Automation and Control of FabryâPérot ... - Semantic Scholar
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Automation and Control of Fabry–Pérot Interferometers Enrico Canuto, Member, IEEE, Fabio Musso, and Luca Massotti
Abstract—Fabry–Pérot interferometry (FPI), which was originally invented for spectroscopy, is now evolving as a basic technology for ultrafine dimensional stabilization and measurement. To this end, the light path length of an optical cavity and the wavelength of a laser source injected into the cavity have to be tuned to each other through a set of frequency and/or displacement actuators driven by a sharp and narrow signal-encoding total-cavity detuning. Digital control is essential in facilitating and automating FPI use in view of space applications and routine instrumentation. This paper shows how embedded model control (EMC) technology, which was developed by one of the authors, allows to systematically proceed from fine dynamics and requirements to the EM, which is the core of control design and algorithms. In this framework, all critical control issues have a coordinated solution: disturbance estimation and rejection, command constraints and multiplicity, hybrid dynamics, constraints due to unmodeled dynamics, and performance analysis. Several experimental results are illustrated and discussed in the light of the methodology. Index Terms—Automation, digital control, embedded model control (EMC), Fabry–Pérot interferometry (FPI).
I. I NTRODUCTION
F
ABRY–PÉROT interferometry (FPI) was invented by the French physicists Fabry and Pérot at the end of the nineteenth century (1899) to enable high-resolution observation of spectral lines. Multiple light reflection between two plane surfaces creates interference patterns, which, being very sensible to wavelength and optical path length, were employed in the course of the 20th century as high-finesse scanning spectrometers. A significant improvement was made after 1970: the FPI optical length was automatically stabilized by means of piezoelectric ceramics and capacitance micrometers [1]. Soon after, the micrometer-to-piezo loop was computerized [2], and in 1984, the same technique was employed for measuring small displacements. In early developments, the FPI signal was not employed for detecting tuning error and driving piezoceramics. A breakthrough was made possible after 1980 by Manuscript received May 26, 2005; revised December 8, 2006. Abstract published on the Internet January 14, 2007. This work was supported in part by grants from Alcatel Alenia Space Italia, Turin, Italy, within research projects supported by the European Space Agency. An earlier version of this paper was presented at the 10th IEEE International Conference on Emerging Technologies and Factory Automation, Catania, Italy, September 19–22, 2005. E. Canuto and F. Musso are with the Dipartimento di Automatica e Informatica, Politecnico di Torino, 10129 Turin, Italy (e-mail: [email protected]; [email protected]). L. Massotti is with the Earth Observation Program, Future Project Division, European Space Agency-European Space Research and Technology Centre, 2200 Noordwijk, The Netherlands (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2007.892002
laser sources and the Pound–Drever–Hall (PDH) technique [3], which applied the same frequency stabilization method used in the microwave field [4] to Fabry–Pérot interferometers. The light beam of a laser source is frequency modulated before being injected into a Fabry–Pérot cavity made by two highly reflecting mirrors. The reflected light intensity collected by a photodiode and then demodulated becomes proportional to the cavity detuning from a frequency-length resonance condition. The corresponding signal can then be employed as a control error moving or keeping the cavity close to resonance. The PDH technique opened the way to a pair of applications: ultrafine laser frequency stabilization [3] and subnanometerlength measurement and stabilization [5]. In the former case, PDH signals are employed for tuning laser frequency to the optical length of a dimensionally stable cavity. In this way, small vacuum cavities (less than 20 cm long), which are highly stable in length, become frequency references having high spectral purity. Length stability is achieved by an athermic glass such as ULE having ultralow coefficient of thermal expansion (CTE), by a stable vacuum thermal bath and suitable suspensions [6]. Frequency references of this kind are expected to become a key technology for space interferometers (LISA and its precursor [7], DARWIN [8], earth’s gravity field from satellite-to-satellite interferometry [9]). Very long (up to 4 km) Fabry–Pérot vacuum cavities are employed as gravitational wave antennas in VIRGO [10] and LIGO [11] ground observatories. Much shorter cavities (1 m long) have been suggested as metrology lines for measuring and stabilizing the linear and angular dimensions of space telescopes such as GAIA [12]. High sensitivity to displacement and high resolution of frequency measurements have recently suggested employing FPI in submicronewton thrust stands such as the Nanobalance [13]. Submicronewton accuracy is essential for qualifying microthrusters to be employed in drag-free space missions such as GAIA, LISA, and DARWIN. One of the authors, who has been contributing to this technology since 1997, has implemented several digital control loops and automation to improve and facilitate FPI, as reported in [12]–[16]. An early application of computer control to scanning spectrometers has already been cited [2]. Earlier and recent applications to laser frequency stabilization are referenced in [15]. Because of interferometer complexity, computer control was imperative in VIRGO [17] and LIGO [18]. Notwithstanding such applications, the authors’ impression is that computer control is being adopted more for exploiting information technology than modern control
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methodologies [19]. The proof comes by taking insight into relevant design theories as provided by [20] and [21]. 1) Design stems from continuous-time transfer functions and not from discrete-time (DT) state equations. The latter ones allow designing all control phases in a unified manner, which is imperative for complex systems and, in this case, for the detection and recovery of FPI resonance conditions, as in Section III-B. At the same time, they favor multivariate control, as in Section III-D, and facilitate casting specific problems into general classes, as pursued in this paper. 2) No explicit interplay between the fine and design models is mentioned. The opposite would allow explicitly and iteratively tuning control parameters and performance as a function of the fine-to-design discrepancies, which are the so-called unmodeled dynamics (see Section III-G and [15]). 3) No explicit dynamics is assigned to each disturbance source, which is different from that in Section III-C, hence, uniquely relying on first-order integrative laws for their rejection, as already mentioned in [15]. The goal of this paper is to apply the embedded model control (EMC) technology [22] inspired by [19] to solve PFI control problems in a unified manner. Section II will provide a brief illustration of the FPI principles, of their applications, and of the output signal to be fed back. Section III will present a unified treatment of the FPI automation and control problems, assuming a single measure and multiple actuators. Section IV will present and discuss experimental results. Formulation will be free of theorem and proofs for the sake of brevity and readability. II. F ABRY –P ÉROT I NTERFEROMETRY A. Basic Principles and Applications The Fabry–Pérot interferometers treated in this paper are optical cavities [23] made by two aligned and highly reflecting mirrors, where a laser lightwave is injected and a standing reflecting wave is created and kept either by regulating the incident wave frequency and/or by fine moving either mirror. As a result, the cavity becomes an optical resonator as the injected light intensity is amplified inside the cavity under very low losses, as in a vacuum. Intensity amplification is measured by cavity finesse F [23], which, depending on mirror reflectivity, can reach values greater than 105 . Any frequency perturbation of the incident lightwave as well as any perturbation of the optical path inside the cavity tends to destroy the standing reflecting wave or resonant condition by varying the intensity of the lightwave emitted by the cavity itself, in which the wave may be arranged to encode perturbation sign and amplitude, the latter scaled by cavity finesse. Let fl = c/λl be the incident light frequency, which typically ranges from 100 to 1000 THz (from near infrared to violet), and is related to wavelength λl by the speed of light in vacuum c. Then, denote cavity length as L, which typically ranges from 0.01 to 4000 m when measured along the optical axis joining the spot centers of the laser beam on each mirror.
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The standing wave condition implies an integer ratio between the length and wavelength given by L = N λl /2 ⇒ fl L = N c/2,
N integer.
(1)
A pair (fl , L) satisfying the preceding condition is called a resonant pair and will be denoted as (f0 (N ), L0 (N )). From (1), it is immediate that two successive resonant pairs indexed N , N ± 1 are separated in length by half-wavelength λ0 /2, where λ0 f0 = c, and in frequency by the so-called free spectral range F0 = 0.5c/L0 = 1/τ0
(2)
where τ0 is the light travel time from mirror to mirror inside the cavity and F0 150 MHz at L0 = 1 m. Actually, (1) cannot be exactly met but only approached, by forcing the frequency and length to fluctuate around a resonant pair through active control. Thus, by denoting length and frequency fluctuations (often called detuning) as ∆L = L − L0 (N ) and ∆f = fl − f0 (N ), respectively, and fractional detuning as η and by rewriting (1) as (f0 + ∆f )(L0 + ∆L) = f0 L0 (1 + η)
(3)
a linear differential equation relating length and frequency detuning less the cavity detuning e can be written, i.e., ∆f + (f0 /L0 )∆L = e,
e = f0 η.
(4)
The preceding relation, which is called the lock-in condition, holds only under sufficiently small detuning e, where small is dictated by cavity finesse F. As the length-to-frequency scale k0 = f0 /L0 is usually much greater than 1012 and frequency detuning ∆f can be accurately measured below 1 Hz, FPI may reveal a subatomic length detuning that is well below Bohr radius (50 pm). This is achieved by revealing e, which is encoded in the emitted lightwave, through the PDH optoelectronics [3] and by keeping e as small as possible through a control loop. Accordingly, three main applications can be conceived. 1) Length stabilization. ∆f is kept negligible by a frequency-stabilized laser source, which implies (4) to reduce to ∆L = L0 e/f0 and to keep ∆L to zero by piezoelectric actuators, for instance, lead–zirconate–titanate (PZT) ceramics as in [12] and [14]. 2) Frequency stabilization. ∆L is kept negligible, for instance, by thermal control and ultralow CTE glass, which implies (4) to reduce to ∆f = e. Then, ∆f is kept to zero by locking the incident laser frequency to cavity length by means of suitable frequency actuators, PZT ceramics, and a thermoelectric cooler as in [15]. 3) Length measurement and tracking. A frequencystabilized laser wave is detuned by a frequency actuator prior to be injected in the cavity so as to zero e in the presence of length detuning ∆L. The actuated ∆f is made proportional to ∆L less residual detuning e, laser frequency instability, and actuator noise. The scale factor
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Fig. 2.
Typical response of the PDH chain versus frequency detuning.
For control purposes, the normalized frequency-to-frequency response is defined as H(e) = h(e)/kmax , Fig. 1. stand.
Essential components of the PDH technique in the Nanobalance thrust
f0 /L0 needs only to be calibrated for what concerns L0 , as f0 is ensured by laser frequency stabilization. As ∆L may result from natural and artificial causes as microthrusts in the Nanobalance thrust stand ([13], [16]), FPI can be made to reveal microscopic phenomena. In all applications, cavity detuning e plays the role of the “true” performance variable to be kept close to zero and below a predefined tolerance within a frequency bandwidth. B. Response of PDH Chain and FPI Performance An introduction to PDH technique can be found in [24]. The essential components are shown in Fig. 1, with reference to the Nanobalance thrust stand [13], which consists of two tilting plates carrying the reflecting mirrors of the cavity. The microthruster mounted on the active plate, when fired, shrinks the cavity length, thus forcing the control system to detune the laser frequency by means of a fast PZT and a slow actuator (thermoelectric cooler). The tilting plates must be equal in mass and geometry, which requires a dummy thruster to balance the active one. The detuned laser frequency is then beaten with a reference laser (not in Fig. 1) and then measured by a frequency counter. The light beam of an actuated laser source is frequency modulated by an electrooptic modulator driven by an RF oscillator, with the modulation frequency fm being typically ≤ 0.1F0 . The modulated lightwave, which is to be injected into the vacuum cavity, and the reflected one are separated by a beam splitter acting as an optical isolator. The reflected intensity is collected by a photodiode, and the output signal is demodulated in an RF mixer so as to provide the amplitude of the first modulation harmonics. To dispose of a sharp zero crossing at resonant condition (1), the demodulated amplitude V of sin(2πfm ) is employed as the FPI measurement. The corresponding frequency-to-voltage static response V = h(e) is highly nonlinear (Fig. 2) and periodic, with the period being equal to F0 [see (2)].
kmax = 2Vmax /emax
(5)
where Vmax denotes the peak voltage before and after a zero crossing corresponding to a resonant condition and emax 0.5 F0 /F indicates the corresponding frequency detuning. Only in the useful region centered on a zero crossing and large 2fm is the H(e) different from zero and is a sector nonlinearity [0, 1], i.e., it lies between the horizontal axis and the unitary slope line. In a smaller region, i.e., in the 2emax -large linear region, H(e) becomes monotonic, undergoes a sharp zero crossing suitable for regulation and tracking, and holds |e/emax | ≤ 1. (6) H(e) = kmax e/ 1 + (e/emax )2 , The values of the Nanobalance thrust stand [13], i.e., L0 = 0.195 m and F ∼ = 110, yield emax ∼ = 3.5 MHz, which, for a green laser frequency fl = 563 THz, corresponds to an upper bound eL,max 1.2 nm, if expressed in length units. e must be kept confined by active control well below such bounds. By neglecting PDH dynamics, the measurement yf in frequency units is related to e through the measurement error v and the response H(e) as follows: yf (t) = H(e) + v(t).
(7)
Solving (7) for eL = e/k0 , which is expressed in length units, inside the linear region and making the cavity finesse F explicit yields eL (t) λ0 (yf (t) − v(t)) /(4Femax ),
λ0 = c/f0 .
(8)
FPI performance depends on (8). 1) yf (t)/emax , which is the fractional control jitter, depends on the closed-loop control bandwidth and on the residual noise, which si usually due to mechanical vibrations. 2) v(t)/emax , which is the fractional PDH measurement error, depends on the PDH optoelectronics quality. 3) λ0 , from visible to near infrared, ranges from about 0. 3 µm (violet) to 3 µm and depends on the application; 4) ceteris paribus, e is attenuated by increasing F. As an example, the space astrometry mission GAIA [12] asks for an average length detuning of less than 1 pm root
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Fig. 4. Essential diagram of the lock-in detection and recovery automaton.
Fig. 3.
Arduous lock-in detection in the Nanobalance cavity.
mean square over a length L0 1 m, where the average must be taken over τ = 3 s. Then, by assuming an infrared wavelength of λ0 1.3 µm and low finesse of F 150, the target performance requires the average fractional jitter and the PDH noise to be less than 0.1%, which is a standard target. III. U NIFIED A UTOMATION AND D IGITAL C ONTROL F ORMULATION A. Automation and Control Tasks With reference to (5) and (6), two tasks are needed. 1) Lock-in detection and recovery detects the prolonged presence of e outside the useful range due to onset and shocks and returns it to the linear range. Fig. 3 shows an arduous lock-in detection in the Nanobalance thrust stand [16] because of the plate differential motion of about 13 Hz (beat) due to ground noise and plate imbalance. The beat rate can be larger than 10 MHz/ms. 2) The linear controller operates under quasi-linear conditions, which are established as soon as e enters the linear range, i.e., |e(t)| ≤ η¯emax , η¯ < 1. Actually, the linear controller may also operate in a stable way within the useful region (Fig. 3), due to sector nonlinearity [0, 1], and it may be adjusted to improve stability when |e(t)| ≥ η¯emax . FPI may start to operate from arbitrary initial detuning e(0) within the free spectral range, which is very likely outside the useful region where the expected PDH signal is close to zero; which asks for lock-in detection.
Control strategy has been designed so as to avoid false lockin detection, which may occur due to parasitic zero crossing, as well as to reduce lock-in detection time in the order of seconds. The key strategy is secure detection of the background signal, after which the extended region can be discovered. To this end, the BACKGROUND state has been split into ZERO, where the background signal has to be confirmed, and SEARCH, where the extended region is looked for. Switching is driven either by a time counter (doubly circled states in Fig. 4) as from ZERO to SEARCH and/or by thresholds on the PDH measure yf . The output signals are the automaton state a and the commanded frequency vector u to the actuators, whose rate may be modulated by the automaton state itself. LOCK-IN triggers the linear controller. Escape from LOCK-IN is triggered by yf and by the estimate eˆ of the frequency detuning e defined in (9). C. Embedded Model (EM) The EM is the design model to be embedded as the core of the linear regulator. It is written as a DT dynamics starting from (4) and (7) and by decomposing e, as in [15], into a single disturbance de (i) and multiple command components cj (i), j = 1, . . . , m, with i being the generic DT. The underlying time unit T is the command sampling step to be designed. It has been fixed to T = 0.1 ms in [15] and in the Nanobalance thrust stand. It yields e(i) = de (i) + c(i) m c(i) = cj (i) j=1
ye (i) = Hm (e(i)) + v(i). B. Lock-in Detection and Recovery Lock-in detection and recovery is implemented by a hybrid closed-loop strategy based on a stylized shape of the PDH response as in Fig. 2, i.e., a doublet made by four parts corresponding to the values of the automaton state a (encircled capital letters in Fig. 4), which are given as follows: 1) unknown signal (DEFAULT); 2) zero or background signal (BACKGROUND) to be further split into a pair of states; 3) signal inside extended region (SIGNAL); 4) zero crossing or linear region (LOCK-IN).
(9)
with Hm being an approximation of the true H, to be estimated and coded as a lookup table, while v accounts for PDH noise and aliasing errors due to sampling. Equation (9) must be completed with disturbance, actuator, and sensor dynamics. Disturbance dynamics models the disturbance class D, which has to be rejected in order to guarantee the target performance. As in [15], it is synthesized starting from the envelope of experimental and simulated spectral densities as the linear combination of white noise, random drifts, and narrowband noise tuned to specific frequencies. To guarantee robust observability, the corresponding DT dynamics has to be written
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Fig. 6.
Fig. 5. stand.
Environment disturbance and control jitter of the Nanobalance thrust
as a series of elementary dynamics driven by “arbitrary” DT signals wk (i)—They may be interpreted as DT white noise processes—to be collected into a vector w(i), as follows: de = w0 + D · wd = w0 + D1 · (w1 + D2 · (w2 + · · · + Dr · wr ))
(10)
where wT = [w0 wdT ] and D · wd denotes a linear DT operator acting on the causal sequence wd = {wd (i − k), k ≥ 0}. The relevant Z transform is denoted by D(z)w(z). The elementary dynamics Dk may assume two forms, i.e., (1)
D
(2)
D
−1
(z) = (z − 1) −1 (z) = αk (z − 1)2 + αk (z − 1) + αk
(11)
depending on the drift or narrowband assumption. For instance, the upper plot in Fig. 5, showing the environment disturbance on the Nanobalance [13], has been modeled as (12) de = w0 + D(1) · w1 + D(1) · w2 + D(2) · w3 to include low-frequency drifts and the differential plate tilting excited by ground noise and plate imbalance. The sequence of actuator dynamics Mj will be ordered in the same way as their voltage-to-frequency low-frequency gains bj < ∞, i.e., an increasing gain bj from j = 1 to m corresponds to a decreasing bandwidth βj . The Z transform of Mj , assuming first-order dynamics—but second order may apply as in [16]—can be written, less actuator noise and modeling errors, as cj (z) = Mj (z)uj (z) = bj βj (z − 1 − βj )−1 uj (z).
(13)
The fastest actuator b1 = minj {bj } is assumed to be dynamic free, i.e., β1 = 1 in (13), and the resulting unit delay is neglected. PDH dynamics can be usually neglected, but antialiasing filters have to be added in order to avoid the spread of wideband
EM block diagram.
disturbances, i.e., ground and acoustic noise, through feedback. Experience [12] and formal proof suggests a decimated firstorder digital antialiasing filter to be adopted. Denote the PDH sampling step with Ty = T /N , N ≥ 1 and the corresponding DT with k. The antialiased PDH measure y(i) can be written as y(i + 1) = y(i) +
k+N −1
yf (k)/N.
(14)
h=k
Among other advantages, the filter can be simply replicated into the EM as a DT integrator, thus avoiding unmodeled dynamics. It means that the EM output ym is related to the PDH output ye , which is defined in (9), through the noise-free input–output unstable dynamics ym = M · ye ,
M(z) = (z − 1)−1 .
(15)
Let us restrict the following analysis to the linear range of the PDH response (see Section II-B); let us employ the approximation Hm (e) = e and define the total disturbance d = de + v, which includes the sum v of the PDH noise and of the aliasing errors. The EM made by (9), (10), (13), and (15) can be rewritten as m Mj · uj + d ym = M · b1 u1 + j=2
d = w0 + D · wd + v.
(16)
The EM block diagram is shown in Fig. 6. A sequence of three actuators j = 1, 2, 3, from the fastest to the slowest, has been included. A cloud denotes a driving noise class to be usually interpreted as DT white noise. 2-D boxes denote the modeled dynamics written as a Z transform. 3-D boxes denote the unmodeled dynamics in the form of a fractional dynamics driven by EM output variables. For instance, the output y(i) of the chain including PDH electronics and antialiasing filter is related to ym [see (15)] by y = ym + ∆y = ym + ∂P(ym , we )
(17)
where ∂P(·) denotes a DT dynamic operator, which is usually unknown and not realizable as a state equation but driven by the causal sequence ym = {ym (i − k), k ≥ 0} less some unknown input signal we independent on ym and, therefore, on the
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EM command. Each fractional unmodeled dynamics ∂P(·) can be given an analytic, linear, and time-invariant approximation ∂P(z; p)ym (z) tunable by a parameter vector p to the elements of an uncertainty class. Examples are reported in [16]. D. Command Law As in [15], multiple commands will be exploited for apportioning disturbance rejection among different actuator capabilities. Denote with dˆ an estimate of d provided by the measurement law (Section III-E) and denote with dˆj the portion to be tracked by the actuator j subject to the constraint dˆm = ˆ Any actuator j will track the low-frequency components d. of dˆj defined by a low-pass filter, which is referred to as tracking dynamics, to be designed and denoted by Vj . As a consequence, the following signal chain builds up: dˆm = dˆ ˆ dj−1 = (I − Vj ) · dˆj , Sj = I − Vj
j = m, . . . , 2 (18)
where Sj is the sensitivity proper to Vj . To match actuator capabilities, gain, and bandwidth, the sequence of sensitivities {Sm , . . . , S2 } will be designed to have increasing bandwidth as in [15]. This is done by appropriate tuning the eigenvalue set Λj of Vj . The tracking dynamics is designed as a dynamic input– output feedback Hj around the actuator model Mj , where Hj must include a DT integrator to allow perfect tracking of dc components. In the first-order case as in (13), the simplest feedback can be shown to hold, i.e., Hj (z) = (z − 1)−1 h0j + h1j (z − 1 − γj )−1 .
(19)
It gives rise to a third-order low-pass filter Vj , whose gains {γj , h0j , h1j } are easily computed from the assigned Λj . Each tracking dynamics may be interpreted as a feedforward controller [25] driven by the portion of estimated disturbance allocated to each actuator j. The chain (18) has to be completed with the command apportionment equation uj = uj − Hj · Sj · dˆj = uj − Hj · dˆj−1 ,
j = m, . . . , 2 (20)
Fig. 7. Block diagram of the command and measurement laws.
is expressed in frequency units, from the model output estimate yˆ (see Section III-E) is sufficient for the purpose, thus providing the command law y + dˆ1 )/b1 . u1 = u1 − (kˆ
The complete command law is made by (18), (20), and (22). The law must be completed with the command digitization ˜ (i) to the plant digitalproviding the digital command vector u to-analog converter. Fig. 7 shows the block diagram of the command and measurement laws. The antialiasing filter is denoted by a DT integrator—a boxed Σ—followed by an ideal sampler representing decimation. E. Measurement Law Define the model error or output innovation eˆy (i) = y(i) − yˆ(i)
with the negative sign forcing disturbance rejection and uj denoting the command computed by the lock-in detection and recovery task (see Section III-B). The fastest actuator must complete disturbance rejection with internal stability properties. To this end, it must force the effects of the bounded residual disturbance eˆd = d − dˆ
(21)
to be bounded all over the feedback chain. This is achieved by stabilizing the antialiasing filter (15) through state feedback so as to avoid integration of eˆd into a drift. Owing to the firsty , which order dynamics in (15), the scalar feedback cCL = −kˆ
(22)
(23)
as the discrepancy between the antialiased decimated PDH measure y in (14) and the estimate yˆ of the model output ym in (16). The measurement law aims to estimate the driving noise w(i) of the class D, defined by (12), under causality ¯ which depends on constraints. First, only a causal estimate w(i) eˆy (i) and, therefore, is correlated to past occurrences w(i − k), k > 0, can be obtained, which, since the Kalman filter, has been shown to be sufficient for updating dˆ and guaranteeing the ˆ must residual eˆd in (21) to be statistically bounded. Second, d(i) ˆ ˆ ¯ − 1)) to be estimated as a one-step prediction d(i) = d(i/w(i allow mechanization of the command law one step in advance,
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which implies that dˆ only depends on the strictly causal part in (10), i.e.,
ˆ By introducing the command law where eˆd (i) = d(i) − d(i). feedback operator
ˆ = D(z)w ¯ d (z). d(z)
VCL = (I + kM)−1 kM = k(z − 1 − k)−1
(24)
This entails the measurement law to separate into a pair of feedback laws around EM (16), which are given as follows: 1) a dynamic feedback driven by eˆy to the predictable disturbance d: ˆ = D(z)w ¯ d (z) = D(z)Ld (z)ˆ ey (z) = Hd (z)ˆ ey (z) d(z) (25) 2) a dynamic feedback driven by eˆy to the unpredictable noise w0 : w ¯0 (z) = Hw eˆy (z).
(26)
Then, model output yˆ, model error eˆy , and residual disturbance eˆd can be expressed by eˆy = Sw · (y − M · uCL ) = Sw · M · eˆd eˆd = d − dˆ = Sd · d yˆ = y − eˆy
one obtains e = −v + I − VCL · M−1 · (I − Sw ) · M · Sd · d
(33)
which, in the single-input–single-output case and assuming VCL I, simplifies into e −v + SML · d.
(34)
Finally, by reusing the definition d = de + v, one obtains the classical equation e −VML · v + SML · de
(35)
where v(i) and de (i) are aliasing free, being the time average of v(t) and de (t). Equation (35) shows the performance to be limited by the low-frequency components of v, which may be affected by nonstationary fluctuations such as flicker noise due to PDH electronics, and by the bandwidth of VML .
(27) G. Robust Stability Versus Unmodeled Dynamics
upon definition of a pair of sensitivity operators Sw = (I + Hw · M)−1 Sd = (I + Hd · S0 · M)−1
(28)
which may be combined into the overall sensitivity SML = Sw · Sd = (I + (Hw + Hd ) · M)−1 .
(29)
It can be shown from (18), (20), and (22) that the EM (16) can be mechanized in the control unit as yˆ = M · (cCL + Hw · eˆy ).
(30)
Equation (30) and the feedback laws (25) and (26) constitute a state predictor driven by the scalar feedback cCL defined in Section III-D and the model error eˆy . The gains of the feedback laws encoded in Hw and Ld are tuned by assigning the eigenvalue set ΛML of the predictor complementary sensitivity VML = I − SML . As the antialiased measure y(i) is noise free, eigenvalue tuning is only constrained by the unmodeled dynamics (see Section III-G) and not by the measurement noise as in Kalman filter design. F. Average Performance Let e(i) be the sampled average of the performance variable e(t) over T = N Ty , where Ty is the sampling time defined in Section III-C. From (9), (16), and the definition of d, it follows that e(i) = b1 u1 (i) +
(32)
m
(Mj · uj )(i) + d(i) − v(i)
j=2
e(i) = − kˆ y (i) + eˆd (i) − v(i)
(31)
Performance and stability are affected by the unmodeled dynamics ∂P(·), which is not explicitly formulated for sake of brevity (see Fig. 6 and [16]). For instance, the residual disturbance eˆd may be affected by an oversimplified dynamics Mj in the frequency region close to the control rate fc = 1/T . In which case, internal stability is no more guaranteed by (22), as eˆd , which is now command-dependent, creates a new loop that is out-of-design and, worse, uncertain. Stability and performance are recovered as in [15], by constraining the eigenvalues of the tracking dynamics Vj and of the predictor sensitivity SML to free eˆd from the effects of the unmodeled dynamics below a certain threshold. This is first obtained through analytic approximations of the unmodeled dynamics and then refined through simulated experiments and in-field tests. Of course, eigenvalue constraints will restrict the bandwidth of Vj and VML , thus degrading disturbance rejection. Degradation may only be overcome by careful disturbance modeling close to such frequency limits. IV. E XPERIMENTAL R ESULTS Experimental results refer to the following cases. 1) The first is an in-vacuum ground test bed for demonstrating subnanometric dimensional stabilization of space optics with mass < 10 kg and size ≤ 1 m, as in [12] and [14]. 2) The second is digital frequency stabilization of a monolithic laser source through a set of coordinated actuators: acoustooptic modulator, PZT ceramics [26], and thermoelectric cooler [15]. 3) Third is a thrust stand for space microthruster qualification, where the laser frequency is actuated by a PZT
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TABLE I PERFORMANCE REQUIREMENTS
TABLE II EXPERIMENTAL PERFORMANCE
ceramics and a thermoelectric cooler to track and measure the length detuning of a Fabry–Pérot cavity due to microthruster firing [13]. To facilitate cavity lock-in in the presence of severe ground noise (see Fig. 3), the plate differential motion—the plate beat—can be zeroed by directly acting on the voice coils mounted at the bottom of each plate (Fig. 1). This relieves the limited range PZT ceramics from compensating the plate beat [16]. 4) The fourth is the ground breadboard of a metrology line for measuring and stabilizing thermoelastic deformations of the GAIA telescope at space temperatures of about 160 K [27]. Metrology lines are the development of the test-bed mentioned previously and in [14]. A stabilized laser frequency is detuned by an acoustooptic modulator, which acts as a variable frequency shifter, to track and measure the length detuning of a cavity mounted on the optics. Actually, to compensate slow on-ground thermal deformations, which are much larger than those in space, the cavity length was controlled by a PZT ceramics to be coordinated with the frequency actuator. Table I shows the performance requirements of the previous experiments. Performance concerns the “true” cavity detuning e defined in (4) or some related variable as the applied thrust in the Nanobalance. Performance is expressed as the root of the unilateral power spectral density (PSD) over a finite-frequency region. In case of frequency stabilization, requirements are usually expressed in fractional terms with respect to the average frequency; here, f0 = 563 THz. Table II shows the relevant experimental performance. A pair of performance variables are compared: 1) the control jitter,
which is actually the PDH signal yf converted to appropriate units, and 2) the “true” residual detuning e. Measurement of the latter one is not straightforward. In laser frequency stabilization, it is obtained by beating the lightwave of two stabilized sources. The same occurs in the Nanobalance (see Fig. 1), where the detuned laser frequency is beaten with a stable laser source. In the metrology lines, the frequency-shifter command is the “true” residual detuning less frequency-shifter errors. The actual performance measure may be corrupted by two classes of errors, which usually manifest as low-frequency drifts, i.e., 1) uncontrollable errors like basement tilt in the Nanobalance; they ask for additional control systems as in the upgraded Nanobalance under construction. 2) measurement and actuator errors inside the control loop: a) flicker noise of the measurement electronics and b) quantization and electronics noise of the frequency shifter (see [27]). In summary, control jitter fully satisfies requirements differently than the “true” detuning e. It means that the underlying plant, from actuators to sensors, needs to be improved. Fig. 5 refers to the Nanobalance. The upper plot shows the residual cavity detuning under zero thrust, which is expressed in force units, corresponding to the instrument noise. The higher peak at 13 Hz is due to the plate beat under ground noise and plate imbalance. The lower peak at about 1.5 Hz is due to suspensions. The instrument noise respects requirements from 0.01 to 1 Hz. Below 0.01 Hz, it is affected by basement tilt to be cancelled by active control. The lower plot is the control jitter, which is completely free of drift and resonance peaks and thus implies the control bandwidth to be larger than 100 Hz
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Fig. 10.
Command apportionment in the Nanobalance thrust-stand.
Fig. 8. Nanobalance raw measures and estimated thrust profile.
specifically, to an operating condition in which the differential tilt at 13 Hz is suppressed by voice coils mounted at the bottom of each tilting plate (Fig. 1). Vibration suppression must not corrupt the measurement bandwidth of < 1 Hz, where only the laser frequency actuators, PZT, and thermoelectric cooler, must intervene. This is obtained by appropriate command shaping, as explained in Section III-D. As pointed out in [15], although a single command rate fc = 1/T = 10 kHz is adopted to simplify mechanization, multirate actuation naturally arises due to progressive bandwidth restriction and resolution relaxation from faster to slower actuators. V. C ONCLUSION
Fig. 9. Metrology line: estimated detuning and control jitter at 160 K and 300 K.
and capable of attenuation of −80 dB at 13 Hz because of the disturbance model (11). A bandwidth larger than the measured one (< 1 Hz) is needed for calibration and adjustment as the 13-Hz resonance reveals imbalance. The price to be paid is the corruption of the raw frequency measurements as Fig. 8 shows. However, the contribution of resonance peaks of > 1 Hz can be filtered out to reveal the actuated thrust profile less a noise of < 1 µN. The frequency-to-force scale factor in Fig. 8 holds 2.5 µN/MHz. Fig. 9 refers to metrology lines. It shows the total line detuning and the control jitter, which is in vacuum, at different temperatures: 160 K corresponds to the satellite onboard temperature. Notwithstanding the great detuning increase from 300 to 160 K, control jitter remains quite the same and fully respects requirements. Detuning increase was mainly due to the thermal control of the vacuum chamber at 160 K. Experimental results showing command apportionment can be found in [15] and [16]. Fig. 10 refers to the Nanobalance and,
This paper has presented a unified design for a class of control problems, taking advantage of the high sensitivity of Fabry–Pérot interferometers to length and frequency detuning. The design follows the framework of the EMC, where a stylized dynamic model of the input–output dynamics is real-time updated in the control unit. The model state is step-by-step updated by the commands themselves and by a noise vector to be extracted from model error, and expressing what new in plant and environment has just occurred. In this way, unknown disturbances can be accurately estimated and then rejected by the command law. Several experiments are reported, showing that control algorithms always respect requirements for what concerns control jitter. When requirements are not respected by the “true” performance variable, some part of the plant has to be improved, or other control actions have to be added. R EFERENCES [1] T. R. Hicks, N. K. Reay, and R. J. Scaddan, “A servo-controlled Fabry–Pérot interferometer using capacitance micrometers for error detection,” J. Phys. E, Sci. Instrum., vol. 7, no. 1, pp. 27–30, Jan. 1974. [2] T. R. Hicks, N. K. Reay, and C. L. Stephens, “A servo-controlled Fabry–Pérot interferometer with on-line computer control,” Astron. Astrophys., vol. 51, no. 3, pp. 367–374, Sep. 1976. [3] R. W. Drever, J. L. Hall, F. W. Kowalski, J. Hough, G. M. Ford, A. J. Munley et al., “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B, Photophys. Laser Chem., vol. 31, no. 2, pp. 97–105, 1983.
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[4] R. V. Pound, “Electronic frequency stabilization of microwave oscillators,” Rev. Sci. Instrum., vol. 17, no. 11, pp. 490–505, Nov. 1946. [5] N. Bobroff, “Recent advances in displacement measuring technology,” Meas. Sci. Technol., vol. 4, no. 9, pp. 907–926, 1993. [6] S. A. Webster, M. Oxborrow, and P. Gill, “Subhertz-linewidth Nd:YAG laser,” Opt. Lett., vol. 29, no. 13, pp. 1497–1499, Jul. 2004. [7] S. Vitale et al., “LISA and its in-flight test precursor SMART-2,” Nucl. Phys. B, Proc. Suppl., vol. 110, no. 2, pp. 209–216, 2002. [8] M. Ollivier et al., “Nulling interferometry for the DARWIN space mission,” Comptes Rendus de l’Académie des Sciences, Series IV, Physics, vol. 2, no. 1, pp. 149–156, 2001. [9] S. Nagano et al., “Displacement measuring technique for satellite-tosatellite laser interferometer to determine Earth’s gravity field,” Meas. Sci. Technol., vol. 15, no. 12, pp. 2406–2411, Dec. 2004. [10] B. Caron, A. Dominjon, F. Marion, L. Massonnet, D. Morand, B. Mours et al., “Status of the VIRGO experiment,” Nucl. Instrum. Methods, vol. 360, no. 1, pp. 258–262, 1995. [11] A. Abramovici, W. Althouse, J. Camp, D. Durance, J. A. Giaime, A. Gillespie et al., “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A, vol. 218, no. 8, pp. 157–163, 1996. [12] E. Canuto, “Sub-nanometric optics stabilization in view of the GAIA astrometric mission,” Control Eng. Pract., vol. 11, no. 5, pp. 569–578, 2003. [13] E. Canuto and A. Rolino, “An automated interferometric balance for micro-thrust measurement,” ISA Trans., vol. 43, no. 2, pp. 169–187, 2004. [14] F. Bertinetto and E. Canuto, “Sub-nanometer digital positioning of large bodies by Fabry–Pérot interferometry,” Opt. Eng., vol. 40, no. 1, pp. 76–80, Jan. 2001. [15] E. Canuto and A. Rolino, “Multi-input digital frequency stabilization of monolithic lasers,” Automatica, vol. 40, no. 12, pp. 2139–2147, 2004. [16] E. Canuto, “Active vibration suppression in a suspended Fabry–Pérot cavity,” ISA Trans., vol. 45, no. 3, pp. 329–346, 2006. [17] F. Barone, E. Calloni, L. Di Fiore, A. Grado, L. Milano, and G. Russo, “Digitally controlled interferometer prototype for gravitational wave detection,” Rev. Sci. Instrum., vol. 67, no. 12, pp. 4353–4359, Dec. 1996. [18] R. Abbott and P. King, “Control system design for the LIGO pre-stabilized laser,” in Proc. 8th Int. Conf. Accelerator and Large Exp. Phys. Control Syst., San Jose, CA, 2001, pp. 361–363. [19] F. Donati and M. Vallauri, “Guaranteed control of “almost-linear” plants,” IEEE Trans. Autom. Control, vol. AC-29, no. 1, pp. 34–41, Jan. 1984. [20] F. Bondu and M. Barsuglia, “VIRGO. Laser frequency stabilization topology,” European Gravitational Observatory—VIRGO Project Rep. VIRNOT-OCA-1390-247, Jun. 2003. [21] A. Freise, “The next generation of interferometry: Multi-frequency optical modelling, control concepts and implementation,” Ph.D. dissertation, Universität Hannover, Hannover, Germany, Feb. 2003. [22] E. Canuto, “Embedded model control: Outline of the theory,” ISA Trans., 2007, to be published. [23] A. E. Siegman, Lasers. Sausalito, CA: Univ. Science, 1986. [24] E. D. Black, “An introduction to Pound-Drever-Hall laser frequency stabilization,” Amer. J. Phys., vol. 69, no. 1, pp. 78–87, Jan. 2001. [25] S. Goto and M. Nakamura, “Multidimensional feedforward compensator for industrial systems through pole assignment regulator and observer,” IEEE Trans. Ind. Electron., vol. 53, no. 3, pp. 886–894, Jun. 2006.
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[26] H.-J. Shieh, F.-J. Lin, P.-K. Huang, and L.-T. Teng, “Adaptive displacement control with hysteresis modelling for piezoactuated positioning mechanism,” IEEE Trans. Ind. Electron., vol. 53, no. 3, pp. 905–914, Jun. 2006. [27] E. Canuto and F. Musso, “Digital control of interferometric metrology lines,” Eur. J. Control, vol. 52, no. 4, 2007, to be published.
Enrico Canuto (M’88) was born in Varallo, Italy. He received the “Laurea” degree in electrical engineering from the Politecnico di Torino, Turin, Italy. In 1983, he joined the Dipartimento di Automatica e Informatica, Politecnico di Torino, as an Associate Professor of automatic control. From 1982 to 1997, he contributed to data reduction of the European astrometric mission Hipparcos. Technological studies in view of scientific and drag-free space missions, such as GAIA, GOCE, and LISA, were the occasion of applying embedded model control to electrooptics. His research interests include challenging control problems due to complexity, uncertainty, and precision.
Fabio Musso was born in Casale Monferrato, Italy. He received the “Laurea” degree in electronic engineering from the Politecnico di Torino, Turin, Italy. He is currently working toward the Ph.D. degree in information and system engineering at the same university. His research interests include automation, control, and data elaboration in the electrooptics field.
Luca Massotti was born in Grosseto, Italy. He received the “Laurea” and Ph.D. degrees in aerospace engineering from the Politecnico di Torino, Turin, Italy. As a Ph.D. student, he attended West Virginia University, Morgantown, working in aircraft modeling and neural network control. He is currently a Research Fellow with the Future Project Division, European Space Agency-European Space Research and Technology Centre, Noordwijk, The Netherlands, under the Earth Observation Program. His research interests include aircraft and satellite modeling and simulation; nonlinear and adaptive control; neural network techniques; and automation, control, and data elaboration of scientific space missions.
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Automation and Control of Fabry–Pérot Interferometers Enrico Canuto, Member, IEEE, Fabio Musso, and Luca Massotti
Abstract—Fabry–Pérot interferometry (FPI), which was originally invented for spectroscopy, is now evolving as a basic technology for ultrafine dimensional stabilization and measurement. To this end, the light path length of an optical cavity and the wavelength of a laser source injected into the cavity have to be tuned to each other through a set of frequency and/or displacement actuators driven by a sharp and narrow signal-encoding total-cavity detuning. Digital control is essential in facilitating and automating FPI use in view of space applications and routine instrumentation. This paper shows how embedded model control (EMC) technology, which was developed by one of the authors, allows to systematically proceed from fine dynamics and requirements to the EM, which is the core of control design and algorithms. In this framework, all critical control issues have a coordinated solution: disturbance estimation and rejection, command constraints and multiplicity, hybrid dynamics, constraints due to unmodeled dynamics, and performance analysis. Several experimental results are illustrated and discussed in the light of the methodology. Index Terms—Automation, digital control, embedded model control (EMC), Fabry–Pérot interferometry (FPI).
I. I NTRODUCTION
F
ABRY–PÉROT interferometry (FPI) was invented by the French physicists Fabry and Pérot at the end of the nineteenth century (1899) to enable high-resolution observation of spectral lines. Multiple light reflection between two plane surfaces creates interference patterns, which, being very sensible to wavelength and optical path length, were employed in the course of the 20th century as high-finesse scanning spectrometers. A significant improvement was made after 1970: the FPI optical length was automatically stabilized by means of piezoelectric ceramics and capacitance micrometers [1]. Soon after, the micrometer-to-piezo loop was computerized [2], and in 1984, the same technique was employed for measuring small displacements. In early developments, the FPI signal was not employed for detecting tuning error and driving piezoceramics. A breakthrough was made possible after 1980 by Manuscript received May 26, 2005; revised December 8, 2006. Abstract published on the Internet January 14, 2007. This work was supported in part by grants from Alcatel Alenia Space Italia, Turin, Italy, within research projects supported by the European Space Agency. An earlier version of this paper was presented at the 10th IEEE International Conference on Emerging Technologies and Factory Automation, Catania, Italy, September 19–22, 2005. E. Canuto and F. Musso are with the Dipartimento di Automatica e Informatica, Politecnico di Torino, 10129 Turin, Italy (e-mail: [email protected]; [email protected]). L. Massotti is with the Earth Observation Program, Future Project Division, European Space Agency-European Space Research and Technology Centre, 2200 Noordwijk, The Netherlands (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2007.892002
laser sources and the Pound–Drever–Hall (PDH) technique [3], which applied the same frequency stabilization method used in the microwave field [4] to Fabry–Pérot interferometers. The light beam of a laser source is frequency modulated before being injected into a Fabry–Pérot cavity made by two highly reflecting mirrors. The reflected light intensity collected by a photodiode and then demodulated becomes proportional to the cavity detuning from a frequency-length resonance condition. The corresponding signal can then be employed as a control error moving or keeping the cavity close to resonance. The PDH technique opened the way to a pair of applications: ultrafine laser frequency stabilization [3] and subnanometerlength measurement and stabilization [5]. In the former case, PDH signals are employed for tuning laser frequency to the optical length of a dimensionally stable cavity. In this way, small vacuum cavities (less than 20 cm long), which are highly stable in length, become frequency references having high spectral purity. Length stability is achieved by an athermic glass such as ULE having ultralow coefficient of thermal expansion (CTE), by a stable vacuum thermal bath and suitable suspensions [6]. Frequency references of this kind are expected to become a key technology for space interferometers (LISA and its precursor [7], DARWIN [8], earth’s gravity field from satellite-to-satellite interferometry [9]). Very long (up to 4 km) Fabry–Pérot vacuum cavities are employed as gravitational wave antennas in VIRGO [10] and LIGO [11] ground observatories. Much shorter cavities (1 m long) have been suggested as metrology lines for measuring and stabilizing the linear and angular dimensions of space telescopes such as GAIA [12]. High sensitivity to displacement and high resolution of frequency measurements have recently suggested employing FPI in submicronewton thrust stands such as the Nanobalance [13]. Submicronewton accuracy is essential for qualifying microthrusters to be employed in drag-free space missions such as GAIA, LISA, and DARWIN. One of the authors, who has been contributing to this technology since 1997, has implemented several digital control loops and automation to improve and facilitate FPI, as reported in [12]–[16]. An early application of computer control to scanning spectrometers has already been cited [2]. Earlier and recent applications to laser frequency stabilization are referenced in [15]. Because of interferometer complexity, computer control was imperative in VIRGO [17] and LIGO [18]. Notwithstanding such applications, the authors’ impression is that computer control is being adopted more for exploiting information technology than modern control
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methodologies [19]. The proof comes by taking insight into relevant design theories as provided by [20] and [21]. 1) Design stems from continuous-time transfer functions and not from discrete-time (DT) state equations. The latter ones allow designing all control phases in a unified manner, which is imperative for complex systems and, in this case, for the detection and recovery of FPI resonance conditions, as in Section III-B. At the same time, they favor multivariate control, as in Section III-D, and facilitate casting specific problems into general classes, as pursued in this paper. 2) No explicit interplay between the fine and design models is mentioned. The opposite would allow explicitly and iteratively tuning control parameters and performance as a function of the fine-to-design discrepancies, which are the so-called unmodeled dynamics (see Section III-G and [15]). 3) No explicit dynamics is assigned to each disturbance source, which is different from that in Section III-C, hence, uniquely relying on first-order integrative laws for their rejection, as already mentioned in [15]. The goal of this paper is to apply the embedded model control (EMC) technology [22] inspired by [19] to solve PFI control problems in a unified manner. Section II will provide a brief illustration of the FPI principles, of their applications, and of the output signal to be fed back. Section III will present a unified treatment of the FPI automation and control problems, assuming a single measure and multiple actuators. Section IV will present and discuss experimental results. Formulation will be free of theorem and proofs for the sake of brevity and readability. II. F ABRY –P ÉROT I NTERFEROMETRY A. Basic Principles and Applications The Fabry–Pérot interferometers treated in this paper are optical cavities [23] made by two aligned and highly reflecting mirrors, where a laser lightwave is injected and a standing reflecting wave is created and kept either by regulating the incident wave frequency and/or by fine moving either mirror. As a result, the cavity becomes an optical resonator as the injected light intensity is amplified inside the cavity under very low losses, as in a vacuum. Intensity amplification is measured by cavity finesse F [23], which, depending on mirror reflectivity, can reach values greater than 105 . Any frequency perturbation of the incident lightwave as well as any perturbation of the optical path inside the cavity tends to destroy the standing reflecting wave or resonant condition by varying the intensity of the lightwave emitted by the cavity itself, in which the wave may be arranged to encode perturbation sign and amplitude, the latter scaled by cavity finesse. Let fl = c/λl be the incident light frequency, which typically ranges from 100 to 1000 THz (from near infrared to violet), and is related to wavelength λl by the speed of light in vacuum c. Then, denote cavity length as L, which typically ranges from 0.01 to 4000 m when measured along the optical axis joining the spot centers of the laser beam on each mirror.
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The standing wave condition implies an integer ratio between the length and wavelength given by L = N λl /2 ⇒ fl L = N c/2,
N integer.
(1)
A pair (fl , L) satisfying the preceding condition is called a resonant pair and will be denoted as (f0 (N ), L0 (N )). From (1), it is immediate that two successive resonant pairs indexed N , N ± 1 are separated in length by half-wavelength λ0 /2, where λ0 f0 = c, and in frequency by the so-called free spectral range F0 = 0.5c/L0 = 1/τ0
(2)
where τ0 is the light travel time from mirror to mirror inside the cavity and F0 150 MHz at L0 = 1 m. Actually, (1) cannot be exactly met but only approached, by forcing the frequency and length to fluctuate around a resonant pair through active control. Thus, by denoting length and frequency fluctuations (often called detuning) as ∆L = L − L0 (N ) and ∆f = fl − f0 (N ), respectively, and fractional detuning as η and by rewriting (1) as (f0 + ∆f )(L0 + ∆L) = f0 L0 (1 + η)
(3)
a linear differential equation relating length and frequency detuning less the cavity detuning e can be written, i.e., ∆f + (f0 /L0 )∆L = e,
e = f0 η.
(4)
The preceding relation, which is called the lock-in condition, holds only under sufficiently small detuning e, where small is dictated by cavity finesse F. As the length-to-frequency scale k0 = f0 /L0 is usually much greater than 1012 and frequency detuning ∆f can be accurately measured below 1 Hz, FPI may reveal a subatomic length detuning that is well below Bohr radius (50 pm). This is achieved by revealing e, which is encoded in the emitted lightwave, through the PDH optoelectronics [3] and by keeping e as small as possible through a control loop. Accordingly, three main applications can be conceived. 1) Length stabilization. ∆f is kept negligible by a frequency-stabilized laser source, which implies (4) to reduce to ∆L = L0 e/f0 and to keep ∆L to zero by piezoelectric actuators, for instance, lead–zirconate–titanate (PZT) ceramics as in [12] and [14]. 2) Frequency stabilization. ∆L is kept negligible, for instance, by thermal control and ultralow CTE glass, which implies (4) to reduce to ∆f = e. Then, ∆f is kept to zero by locking the incident laser frequency to cavity length by means of suitable frequency actuators, PZT ceramics, and a thermoelectric cooler as in [15]. 3) Length measurement and tracking. A frequencystabilized laser wave is detuned by a frequency actuator prior to be injected in the cavity so as to zero e in the presence of length detuning ∆L. The actuated ∆f is made proportional to ∆L less residual detuning e, laser frequency instability, and actuator noise. The scale factor
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Fig. 2.
Typical response of the PDH chain versus frequency detuning.
For control purposes, the normalized frequency-to-frequency response is defined as H(e) = h(e)/kmax , Fig. 1. stand.
Essential components of the PDH technique in the Nanobalance thrust
f0 /L0 needs only to be calibrated for what concerns L0 , as f0 is ensured by laser frequency stabilization. As ∆L may result from natural and artificial causes as microthrusts in the Nanobalance thrust stand ([13], [16]), FPI can be made to reveal microscopic phenomena. In all applications, cavity detuning e plays the role of the “true” performance variable to be kept close to zero and below a predefined tolerance within a frequency bandwidth. B. Response of PDH Chain and FPI Performance An introduction to PDH technique can be found in [24]. The essential components are shown in Fig. 1, with reference to the Nanobalance thrust stand [13], which consists of two tilting plates carrying the reflecting mirrors of the cavity. The microthruster mounted on the active plate, when fired, shrinks the cavity length, thus forcing the control system to detune the laser frequency by means of a fast PZT and a slow actuator (thermoelectric cooler). The tilting plates must be equal in mass and geometry, which requires a dummy thruster to balance the active one. The detuned laser frequency is then beaten with a reference laser (not in Fig. 1) and then measured by a frequency counter. The light beam of an actuated laser source is frequency modulated by an electrooptic modulator driven by an RF oscillator, with the modulation frequency fm being typically ≤ 0.1F0 . The modulated lightwave, which is to be injected into the vacuum cavity, and the reflected one are separated by a beam splitter acting as an optical isolator. The reflected intensity is collected by a photodiode, and the output signal is demodulated in an RF mixer so as to provide the amplitude of the first modulation harmonics. To dispose of a sharp zero crossing at resonant condition (1), the demodulated amplitude V of sin(2πfm ) is employed as the FPI measurement. The corresponding frequency-to-voltage static response V = h(e) is highly nonlinear (Fig. 2) and periodic, with the period being equal to F0 [see (2)].
kmax = 2Vmax /emax
(5)
where Vmax denotes the peak voltage before and after a zero crossing corresponding to a resonant condition and emax 0.5 F0 /F indicates the corresponding frequency detuning. Only in the useful region centered on a zero crossing and large 2fm is the H(e) different from zero and is a sector nonlinearity [0, 1], i.e., it lies between the horizontal axis and the unitary slope line. In a smaller region, i.e., in the 2emax -large linear region, H(e) becomes monotonic, undergoes a sharp zero crossing suitable for regulation and tracking, and holds |e/emax | ≤ 1. (6) H(e) = kmax e/ 1 + (e/emax )2 , The values of the Nanobalance thrust stand [13], i.e., L0 = 0.195 m and F ∼ = 110, yield emax ∼ = 3.5 MHz, which, for a green laser frequency fl = 563 THz, corresponds to an upper bound eL,max 1.2 nm, if expressed in length units. e must be kept confined by active control well below such bounds. By neglecting PDH dynamics, the measurement yf in frequency units is related to e through the measurement error v and the response H(e) as follows: yf (t) = H(e) + v(t).
(7)
Solving (7) for eL = e/k0 , which is expressed in length units, inside the linear region and making the cavity finesse F explicit yields eL (t) λ0 (yf (t) − v(t)) /(4Femax ),
λ0 = c/f0 .
(8)
FPI performance depends on (8). 1) yf (t)/emax , which is the fractional control jitter, depends on the closed-loop control bandwidth and on the residual noise, which si usually due to mechanical vibrations. 2) v(t)/emax , which is the fractional PDH measurement error, depends on the PDH optoelectronics quality. 3) λ0 , from visible to near infrared, ranges from about 0. 3 µm (violet) to 3 µm and depends on the application; 4) ceteris paribus, e is attenuated by increasing F. As an example, the space astrometry mission GAIA [12] asks for an average length detuning of less than 1 pm root
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Fig. 4. Essential diagram of the lock-in detection and recovery automaton.
Fig. 3.
Arduous lock-in detection in the Nanobalance cavity.
mean square over a length L0 1 m, where the average must be taken over τ = 3 s. Then, by assuming an infrared wavelength of λ0 1.3 µm and low finesse of F 150, the target performance requires the average fractional jitter and the PDH noise to be less than 0.1%, which is a standard target. III. U NIFIED A UTOMATION AND D IGITAL C ONTROL F ORMULATION A. Automation and Control Tasks With reference to (5) and (6), two tasks are needed. 1) Lock-in detection and recovery detects the prolonged presence of e outside the useful range due to onset and shocks and returns it to the linear range. Fig. 3 shows an arduous lock-in detection in the Nanobalance thrust stand [16] because of the plate differential motion of about 13 Hz (beat) due to ground noise and plate imbalance. The beat rate can be larger than 10 MHz/ms. 2) The linear controller operates under quasi-linear conditions, which are established as soon as e enters the linear range, i.e., |e(t)| ≤ η¯emax , η¯ < 1. Actually, the linear controller may also operate in a stable way within the useful region (Fig. 3), due to sector nonlinearity [0, 1], and it may be adjusted to improve stability when |e(t)| ≥ η¯emax . FPI may start to operate from arbitrary initial detuning e(0) within the free spectral range, which is very likely outside the useful region where the expected PDH signal is close to zero; which asks for lock-in detection.
Control strategy has been designed so as to avoid false lockin detection, which may occur due to parasitic zero crossing, as well as to reduce lock-in detection time in the order of seconds. The key strategy is secure detection of the background signal, after which the extended region can be discovered. To this end, the BACKGROUND state has been split into ZERO, where the background signal has to be confirmed, and SEARCH, where the extended region is looked for. Switching is driven either by a time counter (doubly circled states in Fig. 4) as from ZERO to SEARCH and/or by thresholds on the PDH measure yf . The output signals are the automaton state a and the commanded frequency vector u to the actuators, whose rate may be modulated by the automaton state itself. LOCK-IN triggers the linear controller. Escape from LOCK-IN is triggered by yf and by the estimate eˆ of the frequency detuning e defined in (9). C. Embedded Model (EM) The EM is the design model to be embedded as the core of the linear regulator. It is written as a DT dynamics starting from (4) and (7) and by decomposing e, as in [15], into a single disturbance de (i) and multiple command components cj (i), j = 1, . . . , m, with i being the generic DT. The underlying time unit T is the command sampling step to be designed. It has been fixed to T = 0.1 ms in [15] and in the Nanobalance thrust stand. It yields e(i) = de (i) + c(i) m c(i) = cj (i) j=1
ye (i) = Hm (e(i)) + v(i). B. Lock-in Detection and Recovery Lock-in detection and recovery is implemented by a hybrid closed-loop strategy based on a stylized shape of the PDH response as in Fig. 2, i.e., a doublet made by four parts corresponding to the values of the automaton state a (encircled capital letters in Fig. 4), which are given as follows: 1) unknown signal (DEFAULT); 2) zero or background signal (BACKGROUND) to be further split into a pair of states; 3) signal inside extended region (SIGNAL); 4) zero crossing or linear region (LOCK-IN).
(9)
with Hm being an approximation of the true H, to be estimated and coded as a lookup table, while v accounts for PDH noise and aliasing errors due to sampling. Equation (9) must be completed with disturbance, actuator, and sensor dynamics. Disturbance dynamics models the disturbance class D, which has to be rejected in order to guarantee the target performance. As in [15], it is synthesized starting from the envelope of experimental and simulated spectral densities as the linear combination of white noise, random drifts, and narrowband noise tuned to specific frequencies. To guarantee robust observability, the corresponding DT dynamics has to be written
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Fig. 6.
Fig. 5. stand.
Environment disturbance and control jitter of the Nanobalance thrust
as a series of elementary dynamics driven by “arbitrary” DT signals wk (i)—They may be interpreted as DT white noise processes—to be collected into a vector w(i), as follows: de = w0 + D · wd = w0 + D1 · (w1 + D2 · (w2 + · · · + Dr · wr ))
(10)
where wT = [w0 wdT ] and D · wd denotes a linear DT operator acting on the causal sequence wd = {wd (i − k), k ≥ 0}. The relevant Z transform is denoted by D(z)w(z). The elementary dynamics Dk may assume two forms, i.e., (1)
D
(2)
D
−1
(z) = (z − 1) −1 (z) = αk (z − 1)2 + αk (z − 1) + αk
(11)
depending on the drift or narrowband assumption. For instance, the upper plot in Fig. 5, showing the environment disturbance on the Nanobalance [13], has been modeled as (12) de = w0 + D(1) · w1 + D(1) · w2 + D(2) · w3 to include low-frequency drifts and the differential plate tilting excited by ground noise and plate imbalance. The sequence of actuator dynamics Mj will be ordered in the same way as their voltage-to-frequency low-frequency gains bj < ∞, i.e., an increasing gain bj from j = 1 to m corresponds to a decreasing bandwidth βj . The Z transform of Mj , assuming first-order dynamics—but second order may apply as in [16]—can be written, less actuator noise and modeling errors, as cj (z) = Mj (z)uj (z) = bj βj (z − 1 − βj )−1 uj (z).
(13)
The fastest actuator b1 = minj {bj } is assumed to be dynamic free, i.e., β1 = 1 in (13), and the resulting unit delay is neglected. PDH dynamics can be usually neglected, but antialiasing filters have to be added in order to avoid the spread of wideband
EM block diagram.
disturbances, i.e., ground and acoustic noise, through feedback. Experience [12] and formal proof suggests a decimated firstorder digital antialiasing filter to be adopted. Denote the PDH sampling step with Ty = T /N , N ≥ 1 and the corresponding DT with k. The antialiased PDH measure y(i) can be written as y(i + 1) = y(i) +
k+N −1
yf (k)/N.
(14)
h=k
Among other advantages, the filter can be simply replicated into the EM as a DT integrator, thus avoiding unmodeled dynamics. It means that the EM output ym is related to the PDH output ye , which is defined in (9), through the noise-free input–output unstable dynamics ym = M · ye ,
M(z) = (z − 1)−1 .
(15)
Let us restrict the following analysis to the linear range of the PDH response (see Section II-B); let us employ the approximation Hm (e) = e and define the total disturbance d = de + v, which includes the sum v of the PDH noise and of the aliasing errors. The EM made by (9), (10), (13), and (15) can be rewritten as m Mj · uj + d ym = M · b1 u1 + j=2
d = w0 + D · wd + v.
(16)
The EM block diagram is shown in Fig. 6. A sequence of three actuators j = 1, 2, 3, from the fastest to the slowest, has been included. A cloud denotes a driving noise class to be usually interpreted as DT white noise. 2-D boxes denote the modeled dynamics written as a Z transform. 3-D boxes denote the unmodeled dynamics in the form of a fractional dynamics driven by EM output variables. For instance, the output y(i) of the chain including PDH electronics and antialiasing filter is related to ym [see (15)] by y = ym + ∆y = ym + ∂P(ym , we )
(17)
where ∂P(·) denotes a DT dynamic operator, which is usually unknown and not realizable as a state equation but driven by the causal sequence ym = {ym (i − k), k ≥ 0} less some unknown input signal we independent on ym and, therefore, on the
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EM command. Each fractional unmodeled dynamics ∂P(·) can be given an analytic, linear, and time-invariant approximation ∂P(z; p)ym (z) tunable by a parameter vector p to the elements of an uncertainty class. Examples are reported in [16]. D. Command Law As in [15], multiple commands will be exploited for apportioning disturbance rejection among different actuator capabilities. Denote with dˆ an estimate of d provided by the measurement law (Section III-E) and denote with dˆj the portion to be tracked by the actuator j subject to the constraint dˆm = ˆ Any actuator j will track the low-frequency components d. of dˆj defined by a low-pass filter, which is referred to as tracking dynamics, to be designed and denoted by Vj . As a consequence, the following signal chain builds up: dˆm = dˆ ˆ dj−1 = (I − Vj ) · dˆj , Sj = I − Vj
j = m, . . . , 2 (18)
where Sj is the sensitivity proper to Vj . To match actuator capabilities, gain, and bandwidth, the sequence of sensitivities {Sm , . . . , S2 } will be designed to have increasing bandwidth as in [15]. This is done by appropriate tuning the eigenvalue set Λj of Vj . The tracking dynamics is designed as a dynamic input– output feedback Hj around the actuator model Mj , where Hj must include a DT integrator to allow perfect tracking of dc components. In the first-order case as in (13), the simplest feedback can be shown to hold, i.e., Hj (z) = (z − 1)−1 h0j + h1j (z − 1 − γj )−1 .
(19)
It gives rise to a third-order low-pass filter Vj , whose gains {γj , h0j , h1j } are easily computed from the assigned Λj . Each tracking dynamics may be interpreted as a feedforward controller [25] driven by the portion of estimated disturbance allocated to each actuator j. The chain (18) has to be completed with the command apportionment equation uj = uj − Hj · Sj · dˆj = uj − Hj · dˆj−1 ,
j = m, . . . , 2 (20)
Fig. 7. Block diagram of the command and measurement laws.
is expressed in frequency units, from the model output estimate yˆ (see Section III-E) is sufficient for the purpose, thus providing the command law y + dˆ1 )/b1 . u1 = u1 − (kˆ
The complete command law is made by (18), (20), and (22). The law must be completed with the command digitization ˜ (i) to the plant digitalproviding the digital command vector u to-analog converter. Fig. 7 shows the block diagram of the command and measurement laws. The antialiasing filter is denoted by a DT integrator—a boxed Σ—followed by an ideal sampler representing decimation. E. Measurement Law Define the model error or output innovation eˆy (i) = y(i) − yˆ(i)
with the negative sign forcing disturbance rejection and uj denoting the command computed by the lock-in detection and recovery task (see Section III-B). The fastest actuator must complete disturbance rejection with internal stability properties. To this end, it must force the effects of the bounded residual disturbance eˆd = d − dˆ
(21)
to be bounded all over the feedback chain. This is achieved by stabilizing the antialiasing filter (15) through state feedback so as to avoid integration of eˆd into a drift. Owing to the firsty , which order dynamics in (15), the scalar feedback cCL = −kˆ
(22)
(23)
as the discrepancy between the antialiased decimated PDH measure y in (14) and the estimate yˆ of the model output ym in (16). The measurement law aims to estimate the driving noise w(i) of the class D, defined by (12), under causality ¯ which depends on constraints. First, only a causal estimate w(i) eˆy (i) and, therefore, is correlated to past occurrences w(i − k), k > 0, can be obtained, which, since the Kalman filter, has been shown to be sufficient for updating dˆ and guaranteeing the ˆ must residual eˆd in (21) to be statistically bounded. Second, d(i) ˆ ˆ ¯ − 1)) to be estimated as a one-step prediction d(i) = d(i/w(i allow mechanization of the command law one step in advance,
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which implies that dˆ only depends on the strictly causal part in (10), i.e.,
ˆ By introducing the command law where eˆd (i) = d(i) − d(i). feedback operator
ˆ = D(z)w ¯ d (z). d(z)
VCL = (I + kM)−1 kM = k(z − 1 − k)−1
(24)
This entails the measurement law to separate into a pair of feedback laws around EM (16), which are given as follows: 1) a dynamic feedback driven by eˆy to the predictable disturbance d: ˆ = D(z)w ¯ d (z) = D(z)Ld (z)ˆ ey (z) = Hd (z)ˆ ey (z) d(z) (25) 2) a dynamic feedback driven by eˆy to the unpredictable noise w0 : w ¯0 (z) = Hw eˆy (z).
(26)
Then, model output yˆ, model error eˆy , and residual disturbance eˆd can be expressed by eˆy = Sw · (y − M · uCL ) = Sw · M · eˆd eˆd = d − dˆ = Sd · d yˆ = y − eˆy
one obtains e = −v + I − VCL · M−1 · (I − Sw ) · M · Sd · d
(33)
which, in the single-input–single-output case and assuming VCL I, simplifies into e −v + SML · d.
(34)
Finally, by reusing the definition d = de + v, one obtains the classical equation e −VML · v + SML · de
(35)
where v(i) and de (i) are aliasing free, being the time average of v(t) and de (t). Equation (35) shows the performance to be limited by the low-frequency components of v, which may be affected by nonstationary fluctuations such as flicker noise due to PDH electronics, and by the bandwidth of VML .
(27) G. Robust Stability Versus Unmodeled Dynamics
upon definition of a pair of sensitivity operators Sw = (I + Hw · M)−1 Sd = (I + Hd · S0 · M)−1
(28)
which may be combined into the overall sensitivity SML = Sw · Sd = (I + (Hw + Hd ) · M)−1 .
(29)
It can be shown from (18), (20), and (22) that the EM (16) can be mechanized in the control unit as yˆ = M · (cCL + Hw · eˆy ).
(30)
Equation (30) and the feedback laws (25) and (26) constitute a state predictor driven by the scalar feedback cCL defined in Section III-D and the model error eˆy . The gains of the feedback laws encoded in Hw and Ld are tuned by assigning the eigenvalue set ΛML of the predictor complementary sensitivity VML = I − SML . As the antialiased measure y(i) is noise free, eigenvalue tuning is only constrained by the unmodeled dynamics (see Section III-G) and not by the measurement noise as in Kalman filter design. F. Average Performance Let e(i) be the sampled average of the performance variable e(t) over T = N Ty , where Ty is the sampling time defined in Section III-C. From (9), (16), and the definition of d, it follows that e(i) = b1 u1 (i) +
(32)
m
(Mj · uj )(i) + d(i) − v(i)
j=2
e(i) = − kˆ y (i) + eˆd (i) − v(i)
(31)
Performance and stability are affected by the unmodeled dynamics ∂P(·), which is not explicitly formulated for sake of brevity (see Fig. 6 and [16]). For instance, the residual disturbance eˆd may be affected by an oversimplified dynamics Mj in the frequency region close to the control rate fc = 1/T . In which case, internal stability is no more guaranteed by (22), as eˆd , which is now command-dependent, creates a new loop that is out-of-design and, worse, uncertain. Stability and performance are recovered as in [15], by constraining the eigenvalues of the tracking dynamics Vj and of the predictor sensitivity SML to free eˆd from the effects of the unmodeled dynamics below a certain threshold. This is first obtained through analytic approximations of the unmodeled dynamics and then refined through simulated experiments and in-field tests. Of course, eigenvalue constraints will restrict the bandwidth of Vj and VML , thus degrading disturbance rejection. Degradation may only be overcome by careful disturbance modeling close to such frequency limits. IV. E XPERIMENTAL R ESULTS Experimental results refer to the following cases. 1) The first is an in-vacuum ground test bed for demonstrating subnanometric dimensional stabilization of space optics with mass < 10 kg and size ≤ 1 m, as in [12] and [14]. 2) The second is digital frequency stabilization of a monolithic laser source through a set of coordinated actuators: acoustooptic modulator, PZT ceramics [26], and thermoelectric cooler [15]. 3) Third is a thrust stand for space microthruster qualification, where the laser frequency is actuated by a PZT
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TABLE I PERFORMANCE REQUIREMENTS
TABLE II EXPERIMENTAL PERFORMANCE
ceramics and a thermoelectric cooler to track and measure the length detuning of a Fabry–Pérot cavity due to microthruster firing [13]. To facilitate cavity lock-in in the presence of severe ground noise (see Fig. 3), the plate differential motion—the plate beat—can be zeroed by directly acting on the voice coils mounted at the bottom of each plate (Fig. 1). This relieves the limited range PZT ceramics from compensating the plate beat [16]. 4) The fourth is the ground breadboard of a metrology line for measuring and stabilizing thermoelastic deformations of the GAIA telescope at space temperatures of about 160 K [27]. Metrology lines are the development of the test-bed mentioned previously and in [14]. A stabilized laser frequency is detuned by an acoustooptic modulator, which acts as a variable frequency shifter, to track and measure the length detuning of a cavity mounted on the optics. Actually, to compensate slow on-ground thermal deformations, which are much larger than those in space, the cavity length was controlled by a PZT ceramics to be coordinated with the frequency actuator. Table I shows the performance requirements of the previous experiments. Performance concerns the “true” cavity detuning e defined in (4) or some related variable as the applied thrust in the Nanobalance. Performance is expressed as the root of the unilateral power spectral density (PSD) over a finite-frequency region. In case of frequency stabilization, requirements are usually expressed in fractional terms with respect to the average frequency; here, f0 = 563 THz. Table II shows the relevant experimental performance. A pair of performance variables are compared: 1) the control jitter,
which is actually the PDH signal yf converted to appropriate units, and 2) the “true” residual detuning e. Measurement of the latter one is not straightforward. In laser frequency stabilization, it is obtained by beating the lightwave of two stabilized sources. The same occurs in the Nanobalance (see Fig. 1), where the detuned laser frequency is beaten with a stable laser source. In the metrology lines, the frequency-shifter command is the “true” residual detuning less frequency-shifter errors. The actual performance measure may be corrupted by two classes of errors, which usually manifest as low-frequency drifts, i.e., 1) uncontrollable errors like basement tilt in the Nanobalance; they ask for additional control systems as in the upgraded Nanobalance under construction. 2) measurement and actuator errors inside the control loop: a) flicker noise of the measurement electronics and b) quantization and electronics noise of the frequency shifter (see [27]). In summary, control jitter fully satisfies requirements differently than the “true” detuning e. It means that the underlying plant, from actuators to sensors, needs to be improved. Fig. 5 refers to the Nanobalance. The upper plot shows the residual cavity detuning under zero thrust, which is expressed in force units, corresponding to the instrument noise. The higher peak at 13 Hz is due to the plate beat under ground noise and plate imbalance. The lower peak at about 1.5 Hz is due to suspensions. The instrument noise respects requirements from 0.01 to 1 Hz. Below 0.01 Hz, it is affected by basement tilt to be cancelled by active control. The lower plot is the control jitter, which is completely free of drift and resonance peaks and thus implies the control bandwidth to be larger than 100 Hz
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Fig. 10.
Command apportionment in the Nanobalance thrust-stand.
Fig. 8. Nanobalance raw measures and estimated thrust profile.
specifically, to an operating condition in which the differential tilt at 13 Hz is suppressed by voice coils mounted at the bottom of each tilting plate (Fig. 1). Vibration suppression must not corrupt the measurement bandwidth of < 1 Hz, where only the laser frequency actuators, PZT, and thermoelectric cooler, must intervene. This is obtained by appropriate command shaping, as explained in Section III-D. As pointed out in [15], although a single command rate fc = 1/T = 10 kHz is adopted to simplify mechanization, multirate actuation naturally arises due to progressive bandwidth restriction and resolution relaxation from faster to slower actuators. V. C ONCLUSION
Fig. 9. Metrology line: estimated detuning and control jitter at 160 K and 300 K.
and capable of attenuation of −80 dB at 13 Hz because of the disturbance model (11). A bandwidth larger than the measured one (< 1 Hz) is needed for calibration and adjustment as the 13-Hz resonance reveals imbalance. The price to be paid is the corruption of the raw frequency measurements as Fig. 8 shows. However, the contribution of resonance peaks of > 1 Hz can be filtered out to reveal the actuated thrust profile less a noise of < 1 µN. The frequency-to-force scale factor in Fig. 8 holds 2.5 µN/MHz. Fig. 9 refers to metrology lines. It shows the total line detuning and the control jitter, which is in vacuum, at different temperatures: 160 K corresponds to the satellite onboard temperature. Notwithstanding the great detuning increase from 300 to 160 K, control jitter remains quite the same and fully respects requirements. Detuning increase was mainly due to the thermal control of the vacuum chamber at 160 K. Experimental results showing command apportionment can be found in [15] and [16]. Fig. 10 refers to the Nanobalance and,
This paper has presented a unified design for a class of control problems, taking advantage of the high sensitivity of Fabry–Pérot interferometers to length and frequency detuning. The design follows the framework of the EMC, where a stylized dynamic model of the input–output dynamics is real-time updated in the control unit. The model state is step-by-step updated by the commands themselves and by a noise vector to be extracted from model error, and expressing what new in plant and environment has just occurred. In this way, unknown disturbances can be accurately estimated and then rejected by the command law. Several experiments are reported, showing that control algorithms always respect requirements for what concerns control jitter. When requirements are not respected by the “true” performance variable, some part of the plant has to be improved, or other control actions have to be added. R EFERENCES [1] T. R. Hicks, N. K. Reay, and R. J. Scaddan, “A servo-controlled Fabry–Pérot interferometer using capacitance micrometers for error detection,” J. Phys. E, Sci. Instrum., vol. 7, no. 1, pp. 27–30, Jan. 1974. [2] T. R. Hicks, N. K. Reay, and C. L. Stephens, “A servo-controlled Fabry–Pérot interferometer with on-line computer control,” Astron. Astrophys., vol. 51, no. 3, pp. 367–374, Sep. 1976. [3] R. W. Drever, J. L. Hall, F. W. Kowalski, J. Hough, G. M. Ford, A. J. Munley et al., “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B, Photophys. Laser Chem., vol. 31, no. 2, pp. 97–105, 1983.
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[4] R. V. Pound, “Electronic frequency stabilization of microwave oscillators,” Rev. Sci. Instrum., vol. 17, no. 11, pp. 490–505, Nov. 1946. [5] N. Bobroff, “Recent advances in displacement measuring technology,” Meas. Sci. Technol., vol. 4, no. 9, pp. 907–926, 1993. [6] S. A. Webster, M. Oxborrow, and P. Gill, “Subhertz-linewidth Nd:YAG laser,” Opt. Lett., vol. 29, no. 13, pp. 1497–1499, Jul. 2004. [7] S. Vitale et al., “LISA and its in-flight test precursor SMART-2,” Nucl. Phys. B, Proc. Suppl., vol. 110, no. 2, pp. 209–216, 2002. [8] M. Ollivier et al., “Nulling interferometry for the DARWIN space mission,” Comptes Rendus de l’Académie des Sciences, Series IV, Physics, vol. 2, no. 1, pp. 149–156, 2001. [9] S. Nagano et al., “Displacement measuring technique for satellite-tosatellite laser interferometer to determine Earth’s gravity field,” Meas. Sci. Technol., vol. 15, no. 12, pp. 2406–2411, Dec. 2004. [10] B. Caron, A. Dominjon, F. Marion, L. Massonnet, D. Morand, B. Mours et al., “Status of the VIRGO experiment,” Nucl. Instrum. Methods, vol. 360, no. 1, pp. 258–262, 1995. [11] A. Abramovici, W. Althouse, J. Camp, D. Durance, J. A. Giaime, A. Gillespie et al., “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A, vol. 218, no. 8, pp. 157–163, 1996. [12] E. Canuto, “Sub-nanometric optics stabilization in view of the GAIA astrometric mission,” Control Eng. Pract., vol. 11, no. 5, pp. 569–578, 2003. [13] E. Canuto and A. Rolino, “An automated interferometric balance for micro-thrust measurement,” ISA Trans., vol. 43, no. 2, pp. 169–187, 2004. [14] F. Bertinetto and E. Canuto, “Sub-nanometer digital positioning of large bodies by Fabry–Pérot interferometry,” Opt. Eng., vol. 40, no. 1, pp. 76–80, Jan. 2001. [15] E. Canuto and A. Rolino, “Multi-input digital frequency stabilization of monolithic lasers,” Automatica, vol. 40, no. 12, pp. 2139–2147, 2004. [16] E. Canuto, “Active vibration suppression in a suspended Fabry–Pérot cavity,” ISA Trans., vol. 45, no. 3, pp. 329–346, 2006. [17] F. Barone, E. Calloni, L. Di Fiore, A. Grado, L. Milano, and G. Russo, “Digitally controlled interferometer prototype for gravitational wave detection,” Rev. Sci. Instrum., vol. 67, no. 12, pp. 4353–4359, Dec. 1996. [18] R. Abbott and P. King, “Control system design for the LIGO pre-stabilized laser,” in Proc. 8th Int. Conf. Accelerator and Large Exp. Phys. Control Syst., San Jose, CA, 2001, pp. 361–363. [19] F. Donati and M. Vallauri, “Guaranteed control of “almost-linear” plants,” IEEE Trans. Autom. Control, vol. AC-29, no. 1, pp. 34–41, Jan. 1984. [20] F. Bondu and M. Barsuglia, “VIRGO. Laser frequency stabilization topology,” European Gravitational Observatory—VIRGO Project Rep. VIRNOT-OCA-1390-247, Jun. 2003. [21] A. Freise, “The next generation of interferometry: Multi-frequency optical modelling, control concepts and implementation,” Ph.D. dissertation, Universität Hannover, Hannover, Germany, Feb. 2003. [22] E. Canuto, “Embedded model control: Outline of the theory,” ISA Trans., 2007, to be published. [23] A. E. Siegman, Lasers. Sausalito, CA: Univ. Science, 1986. [24] E. D. Black, “An introduction to Pound-Drever-Hall laser frequency stabilization,” Amer. J. Phys., vol. 69, no. 1, pp. 78–87, Jan. 2001. [25] S. Goto and M. Nakamura, “Multidimensional feedforward compensator for industrial systems through pole assignment regulator and observer,” IEEE Trans. Ind. Electron., vol. 53, no. 3, pp. 886–894, Jun. 2006.
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[26] H.-J. Shieh, F.-J. Lin, P.-K. Huang, and L.-T. Teng, “Adaptive displacement control with hysteresis modelling for piezoactuated positioning mechanism,” IEEE Trans. Ind. Electron., vol. 53, no. 3, pp. 905–914, Jun. 2006. [27] E. Canuto and F. Musso, “Digital control of interferometric metrology lines,” Eur. J. Control, vol. 52, no. 4, 2007, to be published.
Enrico Canuto (M’88) was born in Varallo, Italy. He received the “Laurea” degree in electrical engineering from the Politecnico di Torino, Turin, Italy. In 1983, he joined the Dipartimento di Automatica e Informatica, Politecnico di Torino, as an Associate Professor of automatic control. From 1982 to 1997, he contributed to data reduction of the European astrometric mission Hipparcos. Technological studies in view of scientific and drag-free space missions, such as GAIA, GOCE, and LISA, were the occasion of applying embedded model control to electrooptics. His research interests include challenging control problems due to complexity, uncertainty, and precision.
Fabio Musso was born in Casale Monferrato, Italy. He received the “Laurea” degree in electronic engineering from the Politecnico di Torino, Turin, Italy. He is currently working toward the Ph.D. degree in information and system engineering at the same university. His research interests include automation, control, and data elaboration in the electrooptics field.
Luca Massotti was born in Grosseto, Italy. He received the “Laurea” and Ph.D. degrees in aerospace engineering from the Politecnico di Torino, Turin, Italy. As a Ph.D. student, he attended West Virginia University, Morgantown, working in aircraft modeling and neural network control. He is currently a Research Fellow with the Future Project Division, European Space Agency-European Space Research and Technology Centre, Noordwijk, The Netherlands, under the Earth Observation Program. His research interests include aircraft and satellite modeling and simulation; nonlinear and adaptive control; neural network techniques; and automation, control, and data elaboration of scientific space missions.
