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UBC Theses and Dissertations

Nanovibration control Greenall, Russ 2004

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NANOVIBRATION C O N T R O L by RUSS GREENALL  B . S c , Simon Fraser University, 2001 A T H E S I S S U B M I T T E D IN P A R T I A L F U L L F I L L M E N T O F THE REQUIREMENTS FOR THE DEGREE OF Master of Science In  The Faculty of Graduate Studies ^ Department c  ' •=  Physics^  University of British Columbia  W e accept this thesjs as conforming to the required standard  T H E U N I V E R S I T Y O F BRITISH C O L U M B I A 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.  Name of Author (please print)  Title of Thesis:  D  e  9  r  e  e  :  /\J ^  n  nOn.cK,',  Date (dd/mm/yyyy)  n wf  e  f  Department of The University of British Columbia Vancouver, B C Canada  L rr,  fSo„  P / w . „ ,  c  C „  Year:  (  Z O O 4-  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. A n interferometric instrument design for remote distance measurement is discussed and measurement results reflecting an accuracy of 0.2nm R M S 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 D S P 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 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 R M S vibration of our test system over our simple PID control.  ii  mMm @ff €®trafteffifts ABSTRACT  ii  TABLE OF CONTENTS  iii  TABLE OF FIGURES  v  1.  INTRODUCTION  1  2.  CONTROL THEORY FOR PHYSICISTS  3  2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 3.  3 5 9 13 16 19 23 27  .-  APPARATUS 3.1. 3.2. 3.3. 3.4. 3.5.  4.  SIMPLE CONTINUOUS MODEL CONTINUOUS CONTROL BODE PLOTS AND TRANSFER FUNCTIONS.. STABILITY IN CONTINUOUS FEEDBACK SYSTEMS DISCRETE TIME DISCRETE CONTROL CONTROL OF TWO-RESONANCE SYSTEM IN DISCRETE TIME CONCLUSION  29  INTERFEROMETER ELECTRONICS DATA ACQUISITION LOCK-IN AMPLIFIER POSITION CONTROL TEST PLATFORM  29 30 31 32 33  INTERFEROMETER 4.1.  INTERFEROMETER SIGNAL MODEL  4.1.1. Physics  37 37  ;  37  4.1.2. Direct Parameterization  39  4.1.3. Calibration 4.1.4. Additional Algorithm Design Criteria  40  4.2.  5.  GEOMETRIC PARAMETERIZATION 4.2.1. Geometric Parameterization Validity 4.2.2. Parameter Iterations in Geometric Model 4.2.3. AIgo rithm Tests 4.2.4. Non-Correlated Noise Test  43  CONTROL TESTS  60  5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8. 6.  PLATFORM RESPONSE GROUND-MOTION CHARACTERISTICS PID CONTROL MODEL RESULTS EXPLORATION OF GAIN LIMITS PID CONTROL WITH RESONANCE NOTCHING COHERENT DISTURBANCE CORRECTION BEST COMPOSITE PERFORMANCE  60 62 66 67 70 72 73 76 77  iii  44 47  56 58  -.  CONCLUSIONS  41  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 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure  1: Simplified Model of a Single Mass with Position Control 2: Sample Evolution of the Simplified Model 3: Evolution of "Stiffened" System Kp=10 4: Damped System 5: Simple Mass on a Spring Viewed as a Transfer Function 6: Bode Plot for Simple Mass on Spring 7: Mass on a Spring Control Schematic 8: Bode Plot for Differentially Controlled Mass 9: Unstable System 10: Two Resonance System 11: Discrete Evolution Sample of the Simplified Model 12: Gain Stability Contours . 13: Simplified Model with P D Gains Set at 50% of Deadbeat 14: Simplified Model with P D Gains Set at 100% of Deadbeat 15: Modified Model Adding a Single Resonance 16: Two-Resonance Model Gain Stability 17: Modified Model at Previously Stable Gains 18: Two-Resonance Model for 4-Dimensional Gain Stability 19: Interferometer Setup 20: Piezo Driver Circuit Schematic 21: Lock-in Amplifier Block Schematic 22: Test Platform Isometric Drawing 23: Test Platform Photograph 24: Actuator Detail 25: Interferometer Setup 26: Characteristic Interferometric Measurement Data. 27: Table of System Parameters 28: Lissajous Pattern 29: Symmetric Matrix Operation 30: Circularization 31: First Order Iteration to Find Centre 32: Orthogonal Iteration to Find Centre 33: Centre Finding Operation 34: Interferometer and Algorithm Test 35: Non-correlated Noise Test 36: System Bode Plot 37: Uncontrolled System Vibration Spectrum 38: Measurement Model (No Control) 39: Ground Spectrum Approximation v  ,  .3 4 6 7 9 10 11 12 14 15 18 20 21 22 23 24 25 26 29 31 32 33 .34 35 37 39 40 43 48 50 51 52 54 57 58 61 62 63 65  Figure Figure Figure Figure Figure Figure Figure  40: 41: 42: 43: 44: 45: 46:  Control System Model Model Response PID Control System Results PID Control System Bode Plots Gain Increase due to Notch Filters Coherent Disturbance Elimination Best Control Achieved  66 68 69 70 72 74 76  vi  Particle accelerators have been increasing in power since their inception i n the 1930's. Since the advent o f two-beam particle colliders i n the 1950's, beam diameter has been decreasing to increase luminosity. The proposed "Next Linear C o l l i d e r " project proposes a beam diameter as small as 2 X 200 nanometers small dimension.  [1]  or just about twenty times the diameter o f an atom i n the  A t this diameter, stabilization o f final focusing magnets against  environmental vibration becomes critical. Acceptable levels o f stabilization w i l l be on the order o f one nanometer. Some concern has been raised regarding the magnitude o f the ground motion which w i l l be encountered i n the N L C environment ' ' . [2]  [3]  These sources indicate an overall expected  [4]  vibration level on the order o f 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 o f up to hundreds o f nanometers i n 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 o f 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 o f the spot size at the beam rate o f 120Hz. This beam deflection provides the only long-term measure o f the relative positions o f the beams. A variety o f feedback algorithms can be used with the beam beam deflection data, with the selection based on the trade o f f between l o w frequency attenuation, and high frequency amplification o f 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 "fixed stars". A t low frequencies the position noise o f 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 i n chapter 4, (page 37). 2. Active control o f one dimension o f a large mass to nanometer precision using the paradigm o f "ground tracking" or "optical anchor". First principles analysis would imply that position control o f 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 o f 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 o f this order invariably excite dozens o f internal modes o f the large mass and/or o f the support structure which are N O T negligible. This is explored i n chapter 5, "Control Tests" on starting page 60.  2  2. Control T h e o r y for P h y s i c i s t s 2.1.  Simple Continuous Model  We shall start with the simplest conceivable model for nano-position control. Reduced to simplest terms o f one degree o f freedom and a monolithic mass the system is as shown i n the figure below.  \\\\\\\\ Stiff Spring 1N/um  Measurement  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 o f the mixed magnitudes. That is, perform a numeric example where our mass is considered to be 1kg and yet we are interested i n 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. W e just have a mass with one degree o f 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 o f motion is as follows from a simple force balance (y measured from equilibrium):  d y(t)  Eqn(l)  2  0=m  dt  2  + ky{t) - ku(t)  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 o f natural evolution and the control function is y(t). Since we take the damping i n this system to be negligible, this system w i 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. B e l o w is a plot o f the system evolution generated using numerical methods. The plot shows an initial small oscillation o f 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 R e s p o n s e 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  Overall, 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.  Continuous Control  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 o f physical equivalence. That is, the control function can be made to artificially "stiffen" and " d a m p " 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. W e would simply apply this pure frequency to the actuator (u(i)) and k n o w that the system response would be to respond at exactly the same frequency. W e 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) = -K y(t) + r(t)  E c  p  l ( ) n  2  K is called the proportional gain. The negative sign is included since by convention, the state o f negative feedback is considered the norm and K is expected to be positive. The term r(t) p  p  5  is included here as the step function shown in red on all o f the plots. It is similar in function to what would normally be referred to as the setpoint or demand function i n 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:  d y(t) 2  0 =m  dt  2  + k(\ +  K )y(t)-kr(t)  Eqn(3)  p  Examining equation 3 above, we can see that the effective spring constant for the system is now k(\+Kp). W i t h 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 i n 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 K p = 1 0 The red step function shows the demand or setpoint function and the system response is shown in blue.  Note that i n this resultant system, what is known as positive feedback would result from a negative K (actuator action which does not oppose the error). Positive feedback up to a point may be applied to "soften" the system. A s K approaches -1 the system formula becomes that p  p  6  of an unconstrained mass. This positive feedback would be limited to K - -1 at which point the natural spring would be overcome by the control and the system would become unstable. p  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  Eqn(4) at  This relation establishes derivative control where Kd is defined as the derivative gain. This leads to a system equation as follows:  dt  Eqn(5)  di  1  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  A-  0.8  !  0.6  55 0.4 o Q_  0.2  02  -0.01  Time (sec)  0.01  Figure 4: Damped System 7  0.02  If we include a combination o f both proportional and derivative control, our new equation o f motion becomes:  d y(t) 2  0 =m  dt  2  dy(t) dt  +  k(\ +  K )y(t)-kr(t)  Eqn(6)  p  This standard form o f damped simple harmonic motion has a well known condition for critical damping about the equilibrium point which is:  Eqn(7)  In our previous figure (4) showing a damped response, Kd was set to 0.001 with no proportional gain. Critical 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 ^ = 9 9 9 9 . This is not to say that a real system can be controlled infinitely well by applying infinite feedback gain. W e 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 i n digital implementations. Another form o f 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 ( D C ) requirements and therefore we w i l l not cover this type o f control here. Together, proportional, integral and derivative control are known as P I D control and are used extensively both industrially and i n the laboratory as an excellent generalized first approach to a control problem.  8  2.3.  Bode Plots and Transfer Functions  It is useful to analyze systems i n terms o f transfer functions. A transfer function is simply the amplitude and phase relationship between the input and output o f a system. For example, i n 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 o f time. This is why the diagram shows the funtions u(co) and y(co) as functions o f frequency co. The system response (often referred to as the " P l a n t " 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 o f 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 o f a reference to frequency, transfer functions are assumed to apply to Fourier space exclusively. Application o f a signal to a transfer function can be explicitly achieved simply through complex multiplication at each frequency. 9  ^  The presentation of a transfer function can be made through use o f 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  -Hi)  r(o)  setpoint (demand)  Piezo Signal  u(o>)  >'(©)  SH System response toPtezo  Controller  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 implicitly on itself through the difference node output (r-y). The resulting system equation is: Eqn(lO)  y = (r-y)CP  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_ r  Eqn(ll)  CP 1 + CP  This equation only applies i f the system is stable.  11  A n ideal control system has an overall closed-loop transfer function o f 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 o f control using only derivative control to influence our mass on a spring. A g a i n applying only .£^=0.001 (sub critical damping ), the following plots reflect the closed loop response o f the same system as shown figure 4. The plot is no longer logarithmic i n frequency as there is a region o f interest near the open-loop resonance to examine i n detail.  100  150  200  250  350  Frequency (Hz)  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 o f unity at zero phase from about 100 H z to about 220 H z . Yet over this region, where the system has an open loop resonance, the control response is good. Differential control alone o f a simple second order system performs well 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.  Stability in Continuous Feedback 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 o f 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. W e see that the open loop phase ( C P in red) crosses -180 degrees at the resonant frequency, and the magnitude o f 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 [02  -0.01  0 Time (sec)  0.01  CD 0.1  0  ® CO  03 J=  -180  CL  0.02  100  150  200  250  Frequency (Hz)  F i g u r e 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 o f C P be less than 1 at any frequency where the phase o f C P is -180 degrees (or more generally, 180 + n x 360 degrees). Control engineers call the reciprocal o f 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. Typically, 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 o f C P finally falls below 1. The system shown in figure 7 (with Bode plot shown i n 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 i n 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 i n figure 10 below shows a system with two strong resonant responses at 100 H z and 300 H z . The controller has been set to provide a derivative gain appropriate to control the first peak. Note however that the phase w i l l always cross - 1 8 0 degrees at the second peak and that the system w i l l be unstable with this gain setting. The open loop response crosses - 1 8 0 degrees at 300 H z and the amplitude is high at this point since it is a resonant point. Thus the system w i l l oscillate at the second resonant frequency even though the system is well designed to control the first resonance.  ISO  o  --  1  L  - —  :  - -  - —  r  t  400  450  •  -180  -360 50  100  150  200  250  300  350  500  F r e q u e n c y (Hz)  F i g u r e 10: T w o 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.  Discrete Time  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 i n figure 1 and i n equation 1 can be represented as follows: 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 .  Eqn(13)  where y is the test mass position scalar. Our specific numerical example from figure 1 is defined as follows: a=  (  V  0  -klm f  k/  V /m)  Eqn(14)  ) 0,  l  -\x\0  K  b=  ' °  6  0 ^  Eqn(15)  1x10'  Where k and m are (as previously noted) the spring constant scalar and the mass scalar. Loosely based upon the derivation i n Dutton et a l . ^ , a continuous system can be translated into a discrete parameterization as follows: [5  Firstly equation 12 may be integrated to solve for an evolving state x(t) as follows.  x(f)  =  exp(a0*(0) + jj exp(a(/ - T))hu{r)dr  16  Eqn(l6)  If we assume that u(t) is constant for the time interval h, this implies:  x{h) = exp(a/*)x(0) + hu  Eqn(17)  exp(a(/z - r))dr  = exp(a/?)x(0) + bwa exp(a(/z - T)) | J 1  =0  = exp(a/*)x(0) + bwa (exp(a(/z) -1)) 1  W e now define the free and forced matrices:  A = exp(a/z)  Eqn(18) Eqn(19)  B = a[A-I]b A n d we can then write  Eqn(20)  x(h) = Ax(0) + Bu If u takes on a different value Uk for each time interval o f duration h:  ki  x  =  A  x  +  k +  Eqn(21)  Ku  k  Turning back to our simple numerical example, let us assume a sampling frequency o f 1 k H z . This time interval o f 0.001 seconds results i n the following parameters for our discrete system description:  A =  B  0.5403  Eqn(22)  0.0008415 0.5403  -841.5 0.4597  Eqn(23)  841.5  Where A is the discrete system propagation operator and B is the discrete effect o f unit output. The solution o f A was found by eigenvalue decomposition o f the system explicitly using the M a p l e mathematics package. Alternatively, M a t l a b offers a direct continuous to discrete system conversion command in its "Control System T o o l b o x " package. [ 1 7 ]  [18]  U s i n g the above discrete system description, we can generate discrete system steps. The figure below shows such a series o f 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  -0 02 5  -0.01  0 Time (sec)  0.01  0.02  Figure 11: Discrete Evolution Sample of the Simplified Model The black symbols represent the discrete evolution of the system at 0.001 second intervals. continuous system response is the blue curve. The actuator position is in red.  The  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  D i s c r e t e Control  2.6.  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) = -K y(t)-K -+ p  d  Eqn(24)  + r(A + r(t)  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  Uy^  u =~K y -K k  p  k  d  ydt  Eqn(25)  j  k  W e can define a (1x2) gain matrix: K = [-K  Eqn(26)  -K ]  p  d  W h i c h 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:  k+\  x  =  +  B  w  *  Eqn(28)  = Ax* + BKx = (A + BK)x k  k  The stability properties of the system are determined by the eigenvalues and eigenvectors of the matrix: Eqn(29)  P = A + BK  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 following plots contours o f greatest eigenvalue for the P matrix against P D gains for our numerical example. ,x 10"'  T w o - m o d e Model G a i n Stability  1.5  |  0.5  (O  Q  -0.5  0  1 Proportional  2  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. G a i n values outside o f this region are unstable. G a i n values inside o f 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 o f proportional gain K 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 i n the discrete model introduces some lag. The time lag due to discrete sampling pushes the phase past -180°. p  Additionally, unlike the continuous case, there is a maximum stable value o f proportional and derivative gain. Physically, this corresponds to applying a force sufficient to more than reverse the velocity during a single time-step. O n the next step, an even larger force i n the opposite direction w i l l be applied, causing a divergent oscillation. The discrete time steps introduce a phase lag that makes the system unstable i n the absence o f natural or artificial damping.  20  W i t h a reasonable amount o f 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 K =0.\ and A^=1.7xl0" . 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 G a i n ) . 3  p  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. W e shall explore the stability region in gain space just a bit. The following plot is the system response with gain set at just 5 0 % of the deadbeat gain activated at t=0 with a setpoint of 1. G a i n at 1 / 2 Deadbeat 16  -&O2  0  -0.01  O Time (sec)  0.01  0.02  F i g u r e 13: S i m p l i f i e d M o d e l w i t h P D G a i n s Set at 5 0 % of Deadbeat T h e c o n t i n u o u s response is s h o w n w i t h the s m o o t h c u r v e (blue) a n d the discrete positions are s h o w n at the t i c k m a r k s . T h e step f u n c t i o n (red) is the ideal response.  21  The response approximates an underdamped second order response. This would be exactly true i n the case o f continuously controlled output. However, since the output is controlled i n 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  0.02  Time (sec)  Figure 14: Simplified Model with P D 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 o f 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 o f a single-mode system. However, for continuous time there was no limit to h o w 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 i n Continuous Feedback Systems) we found that the addition o f a second resonance could constrain the range o f stable gains, but that discussion was only qualitative. In the previous section, we found that discrete time steps limited the range o f stable gains. N o w let us combine the constraints o f discrete time and o f a second resonance, and i n a quantitative way. The one-mass system as previously described is exceedingly easy to control well. The transponder mount to the bulkhead is extremely stiff at l N / u m . There is only one system resonance. However, this simple model o f a monolithic mass and infinitely stiff bulkheads is only adequate to explain and control the response o f a real system up to perhaps 20Hz.. Beyond this limit, 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 i n the mass. Let us add a spring and thus an additional vibration mode to the system.  Modified to Add Block Resonance  \\\\\\\\ Stiff Spring 1N/um  Measurement  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.  W e have split the mass in half, and added a spring i n the middle identical to the original single spring. N o w there w i l l be two modes o f oscillation. One w i l l have the two masses moving roughly as a single unit with about the original frequency. For the other mode, the lower halfmass w i l l be approximately fixed, and the upper mass w i 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:  _{k +k )/ x  2  Q  k/  {° 1  Q  2  And  x=  V  ,b =  Eqn(30)  0 0  I  )  0 )  Let us now apply proportional and derivative control using the position and velocity o f mass 2 (the lower one i n figure 15). The P matrix is P=A+Bw where u=(0,0,K ,Kd) and A and B are exponentiated from a and b. Figure 16 shows the contours o f the largest eigenvalue as a function o f K and Kd. The contour for the 2-state problem are also shown. p  p  -3Two-Resonance M o d e l G a i n Stability  -  2  -  1  0 1 Proportional  2  3  4  F i g u r e 16: T w o - R e s o n a n c e M o d e l G a i n Stability Locus o f maximum eigenvalue o f P V s 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 o f 3 smaller. The "deadbeat" solution for the single-resonance system is a bit more than a factor o f 2 outside the maximum stable gains o f the two-resonance system. 24  Since the new mode o f the system has about twice the frequency, it is not surprising that the maximum stable gains are reduced by a factor o f about two. A l s o , the maximum eigenvalue o f the P matrix is never zero, so there is no "deadbeat" solution that w i l l bring the mass to a dead stop i n two time steps.  1.5  I  1  S  0.5  :o2  0.01  -0.01  0.02  Time (sec)  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 i n figure 13, which are slightly outside the stable region o f figure 17. For t<0, the setpoint is r=0, and the mass shows motion i n 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 i n 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 i n 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 o f 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  W e can plot multi-dimensional stability contours by projecting the maximum eigenvalue found in all non-plotted axes onto two axes which can be plotted. S i x plots then can map the stability zone onto all possible gain pairs for a four-state system. Let us now plot the stability region o f our four state model allowing "perfect" information from our two masses (position and velocity o f 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 o f 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 all 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 well. 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 l s o , the positions and velocities may take on extreme values during the N time steps o f the solution.  2.8.  Conclusion  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 i n such a simple system. The response o f 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 o f 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 w i 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 singleresonance 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 i n Figure 19 below. It is i n 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 o f 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 piezoelectric actuator to allow the optical path length o f 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. [19]  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 H e N e 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 silicon 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 i n 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 V o l 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 p m 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 milliamps ( R M S ) . This created a slewrate issue with respect to our control system. Over the full scale of 100V, with a lOuF piezo, slew time at 6 0 m A is 17mS. Our digital control system data-acquisition/control loop time is 0.2mS. E v e n a 1% of full 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 full voltage range of our piezo, we designed a 2 4 V push-pull output amplifier which was capable of delivering 2 A . 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 A d w i n - G o l d real-time computer system . The o n board D S P is programmable i n 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. 1141  Data presentation is made using the " I g o r " data manipulation package from Wavemetrics. Interface software between the A d w i n - G o l d and I G O R is provided by R T S Consulting L t d . [15]  [ 1 6 ]  Real-time oscilloscope functions as well as control algorithm communications using window tools such as sliders is made relatively straightforward. Real-time determinacy is guaranteed since the A d w i n - G o l d is a dedicated real-time module and the communications to and from the module are relegated to background process within the A d w i n . U s i n g the A d w i n - G o l d i n 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 o f 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 o f this instrument is laid out in figure 21 following.  Lock-in Amp/ Freq Generator DSP Fast Loop  Executive Control Frequency  GUI Settings Min Freq Max Freq Freq-Step Amplitude  Frequency  Sine D a t a S t r e a m  ^  D/A  Generator  Output  Cosine Data Stream  Sequencing And/  Settling rime Acquire Time  -2H  D a t a a t e set a t  loop  rat*  (10kHz)  Cosine  Status Bode Plot (in progress)  Correlator  • Measurement Sine  he  A/D Input  Correlator  Communications rate Set bf Setting-time plus Acqure-time ( w e r y stow!  F i g u r e 2 1 : L o c k - i n A m p l i f i e r B l o c k Schematic  Executive C o n t r o 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 o f 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 o f 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 l o w frequencies to be analyzed.  3.5.  Position Control Test Platform  The test platform assembly is intended to allow a 10-100 k g 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 i n figure 22 below. U B C 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 o f all but one degree o f 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 i g u r e 23: Test P l a t f o r m 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 Physik-Instrument reference arm o f (relatively) rigidly  end post has an optical mount used to adjust the gap and parallelism o f a model D100 capacitive position sensor. It also has a fixed mirror for the the interferometer, and an anchor point to f i x the moving platform 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 o f 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.  F i g u r e 24: A c t u a t o r Detail 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. 4.1.1.  Interferometer Signal Model 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 field at photodiode / may be written  E = Ft t  exp  Eqn(31)  i(k • 3c, -eot + <p)  where F is the transverse intensity and polarization vector o f the beam at photodiode i, and the subscript allows for the fact that the intensity envelopes o f the beams vary across the detector plane. The beam wavevector is k, x, is the coordinate o f photodiode i, co is the frequency, and cp is a phase offset. T  W e want to measure the phase difference between two beams after they have traversed the two interferometer arms. The beams have the same frequency and thus the same magnitude \k\, but slightly different directions. The polarization vectors are essentially parallel but the intensity envelopes may not be identical. The sum o f beams A and B may be written  e  =  P •  x  [ P t A  x, +<P )] + FIB P\_KK •  +% ) ] }e x p ( - / G r f )  ex  A  *i  Eqn(32)  The measured light intensity is proportional to this dotted with its complex conjugate  e x p \ i [ ( k - k ) - x , + (<P - V ))] A  J,^F .F +F -F iA  iA  iB  iB  +  iA  +  iB  A  B  Eqn(33)  iB  \+ exp[-i((k iA  B  F -F  + if,A • iB)2cos[(k F  A  -k )-x,+  A  (q> -<p ))]  B  -k )-x,+  A  B  (<p - <p )]  B  A  B  The intensity at a given diode is the sum of the two beam intensities plus an interference term. The interference term has a phase contribution (k -£ )»5c, that is different for each diode and a contribution that is the same for all diodes that depends on the phase difference between the two beams q>A-(pBa  A  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 c is due to \FJA\ + \FJB\ plus electronic offset and any background light, b is due to 2(F -F )md Si is (k -k )-x-nll t  IA  t  iB  A  B  If we know the constants c , b , and 8 , we can take a single voltage measurement F, and solve for There are multiple solutions, differing by multiples o f 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 S , we can eliminate the opposite-slope ambiguity. If we take a t  t  t  t  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 o f the interferometer end mirror.  4.1.2. Direct Parameterization Signals from such a system have an appearance characteristically shown i n Figure 26 (below). This figure shows the signals expected i n two detectors for a bit more than a quarter cycle o f 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 i n the system remain constant and the only variation with target position is the phase o f the measurement arm. The explicit model then is given below where for our purposes we w i 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 o f equation 34. Where v, is the measured voltage, c , b are arbitrary gain coefficients, cp is the absolute phase due to mirror position, S is the relative phase for different detectors (Si is arbitrarily set to zero) and where i is the detector index. t  t  Position  0  0.1  02 .  0.3  0.4  05 .  06 .  0.7  08 .  0.9  Figure 26: Characteristic Interferometric Measurement Data  1  Top is slow sinusoidal position vs. time. Bottom is the photodetector signals vs. time for typical parameters.  Measurements  0  0.1  02 .  0.3  04  05 .  O.i  07 .  0.J  0.9  1  T (sec)  39  t  W i t h 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 i n the model from equation 35. These are summarized below. 1 2 3 4 5  Cl  b, b  2  5  Detector Detector Detector Detector Detector  1 voltage offset (v) 1 amplitude (v) 2 voltage offset (v) 2 amplitude (v) 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 i n designing a detection algorithm. The principle design requirement is to calculate a continuing series o f position measurements from a continuing series o f light-intensity detector-pair readings. This sounds very straightforward. A s a static analysis o f lab data, it would be quite simple. G i v e n 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 o f time where the position is exploring a large-scale trajectory. If such a time span exists where the position is traversing a region o f the order o f a half wavelength, then a first estimate o f parameter values can be made simply by extraction o f 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 . 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. [7]  4.1.4.  A d d i t i o n a l A l g o r i t h m Design C r i t e r i a  I wanted to explore a different approach which allows for some improved operational criteria. Additional criteria are:  •  N o explicit calibration procedure.  •  M i n i m a l assumptions about calibration parameter values at startup.  •  The calibration parameter estimates should monotonically improve i n 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 o f 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 valid selection between 180 and 360 degrees w i l l simply result in a velocity inversion o f any calculated position.  The final design criteria expressly allows a stalled trajectory. That is, the algorithm has no control nor a-priori knowledge o f whether the trajectory is stalled. N o r does it have any knowledge about the length o f 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 o f 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 w i l l converge correctly from any starting location. For non-linear systems i n 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 o f 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 o f the two detector signals on an oscilloscope plotted against each other i n 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 o f the position resulted i n one traverse o f the ellipse i n the pattern. For a phase difference o f 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 asfigure26. The pattern is actually traced once for every half-wavelength position change (this is not evident in the plot). Data fromfigure26 is replicated on the left for reference.  The main idea o f 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 o f 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 o f equation 35 (pg 39). The answer to this lies i n 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. H o w are the measurements normalized into a circular Lissajous pattern? 2. What are the parameters o f 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. A n y 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 i n matrix form as follows:  Eqn(36)  p(0 + t//) = S(v(<p)-c) = Sr(<p) where we define  Eqn(37)  r((p + y/) = v((p)-c  Where p is the voltage vector (1x2) mapped onto a unit circle, v is the measured voltage vector (1x2) i n volts, S is a symmetric matrix operator used to circularize the Lissajous pattern, (p is the absolute phase due to mirror position, ^/is a phase matching constant, c is the centre o f the Lissajous pattern (1x2) i n volts. A l s o , where we introduce r as a centre adjusted voltage measurement. The phase parameter (our inferred distance indicator) is explicitly related to p as follows ^costf^  Eqn(38)  sin#  44  (3) Equivalence of Geometric Angle & Physical Light Phase: The geometrical fact that a symmetric matrix operation can be used to circularize an ellipse does not guarantee that the geometric angle and the physical light phase are identical. This needs to be demonstrated. There is an analogy argument to intuitively explain the validity of the geometric parameterization. This argument relies upon a familiarity with light polarization. I must credit discussions with A l i M o h a z a b , who identified this argument in discussions held in A p r i l o f 2003 at the University of British Columbia. [13]  Elliptically polarized light may be defined to have the same parametric representation o f electric field as our intensity functions have of voltage given by equation 35 (pg 39) with two exceptions: • •  Elliptically polarized light does not have an offset parameterization (c; in the equations). The fundamental phase of light is driven by time whereas our system phase is driven by position.  W e can neglect the first concern because we w i l l subtract the offset from our system before we take any arc-tan position measurements. The second point is critical to the argument as follows. Elliptically polarized light can be considered the sum of left and rightpolarized light. In circularizing the ellipse, we are isolating one of the left or right polarizations. That is, in applying a linear operation to circularize the ellipse, you are simply taking the projection of your polarization onto either pure left or pure right circular polarizations. A circular polarization guarantees constant angular speed because that is part o f its fundamental definition. Constant angular speed for the polarized light implies proportional position versus angle for our problem. Alternatively, and less abstractly, we can show explicitly that the geometric parameterization maps onto the direct parameterization as follows: -Expand S into explicit parameters A,B,D -Substitute the initial model from equation 35 (pg 39) into r. -Expand p into components.  A B  b sin(#?)  Eqn(39)  x  D  b sin(g> + S) 2  cos(^ + ^ )  y  45  Where Z>/are the measurement amplitudes, A,B,D are the symmetric tensor (S) parameters, <p is the system phase, S is the detector phase difference and if/ is the arbitrary phase match. Since this equation can be solved i n general, the parameters which circularize the pattern are A L S O shown to meet the mapping rate criterion o f question (3). The general solution is:  A=  2  b sin(S)^b  + 2b b sm(S) + b  2  x  B=  Eqn(40)  b +b,sm(S)  .  - °  Jb  C  2  x  2  Eqn(41)  t { S )  + 2b b sm(S) + b x  2  2 2  b\+b sin(c?) 2  b sin(£)^ 2  2 2  + 2b b sin(£) + b  Eqn(42)  2  2  x  . . h + b-, sin(c?) cot(»= ; / b cos(d) 1  ^ ,„„ Eqn(43)  2  2  Incidentally, the solutions above can be used to verify values for the symmetric matrix S i f parameters from equation 29 (pg 32) are known.  46  N  4.2.2. Parameter Iterations in Geometric Model F r o m the previous section we know explicitly how to convert from a direct parameterization to the geometric parameterization. The geometric model allows for geometric visualization o f parameter iteration. However, the motivation for designing the geometric model o f the system is that it must be directly derivable from the measurement stream. There are two conceptually separate stages to obtaining the geometric parameterization from the measurement stream. The first involves "circularization" o f the Lissajous pattern (derivation o f the S matrix). The second involves finding the centre o f the Lissajous pattern (derivation o f the c vector). Operationally, we need a method which generates our measurement without too great a computational load since we would like to implement this algorithm with a moderate power D S P (digital signal processor). Fewer number o f computations per measurement cycle w i l l result i n a better algorithm. Certainly, we would like the number o f computations to be on the order o f less than 100 floating-point additions and multiplications. The algorithm w i l l have a relatively high measurement cycle frequency (~ 10 k H z ) and thus an iterative scheme with only a small improvement per cycle could still provide a highly effective estimate o f the geometric parameters. Functionally, an iteration method which provides some small adjustment to parameters w i l l perform w e l l as long as the iteration adjustment can be shown to be i n the "correct" direction. Symmetric M a t r i x Operation: If we assume the centre o f the Lissajous pattern is exactly correct then we have previously shown that there is some matrix S which correctly converts the elliptical Lissajous pattern into a unit circle. If we are given just one measurement (pair o f detector voltages), what could we do to improve our estimate o f S? If our single measurement is outside the unit circle, we know that our correction S must decrease signals i n our measurement's direction. If our measurement is inside the unit circle, we know that S should apply a correction to increase signals i n our measurement direction. For the direction orthogonal to our measurement, we haven't gained any information and we shouldn't adjust the matrix S i n that direction.  47  A n operation to achieve this result from a single measurement is shown below:  Symmetric Matrix Operation  full Lissajous pattern  Figure 29: Symmetric Matrix Operation The operation of the operation T as calculated from a single measurement r is shown on two different patterns. The red ellipse shows how T operates to force the circle of radius r onto the unit circle in the direction of r. The large green ellipse is a sample Lissajous pattern. The smaller green ellipse shows the operation of T operating on the lissajous pattern. Again the new pattern is constrained to intersect the unit circle in the direction of r.  In the above diagram, the symmetric operator T has an eigenvector in the direction o f r with an eigenvalue o f \l\r\. The orthogonal eigenvalue is unity. That is, we have no information i n that direction. Clearly to gain complete information, measurements are required i n more than a single direction.  48  A symmetric operator with an eigenvector o f value a i n the direction o f r(x,y) and an orthogonal eigenvector o f unity is given by:  ^ax +y 2  T=  x y 2  (a-Y)xy^  2  (a - Y)xy  2  +  Eqn(44)  x + ay 2  2  Where T is the symmetric operator, a is the eigenvalue o f operator i n direction o f r(x,y) and is set i n the diagram to be l/|r|, x is the x value o f point r onto the unit circle and y is the y value o f r(x,y). The formula above can be simply demonstrated by showing  T  (x) (x\ \ =a  T  and  X  W  I  J  x  = K  x  J  Explicit S Iteration: In operation, the full correction T should not be applied because the Lissajous pattern is simultaneously being adjusted to find its centre and this can cause some instability. Additionally, a smaller correction per cycle w i l l reduce the effect o f measurement errors. A description o f the iteration with a small gain factor is as follows:  Eqn(45)  S =I 0  ( !V "  a, =  «1 + r  Eqn(46)  2w  vi'iy  {a -\)x y l  l  x +a y 2  l  2  t  Eqn(47)  j  Eqn(48)  S, = t,.S Where S is the symmetric operator, a is the eigenvalue o f operator i n direction o f r(x,y), r is the measurement vector with components (x,y), t is the symmetric iterant. The symmetric operator S is initiated with the identity operator. Each iteration multiplies S with the iterant operator t. The convergence rate is set by n where as n approaches infinity, t approaches the identity matrix and as n approaches 1, t applies a full circularization to each measurement forcing r onto the unit circle in one iteration. 49  Note that i f r is on the unit circle, t is the identity matrix and thus S is stationary. Thus i f S is the correct value to "circularize" the Lissajous pattern, the value o f S w i l l remain stationary. This is a necessary condition for convergence. Since this iteration requires just one measurement, adjustments to S take place even when the trajectory is stalled. However, overall system performance is not compromised. A sample o f the symmetric matrix iteration scheme operation is shown i n the figure below. The non-stalled trajectory is causing the raw Lissajous pattern to traced as a track clockwise and counter clockwise around the ellipse (outer ellipse i n red). Subsequent iterations to the symmetric operator S cause the modified trace to spiral inwards towards a unit circle. After only a few full oscillations o f the Lissajous pattern, the modified pattern is confined to the unit circle with negligible error. Convergence is monotonic and trend is exponentially asymptotic. Since the pattern is converging to a unit circle, you can see the actual trajectory reverse direction as the platform velocity reverses direction.  L i s s a j o u s Pattern 8  r  6 4 2 CM  •s 0 .2 -4 -6 "•Su  -  -5  0 Det 1  5  ~~10  F i g u r e 30: C i r c u l a r i z a t i o n The direct detector measurements are shown in red and describe the usual Lissajous pattern. The blue trace represents the modified trace as S iterates such that the pattern becomes a unit circle.  50  Finding the Lissajous pattern centre: A first order iteration to converge towards the centre o f a circle from a single new measurement is as follows: pull the existing estimate o f the centre along the vector between the estimate and the measurement to the distance o f the known radius. This is illustrated i n Figure 31 (below).  Single Measurement Iteration Illustration  Figure 31: First Order Iteration to Find Centre  This figure illustrates graphically a construction that might be used to calculate a correction to an estimated centre given a single measurement. Only a small percentage o f the correction vector is applied for each measurement. This scheme converges rapidly for estimates within the pattern circle. A fatal problem with the scheme is i f the estimated centre is in great error and O U T S I D E the Lissajous pattern, then the scheme does not converge at all. Furthermore, the scheme adjusts the centre i n the direction o f the measurement. This is the same direction o f adjustment as the symmetric matrix iteration. Since both algorithms have to operate simultaneously, they must behave as independently as possible. They should not operate i n the same direction. W e have to find an iteration which adjusts the centre estimate i n a direction orthogonal to the measurement direction.  5 1  Orthogonal Iteration to find pattern centre: A n iteration involving two subsequent measurements can be derived which makes adjustment to the estimated centre in a direction orthogonal to the single measurement iteration. This is illustrated i n Figure 32 (below).  Orthogonal Iteration Illustration -Circularized Signal Locus  Figure 32: Orthogonal Iteration to Find Centre  The illustration for the orthogonal iteration scheme illustrates graphically the construction used to calculate a correction to an estimated centre given a two measurements. Only a small percentage o f the correction vector is applied for each measurement. This scheme converges correctly for all initial estimates inside or O U T S I D E (or actually on) the pattern circle. Furthermore, the pattern radius is not required. The correction vector can be found with vector arithmetic to be:  d = p,- _  Eqn(49)  a = El±£iL±  Eqn(50)  Pl  x  52  Eqn(51)  Ua-c)-d)d  d  j  1  Where d is the difference vector, a is the mid-point between measurements, c is the current estimated centre a n d / i s the full adjustment vector. Real data w i l l have errors. When \d\ is small the direction of d is poorly defined. This is a common condition which w i l l happen whenever the mirror position is nearly stationary. The error in the correction vector / is proportional to the measurement error and inversely proportional to the distance between measurements (length of d). Thus we should weight the correction proportional to the inverse of the error squared which is a measure o f the inverse variance of the adjustment. Thus the weighting for the correction can be achieved by multiplying by d . This makes the algorithm even more efficient since the derivation above involves division by d . Elegantly, we simply do not divide by d in equation 51. 2  2  2  Additionally, as in the case of the symmetric matrix iteration, only a small portion o f the weighted correction should be used to aid in accuracy against measurement noise and to aid in stability against algorithm interaction as both algorithms act simultaneously. This orthogonal iteration scheme meets all of the criteria required for our algorithm: •  The correction vector estimate monotonically improves the centre estimate no matter where the initial estimate is located.  •  The radius of the Lissajous pattern is not required for the iteration scheme.  •  If the Lissajous pattern is non-circular, the centre estimate w i l l not converge correctly to the pattern centre however it w i l l not disastrously diverge either.  •  When weighted the adjustments are very small for stalled trajectories.  \  53  The operation o f the orthogonal iteration can be seen i n the figure below where the measurement stream reflects a circular Lissajous pattern not centred at the origin.  Lissajous Pattern  Det  Figure 3 3 : Centre Finding Operation The direct detector measurements are shown in red and describe a circular Lissajous pattern which is NOT centred at the origin. The blue trace represents the modified trace as the centre finding iteration operates such that the pattern becomes centred at the origin  Assembled Algorithm: The overall operation of the system requires that each algorithm iteration performs both operations: an adjustment to the symmetric matrix S and an adjustment to the centre. Note that when the symmetric algorithm and the centre finding algorithm are operated sequentially in a single measurement iteration, the centre adjustment algorithm is N O T receiving data guaranteed to lie on a circle. A n y error in S w i l l allow some eccentricity to the Lissajous pattern. Simulations support this adulteration in two ways. Firstly, the centre finding algorithm actually works quite well on elliptical Lissajous patterns as long as the domain is well explored. Further, the centre finding algorithm is well behaved (does not diverge) even when the domain is not well explored (stalled position). Secondly, under normal operational limits of a domain from 0 to 10 volts with an eccentricity greater than 0.1, the symmetric operator w i l l almost always reduce eccentricity o f the ellipse even when the centre is poorly estimated (or even outside the locus of the pattern). Thus simulations and operational tests which demonstrate convergence indicate a strong degree o f independence between the symmetric iteration and the centre-finding iteration even when coupled as above. Effective operation of the algorithm require that the gain factors be set to iterate only very small steps per iteration. Correct operation of actual simulations indicate gain factors on the order o f one part per 100000 provide a good margin against instability between the two interacting algorithms while still providing a reasonable convergence rate. Gains on the order of one part in 10000 did on occasion demonstrate instability. The final calculation of the position involves taking a four quadrant arctan of the p vector. This operation can be performed simply i f your system has a 4-quadrant arctan function. In our case, our arctan function required logic to identify quadrant. In either case, the number of rotations must be accumulated to integrate a total distance result.  55  4.2.3. Algorithm Tests Unfortunately, there wasn't any independent position measurement o f sufficient accuracy available to measure the accuracy of the algorithm-interferometer setup. Initial tests and expectations from previous work Gray et al ^ indicated we could expect errors on the order of less than one nanometer. Taking a tip from Gray et al, we decided to use duplicate detector sets to determine accuracy. That is, we ran two independent detector pairs/algorithms and compared the measurements between the two systems to determine accuracy. T w o independent pairs of detectors and algorithms were then used to infer position information with respect to the measurement arm. In comparison of these two measurements, some inference can be made as to the accuracy and precision of the detector system and the algorithm. For our tests, we acknowledge that systematic errors in common between our two detector pairs/algorithms w i l l not be illuminated. However, the interferometric phase difference between the two systems was significant as the detector pairs were physically separated in the interference pattern. This should ensure that systematic errors due to geometry (geometry in the sense of the geometrical interpretation of the algorithm) w i l l not be in-phase correlated and should manifest themselves in the measurements. The end reflection mirror in the reference arm was fitted with a piezo which would vary the reference arm distance of our Michelson interferometer. A triangular waveform was fed to this piezo which provided a dynamic measurement for our detector system. Optimal tuning included adjusting interferometer geometry such that the intensity amplitude for all four detectors exceeded 5 0 % o f full scale. Further, 90 degree phase separation between detectors in each pair was tuned as well as possible by eye with only the Lissajous pattern as a guide. The algorithm/data loop rate was 5 K H z . Detector digitization was conducted for each pair using two D / A converters allowing simultaneous readings from both detectors. D/A digitization was 16 bit over a - 1 0 V to 10V range. Mechanical tuning was specified to provide a maximum to minimum swing of greater than 4 V and a maximum o f greater than 6 V . For the most part this was achieved; although obtaining a setup which most closely met these specifications for all four required detectors required some signals to be marginally below these levels.  56  The position measurements are shown below. Only one trace is shown because the two systems/algorithms matched to such a degree that the difference could not be seen on the full scale plot. A l s o shown below is the difference between the two systems/algorithms. Overall, the R M S difference between the two systems was 0.22nm over a 120 second interval test. For random correlations, this would imply an R M S error for a single system o f 0.16 nm. Clearly there is an element o f the difference-plot which is harmonically related to the position phase. This is indicative o f the magnitude o f the systematic error for the systern/algorithm. It is larger than the random fluctuations visible on the same plot and thus dominates the R M S result.  250  H  200  H  150  H  100  H  50  H 1 0  r 2  r  i 4  6  1 8  1 10  sec  Difference Between Systems 0.5  E  H  0.0-  -0.5  H  -1.0 4  10  Figure 34: Interferometer and Algorithm Test The top figure shows the position result from a 2Hz triangle test signal applied to a test mirror. Both systems plot exactly the same on the scale shown here. The bottom figure shows the signal difference between two separate systems.  However, the dominant error is clearly not randomly correlated. The actual error o f a single system w i l l depend upon the correlation between systematic errors on the two systems. This is unknown. However, the physical separation between the two systems' detector pairs should have ensured a phase shift (phase n o w referring to the test mirror position triangle 57  wave) between the two systems and thus lead us to believe that the overall error for a single system is certainly on the order o f 0.22nm. That is, we do not believe that there is a strong i n phase correlation between the systematic errors (in phase with respect to the triangle test signal). When the systems were not tuned optimally, the R M S difference between the signals varied between 0.3nm and 1.5nm. This non optimal tuning included voltage amplitudes o f less than 2 volts, phase differences o f as little as 40 degrees between detectors and l o w signals o f less than 3 0 % o f maximum. It should also be noted that R M S results were uniformly above l n m for a n y mechanical tuning unless a shroud was placed over the assembly to block air currents.  4.2.4. Non-Correlated Noise Test A n estimate o f the white noise level o f the system was generated by analyzing the magnitude o f the difference between subsequent positions as output by the algorithm when the mirror piezo was held at a fixed voltage. A sample o f this data is shown below:  F i g u r e 3 5 : Non-correlated Noise Test A measurement of the non-correlated noise was taken by subtracting the position fix from its subsequent value. This particular sample was takenfroma static system test with no applied mirror motion. However, plot appearance is identical to within expected sample statistical variance for dynamic mirror motion.  58  This figure is taken from a test run with no applied mirror motion. However, analysis of the dynamic data (from previous test) show the same R M S result of 0.07nm. Under the assumption that fix-to-fix variations are uncorrelated, the single fix R M S noise can be inferred to be 0.05nm (0.07/V2).  59  5. 5.1.  Control T e s t s Platform Response  A s discussed in the apparatus section 3.5, Position Control Test Platform, we undertake to control a macroscopic mass (approximately 10 K g ) of aluminum along one degree o f freedom i n an attempt to " l o c k " the mass to ground motion. To accomplish this, we need to understand the response of the platform to our control piezo, and to understand the ground motion to which we are trying to respond. To obtain the system response in isolation, we must ensure that the vibration signals from ground motion do not influence our measurement. W e achieved this in two separate tests. In one test, we excite our piezo actuator with a 50 nm sinusoidal signal. W e use a lockinamplifier setup and sweep the frequencies from 1 H z to 1000 H z in 0.5 H z steps. Each individual frequency test was held for 30 cycles or 5 seconds (whichever is greater) before sampling begins to allow any transients to damp away. Then a 10 second average response is measured providing an explicit amplitude and phase response (with an approximate 0.1 H z frequency discrimination). The lock-in nature of the measurement ensures that only the frequency component exactly matching the excitation frequency w i l l be measured. This reduces measurement effects from ground vibration since ground vibrations contributions are confined to bandwidths of the order of 0.1 H z . This system measures response with an error independent of frequency. Another test involves taking an average response to a step (square-wave) excitation. In this case the average was taken over 5 hours. Then a Fourier transform is taken of this average response. Taking odd harmonics only (since a square wave has no even harmonics) and dividing by the theoretical square wave transform results in the Bode plot shown in the (thin) red trace in the figure. This method has an error proportional to the frequency squared. The error at the high frequency end is clearly visible in the fluctuations of the red trace. Both methods provide essentially the same measurement of system response shown in figure 36 following.  60  v  Figure 36: System Bode Plot The Bode plot for the horizontal position control system is shown in this figure. Thick (black) traces represent a lock-in data acquisition and the thin (red) line represents a Fourier transform of an average square-wave response. These two independent methods conform very nicely.  The system as designed presents a control-system challenge since there are obviously many resonant modes i n the system. For example, notice the greater-than-unity amplitude response not only at the slow fundamental resonance o f around 20 H z , but also other resonances at 460 H z and 650 to 700 H z . These higher frequency resonances are easily excited. The complexity o f the platform response and i n particular the highly excitable resonances could be resolved with a modified mechanical design (refer to Appendix A , N L C Vibration Control Editorial). Notwithstanding, actual N L C design might easily encounter difficult to control mechanical systems and it is instructive to attempt to control such systems. A s such we continue i n this chapter to examine some rudimentary control strategies for this test platform.  61  5.2. Ground-Motion Characteristics The frame and bulkheads o f our test system rest in direct contact with the ground and vibrate in response to vibrations i n the ground itself. Without control, our test system measurement indicates a vibration level averaging about 50 nm R M S . This is shown i n the figure below.  1  0  0  50  100 150 Frequency (Hz)  200  Figure 37: Uncontrolled System Vibration Spectrum  62  250  The vibration spectrum is calculated from a time data series o f 1,000,000 samples over 300 seconds. D F T ' s over four second intervals o f the time series are calculated. This results i n frequency values spaced at 0.25 H z intervals. Multiple D F T ' s are calculated starting i n the time series at intervals o f every 0.15 seconds. This results i n 1974 overlapping D F T regions which are subsequently averaged as to their power spectra. The individual D F T ' s are calculated using a Hanning window and thus have an approximately .5 H z bandwidth (2 times 0.25 due to Hanning). Once combined into an overall average power spectrum, alternate values are again averaged to produce a power spectrum with values spaced at 0.5 H z with a bandwidth also o f approximately 0.5 H z .  (System Response to Ground Motion) (ground vibration) M (measurement) Figure 38: Measurement Model (No Control) Our system measurement measures the distance between the bulkhead and the test mass providing the difference function..  The system vibration spectrum shows two effects o f interest. Firstly, the spectrum shows some very steep peaks which represent highly coherent man-made vibrations from rotating equipment. In particular, 60 H z , 90 H z and 120 H z vibrations are apparent which clearly indicate synchronous vibrations to the North American power standard o f 60 H z . Secondly, the spectrum shows its most pronounced vibration at about 20 H z . This is the fundamental resonance o f our system. Thus this spectrum shows a combination o f actual ground motion and our system response. The power spectrum of the actual ground motion exciting the system is unknown. However, we can arrive at a good approximation o f a ground motion component in the direction o f our single degree o f freedom. To do this we appeal to the system diagram below.  63  Ground vibration G moves the measurement bulkhead directly and also acts through the system response P ' to move the test mass. W e measure the difference i n position between the bulkhead and the test mass. Assuming that the ground vibration couples perfectly to the bulkhead, we see that our measurement implies M = G-P'G  E (  l ( n  5 2  )  Where M is our measurement, G is our ground motion and P ' is the system transfer function. If the test mass was perfectly attached to the bulkhead, P ' would equal unity and the measurement M = G - P ' G would be zero. If the mass were inertially floating, P ' would be zero and M = G . Both these limits bear scrutiny to their logical extremes. W e trivially manipulate the equation to obtain an equation for the ground motion.  G  =  _M_  E  qn(53)  1-F  But P ' is the system transfer function from ground motion to test mass motion. W e have no direct method o f obtaining this transfer function. However, we can make the assumption that the piezo excitation transfer function (see figure 36) is very similar to P ' . The piezo transfer function from figure 36 provides an excitation to the test mass identical to that o f displacement o f the reaction bulkhead. O n the other hand, P ' is the transfer function o f the entire frame including the reaction bulkhead to the test mass. W e expect that the system transfer function and P ' should be identical to figure 36 at least at low frequencies where modes o f the frame and bulkhead w i l l not yet appear. Thus a reasonable assumption to obtain an approximation o f the ground motion is that P ' is the same as the transfer function calculated i n section 5.1, Platform Response. Thus we have a method to estimate the ground motion component in the direction o f our degree o f freedom through direct manipulation o f system response data and uncontrolled time-sequence samples. Transfer functions are complex functions i n frequency space and as such each, and every raw D F T from our time sample must be treated against the complex transfer function P before being averaged into a power spectrum for ground motion.  64  The following plot indicates an approximate ground motion spectrum deduced using this method.  1  o  I  0  i  50  i  i  100  150  I  200  i  250  Frequency (Hz)  Figure 3 9 : Ground Spectrum Approximation This ground spectrum shows the frequency distribution for an overall 23.5 nm RMS vibration.  This procedure essentially eliminates the 20 hz peak due to the platform resonance. Additionally, there is a strong vibration indicated near 0 H z attenuating as frequency increases which is a common indicator o f natural ground vibration. These two observations support the validity o f using the system response transfer function for P ' i n equation 53. The approximated ground motion has an overall amplitude o f 23.5 nm R M S . This is less than half o f our uncontrolled system amplitude. This effect w i l l occur for any suspended structure employed i n the actual N L C project where the structure w i l l resonate from ground vibration. For a control strategy attempting to harden support structures, ground vibration represents the demand function or setpoint for the control system. Our test mass must be accelerated to match this vibration. 65  5.3.  PID Control Model  The physics o f our system are different from a standard application o f control. In a standard application there is a setpoint signal (which is often a constant) and an output value signal. Both these signals are generally known directly. These two signals are subtracted to produce an error signal. In our system, the setpoint and the output from P are not measured nor known directly. Rather, only the error is known (labeled M for measurement below). The setpoint can be considered to be the difference between the ground vibration and the natural mass response to the ground vibration. The desired result for our system design is to obtain a transfer function from G to M equal to zero. Normal control system design discusses the transfer function at the output o f P as driven by the setpoint and strives for an ideal value of unity. These performance criteria are actually the same. That is, we are still striving for a closed loop transfer function of unity. (System Response to Ground Motion)  P* (ground vibration)  (Measurement) 1  _J  c  .  (Controller)  P (System response to Piezo)  F i g u r e 40: C o n t r o l System M o d e l  F r o m the diagram, the measurement M can be seen to be: Eqn(54)  M = G-P'G-MCP  Where C is the controller transfer function and P is the piezo to measurement transfer function (see figure 36),  66  This can be manipulated to obtain the attenuation factor for the transfer function from G to M M  1-F  Eqn(55)  G " I + C P  This factor allows us to calculate expected attenuation of ground motion to measurement. A design criteria which sets our problem apart from most control applications is that the setpoint signal is so complicated. The setpoint is not known a priori, rather, the system is expected to track the ground motion vibrations which can be very intricate. Initially, we coded the controller with classical P I D capability. The differential signal is provided with a second order low pass filter with a configurable rolloff frequency and the proportional control is applied without a filter. Our system goal o f vibration elimination has no requirement for absolute positioning and as such integral control is completely unnecessary. Although included in our control algorithms, we never use integral control. Our system loop time is set at 5000 H z giving a control Nyquist frequency of 2500 H z . A n examination of our system Bode plot shows that this loop time is fast enough to consider our control system from a classically continuous perspective. In support of fast loop times, steps were taken to ensure that the piezo would respond quickly to output signal variations (refer to section 3.3, Data Acquisition).  5.4. Results Initial attempts at simple proportional control were dismal. Effectively no proportional gain at all could be applied. Turning to the system Bode plot (see figure 36 ) we see that the open loop system response is above unity at 450 H z and also between 650 and 750 H z . Phase for these regions is broad and in combination covers over 270 degrees. Proportional gains o f as little as 0.2 nm/nm are enough to cause oscillation under these conditions and l o w gains like this of very little utility for control. Subsequent attempts using only differential control gave moderately better performance.  67  W i t h the low-pass filter rolloff set at 65 H z , a differential gain o f 9.7 ms was reached before oscillation occurred. This provided stable performance and complete elimination o f the fundamental system resonance. The expected attenuation for this configuration is calculated using equation 55 (again assuming P'=P). W e are striving for a response o f zero. The response we obtain o f greater than unity at frequencies above 25 H z is not particularly impressive.  V  A  1  0  50  100  150  200  Frequency (Hz)  F i g u r e 4 1 : M o d e l Response The blue plot (thin) represents the system measurement response to ground vibration without any control. The red plot (thick) represents the system measurement response to ground vibration with differential control set at 97ms and a low-pass filter set at 65 Hz rolloff.  O n the other hand, the application o f the control system does succeed i n eliminating the fundamental resonant response. In the logarithmic plot, the area under that resonant peak is significant. Imagine our system exposed to ground vibrations consisting o f a white noise source with complete cutoff at 250 H z and an R M S value o f l n m . The uncontrolled system would average 7.9 n m R M S whereas the controlled system would average 1.13 n m R M S . This would be a seven fold improvement i n R M S i f the ground vibrations were white noise. However, as we have seen they are not white noise.  68  We are now presented with an interesting opportunity to test our approximations in modeling the control system and the ground vibrations. By applying the attenuation as calculated (refer figure 41) to the ground motion as inferred (refer figure 39) we can obtain a predicted power spectrum for the system under control. We can compare this with an actual power spectrum taken from time sequence data of the system while actually controlling. These two spectra are overlaid in the plot below.  Frequency (Hz)  Figure 42: PID Control System Results The red plot (thin) represents a prediction of the measurement spectrum. The blue (thick) plot represents an actual 300 second sample.  The similarities in both magnitude and spectrum artifacts is very good. Notable are certain lines such as at around 115 Hz which are in one plot but not the other. This represents rotating equipment which was on or off at the two different 300 second time samples. (The prediction is still based upon an actual time sample but of the system not under control). Overall, even a marginal implementation of PID control has predicts the system measurement will reduce from from 48 nm R M S to 12.1 nm R M S . Simple PID implementation has reduced vibration by a factor of four from an uncontrolled system. Actual tests resulted in an R M S measurement of 11.6 nm.  69  5.5. Exploration of Gain Limits But w h y is the gain so limited? W h y can we not get a better performance with our standard controller? The following enlarged Bode plot shows the difficulty.  Figure 43: PID Control System Bode Plots -Green: -Cyan: -Blue: -Red:  Mass response to piezo.. Controller at 9.7 ms gain and 65 Hz 2 order low-pass filter. Open-loop system (product of mass response and controller) Closed loop response. nd  70  In region A , around 100 H z , the phase of the product o f the gain and system response (the open-loop response) crosses through -180 degrees. However the magnitude of the open-loop response is below unity, so instability is not expected here. The phase crosses -540 degrees in region B (290 H z ) , but the magnitude is far below unity, so this is also predicted to be stable. In region C (450 H z ) the magnitude is greater than unity but the phase has some margin from the unstable value of -900 degrees. In region D (650-700 H z ) there are two strong resonances, with platform response greater than unity, and a phase swing o f 360 degrees. The second-order low-pass filter on the velocity signal makes the controller gain less than unity here, but the product of controller and platform gain is still greater than unity. Careful examination of the phase shows that the dangerous values of -900 and -1260 degrees occur where the open-loop magnitude is less than 1. Region E (840 H z ) is not a problem, even though the phase remains near -1260, because the magnitude is much less than unity. Since the gain used for this plot is one that was just at the edge of stability, we expect that the Bode plot w i l l predict that the stability margins are low. This appears to be the case i n region A , where a small increase in derivative gain would cause oscillation at 110 H z . A larger derivative gain increase would cause problems at B , D, and E . Derivative gain would not necessarily cause problems at Point C because the phase is not dangerous. The system shows some significant challenges to control using standard P I D techniques. Notwithstanding, proportional gain could be applied with a low-pass filter and thus be of greater impact below 100 H z . A l s o , the response at the trouble points can be controlled with notch filters. These approaches are discussed in the following section.  71  5.6.  PID C o n t r o l w i t h R e s o n a n c e N o t c h i n g  Resonance notching is a simple concept to implement. The recipe is as follows: Increase gains of the underlying PID controller until oscillation is generated. Then tune a notch filter at that frequency and remove the frequency from the measurement stream thus preventing oscillation. That is, apply notch filters to the measurement stream with adjustable frequencies and bandwidths at the appropriate frequencies to eliminate oscillations. Appendix C details the algorithm used to digitally implement a notch filter and it is a straightforward algorithm to implement. However, in our case, the results only allow a marginal increase to the PID gains. The differential gain increases from 9.7 ms to 10.3 ms and we manage to apply a proportional gain of 0.48 whereas no appreciable proportional gain at all could be applied previously without resonance notching. Four notches were applied at 600 Hz (width 100 Hz), 298 Hz (width 10 Hz), 840 Hz (width 10 Hz) and 113 Hz (width 8 Hz). The improved attenuation envelope is shown in the figure below. Only the frequencies below 100 Hz are shown to highlight the area where there is a difference in attenuation.  0  10  20  30  40  50  60  70  80  90  100  Frequency (Hz)  Figure 44: Gain Increase due to Notch Filters The blue plot shows the maginally improved performance due to an increase in gain due to the notch filters. The red plot represents results from derivative alone.  The thin green plot shows the totally uncontrolled  response.  Measurement vibrations are now reduced to 9.3 nm R M S which now reflects a 5-fold reduction in RMS vibration from a totally uncontrolled system. 72  5.7.  Coherent Disturbance Correction  The internal details of the numerical notch filters inspired us to perform an interesting technique for vibration elimination. Specifically, after P I D application, the largest components of the ground vibration are now observed to consist of narrow spectral spikes. Tracking of these spikes with resonators provide a real-time estimate of their amplitudes. This information proved useful. To improve the system performance now, we model the vibrations themselves rather than trying to improve feedback performance. Imagine we want to eliminate a coherent vibration with exactly known frequency, amplitude and phase. If we added a signal of this exact frequency to the piezo independently of any feedback, the system would respond exactly as given by the detailed system Bode plot (refer figure 36). Thus we could adjust the phase and amplitude of our signal using the prescription o f the Bode plot to cancel the vibration from the measurement entirely. This is one approach to what is known as feedforward control. Unfortunately we do not know the exact frequency, amplitude and phase of our fairly coherent ground vibrations. However, we do know from measurements the narrow frequency bands which be contribute extensively to our spectrum. U s i n g only the frequency and width of our suspected noise peaks we can extract the amplitude and phase of our vibrations by applying the measurement to a resonator tuned to that frequency. Digital signal generators can then feedforward these signals to the piezo and individual sharp noise spikes can be eliminated from the measurement result. The amplitude and phase of individual noise spikes can be ascertained and tracked by what I call accumulating resonators which are simply resonating fdters without damping that increase in amplitude at a constant rate (and indefinitely) when fed with a signal with a resonant frequency component of constant amplitude. This is required since the iteration by iteration signal fed to the piezo may be considered to be feedforward, the amplitude and phase of the actual vibration level (at a given resonator frequency) must be ascertained by slow tracking with integral feedback. The integrating version of a resonator can be created by eliminating the estimation from equation 72 for the resonator algorithm in appendix C as follows:  = m  Eqn(56)  k  73  This alteration eliminates the self-limiting amplitude of the resonator. The residual is simply the measurement. When fed with a signal with any component of resonant signal, the resonator amplitude will grow without limit. The application of the resonator signal ( X i ) to piezo output only requires a phase as appropriate as shown in figure 36. The resonator will grow in amplitude until the applied correction to the piezo totally eliminates the spectral contribution. The phase extracted from the Bode plot does not have to be exact to achieve this result and indeed anywhere within 90 degrees of the correct phase will provide some vibration elimination. r e a  Several resonators may be applied simultaneously to different spectral lines of the normal vibration spectrum and the integral tracking means that variation in the excitation amplitudes is quickly tracked by the system. Indeed, for man-made coherent vibration from equipment such as motors, the vibration may be completely absent at times when the equipment is off; this does not present any difficulty to our system except during the fast change in amplitude where the tracking of amplitude lags behind any actual change in amplitude.  Figure 45: Coherent Disturbance Elimination This figure shows a detail spectrum near the 89Hz coherent excitation. The different traces represent different line widths applies to the resonator. With the width of the line set at 2Hz or 4Hz, the line is totally removed from the spectrum. In this particular case, this represents a six-fold improvement at 89 Hz. Greater improvement is possible for higher amplitude lines.  74  A m o n g the most successful spectral line eliminations using this new technique was a line at 89 Hertz where we achieved a fivefold amplitude reduction at this frequency. Effective elimination o f an individual noise peak at 89 H z is detailed previously i n figure 45. Note that the resonator improves response from 90 to 95 H z and yet does not affect the peak at 86 H z at all. This can be understood by referring to the system Bode plot which shows a rapid phase shift between 85Hz and 89Hz. The corrective phase prescription is simply incorrect for the 86 H z peak. This is an exciting improvement in that it allows us to control and eliminate vibrations at frequencies much higher than could be approached with our P I D feedback. Notwithstanding that there are more advanced feedback techniques available to control than our P I D setup, the method proved very robust and effective for sharp spectrally narrow disturbances. Any system where disturbances are highly coherent with strong spectral lines may benefit from coherent disturbance modeling. Indeed, I feel that any non-flat structure to a vibration spectrum could be reduced or eliminated using analogous techniques. Certainly for our sharply defined spectral lines of vibration, the coherent disturbance modeling provided good improvement.  75  5.8.  Best Composite Performance  Using best result empirical tuning for P I D gains, notch resonators and disturbance resonators, the overall R M S residual is reduced to 4.1 n m . This represents our best performance o f a factor o f 12 improvement over an uncontrolled system. The spectrum for this best performance is shown below: Best Control (all disturbances active) Aug 15 Fine (black) PID only Wide (red) PID, Notches, Disturbance Resonators  Figure 46: Best Control Achieved Best performance shown in thick trace (red) and best "PID only" performance shown in thin trace (black).  Notice the excellent elimination o f the 40 H z noise spike using an accumulating resonator. W e had 10 u m o f control authority which was 8 0 % utilized i n the above control tests. Ground disturbance resonators were applied at 42, 89, 120, 105.5 and 59.5 H z . Note large spectral disturbance eliminations at all o f these frequencies. Indeed, the best performance spectrum lies completely within the envelope o f the " P I D only" below 120 H z . The poor perfromance above 120Hz is related to larger P I D gains without completely compensating notches. Notwithstanding, overall contribution to residual is negligible above 120 H z for both spectrums. The measured best performance was 4.1nm R M S  76  6.  Conclusions  Two separate aspects of vibration control have been investigated: an alternate algorithm for the "optical anchor" interferometer distance reference and direct control to nanometer precision of a large mass with a piezo-electric actuator. The "optical anchor" work consisted of algorithmic design to conform to a slightly more constrained design criteria than had been previously achieved. Previous work had achieved excellent position measurement performance.  However, the previous work maintained a  separate calibration and measurement procedure which might be cumbersome in some applications. The new algorithm was designed to operate without any explicit calibration procedure. The new algorithm takes a geometrical interpretation of the intensity information from two detectors within the interferometric pattern.  The geometrical interpretation prevents non-  linearities in the parameterization from forming local minima at incorrect values of calibration parameters. Thus the calibration proceeds monotonically as data is received. If data explores the parameter space, then the parameters converge rapidly. For fully converged calibration parameters, the random uncorrelated error from measurement to measurement is 0.05 nm with a systematic error related to modeling imperfection of 0.22 nm (reported as rms but actually correlated to the interference phase). This results in an overall accuracy of about 0.23 nm. This accuracy is sufficient to provide the principle measurement to a vibration control system with a target accuracy of 1 nm. However, measurement arms of the interferometer were short and protected from air currents. Offhand observation of longer measurement arms shows that for  long-arm interferometric measurements, air currents must be eliminated and the  measurement legs will have to be enclosed in piping. Control tests on a one degree-of-freedom  experimental platform are performed using  proportional and differential control. Performance did not meet the lnm RMS goal for magnet position residual set by N L C design specifications.  Artificially stiffening a mechanical  support using classic feedback techniques and a piezo-electric actuator proved disappointing. 77  The fundamental resonant reaction was eliminated but overall vibration attenuation was only a factor of 4 improvement over the uncontrolled reaction to 12.1 nm R M S . The extension o f classic P I D with resonance notching served to marginally increase gains which served to reduce R M S vibrations to 9.3 nm. M o r e advanced control techniques would hopefully yield better performance with regard to feedback.  However, once vibrations have been limited such that the principle remaining  vibration is caused by narrow frequency spikes, vibration modeling with accumulating resonators is capable of eliminating on the order of 5 0 % of remaining R M S levels. In our case such modeling reduced our best R M S result to 4.1nm.  78  Appendix A NLC Vibration Control Editorial Let us look at the problem posed by the N L C requirements: •  The final focus magnets contribute the largest component of beam position to the particle beam prior to collision. The final focus magnets are subject to movement due to environmental motion. The motion is a function of the magnet support design combined with the magnitude and frequency of the man-made and natural vibrations. The aim of the particle beam needs to be accurate to approximately 1 nanometer RMS at the point of beam collision.  Most obviously, the man-made vibrations can be reduced. F. Le Pimpec et a l . vibrational stability of N L C Linac Accelerating Structure. F. Asiri  [4]  [ 2 ]  examines the  et al is a study of near-  field vibration sources for the N L C linac components. Conclusions are twofold: one, the final magnets should probably not be fluid cooled and are therefore required to be permanent magnets  [2]  ; two, near-field vibrations can and should be reduced through careful  consideration of vibration during facility engineering. However, vibrations should be considered to still be on the order of an urban university lab to provide a safety design margin . [3]  A n exacerbating point regarding the expected vibration levels is that the ground motion is only the disturbance to any support structure. Actual vibration experienced by the focussing magnets will be much larger than the raw ground motion since it will include the resonance response of the support structure.  The "pessimistic" model of ground motion may not be  pessimistic enough to encompass the actual vibration encountered by the immediate support of the final quadrupole magnets. 79  M y recommendation to the actual overall problem is as follows: 1. Eliminate vibration at its source to the limit of facility design. 2. Design the support structure as stiffly as possible to an upper limit of perhaps 15 H z main resonance. Incorporate maximum passive damping in this design. D o not exceed 15Hz for the principle resonance; this w i l l insure that rotating machinery vibrations w i l l not resonate with the support structure (generally, rotating machinery w i l l resonate at 30 H z or more). 3. Measure vibration inertially of the support structure near the magnetic supports. 4. Use the inertial vibration measurement of the support structure in a feedforward manner to the piezo supports for the quadrupole magnet. 5. Measure vibration remaining with an accelerometer mounted on the magnet itself. Use a feedback signal into the same piezo supports. 6. U s i n g an optimal K a l m a n estimator of any residual vibrations to feedforward into electromagnetic dipole steering magnets. 7. Inertial measurements may be made directly i f a magnetic-field resistant device can be purchased or manufactured. Alternately, an inertial standard can be passively supported on the ground and referenced using optical-anchor technology.  80  An Argument for Inertial "Hard" Support Vibration Control:  W h e n first given this problem, ones first instinct is to apply isolation to the system. Then try to use suspension systems and ultra-low frequency support schemes: fundamentally passive systems. Indeed, S L A C has undertaken a project designed to produce just such a system for this problem. U s i n g a suspension system draws  upon the one reference  frame  easily accessible  technologically to both magnets: the inertial reference frame. Furthermore, by controlling the magnets to the inertial frame, no high order modes o f vibration are induced i n the magnets since the purpose o f suspension is to eliminate all accelerations. However, i n m y view, It is the inertial reference frame that provides the functional key to the solution, not the soft suspension. One misconception that can easily be made is assuming that a system using an inertial reference as a position reference also should include a soft passive support system. It is a natural assumption since the soft support system inherently produces an inertially quiet system. However, when dealing with large masses, possible vacuum forces and large magnetic forces, a soft support system becomes difficult to implement because it cannot deal with large static forces easily. M y comment on this matter is that the obvious actual engineering solution to the magnetic support system is to establish an inertial setpoint but control using hard supports such as piezo elements which can withstand very large static forces.  81  Appendix B deadbeat Gain Deadbeat gain is the name given to the gain which results in a minimum time to correct all state values to zero from an arbitrary starting condition. It has a unique non-iterative solution. Starting with a common discrete system definition given below: Eqn(57)  x , = AJC + Bw But with perfect knowledge of the system states:  Eqn(58)  x A B u y  State Vector System Propagation Matrix System Response to Output Operator System Output Vector Measurement vector  A n d where the system output is defined with a constant gain Operator: Eqn(59)  K  Gain R o w Vector  Combining gives: Eqn(60)  *„ ,=(A + B K K +  Define the Propagation Operator P as: Eqn(61)  P=A+BK  The deadbeat gain operator Kj is defined as the gain vector K which produces a propagation operator P with A L L E I G E N V A L U E S E Q U A L T O Z E R O . This w i l l produce an identically zero system state from any initial state within m iterations where m is the order of the system. 82  The following procedure solves for the deadbeat gains assuming a controllable system when the output, u, is scalar. Generate a series of states starting from the zero state but excited with a unit output: x =0  E  0  x  „  +  i -  A  x  n  +  ^  6  2  ^  Eqn(63)  B  Continue the process until there are m states generated where m is the order of the system. Construct a matrix with columns consisting of these constructed states: T = (x,  |  x  2  | x,  | ...  | x„)  Eqn(64)  Then the deadbeat gain vector can be explicitly calculated as:  K  = bottom - row - o / ( T A ) _ 1  d  T Kd  Eqn(65)  m  Excitation Series Matrix Deadbeat Gain Vector  When stability of a system is evaluated by the largest eigenvalue o f the propagation matrix, P, then the deadbeat solution is the most stable system possible. However, the deadbeat solution is "similar" to an L Q R solution in the limit as the cost-ofoutput, (usually known as R ) , goes to zero. The deadbeat solution is thus just as unreasonable for actual application in most cases as a low cost-of-output L Q R gain. " S i m i l a r " gain in this case can be taken to mean gains whose elements are o f the same close order of magnitude and same sign.  83  ppendix C Notch Filter Algorithm We implemented our notch fdter with a 'resonator' which is a numerical technique using a state vector designed to resonate at a single frequency excited by an input signal. This resonator vector is a numerical representation of a sine and cosine component (or real and imaginary component) of a single frequency amplitude.  x  X.real  Eqn(66)  X ,imag  The vector x is updated every iteration of a numerical system through a rotation operation and an innovation as follows: Eqn(67)  Where G and A are defined shortly and where the matrix R is a standard rotation matrix:  R =  cos(or)  -sin(a)  sin(a)  cos(a)  Eqn(68)  The angle alpha is defined by the algorithm iteration time to be the angle of phase transited during a single algorithm loop time at the resonant frequency: Eqn(69)  a = 2njT  Where / is the frequency of the resonator in hertz and T is the loop time for the algorithm in seconds. Both R and a are constant and don't have to be calculated every loop iteration. The matrix G is the gain matrix which allows the measurement to affect the filter amplitude and is given by:  Eqn(70)  G=  84  Where w defines the width o f the notch filter as follows.  W = 27tfJ Where f  w  Eqn(71)  is the width of the resonator response (or notch filter) i n hertz.  The residual is defined as the measurement minus an estimate of the resonator amplitude as follows:  A =m -(0 k  k  l)Rx  Eqn(72)  t  Where nik is the measurement. The notched measurement stream is simply the first element of the residual A*. The measurement minus the predicted amplitude of the resonator is the notched result. In short, the residual is the notched measurement stream. To obtain mutiple resonant notches, every resonator prediction must be subtracted from the measurement to obtain an overall residual: Eon(73^  MaxResonators  Jk 7=1  The real component of Xj can be used as a real-time measurement o f a single spectral component of the signal. A n overall amplitude of the signal can be obtained by taking the norm o f the vector xj. Thus the entire algorithm takes about five multiplication operations and about the same number of additions per iteration to provide either a specral line amplitude monitor or an effective digital notch filter.  85  References 1) J . Frisch et al., Active Vibration Suppression R&D for the NLC i n Proceedings o f the 8th International Conference on Accelerator and Large Experimental Physics Control Systems, 2001, San Jose, California (published electronically) Retrieved June 2, 2004 from http://epaper.kek.jp/ica01/proceedings.html 2) F. L e Pimpec et al., Vibrational Stability of NLC Linac Acceleration Structure i n proceedings o f E P A C 2002,Paris (published electronically) Retrieved June 2, 2004 from http://accelconf.web.cem.ch/AccelConf/e02/KEYWORDS/K109.htm 3) S. A l l i s o n et al., Active Vibration Suppression R+D for the Next Linear Collider i n Chicago 2001, Particle accelerator 1261-1263 4) F. A s i r i , F. L e Pimpec, A . 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Robust and Optimal Control (Prentice-Hall, Upper Saddle River, N e w Jersey, 1996) I S B N 0-13-456567-3 10) Pedrotti, Pedrotti. Introduction to Optics Second Edition (Prentice H a l l , Upper Saddle River, N e w Jersey, 1993) I S B N 0-13-501545-6 11) Franklin, Powell, Workman. Digital Control of Dynamic Systems (Third Edition, Addison-Wesley U S A , 1998) I S B N 0-201-82054-4 YI) Modern Control Engineering (Third Edition, Ogata, Prentice H a l l , Upper Saddle River, N e w Jersey, 1997) I S B N 0-13-227307-1 13) A Mohazab (private communication A p r i l 2003)  Corporate References 14) Jager Computergeu MesstechnikGmbH, Rheinstr. 4,64653 Lorsch, Germany Manufacturer o f A d w i n - G o l d realtime computer system 15) WaveMetrics, Inc., P.O. B o x 2088, Lake Oswego, O R 97035, U S A Manufacturer o f " I G O R " data aquistion/ manipulation software 16) R T S Consulting Ltd, 1 Noble Court, Port M o o d y , B . C . V 3 H - 3 Z 5 Canada. Manufacturer o f A D W L N - I G O R connection driver software. 17) Waterloo Maple Inc. 450 Philip Street, Waterloo Ontario Canada N21 5J2 18) The Mathworks Inc. (Publishers o f M A T L A B ) 3 Apple H i l l Drive Natick, M A 01760-2098 UNITED STATES 19) Physik Instrumente (PI) G m b H & C o . K G A u f der Roemerstrasse D-76228 Karlsruhe/Palmbach Germany  87  I n d e x accuracy, 57 actuator, 3 A d w i n , 31 Algorithm Tests, 57 Apparatus, 29  interferometric instrument, i i lissajous, 43, 44, 50, 51, 53, 57 lock-in amplifier, 32, 33 luminosity, 1 Mechanical tuning, 57 modes, 2 Mohazab, 45 monlithic mass, 23 natural system evolution, 16 Next Linear Collider, 1 Non-Correlated Noise Test, 59 non-linear systems, 43 Notation convention, 16 optical anchor, i i , 1,2 Optimal tuning, 57 Parameter Iterations, 47 Parameters o f the geometric model, 44 Particle accelerators, 1 phase-locked, 32 PID, 8 piezo, 33, 34, 58 propagation matrix, 17 quadrature, 40 second order response, 5 Simple Continuous M o d e l , 3 spectral system response, 32 stability, 4 stability contour, 21 stalled trajectories, 53, 54 symmetric matrix, 50 system evolution, 4 System M o d e l , 5, 9, 13 system response, 4 systematic error, 57 test mirror, 58  Beam Based Systems, 1 beam deflection, 1 beam diameter, 1 Coherent Disturbance Correction, 74 coherent ground disturbance, i i control response, 16 control term, 16 control transducer, 34 Convergence, 50 critical damping, 8 damping, 4, 6 Data Acquisition, 32 deadbeat gain, 21 degrees o f freedom, 45 Direct Parameterization, 37, 39 discrete parameterization, 16 effective control, 5 explicit model, 39 Explicit S Iteration, 49 feedback, 5 feedforward, 5 first order iterant, 48 flexure mode, 23 geometric parameterization, 45 Geometric Parameterization, 43 Geometric Parameterization Validity, 44, 47  Inertial Based Systems, 2 Interferometer, 37  Interferometer Based Systems, 1 Interferometer Setup, 37  88  


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