Bode Plot for Differentially Controlled Mass 12 Figure 9: Unstable System 14 Figure 10: Two Resonance System 15 Figure 11: Discrete Evolution Sample of the Simplified Model 18 Figure 12: Gain Stability Contours . 20 Figure 13: Simplified Model with PD Gains Set at 50% of Deadbeat , 21 Figure 14: Simplified Model with PD Gains Set at 100% of Deadbeat 22 Figure 15: Modified Model Adding a Single Resonance 23 Figure 16: Two-Resonance Model Gain Stability 24 Figure 17: Modified Model at Previously Stable Gains 25 Figure 18: Two-Resonance Model for 4-Dimensional Gain Stability 26 Figure 19: Interferometer Setup 29 Figure 20: Piezo Driver Circuit Schematic 31 Figure 21: Lock-in Amplifier Block Schematic 32 Figure 22: Test Platform Isometric Drawing 33 Figure 23: Test Platform Photograph .34 Figure 24: Actuator Detail 35 Figure 25: Interferometer Setup 37 Figure 26: Characteristic Interferometric Measurement Data. 39 Figure 27: Table of System Parameters 40 Figure 28: Lissajous Pattern 43 Figure 29: Symmetric Matrix Operation 48 Figure 30: Circularization 50 Figure 31: First Order Iteration to Find Centre 51 Figure 32: Orthogonal Iteration to Find Centre 52 Figure 33: Centre Finding Operation 54 Figure 34: Interferometer and Algorithm Test 57 Figure 35: Non-correlated Noise Test 58 Figure 36: System Bode Plot 61 Figure 37: Uncontrolled System Vibration Spectrum 62 Figure 38: Measurement Model (No Control) 63 Figure 39: Ground Spectrum Approximation 65 v Figure 40: Control System Model 66 Figure 41: Model Response 68 Figure 42: PID Control System Results 69 Figure 43: PID Control System Bode Plots 70 Figure 44: Gain Increase due to Notch Filters 72 Figure 45: Coherent Disturbance Elimination 74 Figure 46: Best Control Achieved 76 vi Particle accelerators have been increasing in power since their inception in the 1930's. Since the advent of two-beam particle colliders in the 1950's, beam diameter has been decreasing to increase luminosity. The proposed \"Next Linear Col l ider\" project proposes a beam diameter as small as 2 X 200 nanometers [ 1 ] or just about twenty times the diameter of an atom in the small dimension. A t this diameter, stabilization of final focusing magnets against environmental vibration becomes critical. Acceptable levels of stabilization w i l l be on the order of one nanometer. Some concern has been raised regarding the magnitude of the ground motion which w i l l be encountered in the N L C environment [ 2 ] ' [ 3 ] ' [ 4 ] . These sources indicate an overall expected vibration level on the order of lOOnm R M S with no effective vibration above 300Hz. Further, narrow band near-field excitations from rotating machinery w i l l cause narrow band peaks of up to hundreds o f nanometers in amplitude at frequencies anywhere from 30 H z to 200 H z [ 4 ] . J. Frisch et al. [1] discuss stabilization technologies with respect to the N L C project as o f 2001. These are broken down into three categories: 1) Beam Based Systems. The interaction of the electron and positron beams at the IP (intersection point) causes a beam deflection that is related to the beam offset. This allows the offset to be measured to a fraction of the spot size at the beam rate of 120Hz. This beam deflection provides the only long-term measure of the relative positions of the beams. A variety of feedback algorithms can be used with the beam -beam deflection data, with the selection based on the trade off between low frequency attenuation, and high frequency amplification of noise. 2) Interferometer Based Systems. Optical interferometers can be used to measure the distance between the final focus magnets and an external reference which may or may not be rigidly attached to the ground. These are termed \"optical anchor\" systems. 1 3) Inertial Based Systems. Inertial sensors can be used to measure the motion o f the final quadrupoles relative to the \"f ixed stars\". A t low frequencies the position noise of an inertial sensor increases and a transition to a beam based system must be made. A n inertial vibration stabilization system has been constructed and is being used to test feedback hardware and algorithms [1]. This thesis is dedicated to investigating two aspects of: 2) Interferometer Based Systems: 1. Instrument Design for an accurate and relatively inexpensive interferometric distance measurement device. This is explored in chapter 4, (page 37). 2. Act ive control of one dimension of a large mass to nanometer precision using the paradigm of \"ground tracking\" or \"optical anchor\". First principles analysis would imply that position control of a large mass should be straightforward (refer to the theory chapter section 2.1: \"Simple Continuous M o d e l \" on page 3). Actual implementation proved very elusive however. The paradigm of the \"optical anchor\" requires that the large mass must be accelerated to. undergo a trajectory which matches the ground-based reference. This is an extremely difficult proposition as accelerations of this order invariably excite dozens o f internal modes of the large mass and/or of the support structure which are N O T negligible. This is explored in chapter 5, \"Control Tests\" on starting page 60. 2 2 . C o n t r o l T h e o r y f o r P h y s i c i s t s 2 . 1 . S i m p l e C o n t i n u o u s M o d e l We shall start with the simplest conceivable model for nano-position control. Reduced to simplest terms of one degree of freedom and a monolithic mass the system is as shown in the figure below. \\ \\ \\ \\ \\ \\ \\ \\ Measurement Stiff Spring 1N/um Actuator (Length Adjust) 1 kg Test Mass Figure 1: Simplified Model of a Single Mass with Position Control Let us carry through a numeric example allowing illustration of the mixed magnitudes. That is, perform a numeric example where our mass is considered to be 1kg and yet we are interested in nanometer motions. We shall ignore damping, because natural damping has a negligible effect on system gain limitations and it adds a term to our equations with which we need not deal. Similarly, gravity is ignored in this model. We just have a mass with one degree of freedom mounted to a bulkhead through a stiff spring and an actuator which can change its length under a control system's command. 3 The system has no damping (infinite \" Q \" ) , thus the differential equation of motion is as follows from a simple force balance (y measured from equilibrium): 0 = m d2y(t) dt2 + ky{t) - ku(t) Eqn( l ) Where m is the mass, y(t) is the test mass position from equilibrium, u(t) is the actuator length (measured from null) and k is the spring stiffness. Note that u(t) is an arbitrary function which we can specify. The resultant position function from the combination of natural evolution and the control function is y(t). Since we take the damping in this system to be negligible, this system wi l l be on the border o f stability without active feedback. Once set oscillating, this system as described w i l l continue to oscillate forever. Below is a plot of the system evolution generated using numerical methods. The plot shows an initial small oscillation of 0.1 nm ampliuide. A t time t=0, the actuator is stepped from zero to one nm. The resultant system response is shown. System Response to 1 nm Output Step 2.5r 2 1.5 E 1 o 0-0.5 0 -0.01 0 Time (sec) 0.01 0.02 Figure 2 : Sample Evolution of the Simplified Model The actuator is shown in red. The resulting system response is shown in blue. 4 Overal l , the second order response is very simple to understand. One can easily see that adjustment of the actuator length u(t) has a strong coupling to the position and should be capable of very effective control of the system. 2.2 . C o n t i n u o u s C o n t r o l A s mentioned, the function u(t) is under our control. It is there to influence the behavior of the system. Indeed, in the continuous (and perfectly modeled) world, the output u(t) can be set as a function of position and velocity to damp position oscillations arbitrarily fast. There is a host of engineering literature related to control of simple continuous second order systems. We are going to examine continuous control from the point of view of physical equivalence. That is, the control function can be made to artificially \"st i f fen\" and \"damp\" the system. Manipulation of the system evolution sets this analysis apart from many mechanical analyses where one is merely trying to predict system outcomes. Here, we are purposefully interfering with the natural evolution of our system. There are two general classes of control: feedforward and feedback. In feedforward, we adjust u(t) in some arbitrary fashion without regard to system measurements. This mode might be applied if, for example, we wanted to cause the system to oscillate at some fixed frequency. We would simply apply this pure frequency to the actuator (u(i)) and know that the system response would be to respond at exactly the same frequency. We would not offhand know the phase or amplitude. Feedback control is far more common. In feedback, we examine the system measurements and use these measurements to calculate an appropriate actuator length (u(t)). This mode of control is almost universally implied in control theory. So, what are some obvious feedback strategies? For one, there is the \"proportional\" control strategy. If the measurement shows the position too low, retract the actuator, which w i l l apply a restoring force to raise the mass. Similarly, i f the position is too high, extend the actuator. In proportional control, the \"error\" is taken to be the difference between an actual measurement and a setpoint or desired point for that measurement.. The actuator is then set in opposition to the error by an amount proportional to the error. u(t) = -Kpy(t) + r(t) E c l n ( 2 ) Kp is called the proportional gain. The negative sign is included since by convention, the state o f negative feedback is considered the norm and Kp is expected to be positive. The term r(t) 5 is included here as the step function shown in red on all of the plots. It is similar in function to what would normally be referred to as the setpoint or demand function in control literature. The composite system equation when u(t) is set using proportional feedback is shown below. Substituting the proportional control condition equation (2) into the system equation (Eqn 1) yields: 0 = m d2y(t) dt2 + k(\\ + Kp)y(t)-kr(t) Eqn(3) Examining equation 3 above, we can see that the effective spring constant for the system is now k(\\+Kp). With a second order system such as we have modeled, proportional control only serves to \"stiffen\" the system. That is, the system now has a higher resonant response than the natural system but there is still no damping. This can be seen in figure 3 where Kp is stepped from 0 to 10 at time zero. \"Stiffened\" System £ o CL 0.02 Time (sec) Figure 3 : Evolution of \"Stiffened\" System Kp=10 The red step function shows the demand or setpoint function and the system response is shown in blue. Note that in this resultant system, what is known as positive feedback would result from a negative Kp (actuator action which does not oppose the error). Positive feedback up to a point may be applied to \"soften\" the system. A s Kp approaches -1 the system formula becomes that 6 of an unconstrained mass. This positive feedback would be limited to Kp - -1 at which point the natural spring would be overcome by the control and the system would become unstable. Another feedback strategy is called \"derivative\" control where the actuator length is set proportionally to the velocity of the mass. When set in a contrary direction to the velocity, the control serves to dampen the system oscillations. This strategy for u(t) is defined as follows at Eqn(4) This relation establishes derivative control where Kd is defined as the derivative gain. This leads to a system equation as follows: dt1 di Eqn(5) A n example of this control is shown below where the derivative gain factor is set to AT^O.001 sec at t=0. The system now appears damped despite the absence of any natural damping. Damped System 1.2 0.8 ! 0.6 55 0.4 o Q _ 0.2 02 A --0.01 Time (sec) 0.01 0.02 Figure 4: Damped System 7 If we include a combination of both proportional and derivative control, our new equation o f motion becomes: 0 = m d2y(t) dt2 dy(t) dt + k(\\ + Kp)y(t)-kr(t) Eqn(6) This standard form of damped simple harmonic motion has a wel l known condition for critical damping about the equilibrium point which is: In our previous figure (4) showing a damped response, Kd was set to 0.001 with no proportional gain. Crit ical damping according to the relation above would be at A ^ O . 0 0 2 . This is consistent with figure (4), which shows significant damping but is still somewhat under-damped. For the mass and spring system described, we could make the proportional gain arbitrarily large and the derivative gain critically damped and achieve arbitrarily good control o f the mass despite the finite natural frequency. However, to raise the effective frequency by a factor o f 100, we would need ^=9999 . This is not to say that a real system can be controlled infinitely wel l by applying infinite feedback gain. We shall see that the system can become unstable at large proportional or derivative gains due to additional system modes, or to finite time step effects in digital implementations. Another form of feedback called integral control is used to remove all steady-state error from control system output. However our application does not specify any steady-state (DC) requirements and therefore we wi l l not cover this type of control here. Together, proportional, integral and derivative control are known as PID control and are used extensively both industrially and in the laboratory as an excellent generalized first approach to a control problem. Eqn(7) 8 2.3. B o d e P l o t s a n d T r a n s f e r F u n c t i o n s It is useful to analyze systems in terms of transfer functions. A transfer function is simply the amplitude and phase relationship between the input and output of a system. For example, in our simple system, we can regard the control signal u as an input and the resultant position measurement y as an output as follows: (System Response) u(co) — (piezo movement) (measurement) Figure 5: Simple Mass on a Spring Viewed as a Transfer Function The simple mass on a spring can be regarded as a transfer function P which relates the amplitude and phase relationship between the input u(w) and the output y(co) The inputs and outputs are Fourier transformed to be complex functions o f frequency rather than real functions of time. This is why the diagram shows the funtions u(co) and y(co) as functions of frequency co. The system response (often referred to as the \"Plant\" response) is characterized by the transfer function P where y(a>) = P(co)u(oS) Ec*n(8) A n d where the explicit references to the frequencies are invariably omitted. The convention maintained hereafter w i l l be that transfer functions and the associated functions of frequency w i l l be written as non-italic, non-bold typeface. Where this is not definitive, context w i l l hopefully be sufficient for clarity: y=Pu E ^ Notwithstanding the explicit omission of a reference to frequency, transfer functions are assumed to apply to Fourier space exclusively. Application of a signal to a transfer function can be explicit ly achieved simply through complex multiplication at each frequency. 9 The presentation of a transfer function can be made through use of a Bode plot which shows amplitude and phase response in two separate graphs which share a frequency axis. The Bode plot for our simple mass on a spring is shown below. The amplitude actually rises to infinity for the undamped resonance at about 160 H z but quantization o f the plot produces this approximation of the actual response. A t very low frequencies, the mass moves in the same manner as the piezo extension. A s the frequency approaches resonance, the response gets larger reaching an infinite result exactly at resonance (for our perfect undamped system). After resonance the response falls off with increasing frequency. The phase is unchanged at zero degrees until after resonance where it instantly (for an undamped resonance) switches to 180 degrees of lag. Note that the Bode plot does not directly show transient responses but rather shows the steady state output to input relationship only. 10 When u(t) is made to be a function of the measurement y(t) that process is said to be feedback. The configuration is shown schematically below. Error r(o) -Hi) SH setpoint (demand) Piezo Signal u(o>) Controller System response toPtezo >'(©) Mass Position Figure 7: Mass on a Spring Control Schematic Figure 7 requires the system output y to depend upon the plant transfer function P, the controller transfer function C and implicit ly on itself through the difference node output (r-y). The resulting system equation is: y = ( r - y )CP Eqn(lO) Where all o f the symbols are complex functions of frequency and the equation holds for any given value of frequency. This can be manipulated to lead to an explicit expression for the closed-loop transfer function. y _ C P r 1 + C P Eqn(ll) This equation only applies i f the system is stable. 11 A n ideal control system has an overall closed-loop transfer function of identically one for all frequencies. A s can be seen by equation 11, this ideal can only be approached as the controller-plant combination (CP) approaches infinity. Thus, system designers are always seeking to increase gain to improve performance. However, gain may not be increased arbitrarily as we w i l l explore subsequently. Let us examine a non ideal application of control using only derivative control to influence our mass on a spring. Again applying only .£^=0.001 (sub critical damping ), the fol lowing plots reflect the closed loop response o f the same system as shown figure 4. The plot is no longer logarithmic in frequency as there is a region of interest near the open-loop resonance to examine in detail. 100 150 200 250 Frequency (Hz) 350 Figure 8: Bode Plot for Differentially Controlled Mass The blue plot (thick) represents the closed-loop transfer function for the system. The red plot (thin) represents the open-loop transfer function of the control-mass system. Our system only approaches the ideal of unity at zero phase from about 100 H z to about 220 Hz . Yet over this region, where the system has an open loop resonance, the control response is good. Differential control alone of a simple second order system performs wel l near resonance with degraded performance near D C and above resonance. 12 The open-loop plot is the combination of differential control and the fundamental system. Notice that the fundamental resonance is still present but the shape of the amplitude function has changed. This is the nature of \"Bode plot driven design\" of control systems where the open-loop Bode plot is adjusted with controller functions to achieve a desired result. Notice that the closed loop transfer function is very close to unity from 100 Hz to 230 Hz and that the resonance is completely eliminated. 2.4 . S t a b i l i t y i n C o n t i n u o u s F e e d b a c k S y s t e m s The open loop transfer function (CP) can give some indication of stability. Stability criteria for systems can be inferred from the open loop Bode plot. The phase margin is the phase difference from -180 degrees at the frequency where the amplitude plot crosses unity gain. The gain margin is the reciprocal of the amplitude where the phase crosses -180: this must be greater than unity (ie. the amplitude plot must be below unity as phase crosses -180). Stability in a continuous system is explored more fully in the following section. If the complex phase of CP is -180 degrees at some frequency and the magnitude of CP at that frequency is slightly less than 1, the denominator of eqn 11 will be nearly zero, and the closed-loop system response will be very large. Physically, this is just an oscillator with low damping being driven near its resonant frequency. If CP is exactly equal to -1 at some frequency, the closed-loop response is infinite. Equivalently, when CP=-1, any value of y whatsoever is consistent with eqn 10. The physical meaning of this is that the system with feedback activated will oscillate indefinitely at constant amplitude at that frequency, even in the absence of any input. This is known as the Barkhausen criterion for oscillations. If the phase of CP is not exactly -180 degrees at some frequency, the only solution of eqn 10 at that frequency with r=0 is y=0. Self-perpetuating oscillations of a feedback system are not possible at a frequency where the phase of CP is not -180 degrees. If magnitude of CP is greater than 1 at a frequency where the phase of CP is -180 degrees, there is another solution to eqn 10 besides y=0, namely y=infinity. Physically, the plant output causes the controller to command a still larger plant output. The system with feedback will display growing oscillations at that frequency. (Predicting the actual growth rate requires knowledge about the plant and controller response at other frequencies.) 13 This is made concrete in figure 9, a Bode plot of the same system we have been studying, except with negative derivative feedback. Physically, this corresponds to anti-damping, and we expect the system to exhibit growing oscillations at the resonant frequency. We see that the open loop phase (CP in red) crosses -180 degrees at the resonant frequency, and the magnitude of the response is greater than one, so the Barkhausen criterion predicts the oscillations that we intuitively expect. Anti-dam ped System A ft j 1 V CD 0.1 0 ® C O 03 -180 J= CL [02 -0.01 0 0.01 0.02 Time (sec) 100 150 200 250 Frequency (Hz) Figure 9: Unstable System The plot on the left shows a system with slightly negative derivative gain and unstable time-domain response. The figure on the right show a corresponding Bode plot where the Barkhausen criterion is violated at resonance. Stable feedback control requires that the magnitude of C P be less than 1 at any frequency where the phase of C P is -180 degrees (or more generally, 180 + n x 360 degrees). Control engineers call the reciprocal of the magnitude o f C P at such a frequency the \"gain margin\" at that frequency. If the gain margin is less than 1, the gain is greater than 1, and the system is unstable. Higher gain margin is better. Typical ly, the magnitude o f C P falls below 1 at some frequency and continues to fall off at higher frequencies. The \"phase margin\" is the difference between the acuoal phase o f C P and the nearest unstable phase at the frequency where the magnitude of C P finally falls below 1. The system shown in figure 7 (with Bode plot shown in figure 8) is stable, with a phase margin o f 90 degrees. The gain margin is undefined because the phase never crosses -180 degrees. 14 The system shown in figure 9 applies negative derivative gain to demonstrate instability. This was a highly artificial example shown for stark simplicity. In actual practice, systems may show an unstable response to simple control strategies whenever there is more than one resonance. For example, the Bode plot in figure 10 below shows a system with two strong resonant responses at 100 H z and 300 Hz . The controller has been set to provide a derivative gain appropriate to control the first peak. Note however that the phase wi l l always cross -180 degrees at the second peak and that the system wi l l be unstable with this gain setting. The open loop response crosses -180 degrees at 300 H z and the amplitude is high at this point since it is a resonant point. Thus the system wi l l oscillate at the second resonant frequency even though the system is well designed to control the first resonance. ISO L 1 -- - — : - - - — r t o • -180 -360 50 100 150 200 250 300 350 400 450 500 Frequency (Hz) Figure 10: Two Resonance System This bode plots shows the open loop response (CP) in red and the closed loop response in blue. Derivative control response increases with frequency and any resonance above the fundamental may cause instability. For this reason, derivative gain is usually applied with a first or second order low-pass filter with the rolloff frequency below any additional plant resonances. 15 2 .5 . D i s c r e t e T i m e A common practice with higher order differential equations is to separate an n-order differential equation into n first order equations. The standard procedure with control system notation is to separate the control term so that natural system evolution and control response are clearly separated and the system is converted into a matrix vector equation. Notation convention used herein is that a bold typeface indicates a matrix whereas simple, non-bold italics constitute a scalar or vector as noted. A s an example, the system as described in figure 1 and in equation 1 can be represented as fol lows: x = ax + bu E c i n ( 1 2 ) Where a is the continuous system propagation operator, b is the continuous system response to the control actuator, x is the state vector, and u is the actuator length (measured from null). The state vector may be written as: y(t) dy{t) dt . where y is the test mass position scalar. Our specific numerical example from figure 1 is defined as follows: a = ( 0 V ' ° l) K-klm -\\x\\06 0, Eqn(13) Eqn(14) b = k/ V /m) f 0 ^ 1x10' Eqn(15) Where k and m are (as previously noted) the spring constant scalar and the mass scalar. Loosely based upon the derivation in Dutton et a l . [ 5 ^ , a continuous system can be translated into a discrete parameterization as follows: Firstly equation 12 may be integrated to solve for an evolving state x(t) as follows. x(f) = exp(a0*(0) + j j exp(a(/ - T))hu{r)dr Eqn( l6) 16 If we assume that u(t) is constant for the time interval h, this implies: x{h) = exp(a/*)x(0) + hu exp(a(/z - r))dr = exp(a/?)x(0) + bwa1 exp(a(/z - T)) | J = 0 = exp(a/*)x(0) + bwa1 (exp(a(/z) -1)) We now define the free and forced matrices: A = exp(a/z) B = a[A-I]b A n d we can then write x(h) = Ax(0) + Bu Eqn(17) Eqn(18) Eqn(19) Eqn(20) If u takes on a different value Uk for each time interval of duration h: xk+i = A x k + Kuk Eqn(21) Turning back to our simple numerical example, let us assume a sampling frequency o f 1 kHz. This time interval of 0.001 seconds results in the fol lowing parameters for our discrete system description: A = 0.5403 0.0008415 -841.5 0.5403 Eqn(22) B 0.4597 841.5 Eqn(23) Where A is the discrete system propagation operator and B is the discrete effect o f unit output. The solution of A was found by eigenvalue decomposition of the system explicit ly using the M a p l e [ 1 7 ] mathematics package. Alternatively, Ma t lab [ 1 8 ] offers a direct continuous to discrete system conversion command in its \"Control System Toolbox\" package. Using the above discrete system description, we can generate discrete system steps. The figure below shows such a series of discrete positions generated using the matrix values 17 above. These are superimposed upon the continuous system response shown previously in Figure 2. Discretely Generated Positions 2.5r -0502 -0.01 0 0.01 0.02 Time (sec) Figure 11: Discrete Evolution Sample of the Simplified Model The black symbols represent the discrete evolution of the system at 0.001 second intervals. The continuous system response is the blue curve. The actuator position is in red. In the previous figure, we see a set of generated discrete position values superimposed upon the continuously resolved system plot. The discrete values correctly and exactly superimpose on the plot. This would be true even i f the discrete time interval had been taken to be many cycles of the system oscillation. 18 2.6. D i s c r e t e Cont ro l Discrete-time descriptions of dynamical systems are ideally suited to analysis o f control using a computer, where measurement and control signals are updated at regular intervals. Proportional and derivative control o f the mass and spring system can be combined from eqn. 2 and eqn. 4 to be: u(t) = -Kpy(t)-Kd-+ + r(t) ± + r(A Eqn(24) dt The r(t) term allows the control to displace the equilibrium away from the natural y=0. If the sampling rate is high, we can approximate each function as discrete steps Eqn(25) uk=~Kpyk-Kd Uy^ ydt jk We can define a (1x2) gain matrix: K = [-Kp -Kd] Eqn(26) Which allows us to write: u k = K x k + r k Eqn(27) If we only wish to control the dynamics of how the system approaches (or possibly diverges from) the equilibrium, and are not interested in changing the equilibrium position, we can set r*=0. Then we can write the system evolution with feedback control as: xk+\\ = + B w * Eqn(28) = Ax* + BKxk = (A + BK)xk The stability properties of the system are determined by the eigenvalues and eigenvectors of the matrix: P = A + BK Eqn(29) If any eigenvalue of P is greater than unity, then the system is unstable. A n initial state x w i l l grow in magnitude on subsequent iterations (unless it happens to be orthogonal to all the eigenvectors whose eigenvalue is greater than 1). 19 The figure fol lowing plots contours of greatest eigenvalue for the P matrix against P D gains for our numerical example. ,x 10\"' Two-mode Model Ga in Stability 1.5 | 0.5 (O Q -0.5 0 1 2 Proportional Figure 12: Gain Stability Contours System propagation eigenvalue contour (contour interval 0.1). The outer triangle represents the locus o f points where the largest eigenvalue (absolute value) of the propagation matrix is exactly unity. Gain values outside of this region are unstable. Gain values inside of this region show decreasing values o f maximum eigenvalues. In the continuous control case, the system was stable for any positive value o f derivative gain Kd, and any value of proportional gain Kp greater than - 1 . In the discrete-time case the stable region is different. Interestingly, pure positive proportional gain is unstable. This is because the finite time step size keeps the control from reversing sign at exactly the same time the position reverses sign. That is, pure proportional gain in the discrete model introduces some lag. The time lag due to discrete sampling pushes the phase past -180°. Addit ionally, unlike the continuous case, there is a maximum stable value of proportional and derivative gain. Physically, this corresponds to applying a force sufficient to more than reverse the velocity during a single time-step. On the next step, an even larger force in the opposite direction wi l l be applied, causing a divergent oscillation. The discrete time steps introduce a phase lag that makes the system unstable in the absence of natural or artificial damping. 20 With a reasonable amount of derivative gain applied, there is still a maximum stable proportional gain. Physically, this corresponds to applying a force more than sufficient to reverse the position of the mass during a single time step. The allowable proportional gain increases as the derivative gain is increased because the derivative gain (when lower than its own instability threshold) tends to oppose the excessive proportional gain. A n obvious feature of the stability contour is the position near Kp=0.\\ and A^=1.7xl0\" 3 . This position has the interesting property that both eigenvalues are zero for these gains. This is known as the system deadbeat gain and can be explicitly derived (refer to Appendix B Deadbeat Gain) . The maximum stable gains, and the point of optimal gains, depend on the time step compared to the natural frequency o f the system with zero gains. The time step was chosen to be relatively large for this example, about 1/6 of the period at the natural frequency. This makes the optimal proportional gain relatively small. In other numerical experiments not shown here, i f the time step is made smaller, the maximum stable gains increase. Higher proportional gain increases the closed-loop oscillation frequency. The system typically becomes unstable when the closed-loop period is reduced to only a few time steps. We shall explore the stability region in gain space just a bit. The fol lowing plot is the system response with gain set at just 50% of the deadbeat gain activated at t=0 with a setpoint of 1. Gain at 1 / 2 Deadbeat 1 6 0-&O2 - 0 . 0 1 O 0 . 0 1 0 . 0 2 Time (sec) Figure 13: Simpl i f ied M o d e l wi th P D Gains Set at 5 0 % of Deadbeat The cont inuous response is shown w i th the smooth curve (blue) and the discrete posi t ions are shown at the t ick marks. The step funct ion (red) is the ideal response. 21 The response approximates an underdamped second order response. This would be exactly true in the case of continuously controlled output. However, since the output is controlled in steps and remains constant between tick marks, the output actually only approximates an underdamped second order response. Let us look at system response at the exact deadbeat gain settings. The plot below shows this theoretical response. Gain at 1 0 0 % Deadbeat 1.2 1 0 . 8 1 0 . 6 | 0.2 0 . 0 2 Time (sec) Figure 14: Simplified Model with PD Gains Set at 100% of Deadbeat This plot shows the system response to activation of the control system at t=0 sec. The setpoint is set at 1 nm to reflect the similarity with previous plots. The continuous response is shown with the smooth curve (blue) and the discrete positions are shown at the tick marks. The step function (red) is the ideal response. After only two tick times after start of control, the response is exactly flat. This is not an exponential asymptote but rather is exact. Superficially, this might appear superior to the exponential damping achieved in the continuous-time analysis of a single-mode system. However, for continuous time there was no limit to how small the exponential damping time constant could be for the corresponding single mass and spring system, while here we are limited by the sampling time. 22 2.7. Control of Two-Resonance System in Discrete Time In section 2.4 (Stability in Continuous Feedback Systems) we found that the addition of a second resonance could constrain the range of stable gains, but that discussion was only qualitative. In the previous section, we found that discrete time steps limited the range of stable gains. N o w let us combine the constraints of discrete time and of a second resonance, and in a quantitative way. The one-mass system as previously described is exceedingly easy to control wel l . The transponder mount to the bulkhead is extremely stiff at lN /um. There is only one system resonance. However, this simple model of a monolithic mass and infinitely stiff bulkheads is only adequate to explain and control the response of a real system up to perhaps 20Hz.. Beyond this l imit, the model becomes inadequate since the mass and bulkheads begin to exhibit non-ideal responses. Let us simulate one such addition to complexity meant to model a possible internal mode in the mass. Let us add a spring and thus an additional vibration mode to the system. Modified to Add Block Resonance Measurement \\ \\ \\ \\ \\ \\ \\ \\ Stiff Spring 1N/um Transducer (Length Adjust) Stiff Spring 1 N/um 0.5 kg Test Mass Figure 15: Modified Model Adding a Single Resonance This model adds a simple resonance without damping to the system. We have split the mass in half, and added a spring in the middle identical to the original single spring. N o w there w i l l be two modes of oscillation. One wi l l have the two masses moving roughly as a single unit with about the original frequency. For the other mode, the lower half-mass w i l l be approximately f ixed, and the upper mass wi l l move at about twice the original frequency (since it has half the mass, and sees twice the spring-constant). 23 The system is now a four state model represented as follows: x = And _{kx+k2)/ Q k2/ Q 0 { ° 1 V ,b = 0 ) I 0 ) Eqn(30) Let us now apply proportional and derivative control using the position and velocity o f mass 2 (the lower one in figure 15). The P matrix is P=A+Bw where u=(0,0,Kp,Kd) and A and B are exponentiated from a and b. Figure 16 shows the contours of the largest eigenvalue as a function o f Kp and Kd. The contour for the 2-state problem are also shown. -3Two-Resonance Model Ga in Stability - 2 - 1 0 1 2 3 4 Proport ional Figure 16: Two-Resonance M o d e l G a i n Stabi l i ty Locus of maximum eigenvalue of P Vs feedback gains: 1 for outer contour, decreasing by 0.1 per contour. Bold lines are for two-resonance system, finer lines for one-resonance system. In comparing the two-resonance system with the one-resonance system, the stable region is much smaller. The maximum stable gains are about a factor of 3 smaller. The \"deadbeat\" solution for the single-resonance system is a bit more than a factor of 2 outside the maximum stable gains of the two-resonance system. 24 Since the new mode of the system has about twice the frequency, it is not surprising that the maximum stable gains are reduced by a factor of about two. A lso , the maximum eigenvalue of the P matrix is never zero, so there is no \"deadbeat\" solution that w i l l bring the mass to a dead stop in two time steps. 1.5 I 1 S 0.5 :o2 -0.01 Time (sec) 0.01 0.02 Figure 17: Modified Model at Previously Stable Gains The black points show the system at discrete time samples, the blue plot is the continuous motion of the system and the red trace is the ideal response. Figure 17 shows the time response of the lower mass in the two-mode system, using the same gains as in figure 13, which are slightly outside the stable region of figure 17. For t<0, the setpoint is r=0, and the mass shows motion in both modes. A t t=0, the feedback is turned on and the setpoint is changed to r=l . The mass then starts oscillating around the new setpoint, but the oscillations are not damped, and in fact are slowly growing. Interestingly, the discrete time samples (the black crosses) show much less motion than is actually occurring between the samples. It should be noted that the feedback gain in this model is applied to the actual velocity, the slope of the smooth curve, rather than the velocity that might be derived from finite differences of the discrete time samples. A natural question is, can we do better by allowing the actuator setting to be a function o f all o f the elements of the state vector x, using more gain coefficients? For this system, we need to search in a 4-dimensional gain space. 25 We can plot multi-dimensional stability contours by projecting the maximum eigenvalue found in al l non-plotted axes onto two axes which can be plotted. Six plots then can map the stability zone onto al l possible gain pairs for a four-state system. Let us now plot the stability region of our four state model allowing \"perfect\" information from our two masses (position and velocity of each mass) to be used by our control algorithm. Our controller is still described by equation 29 only now the vector K consists o f gains for each of the four system states (mass 1 position, mass 1 velocity, mass 2 position, mass 2 velocity). Figure 18: Two-Resonance Model for 4-Dimensional Gain Stability Locus of maximum eigenvalue of P Vs feedback gains: 1 for outer contour, decreasing by 0.1 per contour. Bold lines are for two-resonance system, finer lines for one-resonance system. Each plot is a projection of the 4 dimensional locus onto two gain axes. 26 It appears that there is still a deadbeat solution, with all 4 eigenvalues of the P matrix being zero. But it requires that the acnaator setting be a function of the position and velocity of both masses. The deadbeat gains are positive damping for both velocities, slightly positive proportional gain for one mass, and nearly -1 proportional gain for the other. The solution does not appear to be fine-tuned in the sense that small changes in any gain parameter still gives reasonably small maximum eigenvalues. However, small changes in the parameters o f the mechanical system can cause large changes in the location of the deadbeat gains. It is shown in Appendix B that there is a deadbeat solution to any order N linear control problem that makes the entire state vector zero. The solution generally requires N time steps to reduce al l elements of an arbitrary initial the state vector to zero. However the solution theoretically still exists even when the time-step is reduced to zero. This would appear to be potentially superior to the continuous-time case, where the introduction of additional modes put a limit on the gain and thus on the exponential time constant. This is not a fair comparison, because the discrete time deadbeat solution uses more gain coefficients than we used for the continuous time two-mode case. It is possible to use more gain coefficients and reduce the exponential time-constant to zero in continuous time as wel l . Numerical experiments indicate that in both the continuous and discrete time cases, large gain values are required to achieve solutions with fast control (short time steps). A lso , the positions and velocities may take on extreme values during the N time steps of the solution. 2 .8 . C o n c l u s i o n For a simple mass and spring mechanical system, with control applied through a piezoelectric actuator, proportional gain increases the resonant frequency, and derivative gain damps the oscillations. Control gains can be determined to artificially raise the resonant frequency while applying artificial critical damping. In continuous time, there is no limit to the speed of control in such a simple system. The response of a system to the control actuator, with or without feedback control, can be described through a Bode plot of magnitude and phase versus frequency. If the magnitude of the response of the system and controller (including the gain coefficients) exceeds 1 at a frequency where the phase is 180+n x 360 degrees, the system wi l l be unstable at that frequency i f feedback is applied. This can greatly limit the maximum stable gain for systems with multiple resonances, i f the control algorithm is limited to use only the value and derivative of the system output. 27 It is possible to describe exactly a continuous time system with actuator settings that are updated at discrete time intervals using a matrix formalism. The maximum stable gain is limited even for a single-resonance system. But there is a gain setting that w i l l bring a single-resonance system to a stop in 2 time steps, no matter how short the time steps are. If there are multiple masses and springs, but the feedback control responds to only a single position and velocity, the maximum stable gain can be greatly reduced. If the feedback control responds to all the positions and velocities, there is again a solution that brings all M masses to a stop in 2 M time steps. However i f the time steps are short, the required gains can become very large. 28 3. A p p a r a t u s 3.1. I n t erf e ro m et er The interferometer for nanometer position measurements was assembled from standard optical laboratory equipment and is shown in Figure 19 below. It is in Michelson configuration, except the arms are made parallel by a 45 degree mirror. This allows the arm lengths to be extended easily, and keeps the light paths close together so they traverse the same air density, reducing some sources of error. Figure 19: Interferometer Setup Photo of Michelson interferometer setup. The reference arm optical path has been reflected to be in the same direction as the measurement path. The red laser has been added with a paint package for illustration since the beam itself does not show in the photo. In actual operation, one optical baseplate has the laser, the beam splitter, and the 45 degree mirror, which is mounted on a Physik-Instrumente [ 1 9 ] piezoelectric actuator to allow the optical path length of one interferometer arm to be varied for calibration and tests. This piezo is far larger than required to move the small mirror, but the large size is convenient for attaching the mirror and for translating it without tilting it. The two end mirrors are located asymmetrically to compensate for the path-length difference introduced by the 45 degree mirror. 29 The light source is a Melles-Griot model STP-901 intensity/frequency-stabilized HeNe laser with power at 1 milliwatt. It is possible to adjust the mirrors so light does not go back into the laser aperture, so an optical isolator is not required. For arm lengths o f less than 10 meters, the divergence of the beam is small enough that a collimator is not required either. The interference pattern is expanded by a cylindrical lens onto a linear si l icon photodiode array. The orientation and spatial period of the fringes can be controlled by adjusting the mirror angles. Intensity control is done by vertical steering of the laser partially off of the sensitive area of the photodiode array. 3.2. Electronics The photodiode detectors were operated in unbiased mode, with current signals travelling in coaxial cables to a multichannel amplifier unit. Each channel had an OP-27 amplifier configured to produce a DC-coupled current to voltage output capable o f producing a saturated 10 Vol t output with moderate laser light incidence upon the detector. For most o f the test-platform control experiments described later, it was convenient to use a simple capacitive position sensor instead of the interferometer. This was a Physik-Instrumente model D100 sensor head pair and model E610-C0 electronics module. This combination has a noise figure of 8 pm per root Hertz, for a position resolution of 0.6 nm at our 5 k H z sampling frequency. For early experiments we used a commercial piezo driver to buffer our digital output to the piezo. Specifically, the model used was Physik-Instrumente model E610-00 with a voltage gain o f 10 and maximum output of 100 volts and 60 mill iamps (RMS) . This created a slew-rate issue with respect to our control system. Over the ful l scale of 100V, with a lOuF piezo, slew time at 60mA is 17mS. Our digital control system data-acquisition/control loop time is 0.2mS. Even a 1% of ful l scale adjustment would require the full loop time to slew. This proved unacceptably slow. To insure that control effects were linear and related to the system and not to slew rate, we needed a much higher current source. Since we didn't actually need the ful l voltage range of our piezo, we designed a 24V push-pull output amplifier which was capable of delivering 2A. Schematic for this amplifier is seen in the figure 20 following: 30 Figure 20: Piezo Driver Circuit Schematic V Monitor (IX I/P) This simple push-pull current driver also provides a X 2 voltage amplification. Cross-over distortion is negligible since the O/P load is capacitive. V monitor can be used to verify actual output voltage and test for slew rate difficulties. 3.3. Data Acquisition Data acquisition and control used an Adwin-Go ld real-time computer system 1 1 4 1 . The on-board D S P is programmable in B A S I C and has a 25nS clock cycle with B A S I C statements taking as little as four clock cycles and as great as 400 clock cycles for floating point math functions such as log. Data presentation is made using the \" Igor\" [ 1 5 ] data manipulation package from Wavemetrics. Interface software between the Adwin-Go ld and I G O R is provided by R T S Consulting L t d [ 1 6 ] . Real-time oscilloscope functions as wel l as control algorithm communications using window tools such as sliders is made relatively straightforward. Real-time determinacy is guaranteed since the Adwin-Go ld is a dedicated real-time module and the communications to and from the module are relegated to background process within the Adwin . Using the Adwin-Go ld in conjunction with the I G O R software, it is quite straightforward to create hybrid instruments, to test control strategies and to perform data acquisition. 31 3.4. Lock-in Amplifier One notable hybrid instrument is the lock-in amplifier used to take spectral system response data. The instrument consists of a sine-wave frequency generator with a phase-locked measurement correlation. A n executive control can sweep the frequency and archive the individual frequency responses. The software architecture of this instrument is laid out in figure 21 following. Lock-in Amp/ Freq Generator Executive Control GUI Settings Min Freq Max Freq Freq-Step Amplitude Settling rime Acquire Time Sequencing And/ Status Bode Plot (in progress) he DSP Fast Loop Frequency -2H Frequency Generator Cosine Data Stream Cosine Correlator Communications rate Set bf Setting-time plus Acqure-time (wery stow! Sine Data Stream ^ D/A Output D a t a a t e set at loop rat* (10kHz) Sine Correlator • Measurement A/D Input Figure 21 : Lock - i n Amp l i f i e r B lock Schematic Execut ive Con t ro l : The executive control is required to sequence through the frequency limits and apply a settling time and acquire time for each individual test frequency. This process is very slow and does not need to be deterministic. However an additional signal (not shown) for end-of-acquisition and change-frequency is required for the D S P fast loop since correlations are only made using complete sine-wave cycles and zero crossing must thus be used as triggers. 32 Frequency Generator: Frequency generation is performed using floating point calculations for both the phase operand o f the trig functions and the trig functions themselves. This allows the system to operate at frequencies without an algorithmic lower bound. Practically, the lower bound is limited to 0.001 H z or so simply due to realistic acquisition-time requirements. Similarly, the floating point phase operand allows frequencies of any non-harmonic (of the loop time) and effectively allows operation right up to the Nyquist frequency o f the loop time (in this case fn=5kHz). Correlators: Correlation is performed at full loop rate and is simply the accumulated sum of the measurement signal multiplied with the sine (or cosine) data stream. Correlation values are initiated and archived at zero crossings resulting in accurate integrations even over very few cycles. This allows very low frequencies to be analyzed. 3.5. Pos i t i on Contro l T e s t P lat form The test platform assembly is intended to allow a 10-100 kg mass to move in one dimension, with force applied passively by an adjustable spring constant, and controlled actively by a piezoelectric actuator, but constrained from moving in the other 5 degrees o f freedom. The concept is illustrated schematically in figure 22 below. UBC Nanometer Vibration Stabilization Test Platform Design Replacable Spring Feedback Piezo Drive Piezo Figure 2 2 : Test Platform Isometric Drawing The design allows for various spring constants to be used for platform connection to the bulkhead through the piezo actuator. The elimination of all but one degree of freedom is achieved from the flexure mounts. Provision is made for both capacitive position sensing and interferometric position sensing. 33 The actual platform was constructed out of aluminum and shown in the photograph below: F igure 23: Test P lat form Photograph The baseplate assembly has mounts for a pair of aluminum or steel flexures to hold the test platform. It also has end posts for controlling and measuring the platform position. The entire assembly has provisions for bolting to the floor, although it was simply resting on the floor or isolated by innertubes for the data presented here. A t high system control gains, it was excitations of mechanical modes within the base structure which provided some of the most intractable resonance responses. Future test platform designs anticipate deliberate construction of a reaction mass rather than a bulkhead for the control actuator (piezo element) to act against. 34 The left baseplate end post has an optical mount used to adjust the gap and parallelism of a Physik-Instrument model D100 capacitive position sensor. It also has a fixed mirror for the reference arm of the interferometer, and an anchor point to f ix the moving platform (relatively) rigidly in place. The test platform to be controlled is a solid aluminum block. A t the right end there is a Physik-Instrumente model P840-10 piezoelectric actuator, and on the upper right surface there is an anchor for a fixture to adjust the tension of piezo preload springs. The right baseplate end post has a slider with another PI model P840-10 piezo, and eyebolts for the preload springs. The detail o f the actuator can be seen in figure 24 below. Figure 24: Actuator Detai l The V shaped part is the main reaction spring with the pretension for this spring supplied by the top pair of tensioning springs. Between the two piezos is a U-shaped aluminum \"spring\" with multiple attachment points. Changing which attachments are used changes the spring constant for the test platform's motion relative to the baseplate. 35 It was intended to be possible to change the mass of the test platform by simply putting more objects on top. A s w i l l be seen, it proved difficult enough to control the platform at minimum mass, and such high-mass experiments have yet to be performed. 36 4. I n t e r f e r o m e t e r 4.1. Interferometer S igna l Mode l 4.1.1. Physics The optical system is a folded Michelson interferometer that was described in section 3.1: Interferometer. Figure 25 shows the classic Michelson configuration The mirrors are adjusted so the two laser beams are not quite parallel at the detector plane, so the beams interfere constructively at some points and destructively at others, forming interference fringes. The fringe pattern migrates across the detector plane i f the length of either interferometer arm changes. The change in path length is twice the amount that the target mirror moves. For a Ffe-Ne laser with 632.8 nm wavelength, 50.35 nm of mirror motion corresponds to a beam phase change of 1 radian. To Digitizers Processor Detectors Interference Region .— Beam Expander Beamsplitter Target (moving) Mirrior « > Laser (777777 Stationary Mirror Figure 25: Interferometer Setup 37 The electric f ield at photodiode / may be written Et = Ft exp i(k • 3c, -eot + A -A-(pB-The photodiode current is proportional to light intensity, and is converted to a voltage by the linear analog electronics. The digitized voltage for photodiode / can thus be parameterized as Eqn(34) where ct is due to \\FJA\\ + \\FJB\\ plus electronic offset and any background light, bt is due to 2(FIA -FiB)md Si is (kA-kB)-x-nll If we know the constants ct, bt, and 8t, we can take a single voltage measurement F, and solve for There are multiple solutions, differing by multiples of 2TI, and also solutions with the same value but opposite slope for the sine function. If we have a second voltage measured by a detector with a different St, we can eliminate the opposite-slope ambiguity. If we take a 38 sequence o f measurements over time, frequent enough that the phase change between measurements is much less than 2n, we can track motions through many wavelengths. In our application, the remaining arbitrary overall phase offset is acceptable. Thus with two light intensity detectors, we should be able to reconstruct the motion of the interferometer end mirror. 4.1.2. Direct Parameterization Signals from such a system have an appearance characteristically shown in Figure 26 (below). This figure shows the signals expected in two detectors for a bit more than a quarter cycle of sinusoidal mirror motion with an amplitude o f a few microns . The intensity measurements vary sinusoidally with target position. But the amplitude, offset and phase shift are arbitrarily set by system optics and detector sensitivity. Individual detectors w i l l have different sinusoid envelope parameters. A model can be constructed which assumes that all angles in the system remain constant and the only variation with target position is the phase of the measurement arm. The explicit model then is given below where for our purposes we wi l l restrict ourselves to two detectors and thus i s {1,2}. v,(p) = c,+&,sin(co + c?,) Eqn(35) This is simply a restatement of equation 34. Where v, is the measured voltage, ct, bt are arbitrary gain coefficients, cp is the absolute phase due to mirror position, St is the relative phase for different detectors (Si is arbitrarily set to zero) and where i is the detector index. Position 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Measurements 0 0.1 0.2 0.3 0 4 0.5 O.i 0.7 0.J 0.9 1 Figure 26: Characteristic Interferometric Measurement Data Top is slow sinusoidal position vs. time. Bottom is the photodetector signals vs. time for typical parameters. T (sec) 39 With only two detectors, it is desirable for accuracy that the interference phase difference between the detectors should be set close to 90°. This is the optimum phase difference for standard \"quadrature\" phase detection. The phase difference can be easily adjusted through various mechanical manipulations such as mirror rotation and detector distance. 4.1.3. Calibration For two detectors, there are five calibration parameters in the model from equation 35. These are summarized below. 1 Cl Detector 1 voltage offset (v) 2 b, Detector 1 amplitude (v) 3 Detector 2 voltage offset (v) 4 b2 Detector 2 amplitude (v) 5 5 Detector 2 phase compared to detector 1 Figure 27: Table of System Parameters Once the five calibration parameters are known, it is quite straightforward to solve for a unique position given the two measurements. That is, there are two equations with one unknown. The redundant information is sufficient to resolve ambiguities which arise when solving individual inverse sine equations. However, the five calibration parameters are N O T known. This causes difficulty in designing a detection algorithm. The principle design requirement is to calculate a continuing series of position measurements from a continuing series of light-intensity detector-pair readings. This sounds very straightforward. A s a static analysis of lab data, it would be quite simple. Given a measurement data set, it is straightforward to calculate, a-posteri, a position data set. Firstly, the measurement data set could be plotted to establish regions of time where the position is exploring a large-scale trajectory. If such a time span exists where the position is traversing a region of the order of a half wavelength, then a first estimate of parameter values can be made simply by extraction of detector maximums and minimums. These first estimates can be refined using a linearized model centered on the initial estimates and then very finely resolved with a least-squares fit. Such a system has limitations i f the parameters are not static but slowly varying. Standard analysis practice would then be to sub-divide the sample into time-slices small enough to 40 regard the parameters as constant. Within these slices, parameters could be independently fit. Statistics relating parameter change rates could be determined and a position data set could be generated for the entire time period. O f course, again, each time slice must be analyzed using data which explores a sufficient region of the sinusoid to enable a fit; and this is just not certain. Other design teams have had excellent success with this problem using multiple detectors (>2) and an explicit calibration procedure [ 7 ]. These algorithms are composite in that the position is actually controlled and thus manipulated to insure a non-stalled state during calibration. This relaxed constraint allows a statistically sound calibration phase to operate with statistically predictable error. 4.1.4. Addi t iona l Algor i thm Design Cr i te r ia I wanted to explore a different approach which allows for some improved operational criteria. Addit ional criteria are: • N o explicit calibration procedure. • Min imal assumptions about calibration parameter values at startup. • The calibration parameter estimates should monotonically improve in accuracy from startup. • The algorithm should be capable of implementation on a moderately priced D S P processor as of 2003 vintage with an algorithm iteration / data rate o f about 5kHz. This precludes large dimensional array inversion procedures. • The calibration parameter estimates must not degrade even i f the position remains essentially constant for long and unpredictable times. 41 The algorithm must be capable o f achieving a good estimate from an initial extreme condition of ignorance. The physical limits are: • The amplitudes and offsets are less than the saturation voltage and more than zero. • The phase angle between 0 and 180 degrees since another val id selection between 180 and 360 degrees w i l l simply result in a velocity inversion of any calculated position. The final design criteria expressly allows a stalled trajectory. That is, the algorithm has no control nor a-priori knowledge of whether the trajectory is stalled. Nor does it have any knowledge about the length of the stalled time interval. Clearly, the algorithm can never resolve a position i f the trajectory is permanently stalled. Therefore, the assumption is that at some time the trajectory W I L L explore a reasonable expanse. The algorithm must be prepared to wait an indefinitely long period of time for this to occur. 42 4.2. Geometric Parameterization In finding a solution to the dilemma presented by startup combined with a stalled trajectory, the principle issue is to find a system which wi l l converge correctly from any starting location. For non-linear systems in general, this entails understanding local minima to which a system might become attracted. Alternatively, i f a system has a simple minimum then the other aspects of the design become tractable. In this design there is an easy-to-visualize system model which is geometric. The origin for this idea came about from observing a plot of the two detector signals on an oscilloscope plotted against each other in X - Y mode. The position was free moving at the time and definitely not stalled. The oscilloscope was presenting a Lissajous ellipse pattern. Every half-wavelength motion of the position resulted in one traverse of the ellipse in the pattern. For a phase difference of 90 degrees and equal amplitude parameters between the two detectors, such a Lissajous pattern traces a circle and the position is simply proportional to the generating vector angle. T (sec> \" Det 1 (v) Figure 28: Lissajous Pattern Graph above on the right shows the Lissajous pattern for the same characteristic data as figure 26. The pattern is actually traced once for every half-wavelength position change (this is not evident in the plot). Data from figure 26 is replicated on the left for reference. The main idea of the new algorithm can be simply stated: • The geometrical model plots the two measurements against each other; then normalizes the measurements into a circle. The position is then taken as the arc-tan of the normalized measurements. A t this point it may not be clear why this has any advantage to the more straightforward model given by the direct parameterization of equation 35 (pg 39). The answer to this lies in 43 the fact that the parameters are easy to visualize (illustrated subsequently) and furthermore, iteration techniques can be designed which always converge for an easily defined starting condition. 4.2.1. Geometric Parameterization Validity There are three questions which must be answered to verify the geometric parameterization's validity: 1. How are the measurements normalized into a circular Lissajous pattern? 2. What are the parameters of the geometric parameterization? 3. Does the normalization procedure map position to angle proportionately for all positions and angles? (1) How measurements are normalized: A linear mapping can be easily achieved by first operating on the measurements to remove the offsets so that the ellipse pattern is centred at the origin. Subsequently, a general matrix operation can be applied to circularize the pattern. Any anti-symmetric component to this matrix would only serve to rotate our measurement vector and is therefore not wanted or required. Hence the matrix can be assumed to be a symmetric operator. (2) Parameters of the geometric model: The geometric parameterization is best illustrated in matrix form as follows: p(0 + t//) = S(v( + S) cos(^ + ^ ) y Eqn(39) 45 Where Z>/are the measurement amplitudes, A,B,D are the symmetric tensor (S) parameters,