WO2007081399A2 - System and method for processing control system signals - Google Patents

Abstract

A system and method for processing one or more signals encoded with angle information that corresponds to a physical quantity is described. In one embodiment, a first and second sinusoidal signals are received that are offset in phase. Each of the sinusoidal signals including a respective DC offset, amplitude and an encoded angle, and the encoded angle represents a physical quantity. A representation of the encoded angle is generated using, at least in part, the DC offsets and the representation of the encoded angle is usable to control the physical quantity.

Description

SYSTEM AND METHOD FOR PROCESSING CONTROL SYSTEM SIGNALS

FIELD OF THE INVENTION

[0001] The present invention relates to control systems. In particular, but not by way of limitation, the present invention relates to systems and methods for processing control- system signals.

BACKGROUND

(0002J In many contexts, precisely measuring and/or controlling the position of an object is critically important. As an example, robotic systems, machine tools, and spectroscopic- analysis equipment all require precise positioning to perform their intended functions. Tn many systems where position control is important, a position of an object is controlled based upon feedback generated from sinusoidally encoded position information.

[0003] As an example, a Fourier transform infrared (FTlR) spectrometer uses the sinusoidal interference pattern generated from coherent light (e.g., a HeNe laser) to control the position of a movable mirror. Specifically, a signal from a photo detector that senses the interference pattern may be AC coupled to a zero crossing detector circuit, which causes a sample spectrograph to be taken at each zero crossing of the detected interference. To achieve a higher resolution, in some implementations, a phase lock loop (PLL) may be used to generate a control signal based upon a multiple of the number of zero crossings.

[0004] In many instances, it is desirable to move an object in discrete steps instead of a continuous motion. Tn some systems, however, the position signal relative to a stationary object is a constant DC signal, which can not be AC coupled to determine zero crossings. As a consequence, the method of using a PLL to increase the frequency will not work because the PLL docs not have an AC reference to lock on to and multiply. In these instances, the resolution of the position control is limited by the frequency of the sinusoidal signal. Although, higher- frequency signal sources may be employed to improve resolution, the implementation of these signal sources may be cost prohibitive.

[0005] In the context of Fourier transform infrared (FTIR) spectrometer, for example, the use of HeNe lasers is pervasive due to their low cost relative to higher frequency sources. As a consequence, in current FTIR systems employing an HeNe laser, the sampling during a step-mode type of scanning is limited to the zero crossings of the HeNe signal, which means the maximum wavenumber that can be resolved is 15798 cm^"1.

[0006J Although present control systems (e.g., FTIR spectrometer control systems), are functional, they are not sufficiently accurate or otherwise satisfactory. Accordingly, a system and method are needed to address the shortfalls of present technology and to provide other new and innovative features.

SUMMARY

[0007] Exemplary embodiments of the present invention that are shown in the drawings are summarized below. These and other embodiments are more fully described in the Detailed Description section. It is to be understood, however, that there is no intention to limit the invention to the forms described in this Summary of the Invention or in the Detailed Description. One skilled in the art can recognize that there are numerous modifications, equivalents and alternative constructions that fall within the spirit and scope of the invention as expressed in the claims. I0008J The present invention can provide a system and method for processing one or more signals encoded with angle information that corresponds to a physical quantity. In one embodiment, a first and second sinusoidal signals are received that are offset in phase. Each of the sinusoidal signals including a respective DC offset, amplitude and an encoded angle, and the encoded angle represents a physical quantity. A representation of the encoded angle is generated using, at least in part, the DC offsets and the representation of the encoded angle is usable to control the physical quantity.

10009] As previously stated, the above-described embodiments and implementations are for illustration purposes only. Numerous other embodiments, implementations, and details of the invention are easily recognized by those of skill in the art from the following descriptions and claims.

BRIEF DESCRIPTION QF THE DRAWINGS

[0010] Various objects and advantages and a more complete understanding of the present invention are apparent and more readily appreciated by reference to the following Detailed Description and to the appended claims when taken in conjunction with the accompanying Drawings wherein:

FIG. 1 is a schematic diagram that depicts an exemplary environment in which embodiments of the present invention may be implemented;

FIG. 2 is a block diagram depicting a typical control system;

FIG. 3 is a block diagram depicting one embodiment of a control system in accordance with the present invention;

FIG. 4 is a block diagram depicting a signal processing module in accordance with an exemplary embodiment; FlG. 5 is a graph depicting a relationship among parameters processed by the signal processing module depicted in FlG. 4;

FIG. 6 is a graph depicting a mapping between detector outputs and projected outputs;

FlG. 7. is a flowchart depicting a method in accordance with an exemplary embodiment;

FlG. 8. is another flowchart depicting a method in accordance with another embodiment; and

FIG. 9 is a schematic representation of an exemplary embodiment of an interface to a digital signal processor that is implemented for the signal synthesizer of FIG. 3.

DETAILED PESCRIPTTON

[0011] Referring now to the drawings, where like or similar elements are designated with identical reference numerals throughout the several views, and referring in particular to FIG. 1 , shown is a schematic diagram depicting an exemplary environment in which embodiments of the present invention maybe implemented. Shown in FlG. 1, is a HeNe mirror-position control system 100, which maybe employed in a spectrometer (e.g., a Fourier transform infrared (FTIR) spectrometer). As depicted in FIG. 1, a moving mirror 102 is moved by an actuator 104, which is driven by a control system 106. The control system in this embodiment uses the feedback from detectors^ and yi to estimate the mirror position. [0012] Although many embodiments of the present invention are described in connection with a spectrometer for exemplary purposes, those of ordinary skill will recognize, in light of this specification, that other embodiments may be implemented in a variety of applications where it is desirable to monitor and/or control a physical quantity. For example, the systems and methodologies described herein may be employed in many types of interferometer position control systems (e.g., FTIR spectrometers or a micro-optical mechanical devices). Moreover, the teachings herein apply to systems for controlling or measuring physical quantities (not necessarily position) that can be sensed by an interferometer. For example, the phase of an electro-optic crystal may be more precisely controlled using embodiments described herein. Similarly, lasers are often stabilized in frequency using an interferometer, and variations of the embodiments described further herein may be used to slowly scan such a laser in frequency.

10013] In addition, embodiments may be utilized in virtually any kind system that carries out position sensing or control using a sensor capable of generating roughly quadrature signals related to position (e.g., where gain drifts and DC offsets are a problem). A polaroid-type resolver, is an example of an optical device of this type. As another example, it is contemplated that embodiments of the present invention may enable a coreless synchro- resolver to be implemented, particularly in microfabricated systems where shielding is hard to achieve and charge coupling are difficult to avoid.

[0014] Yet other embodiments of the present invention may be employed in systems using a gear and sensor (e.g. a Hall effect sensor) to measure angle (e.g., crankshaft angle measurement for engine control) or displacement (e.g., machine tool position control). Tn some of these applications the sign of rotation is known a priori and the techniques described further herein may be applied using just one measured signal.

[0015J As another example, embodiments may be utilized in systems where a phase-locked loop is desired near DC and/or where signal imperfections such as gain and offset drift exist. Specifically, applications including synchronous detection at extremely low-frequencies (e.g., low -frequency excitation of linear variable differential transformers or wheatstone bridges).

[0016] Referring again to FlG. 1 , the theoretical detector outputs, taking into account the extra path length difference for y>₂ may be expressed as: i)_\ = /1( 1 + {Os(27r/;_/,,..v,,/>) )

(1) //., = Λ ( I 4- COS(27ΓJ'_H(..V,.0/ + Δ)) )

where p is the path length difference for detector y\, Δ is the extra path length difference for detector yi, and V_H^_NC is the HeNe wavenumber. In the exemplary system 300, the detectors are offset in phase, i.e., iy, =-- A( I l- COrf(rt))

(2)

(/_> = A( 1 + CΛ)*{ (1> |- Ou) ) _:

where φ is the effective angle corresponding to mirror position and φ₀ is the constant phase difference between detectors y_\ and y-χ, which may be calibrated to be close to ninety degrees. [0017] Referring next to FIG. 2, shown is a block diagram depicting a prior art control scheme 200 for the system depicted in FIG. 1. As shown, the input p_c(t) in FIG. 2. is a mirror-position-control command corresponding to the PC input of FIG. 1. The control system 200 estimates the mirror's position with the position estimator, and generates an error signal e(t) by subtracting the mirror position estimate from the PC position command. The error signal is fed into an amplifier 202, with a gain K(s), which drives the actuator 204, which may be modeled with a single time constant.

(0018J Analysis of the feedback system of FIG. 2 can be considerably simplified with a few reasonable assumptions. First, if the HeNe mirror position control system is assumed to work properly, then the feedback path frompfc) to pit) has a gain of unity, i.e.

Pit) = p(t). (3)

Continuing with this assumption, the open loop gain in the Laplace domain is

The closed loop transfer function from p_c(t) to p(t) is completely determined by the loop gain L(s). From the diagram,

The closed loop transfer function is therefore

P, ( s ) L ( <. )

P( s ) I H- L ( s) ^" (6)

Substituting Equation 4 into Equation 6, the closed-loop transfer function can be rewritten as

WO _ A-(^n

The closed loop transfer function is significant because it determines how the mirror position p(t) responds to a command \apvΛp_t(t). The roots of the denominator of this transfer function are the natural frequencies of the closed loop system. If the natural frequencies are complex

conjugate symmetric, the real part of the natural frequencies is — The step response of the

system, based on this model, should have a damped sinusoidal part decaying with an envelope of e^~'^/2τ . Simulations with constant K(s) and r = 9 ms show substantial agreement with actual measurements taken, and the closed loop model provides a good understanding for the dynamics of the mirror control system. [0019] Referring next to FIG. 3, shown is a block diagram 300 depicting a control scheme in accordance with one embodiment of the present invention. In this embodiment, the control scheme depicted in FIG. 2 is adapted so that easily accessible position-detector outputs (e.g., interferometer detector outputs) from a position detector 302 are intercepted by a signal synthesizer 304 and an estimate φ(t) of the angle φ(t) is computed. As shown, the intercepted signals are substituted with quadrature "synthetic" detector signals with an new angle Nφ{t) where N is a small integer. [0020] The gain from the mirror position p(t) to p(t) is N rather than unity, i.e. p{t) = K_V{i). (8)

Similarly, the loop gain L(s) increases by a factor of N, /,» > -. - '^YA> '" (9)

and the closed loop transfer function becomes

J In terms of the behavior of the control system 300, the ideal closed loop gain in FIG. 3 is

— - , so when N = 1 the mirror position p ideally steps as commanded by p_c(0- If /V = 2,

N then the steps in the actual mirror position p are half that of the command, providing finer spatial sampling and enhanced frequency range.

[00211 It should be recognized that the embodiment depicted FIG. 3, which may be used to retrofit an existing control system, depicts merely one embodiment of the present invention and mat many other embodiments, which may integrated as part of an original design of a control system, are contemplated. In an alternative embodiment, for example, the signals from the position detector 302 may be received and processed to generate an error signal without generating intermediate synthesized signals. It should also be recognized that the control scheme depicted in FlG. 3 is based upon the system described with reference to FIG. 1 , and that in other implementations, two detector signals are unnecessary to establish a representation of φ(f) . [0022 J Referring next to FIG. 4, depicted is an embodiment of a signal processing module 400, which may be employed in the signal synthesizer of FIG. 3 as well as in other implementations. As shown, the signal processing module 400 in this embodiment includes an estimator module 402 and a projection module 404. Also depicted is an optional angle-scaling module 406. The signal processing module 400 may be realized by hardware, software or a combination thereof. In general, the signal processing module 400 is configured to receive one or more signals (e.g., from optical or magnetic detectors) and extract encoded angle information that corresponds to a physical quantity (e.g., a position of an object, a temperature, an angle of a rotatable shaft or a phase delay of a wave front). In many embodiments, for example, the signal processing module 400 is configured to utilize parameters (e.g., DC offsets and gain information) from the received signal(s) to extract the encoded-angle information from the received signal(s). The angle scaling module 406 may be used in the embodiment depicted in FIG. 3 to scale the angle information by N. (0023J The idealized detector outputs in Equation 2 differ from true detector outputs by electronic offsets and gain imbalances between the two sensors. A more realistic model of the electronic detector outputs is

b:

(H)

Vi = Λ'U -on ( o 4- <;>„ ) -T- h₂. where b_/ and &₂ lump the DC term of the ideal response with the electronic offset for each detector, and δ is a gain-mismatch between the detectors.

[0024] Referring briefly to FIG. 5, shown is a graphical representation of the parameters in Equations 1 1. As shown, y_/ is plotted with respect to j/2 for a complete cycle of φ . The elliptical shape is due to the gain-mismatch in the detectors, the rotation of the ellipse is due to φ₀ , and the center of the ellipse represents the DC offset values. All of the parameters

from Equation 1 1 are time varying, and _^4, δ , bj, b₂ and φ₀ change the corresponding

signals at a much slower rate than changes of φ . The angle φ in many implementations

changes on a much faster time scale due to the change in the physical quantity (e.g., object movement) that is associated with φ .

[0025] In some variations the estimator 402 places a constraint upon how the parameters of Equation 11 change with respect to one another. In one embodiment for example, an estimate of the parameters of Equation 11 is made by setting up two coordinate systems: a cartesiancoordinate system and a polar coordinate system. The center of the ellipse, for example, may be estimated on a cartesian coordinate system and the angle φ may be

estimated on a polar coordinate system. This allows the estimates of each coordinate system to evolve at two different rates. The time constants used by the estimator 402 may be selected so that the parameters do not drift around when the mirror is stopped. This type of estimation involves two time constants, one for φ and another time constant for the parameters that

affect relatively slow changes to the signal, A, 5 , b_/, l>₂ and φ₀ .

[0026| In other implementations, this two-time constant approach is avoided because, in many instances, φ (per se) does not have to be estimated. The smoothing effect that the estimation has

on φ may not be required because many control systems are able to tolerate and reject some

noise. Indeed, the phase lag of the estimate would tend to destabilize the control loop. This allows for φ to be calculated directly from the measurements :)// and y^ and reduces the order

of the estimator.

[0027] In the embodiment depicted in FIG. 4, the projection module 404 may be used to project the ellipse onto a circle by mappings_/ and _v^ to a different coordinate system. By recognizing that the cosine term in yj can be written as cos(ø ) and sin( ^ ), Equation 1 1

becomes

/ + (12)

where M contains δ and φ₀ information to rotate and stretch an ideal circle to the rotated

elliptical shape of FIG. 5 and bi and Z»2 shift the center of the ideal circle to the first quadrant, as in FIG. 5. This model can be recast as

/ W y₁ ) X b₂ ) J \ a b ) \ y₂ ) \ rlj

This equation represents the projection of the measurements into an ideal coordinate system where the path difference is mapped to a point q on a circle with radius A, centered on the origin. FIG. 6 shows the mapping graphically between the true detector outputs and the projected outputs. 100281 To reduce the order and allow for a slow varying, single time constant estimator, the estimator 402 may establish an estimate that docs not depend on φ . Since g^τ q — A?,

Λ^~ = ( 1 4- tt² )yf 4- Α/.j — -"''</ 1.2/2 — '2>.c — 'to )."ι +

This equation can be written as a linear identification problem with

I (15)

where

1 4- <*-

/^ l =

Ju

1*2 = b

^■2<1

/'^••i =

^ _{^ f}μ _ A¹

I'b ==

Given sufficient excitation, the parameters μ* can be determined by standard methods, for example, recursive least squares. Similarly, the coefficients of the mapping Equation 13 can be determined algebraically from the μ%- If all the quantities to be estimated vary slowly compared to the physical quantity (e.g., a mirror position), then the parameter update can be done on a convenient time scale. It is only necessary to evaluate Equation 13 in real time to find the latest q from y_\ and jμ^ for purposes of control. [0029] Given q, normalized synthetic detector outputs u for an integer angle multiplication factor N are given by the real and complex parts of the rotation.

|0030] In general, practical values of TV are two or three. Detector and optic limitations generally make further extension of the frequency range impractical. |0031] The signal processing module 400 in many embodiments computes an estimate φ of

the effective angle φ that corresponds to the physical quantity at issue in a system (e.g., an

interferometer path length difference) and the angle scaling module 406 optionally (e.g., in retrofit situations) outputs signals with Nφ to the rest of the control system. While the a

control system (e.g., mirror control system) may be able to reject some zero-mean noise in this feedback path, any offset value or bias in φ would cause an error in the physical quantity (e.g.,

mirror position). Therefore, in many embodiments the angle represented by q in Equation 13 is unbiased with respect to sensor noise in the inputs yj and>>₂. A computational technique known as the bootstrap can be used to handle the complexity of how sensor noise would influence the computation of q through Equations 15, 16, and 13. A bootstrap dataset consists of data plus synthetic sensor noise with some assumed statistical properties. Applying the estimator to many datasets with different noise samples builds an approximation to the distribution of quantities of interest, e.g., φ .

[0032] Estimating the projection parameters and using them to project the measurements, j/i and yx, onto the ideal coordinate system enables the calculation of φ instead of estimating it. This significantly reduces the complexity of the calculation and greatly simplifies the choice of the time constant. And the mapping approach appears to provide an unbiased value of φ .

[0033] Referring next to FIG. 7, shown is a flowchart 700 depicting a method in accordance with one embodiment. As shown, in this embodiment a first and second sinusoidal signals that are offset in phase are received, and each of the signals includes a DC offset, an amplitude, gain information as well as encoded angle information that corresponds to a physical quantity (Blocks 702, 704). Each of the signals may be, for example, from a detector (e.g., a photo diode or Hall-cffcct sensor) in a control system. As discussed, the physical quantity corresponding to the angle is dependent upon the environment, but may include, without limitation, a quantity characterizing a position of an object, a quantity characterizing an angle of a rotatable shaft, a quantity characterizing a temperature or a quantity characterizing a phase delay of a wavefront.

[0034] As depicted in FIG. 7, a representation of the angle is generated using at least the DC offsets from the received signals. As discussed with reference to FIGS. 4 through 6, instead of filtering out DC offset information from received detector signals like prior art approaches, the DC offset information is used in this embodiment to arrive at a representation of the angle information that is encoded within each of the signals. For example, the DC offset information may be used in connection with other parameters (e.g., parameters that affect changes in the detector signals at a slower rate than the encoded angle) to generate a representation of the angle (e.g., in the form of sinusoids or the encoded angle per se). And, once the representation of the angle is generated, it may optionally be used to control (e.g., directly or indirectly) the physical quantity (Block 708). In other embodiments, the angle information maybe used merely for informational purposes, e.g., reporting system information.

[0035] Referring next to FIG. 8, shown is another flowchart 800 depicting a method in accordance with another embodiment of the present invention. As shown, in this embodiment, an object is moved in a desired direction (Block 802), and a sinusoidal signal that includes an encoded angle that corresponds to a position of the object is received (Block 804). As shown, the sinusoidal signal includes DC offset and gain information, and at least the direction of the of the object in connection with the DC offset is used to generate a representation of an angle (Block 806). As a consequence, in this embodiment, by virtue of having information about the direction an object is moving, the representation ot the angle may be generated with a single sinusoidal signal.

[0036J As depicted in FIG. 8, the representation of the angle may then be used to identify the position of the object (Block 808). One application where the present embodiment may be employed is in an engine where encoded-crankshaft-angle information may be obtained from a single detector signal because the sign of rotation is known a priori.

J00371 As one of ordinary skilled in the art will appreciate in light of this disclosure, the methods described with reference to FΪGS. 7 and 8 may be carried out, at least in part, by a digital signal processor (DSP). The signal synthesizer 304 depicted in FIG. 3, for example, may be realized by a DSP module mat includes interface circuitry and a DSP. Specifically, the

DSP may be used to calculate φ and generate the synthetic detector output signals with an

angle of Nψ . Slight adjustments to the control system may be needed to run the DSP module

in an existing control loop (e.g., the control loop depicted in FIG. 2) [0038] One DSP module that may be used includes two printed circuit boards (PCBs) stacked on top of one another. The bottom board, for example, may be a LF2407A eZdsp evaluation board from Texas Instruments, and the DSP may be a TMS320LF2407A 16 bit, 40 MIPS, control oriented DSP with a built in 16 channel, 10 bit analog to digital converter (ADC) and a serial communication interface to easily send data to a digital to analog converter (DAC). The ADC is capable of sampling at 500 kHz, however, it has been found that 80 kHz is an effective sample rate.

[0039] The top PCB is the analog interface board, which has the input circuitry, output circuitry, DAC, and the DSP loop switch. A schematic of the analog interface board is shown in FIG. 9. As depicted in FIG. 9, the input circuitry amplifies and conditions the signals from

and yz for the ADC. The circuitry includes of a cascade of operational amplifiers (op amps) to convert the current from detectors y_\ and j>_2> to a voltage. The first stage is a traditional current to voltage inverting op amp. Tine second stage is a standard inverting op amp which is used to lower the cut off frequency and provide additional gain if needed. The output circuitry uses a Howland current source to convert the signals from the DAC into signals which look like they are coming from detectors y>_\ andj/₂. The section labeled "Output Circuitry" in FIG. 9 shows the schematic diagram for a photodiode "look-alike" circuit. For a typical Howland current source, the resistors are chosen so that the ratio of the resistors connected to the inverting node is equal to the ratio of the resistors connected to the non- inverting node. If this condition is met, the output current is the difference between the input voltage from the switch and the ground node voltage, divided by the input resistor connected to the non-inverting node and the switch, providing a differential input and an output current that does not depend on the load resistance. The typical problem with a Howland current source is that the resistors are not perfectly matched. This means that the input voltage to the Howland current source no longer has the differential advantage and the load resistance can change the gain of the current source. However, this is not a problem in this application due to the fact the current is being fed into the control system electronics with a very low input impedance eliminating the load impedance problem. Moreover, the gain does not have to depend strictly on the input resistor since the gain and offsets are being estimated by the control system. The DAC is a Texas Instrument TLV5638 with 12 bits of resolution, dual output, and a serial input. [0040] The DAC is capable of output sample rate of 625 kHz, however, it has been found that 80 kHz is an effective output sample rate. The DSP loop switch, in FIG. 9, is used to switch the DSP module between two modes. "Pass through" mode allows the detector signals y_\ and J₂ to be passed through as if the DSP was not in the loop. This allows the DSP module to be in the loop without affecting the current operation of the control system. The other mode of operation is N=3 mode, where the DSP is in the loop and the output of the module is the synthetic detector signals Ui and U₂. [0041] To calculate the synthetic detector signals Ui and u?, the DSP first estimates μ from Equation 15, using RLS with a forgetting factor. The basic form of the state update given the measurement model z_k = φ' kU,, is

/U-M = Ui + I\ _r-ιA t - -l/'J- i (18)

and the inverse covariance update is

where λ is the forgetting factor and ranges from 0 to 1. A forgetting factor is commonly used to ensure that the state continues to track the input. Without a forgetting factor, the inverse covariance integrates to a very large number and the covariance goes to zero. The state update stops adding any new information from the measurement z* and the update stops evolving. For a time-varying process, the forgetting factor λ allows the newest measurements to affect the update.

[0042] In one experiment, the detector outputs fromy_/ andy^ were measured over a period of time to find how the parameters of Equation 11 varied with time. To do this experiment, the interferometer was set to continuous scan mode so the output of detectors y_/ and y₂ were 2.2 kHz sine waves. A National instruments PX1-5122 14-bit DAC card, sampling at 200 kHz, was used to measure the outputs from detectors yi and y_∑. The room temperature was also measured to see if it had any effect on the parameters. A two-second interval of temperature and detector output data was collected every five minutes, with a total test run time of 13 hours. During each two-second interval of data collection, the temperature for that interval was estimated using least squares with a constant temperature model. The parameters for the detector outputs were estimated using a nonlinear Gauss-Newton parameter estimation for all the parameters of Equation 11. The model used for the estimate assumes that the mirror is moving at a constant speed. Each two-second interval of detector output data was reduced to the 20,000 samples where the mirror was moving at its most constant velocity (note that when the mirror changes directions, the velocity changes at a high rate and this data should not be used). The Gauss-Newton routine was run for each of the data intervals. Of note, the amplitude in this experiment was found to have a time constant of approximately 45 minutes and changed a maximum of 5%. This is used to determine the forgetting factor for the parameter estimate.

[0043] When deciding the correct forgetting factor, there is a tradeoff between how long the mirror can sit in one place without excitation and being able to adapt to the changing parameters. Lack of excitation requires a slow forgetting factor. If the system parameters change rapidly, a fast forgetting factor is needed. Based experimental requirements, a forgetting factor of 15 minutes is used.

[0044] To implement RLS with the forgetting factor in the DSP, the steady-state solution of the covariance was found so the inverse covariance does not have to be computed at each update. This allows the DSP to only update Equation 18 and makes it possible to compute the estimate of μ in real time. The steady-state value was found by simulating the detector signals yi and JK_? in Matlab and estimating μ for the simulated detector signals until the covariance reached a steady-state value. The steady-state value of the covariance was taken with the forgetting factor of 15 minutes used in this simulation.

[0045] After the DSP has estimated μ, the DSP finds the mapping parameters from Equation 13. The mapping parameters could be solved algebraically, however, due to the speed and quantization error from calculating the parameters, an iterative scheme is used to find the mapping parameters. The iteration is derived from a first order Newton's method for finding the roots of an equation. The Newton update is derived from the Taylor series expansion of a function for perturbations v about a point x, i.e. f{x -4- v) = /(.» J + V.f>Λ⁷V. (20)

Setting f (x H- v) = 0 and solving for v gives v = - (V fin' \ ^lj > > = - ' ¹ Z O- J. (21 ) where J is the Jacobian of f (x) and the step v minimizes the first order Tayloi seπes approximation of Equation 20. This can be written as an update,

/ , __! = ( ! . / ' /^'> / J (22)

where the roots of f (x) arc being estimated. This iteiation scheme is used for inverting and finding the parameters of Equation 16 The equations to solve are

1 - M -

Ii x

2« l>2 T

I² . X¹

The system can be multiplied by b² so that the divide does not have to be carried out. This may be done where the DSP does not have a hardware divider and the software divide takes many cycles. The system of equations becomes

The Jacobian of/(α, b, c, d) is

Substituting Equation 23 and Equation 24 into Equation 22 gives

To avoid having to take the inverse of the Jacobian matrix for each update, the Jacobian can be appioximated with a fixed matrix, i.c

Using this suboptimal gam in Equation 26, the DSP can converge to the correct value within a few samples. The new estimate becomes

Using this result, q is calculated with Equation 17 to give the projection for y/ and y₂ to the ideal coordinate system. The synthetic detector signals it/ and U2 are calculated with Equation 17. The calculated DC offsets are added back to u; and u∑ and sent to the DAC for output. [0046] The interferometry experiments were done using a Bruker ISF 66/S interferometer with the step-scan option. The interferometer is controlled by a PC using Opus 3.0.1 software running on OS/2. The setup begins by inserting the DSP module in between detectors yi and y>2 and the Bruker control system and setting the DSP module to "pass through" mode. With the DSP module in the loop, the Bruker mirror is set to a slow, continuous dither so the output from detectors y_/ and y_∑ is a 1.6 kHz sine wave. This allows the DSP to adapt its coefficients to the correct values. Once the DSP has estimated the correct parameters, the moving mirror is positioned for the start of the step-scan experiment and waits for the start command from the PC. The DSP module is then switched to Λ/=3 mode and a step-scan experiment is started with the PC interface software.

[00471 The Bruker control board has a proportional integral derivative (PID) compensator to change the mirror dynamics in step-scan mode. The gain of each clement of the PID compensator is easily adjusted with trim pots located on the Bruker control PCB. With the DSP in N-3 mode the loop gain of Bruker control system is increased by a factor of three as mentioned earlier. This increase in loop gain reduces the stability of the closed loop system by reducing the phase margin of the system. Another source of instability is the delay that the DSP module adds to the system. The delay also decreases the phase margin of the control system. These effects and other higher-order effects tend to make the control system unstable. To correct for instabilities of the system, the gain of the loop can reduced by a third by reducing all the elements in the PlD controller along with slight tweaking of each element to provide a slight increase in the phase margin.

[00481 To adjust the elements of the PID controller, an oscilloscope is used to monitor the output of detectors yi and y₂. To adjust the mirror dynamics, the DSP module is put into 7V=3 mode and a step-scan experiment is started. If there is any oscillation of the mirror, when the mirror is suppose to be at a fixed position, or if the over shoot and settling time are not good enough, the PID gains of each element are adjusted until the desired step response is achieved. After the dynamics have been adjusted to give a good step response, the mirror can be moved back to the starting position and more step-scan experiments can be run. [0049] How long the mirror sits in one place without any excitation is a potential concern. If the mirror sits at the starting position or any other position long enough, the estimate of the mapping equation in Equation 13 will drift, which will produce erroneous values of u_/ and U2. The period of time spent at each position is short enough that he effects of this drift are minimal for a normal step-scan experiment. [0050] Appendix A includes the open loop and closed loop results of the DSP module in the control loop along will experimental data for a step-scan experiment. [0051] In conclusion, the present invention provides, among other things, a system and method for processing signal information. Those skilled in the art can readily recognize that numerous variations and substitutions may be made in the invention, its use and its configuration to achieve substantially the same results as achieved by the embodiments described herein. Accordingly, there is no intention to limit the invention to the disclosed exemplary forms. Many variations, modifications and alternative constructions fall within the scope and spirit of the disclosed invention as expressed in the claims.

APPENDIX A

EXPERIMENTAL PROCEDURE AND RESULTS

To test the performance of the DSP module in the loop, multiple experiments have been performed using the FTlR spectrometer. All of the experiments were done using the same sample test specimen so the data can be easily compared. The testing consisted of a set of measurements for the sample with the DSP module in "pass through" mode and a set of measurements with the DSP module in N=3 mode. The experiments done in pass through mode are used to show the current performance and measurement capabilities of the interferometer. The other experiments show the performance and advantages of using the M=3 mode of the DSP module.

Closed Loop Mirror Positions

The input signals, y_\ and yi, to the DSP module were measured with a 14-bit National Instruments PXI-5122 DAC card for a step-scan N=3 experiment to test the stopping positions of the closed loop control system. The sample rate of the DAC card was 100k Hz and data was taken for 15 seconds, allowing the mirror to step through 81 mirror positions. The steady state values of each mirror position were taken and used in a least squares estimate to find the gain and offset parameters for the entire data set. To estimate the parameters, it was assumed that the mirror stopped at evenly spaced positions for each mirror step.

The stopping mirror positions for the closed loop system are shown in Fig. 14. The clusters of dots show where the mirror is stopping. The dashed line represents the ideal mirror position as it moves. The cross marks on the graph are generated from the least squares fit and represent ideal, evenly spaced, mirror stopping positions. This figures shows that there is a bias in the fixed mirror positions. It is unclear, in this closed loop experiment, whether the bias comes from the DSP method or other parts of the loop.

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Figure 10: This figure shows the ideal stopping mirror positions along with the actual stopping positions. The cross arrows show the ideal mirror positions and the clusters are where the mirror actually stopped. The dotted line is the ideal mirror path.

DSP Mirror Estimate

The DSP estimation routine was checked by measuring the input and output signals of the DSP module, y\ and u\ respectively, and checking to see if the output signal u_\ represented three times the angle of y_\ . Figure 15 shows a block diagram of the test that was done to check if the DSP was introducing angular error. Signals y_\ and u_\ were measured with the National Instruments PXI-5122 DAC card sampling at 100k HZ for a step-scan N=3 experiment. A Matlab model of an ideal DSP was created generating an output signal ύ_\ with three times the angle of the input signal y_\. Figure 16 shows the results of the computed value u as compared to the actual DSP output «|.

DSP UΛ error MODULE

Modeled DSP UΛ Output

Figure 11 : A block diagram for a test that was done to check if the DSP module was producing biased mirror positions. The DSP module was tested against an ideal DSP model.

The figure shows that the output of the DSP module is three times the frequency of the input signal. The error distribution in Fig. 16 shows the DSP module output minus the modeled DSP output. This value is zero mean, showing that the DSP module is estimating the correct value.

Spectral Experiment

The Bruker interferometer uses a tungsten broad band light source for the interferometer measurements. To do the N=3 step-scan experiment, blue optical filters are used to reduce the intensity of the red and infrared light so a larger aperture could be used without saturating the interferometer's photodetector. The larger aperture and blue optical filters allow more of the higher frequency light into the interferometer, increasing the signal to noise ratio of the higher frequency light.

To run a step-scan experiment, the DSP module is switched to pass through mode and the interferometer's alignment software is run to align the fixed mirror. The aperture is set to 6 mm to allow the maximum amount of light through the interferometer. To do a full experiment, two step-scan experiments are run back to

Figure 12: Measured DSP output and model output in 3x mode. The empirical distribution of errors is very close to zero mean.

back to obtain a spectrum of a sample and a background measurement without a sample. The two measurements are used to generate a transmittance spectrum for the sample of interest. For this experiment, an erbium doped crystal was used for the sample. After the mirror is aligned and the DSP module has adapted to the parameters, the mirror is stopped, the DSP is switched to N=3 mode, the erbium crystal is placed in the sample chamber, and the step-scan experiment is started. Once this experiment has finished, the erbium crystal is removed from the sample chamber and another experiment is done for the background sample.

A static N=3 step-scan experiment was done on an erbium crystal to test the performance of the DSP module. The static experiment was done because it can be compared to a continuous scan experiment, which uses the electronic PLL to measure higher frequencies. The extended spectral features in a TRS experiment, in N=3 mode, can not be compared to anything else since a step-scan experiment done in pass through mode can only measure frequencies up to 15798 cm^"1. Figure 17 shows the spectrogram and transmittance properties for a step-scan N=3 experiment with an erbium crystal. The dashed line marks the unmodified step-scan maximum frequency. The features to the left of the dashed line cannot be resolved with the traditional step-scan techniques. These features would be aliased to the lower frequencies if this experiment was repeated with the DSP in "pass through" mode. A spectrogram and transmittance diagram for a continuous scan is shown for the same erbium crystal, which is used as an ideal spectrogram and transmittance diagram to compare against the step-scan N=3 experiment. The transmittance properties of the step-scan experiment match that of the continuous scan experiment very closely. There is some deviation at the higher frequencies. Note that the photodetector's response falls off at the upper end of the spectrogram for the continuous scan experiment. This is due to the scanning velocity of the experiment and the response of the photodetector. Furthermore, the ends of the transmittance spectrum are noisy due to the low optical power, resulting in a much lower signal to noise ratio in that region.

Figure 13: The lower section of this graph contains four plot lines showing the back-ground spectrograms (the two upper lines) and the erbium crystal sample spectrograms (two lower lines). In the upper section of this graph, the transmittance spectrum of the erbium crystal sample is shown as determined in step and continuous scan modes. The step-scan data was collected in N=3 mode and the continuous scan data was collected utilizing the PLL with the DSP in "pass through" mode.

Claims

WHAT IS CLAIMED IS:

1. A method comprising: receiving a first sinusoidal signal and a second sinusoidal signal, the first sinusoidal signal being offset in phase from the second sinusoidal signal and each of the first and second sinusoidal signals including a respective DC offset, amplitude and an encoded angle, wherein the encoded angle represents a physical quantity, and wherein the DC offsets and amplitudes change the first and second signals at a much slower rate than the encoded angle; generating a representation of the encoded angle using, at least in part, the DC offsets; and using the representation of the encoded angle to control the physical quantity.

2. The method of claim 1, including: scaling the representation of the encoded angle so as to generate a scaled representation of the encoded angle; and controlling the physical quantity using the scaled representation of the encode angle.

3. The method of claim 1, wherein the physical quantity includes a physical quantity selected from the group consisting of: a quantity characterizing a position of an object, a quantity characterizing an angle of a rotatable shaft, a quantity characterizing a temperature, and a quantity characterizing a phase delay of a wavefront.

4. The method of claim 1, wherein the representation of the encoded angle includes two sinusoid signals.

5. The method of claim 1, wherein the representation of the angle includes a scalar representation of the angle.

6. The method of claim 1 , wherein the first sinusoidal signal and a second sinusoidal signal are received from sensors selected from the group consisting of hall effect sensors and optical sensors.

7. A method for controlling a position of a movable object comprising: receiving outputs from a first and second detectors, each of the outputs corresponding to two respective paths of coherent light, wherein one of the paths is defined by a position of the movable object, and wherein each of the outputs includes offset and gain information; generating, using the offset and gain information from each of the outputs, a representation of an angle corresponding to the position of the movable object; scaling the representation of the angle so as to generate a scaled-representation of the angle; and controlling the position of the movable object using the scaled-representation of the angle.

8. The method of claim 7, wherein the movable object is a mirror in a spectrometer.

9. The method of claim 7, wherein the representation of the angle is a pair of sinusoids, and wherein the scaling includes scaling the pair of sinusoids.

10. A method for measuring a position of a moving object, comprising: receiving a sirmsoidal signal, the sinusoidal signal including a DC offset, an amplitude and an angle that corresponds to a position of the object; generating a representation of the angle using the DC offset and, at least, a direction that the object is moving; and using the representation of the angle to identify the position of the object.

11. The method of claim 10, wherein the signal is received from a sensor selected from the group consisting of a hall effect sensor and an optical sensor.

12. The method of claim 10, wherein the generating includes generating the representation of the angle using the DC offset, an amplitude of the signal and gain information of the signal.

13. An apparatus comprising: inputs to receive a first sinusoidal signal and a second sinusoidal signal, the first sinusoidal signal being offset in phase from the second sinusoidal signal and each of the first and second sinusoidal signals including a respective DC offset, amplitude and an encoded angle, wherein the encoded angle represents a physical quantity, and wherein the DC offsets and amplitudes change the first and second signals at a much slower rate than the encoded angle; and a signal processing portion configured to generate a representation of the encoded angle using, at least in part, the DC offsets.

14. A computer readable medium encoded with instructions for processing signals, the instructions including instructions for: receiving a tirst sinusoidal signal and a second sinusoidal signal, me iirst smusoiαai signal being offset in phase from the second sinusoidal signal and each of the first and second sinusoidal signals including a respective DC offset, amplitude and an encoded angle, wherein the encoded angle represents a physical quantity, and wherein the DC offsets and amplitudes change the first and second signals at a much slower rate than the encoded angle; and generating a representation of the encoded angle using, at least in part, the DC offsets.