Abstract

The present work concerns with the modeling, development and application of a novel control strategy based on sliding mode control, for two beam quadrature interferometers, with the high-gain approach. In this case, by reading the control signal the demodulation process does not require phase unwrapping algorithms, i.e., the output signal presents a straight-line relationship with the interferometer total phase shift. This system was implemented in a digital platform to control a bulk Michelson interferometer whose performance was experimentally determined, showing its capability on achieving real-time measurement and presenting wider dynamic range and bandwidth (52.5 rad in low frequencies and 5.8 rad up to 500 Hz) when compared with the literature. Moreover, this performance can be improved even further by simply increasing the feedback gain.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Interferometer is an extremely sensitive measurement device, usually employed for measuring a physical quantity (signal of interest) which induces an optical phase shift between its arms. Since the interferometer characteristic curve is given by a cosine-type function, it exhibits a nonlinear behavior and a small dynamic range (usually smaller than π/2 rad) for operation without demodulation or phase unwrapping methods. In addition, due to spurious environmental disturbances such as, temperature variation, weak mechanical vibration and air turbulence, the interferometer operation point suffers from a quasi-static phase shift variation (below 20 Hz approximately), which can lead to the undesirable signal fading [1]. Both, nonlinearity and fading increases the complexity for the demodulation of the signal of interest. One technique usually employed to solve these problems is the closed-loop interferometry, in particular, the active homodyne method [2–5].

Two different approaches risen from active homodyne method: the low-gain and the high-gain [5–7]. The low-gain approach compensates only the quasi-static phase shift while the signal of interest can be obtained directly from the output of the photodetector in a higher frequency, from approximately 100 Hz up to the photodetector limit (which can reach tens of GHz). However, the dynamic range using low-gain approach and without phase unwrapping or demodulation is limited (smaller than π/2 rad) for real-time operation.

In contrast, the high-gain approach employs a higher gain in order to compensate the total phase shift (the signal of interest plus the quasi-static phase shift), consequently holding a straight-line relationship between the output signal and the input. This means wide dynamic range and real-time operation with low computational cost, since any further phase unwrapping (see [8] for example) or demodulation method are required. In addition, the high-gain approach can keep the sensitivity and accuracy of the interferometer with smaller harmonic distortion, and relative insensitivity to laser output variations [4,5]. Therefore, aiming applications which require these characteristics (for instance refractive index variation in plasmas or gases, measurement of atmospheric phase distortion [4], characterization of large displacement piezoelectric actuators), the high-gain approach was chosen in the present work.

Regarding the literature on high-gain approach, Fisher and Warde [4] described a control system able to compensate the signal of interest within the limits of 3.5π rad of phase amplitude for low frequencies and amplitudes up to π/2 rad for 150 Hz of bandwidth. Fritsch and Adamowsky [2] showed phase shift detection in the order of 500 μrad rms for frequencies up to 500 Hz, however employing a reset circuit, which acts once a minute. Xie et al. [9] proposed an innovative setup to measure long range displacement, which requires reset circuit and although the control loop simplicity, this comes at the cost of increased complexity of the optical hardware. The limitations in dynamic range or bandwidth, the low robustness, and the need of reset circuit in the aforementioned papers, are mainly due to the control system type, usually based on linear control methods [10–12].

On the other hand, a recent work, presented by Martin et al. [13] concerned a non-linear control based on variable structure control with sliding modes applied to stabilize two beam interferometers [13, 14] under low-gain approach. The proposed nonlinear control system presented high robustness, excluded the need of a reset circuit, and was able to keep the interferometer in phase quadrature and to suppress signal fading.

In the present work, inspired in [13], we propose a novel sliding mode control using the high-gain approach to compensate the total phase shift for two beam quadrature interferometers. This system operates in real-time and presents a wide dynamic range, able to measure 52.5 rad in low frequencies and 5.8 rad up to 500 Hz, which is a great improvement in comparison with the literature [2,4,5,9]. In addition, we show the system versatility, since it is possible to increase even further the dynamic range and bandwidth by simply increasing the feedback gain.

2. Theoretical background

2.1. Modified michelson interferometer

A modified Michelson interferometer configuration is presented in Fig. 1, comprising a laser, a negative lens, two neutral beam-splitters (BS 1 and BS 2), two mirrors (M1 and M2), two piezoelectric actuators (PZT 1 and PZT 2), and two photodetectors (PD 1 and PD 2). The beam reflecting in mirror M1 forms the sensor arm, while the beam reflecting in M2, the reference arm. PZT 2, attached to M2, is the feedback actuator, responsible to control the interferometer. PZT 1, attached to M1, is employed to generate a known phase shift and to verify its proper working. PZT 1 can be dismissed when performing a real measurement task.

 

Fig. 1 Modified Michelson interferometer and fringe pattern with different phase shifts.

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The control system requires in-phase and quadrature-phase signals (relative phase difference of π/2 rad), which can be obtained in a simple setup by setting PD 1 and PD 2 at different positions on the fringe pattern [15] (inset of Fig. 1), instead of using polarizing optics [8, 16, 17]. The negative lens provides an expanded optical spot on the photodetectors to facilitate this process.

The output AC voltage signals vac1 and vac2 respectively from the photodetectors PD1 and PD2 (with transimpedance amplifier circuits) are expressed by:

vac1(t)=A1V1cos(Δϕ(t)+ϕ0(t)+ϕc(t)),
vac2(t)=A2V2sin(Δϕ(t)+ϕ0(t)+ϕc(t)).
where the parameters A1 and A2 are voltages associated with the responsivity of the photodetectors and the gain of the transimpedance circuit, while V1 and V2 are the fringe visibilities, which varies between 0 and 1 and takes into account the parallelism between the optical electrical fields of the two combining beams, their intensities, the coherence of the light source, among others [6]. The parameter ϕ0 is related to the quasi-static phase shift generated by the difference between the optical path length of the interferometer arms in the absence of signal inputs, and varies randomly in time due to low-frequency disturbances (< 20 Hz) caused by environmental changes in temperature, air turbulence and mechanical vibration. The signal of interest Δϕ(t) is related with the physical quantity to be measured, and ϕc is the correction phase, introduced by the control system in order to suppress Δϕ and ϕ0, as will be discussed in subsection 2.2.

2.2. Sliding mode control

Variable structure control has as the main idea, a control law switching the dynamic of the closed loop system between different subsystems, in such a way the resulting dynamic can present a different behavior from the individual ones [18]. Under specified circumstances this operating regime may originate the so called sliding mode, which is characterized by a fast switching control action, forcing the system trajectories to certain regions of the state space, where the system will present the desired behavior. These regions are referred to as sliding manifolds or sliding surfaces [19]. The main advantages of such control strategy are: robustness to parametric variations, disturbance rejection, decoupling designing procedure and simple implementation [20]. Let us consider the state space representation of a generic dynamical system given by:

x˙=f(t,x,u),
u={u+(x)ifs(x)>0u(x)ifs(x)<0,
where x ∈ ℝn is the state space n-dimensional vector of the system, = dx/dt, f(t, x, u) ∈ ℝn, u is a scalar control action with discontinuous function, and s(x) is the sliding variable. The sliding surface is a closed space in the state space defined as {x(t) | s(x) = 0}. Therefore, the problem of designing a sliding mode control system consists in choosing the appropriate sliding variable s(x), and after that, design the appropriate control function u to drive the system to the sliding surface.

According to [21], the sliding mode exists on a discontinuous surface whenever the distances to this surface and the velocity of its change are of opposite signs, in the vicinity of the sliding manifold, i.e. when:

lims0+s˙<0andlims0s˙>0.

This condition is achieved if the control action guarantees the reachability criterion, given by:

ddt(12s2(x))=s(x)s˙(x)η|s(x)|,
where η is a positive constant. In other words: s(x)(x) ≤ −η|s(x)|, meaning that the multiplication of the sliding variable s by its temporal derivative results in a negative number, satisfying the existence condition of Eq. (5).

It can be seen from Eqs. (1) and (2) that the interferometer is a zero-order nonlinear dynamic system. Therefore in order to apply the aforementioned control strategy, an integrator is added in the system, relating the phase correction signal and the control input by ϕ̇c = u. The block diagram of the resulting dynamic system is shown in Fig. 2, where ϕt = Δϕ + ϕ0 + ϕc. The main objective of the control system is to keep the interferometer operating in the quadrature points, where the output vac1 = A1V1 cos(Δϕ + ϕ0 + ϕc) = 0. Thus a suitable choice of state space variables, sliding surface and model for this system is:

s(x)=x1(t)=vac1(t),
s˙(x)=x˙1=f(Δϕ+ϕ0+ϕc)(Δϕ˙+ϕ˙0+ϕ˙c),
where fϕ + ϕ0 + ϕc) = −A1V1 sin(Δϕ + ϕ0 + ϕc). Let us now use the equivalent control method described by [18] to obtain the equations of the phase correction signal ϕc with the system confined in the sliding surface, meaning that s(x) = 0 and (x) = 0 as well, then:
{s(x)=x1=0s˙(x)=x˙1=0{ϕc=Δϕϕ0+kπ2ϕ˙c=Δϕ˙ϕ˙0wherekisodd.

 

Fig. 2 Block diagram of the interferometric dynamical system.

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From Eq. (9) it can be seen that, for the system confined into the sliding surface, ϕc will follow the phase shifts between the interferometer arms and therefore, one can measure Δϕ + ϕ0 from ϕc in a proportional relationship. A suitable choice for the control law, in order to guarantee the reachability criterion and to drive the system trajectory to the sliding surface is:

u=Csgn(x1f),
where C is a strictly positive constant, comprising the overall closed loop gain, and sgn(x) is the sign function, defined as sgn(x)={+1ifx>01ifx<0. It is known that the functions |x| and sgn(x) are related by x = |x|sgn(x) and |x| = xsgn(x), for all x ∈ ℝ. Multiplying Eq. (7) and Eq. (8), substituting ϕ̇c = u, for u given by Eq. (10), it is obtained:
ss˙=(x1f)[Δϕ˙+ϕ˙0Csgn(x1f)]=(x1f)(Δϕ˙+ϕ˙0)C|x1f|.

Choosing the overall closed loop gain C in such a way that:

C>Cn+|Δϕ˙+ϕ˙0|,
where Cn > 0, leads to:
ss˙<(x1f)(Δϕ˙+ϕ˙0)Cn|x1f||Δϕ˙+ϕ˙0||x1f|.
It can be seen that the upper bound of the right side of Eq. (13) occurs when (x1f) > 0 and (Δϕ̇ + ϕ̇0) > 0 or when (x1f) < 0 and (Δϕ̇ + ϕ̇0) < 0. Both cases lead to:
ss˙<Cn|f||s|.

From Eq. (14), it can be seen that the reachability criterion is satisfied for every f ≠ 0. In other words, for every initial condition of the argument of the function f such that Δϕ + ϕ0 + ϕc, k ∈ ℤ, the system will reach and remain within the sliding surface.

2.3. Stability analysis for the critical points

In order to analyze the behavior of the system in the vicinity of the points where the function f is equal to zero, let us study the product of the functions f and . If fḟ results in a negative number it can be affirmed that, for the closed loop system, there is attraction to the point where f = 0, whereas if fḟ results in a positive number, there is repulsion in the vicinities of f = 0. Let be given by:

f˙=A1V1cos(Δϕ+ϕ0+ϕc)[Δϕ˙+ϕ˙0Csgn(x1f)],
therefore the product fḟ can be written as:
ff˙=x1f[Δϕ˙+ϕ˙0Csgn(x1f)],
or
ff˙=x1f(Δϕ˙+ϕ˙0)+C|x1f|.

Regarding the condition used to guarantee the reachability criterion, given by Eq. (12), it follows that:

ff˙>x1f(Δϕ˙+ϕ˙0)+Cn|x1f|+|x1f||Δϕ˙+ϕ˙0|,
where the minimum value of fḟ occurs when x1fϕ̇ + ϕ̇0) assumes positive values, leading to:
ff˙Cn|x1||f|.

From this result it can be inferred that for the points close to f = 0, the closed loop system will not attract the state trajectory to f = 0; on the contrary, the trajectory will diverge from these points.

Considering now the hypothesis of the initial condition of the system be Δϕ + ϕ0 + ϕc = for k ∈ ℤ, i.e. f = 0, the system will remain in this position only if is equal to zero to every subsequent time. In this initial condition cos(Δϕ + ϕ0 + ϕc) will be equal to +1 or −1, therefore, from Eq. (15), = −A1V1ϕ̇+ ϕ̇0Csgn(x1f)), for k even and = A1V1ϕ̇+ ϕ̇0Csgn(x1f)), for k odd, therefore it can be written as:

f˙=A1V1(Δϕ˙+ϕ˙0Csgn(x1f)),
which will be zero only if Δϕ̇ + ϕ̇0Csgn(x1f) = 0. Even if such condition happens in an initial state it will not hold for the subsequent time due to the variation of Δϕ̇ + ϕ̇0 and the repulsion around f = 0, therefore when f ≠ 0 the reachability criterion holds and the system converges to the sliding surface.

The attraction to the points where s = x1 = 0 and repulsion from the points where f = 0 can also be intuitively understood using a graphical approach. In Fig. 3 it can be seen the graphics of vac1 = x1 = A1V1 cos(Δϕ + ϕ0 + ϕc) in function of the total phase ϕt = Δϕ + ϕ0 + ϕc in black color, and in red the graph of the control signal u, which can be rewritten as:

u=Csgn[(A1V1)22sin(2ϕt)].

When the relation shown in Eq. (12) holds, it is noted that the increase and decrease of the total phase is dictated by the control signal u.

 

Fig. 3 Equilibrium points in the interferometric characteristic curve.

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Therefore, if u > 0, Δϕ + ϕ0 + ϕc increases, and for u < 0, Δϕ + ϕ0 + ϕc decreases. In Fig. 3 the dotted line represents the transition locals of the control signal u, between positive and negative values. Note that when traversing the ϕt-axis in the increasing direction, the locals marked by a blue circle, where the control signal changes from positive to negative values, constitute attraction points. If ϕt is to the left of this point, the control signal forces the phase correction signal to increase, while if ϕt is to the right, u will force ϕc to decrease. The inverse behavior is observed in the locals marked by the red cross when the control signal transition goes from negative to positive values, i.e, the red crosses indicate repulsion points.

It is noteworthy that the different points where vac1 = x1 = 0 as, for instance, π2 and π2 have opposite sign slopes in the interferometer characteristic curve. However for the operation using the high-gain approach, this behavior does not induce ambiguity in direction, because the system generates a phase correction signal to suppress the phase disturbances at the input of the interferometer. For example, if the system is working in the operation point ϕt=π2, the slope of the interferometric curve is positive and therefore, a disturbance in the positive direction of the phase ϕt = Δϕ + ϕ0 will cause a reaction, decreasing the value of ϕc. Considering now the interferometer working at the operation point ϕt=π2, if any phase disturbance in the positive direction of ϕt = Δϕ + ϕ0 occurs, the control action again will decrease the value of ϕc as it can be seen by the sign of the control signal in the region between π2 and π, in Fig. 3.

2.4. Closed loop simulation

The system comprising the interferometer in closed loop with the designed control law was simulated with the software Simulink. The block diagram is shown in Fig. 4, where the signal generator blocks are responsible to simulate the signals ϕ0 and Δϕ with amplitudes of 4 rad peak and frequencies at 3 Hz and 500 Hz respectively, while the modified Michelson interferometer is represented as the block with sine and cosine outputs together with the −1 gain block, generating the signals x1 = vac1 = A1V1 cos(Δϕ + ϕ0 + ϕc) and f = vac2 = −A2V2 sin(Δϕ + ϕ0 + ϕc), from Eq. (1) and Eq. (2) respectively, with the parameters A1V1 = A2V2 = 1, for simplicity. It is important to highlight that the parameters A2V2 can be adjusted to the approximately the same value as A1V1 by the use of an iris in the experimental setup, however, even if the parameters are different the control law will not be affected once that the expressions x1 and f are inside the argument of the sign function.

 

Fig. 4 Closed loop block diagram in Simulink software.

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The control signal u = −Csgn(x1f) is formed with the blocks Product, Sign and feedback gain −C. The block Sign implements the relay function, responsible for the discontinuous control action and the gain −C represents the total overall feedback gain of the closed loop. In order to obey the reachability condition, the overall feedback gain of the closed loop must be C > |Δϕ̇+ ϕ̇0|, therefore a sufficient condition is C > |Δϕ̇|+|ϕ̇0|. Considering Δϕ = 4sin(2π500t) and ϕ0 = 4sin(2π3t), the overall feedback gain can be a value that meets the condition C > 12642. Thus, for this case, we choose C = 12800. As the amplitude of Δϕ is higher than π rad, the open loop interferometer operates with multiple fringes, which is very suitable for testing the high-gain approach, which makes it unnecessary to use any phase unwrapping method. The simulations where performed in order to verify the convergence to the stable equilibrium points. The behavior in open (brown curve) and closed loop (blue curve) of x1 and f are respectively shown in Figs. 5(a) and 5(c), regarding the input phase shift given by ϕ0 + Δϕ and initial condition chosen as ϕc + ϕ0 + Δϕ = 0.7π radians. As it can be seen, it results in the convergence of the state trajectory to x1 → 0 and f → −1.

 

Fig. 5 Open and closed loop simulations for: (a) x1 for t between 0 and 0.01 s and initial condition 0.7π. (b) x1 for t between 0 and 0.01 s initial condition 1.0001π. (c) f for t between 0 and 0.01 s and initial condition 0.7π. (d) f for t between 0 and 0.01 s and initial condition 1.0001π.

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In addition, the comparison between the phase displacements ϕ0 + Δϕ and the phase correction signal ϕc is shown in Fig. 6(a), where it can be seen that the sum of both signals results in the approximately constant value of π/2. Therefore, the simulation agrees with the stability analysis, since, with the initial value of input phase equals to 0.7π rad, the initial condition lies in the attraction region of the equilibrium point π/2 rad. Repeating the simulations for the initial condition ϕc + ϕ0 + Δϕ = 1.0001π rad results in the convergence of the state trajectory to x1 → 0 and f → 1 as can be seen in Fig. 5(b) and Fig. 5(d), agreeing with the convergence to the equilibrium point 3π/2 rad as expected from the stability analysis, and showed in Fig. 6(b).

 

Fig. 6 Open and closed loop input phases. (a) Simulation for t between 0 and 0.02 s for initial condition 0.7π rad. (b) Simulation for t between 0 and 0.02 s for initial condition 1.0001π rad.

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3. Experimental setup

The overall block diagram of the experimental setup is presented in Fig. 7, comprising the modified Michelson interferometer, responsible for generating the output signals v1 and v2, the implemented controller, assembled within the myRIO-1900 (National Instruments) in digital format (which will be discussed in sub-section 3.1), and a linear voltage amplifier (LVA) that drives the feedback PZT actuator with a suitable control signal. The feedback PZT actuator, in turn, converts the applied voltage signal to a linear displacement of the attached mirror M2 in Fig. 1, inducing the phase correction signal ϕc and closing, therefore, the feedback loop of the control system.

 

Fig. 7 Block diagram of the implemented control system.

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The modified Michelson interferometer (shown schematically in Fig. 1) was assembled in two different versions: the first one, shown in Fig. 8(a) used a cage system from Thorlabs, with the following components: 1. He-Ne laser operating at 632.8 nm with 5 mW power (Electro Optics model LHRP-0501); 2. Negative lens; 3. BS1 (CMI-BS013 from Thorlabs); 4. Feedback piezoelectric actuator, PZT 2 (Control Technics) with mirror M2; 5. BS2; 6. PD 1 (DET36A from Thorlabs); 7. PD 2 (DET10A from Thorlabs); 8. Mirror M1. On the second assembly, shown in Fig. 8(b), numbers 1 to 7 indicate the same optical components of Fig. 8(a), while number 8 is the PZT 1 attached to a tiny mirror M1. The PZT 1 (piezoelectric stack actuator PK2FVF1 from Thorlabs, named from now on PK2 actuator) is capable to produce displacements in the order of 420 μm for applied voltages in the range of 0 – 75V. Therefore, known phase shifts greater than π radians could be induced in frequencies up to 500 Hz, in order to prove the capability of the system to operate in high-gain mode.

 

Fig. 8 Modified Michelson interferometer assemblies. (a) Assembled Modified Michelson interferometer 1. (b) Modified Michelson interferometer with PK2 actuator.

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The quadrature setting was carried out on both assemblies by applying a sinusoidal voltage to the PZT 2 (feedback actuator) while the output signals of the PD 1 and PD 2 were connected to the oscilloscope in the acquisition mode XY, allowing the visualization of a Lissajous figure. Thus, by adjusting the positions of PD 1 and PD 2 in the fringe pattern, and the relative power reaching them (through an iris in PD 1), the quadrature setting was reached when the Lissajous figure becomes approximately a circle, as shown in Fig. 9(a). The yellow and green signals, showed in Fig. 9(b), are respectively from PD 1 and PD 2, while the purple signal is the applied voltage to the PZT 2. In order to verify the quadrature setting in time-domain, the applied voltage to PZT 2 was decreased to low modulation depth and, in Fig. 9(c), it was observed the following: when one of the photodetected signals is in quadrature point, the other is at one point of maximum or minimum of the interferometer curve, therefore presenting a signal with twice the frequency of the applied voltage and attenuated amplitude, as expected for quadrature signals.

 

Fig. 9 Modified Michelson interferometer output signals. (a) Lissajous figure of the quadrature interferometer. (b) Applied voltage to PZT and output signals of the interferometer. (c) Input (in yellow) and output quadrature signals (in purple and green) with low modulation depth.

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3.1. Control system implementation

The shaded region of the block diagram in Fig. 7 was designed within the FPGA of myRIO platform, comprising two analog inputs, sampled at 100 kHz using a voltage resolution of 4.883 mV, two subtractors, a multiplier, the sign function (approximated by a sigmoidal function as will be further explained in the following), a negative gain −Csgn, and an integrator using the Simpson’s rule, allied with an antiwindup strategy and the analog output.

The feedback signal comes from the interferometer outputs (v1 and v2), which enters the controller and are converted to vac1 and vac2 (given by Eqs. (1) and (2)) by subtracting A1 and A2 values (determined experimentally as described in [22]), respectively. The product of vac1 and vac2 (vac1 = x1 and vac2 = f) is performed and the result becomes the argument of the sigmoid function. Thus, after the gain −Csgn, the signal is applied to the input of a digital integrator, implemented with the Simpson’s rule. This signal is then applied to the LVA, which multiplies the input signal by 21 V/V and drives the feedback PZT actuator, inducing the phase correction signal ϕc, and closing the control loop.

The overall gain of the control loop “C” comprises: the gain Csgn of the sign function, the gain CLVA of the linear voltage amplifier (LVA) and the gain of the feedback PZT actuator CPZT, which must be previously known (see discussion in Section 4) resulting in:

C=CsgnCLVACPZT.
One drawback of sliding mode control is the chattering phenomenon, which consists of an oscillation around the sliding surface [23]. To overcome this problem, we exchange the sign function by the sigmoid function (to obtain a smooth approximation in the boundary layer of the sliding surface [24,25]), as follows:
sgn(x1f)(x1f)(|x1f|+),
where is a positive parameter for the sigmoid slope adjustment. For the system with the boundary layer to behave as close as possible to ideal sliding mode, high values of gain and small values of are needed.

The main advantages of the control system implementation on the FPGA platform are the ability to digitally configure the gain Csgn, parameters and topology of the sigmoid function, allowing optimization of chattering while maintaining the disturbance rejection capability. Also with the hardware designed in FPGA, the system does not present the non-idealities intrinsic to analog electronic components.

Before proceeding, it is important to discuss another drawback. When the controller output reaches the saturation level, the feedback loop does not act properly and the calculated values may then drift to undesirable values; this phenomenon is referred to as windup. Control systems submitted to saturation may present two different behaviors, one dictated by the original differential equation of the system model, and another one due to a saturated mode [26]. This could originate premature saturation, hysteresis in the input-output relationship, jump resonance and other anomalous effects, denigrating the steady state performance. Many authors described this problem and proposed ways to overcome it [27–29].

This effect can occur in our system since the output of myRIO-1900 embedded platform is limited in the range of ±10 V. Therefore, considering the interferometer subjected to a phase shift high enough to make the analog output to lie out of this range, the integrator would keep increasing the internally calculated value. Meanwhile the output value would stay fixed in the saturation level and, after the end of disturbance, it would take a long time to the integrator output reaches again values in the range of the analog output. A simple way to avoid such problem is to stop the integration when the calculated value is out of the analog output voltage range and return to normal operation when the result lies within ±10 V. This fact and the high robustness provided by the non-linear control system, makes the controller not to need a reset system like those used in [2,3,9].

4. Results

Initially, the experimental setup of Fig. 8(a) was used to evaluate the properly working of the proposed control technique for a large DC disturbance, which was introduced in the system by exerting pressure on the mirror indicated by the number 8. The disturbance level was high enough to shift several fringes in the interferometric curve, and during this procedure, the control system was turned on, stabilizing the output x1 of the interferometer at quadrature point and, therefore, rejecting the induced disturbance. A video showing the execution of the test is presented in the Visualization 1, where channel 1 (yellow signal) is the x1 output, channel 2 (green signal) is the f output of the interferometer and the purple signal in channel 3 is the control signal applied to the voltage amplifier. Visualization 1 shows the very high robustness of the system, where a mirror on the interferometer was pressed, but the system compensated the phase shift by driving the feedback actuator and therefore, it kept the system operating at one of the quadrature points.

When the low-gain approach is used to detect interferometric signals, the output waveform is measured directly from the photodiode, and the interferometer constitutes a primary standard for measuring mechanical displacement, based on the knowledge of the laser wavelength in air [7]. On the other hand, when using the high-gain approach, the output signal is the voltage applied to the PZT feedback actuator itself, and so, an accurate calibration procedure must be applied by measuring its linear length-to-voltage sensitivity (LLVS, nm/V) for each frequency inside the operation bandwidth [6]. Knowing this voltage and the feedback actuator LLVS, the control phase ϕc can be determined, and so Δϕ + ϕ0. As ϕ0 varies in low frequencies, simple high-pass filtering can be used to separate Δϕ from ϕ0. In this work, the signal coincident method (SCM-International Standard ISO16063-41 [30], an applicable modification of the standard for primary vibration calibration ISO 16063-11 [31]), which is described in detail in [32], was chosen to obtain the calibrated curve for the feedback actuator assembly (see Fig. 7).

With the frequency response of CPZT carefully determined, as shown in Fig. 10, the proposed control system can be used to measure phase shifts Δϕ + ϕ0 in both modulation depths, low and high. Therefore, using the setup of Fig. 8(b), a sinusoidal voltage signal of 10 Hz was applied to PK2 actuator, producing a phase shift of several radians when the system is operating in open loop. The control system was then turned on and the control signal becomes proportional to the applied voltage while the outputs of the interferometer are stabilized. In Figs. 11(a) and 11(b) are presented respectively the open and closed loop output signals x1 and f, given by the yellow and green waveforms; the inverted control signal in pink, and the input applied voltage to the PK2 actuator on the sensor arm, in purple.

 

Fig. 10 Frequency response of the feedback actuator assembly.

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Fig. 11 Interferometer output signals x1 (yellow), f (green), input applied voltage to PK2 actuator (purple) and inverted control signal (pink) with high modulation depth at 10 Hz. (a) Open loop interferometer. (b) Closed loop interferometer.

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It can be seen the multifringe behavior of the interferometer output signals (yellow and green) shown in Fig. 11(a) and, by using fringe counting, it is observed that there are between 8 and 9 fringes within each interval of minimum and maximal applied voltage to the PK2 actuator, therefore the total phase shift can be estimated as being between the range of Δϕcounting ∈ [50.27 56.55] rad. Comparing now with the system operating in closed loop with a gain Csgn = 40000 (Fig. 11(b)), the measured control signal amplitude is 10 V which, multiplied by CLVA = 21 V/V and by CPZT = 0.25 rad/V, shows that the phase shift induced by PK2 actuator is Δϕ = 52.5 rad, corresponding to a linear displacement of [6] Δl=λ4πΔϕ=2.64μm. Both results are in agreement, once that the phase shift measured with the control system lies within the range estimated by the fringe counting method, showing the capability of the system to measure large displacements (large enough to cause multifringe operation) with higher accuracy and without requiring phase-unwrapping algorithms.

The transition from open loop operation to closed loop operation for sinusoidal excitation signal is shown in Fig. 12(a), while a test using an arbitrary excitation signal is shown in Fig. 12(b). It is important to note that for frequency narrow-band signals the system is capable to detect phase shifts with arbitrary shape and not only sinusoidal signals. This is an interesting feature in sensors such as gyroscopes, accelerometers, temperature sensors among others.

 

Fig. 12 Interferometer operating under deep phase-modulation. Output signals x1 (yellow), f (green), control signal (purple) and input applied voltage to PK2 actuator (pink). (a) Transition from open to closed loop. (b) Closed loop interferometer with arbitrary signal.

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In order to verify the phase shift suppression capability of the system for different values of modules Csgn, sinusoidal signals were applied to the PK2 actuator, in different frequencies. While the system was operating in closed-loop, the amplitude of the applied voltage was increased, up to the last value in which the relationship between control signal and the PK2 actuator applied voltage was proportional, therefore presenting the outputs of the interferometer stabilized at the quadrature points. The obtained values of amplitude were multiplied by the gain of the linear voltage amplifier CLVA and after that, by the gain of the feedback PZT actuator CPZT for each respective frequency. It can be seen in Fig. 13 that for higher modules of Csgn, higher amplitudes of phase shift can be suppressed and therefore measured through the control signal in a proportional relationship. It is noteworthy that there is an approximated inverse relationship between the frequency of the phase shift signal and its amplitude, once that for a constant gain Csgn = 35000, in 10 Hz, signals with amplitudes over 43 rad can be measured, while for 500 Hz, amplitudes up to 5.8 rad can be measured. This result is a considerable improvement when compared with the literature using high-gain approach [2,4,5], considering also that the system does not need reset circuitry.

 

Fig. 13 Supressing capability for different values of gain.

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The frequency response of PK2 actuator was obtained for the range from 30 Hz up to 500 Hz, where the phase shift, measured as explained before, was normalized by the amplitude of the voltage signal applied to PK2 actuator. This procedure was carried out on two tests in different days, resulting in the red dotted and traced green lines shown in Fig. 14. The blue, yellow and dashed purple lines were obtained for the same actuator (PK2) using low modulation depth method associated with the control system proposed by [13].

 

Fig. 14 Comparison of frequency response of the PK2 actuator.

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From Fig. 14 one can see the agreement between the frequency responses obtained with high-gain approach and with the low modulation depth, where a PK2 main resonance is observed near to 253 Hz and an anti-resonance near to 280 Hz.

It is noteworthy that the low modulation depth is not suitable for the characterization of a of large displacement actuator as the PK2, since it is hard to avoid multifringe behavior, mainly in resonances. Thus, very careful experiment was carried out when using low modulation depth, which was employed only to corroborate the results obtained with the high-gain approach.

5. Conclusion

In this work, a novel control system based on sliding mode control and high-gain approach, for two beam quadrature interferometers, was proposed and investigated through theoretical modeling, computational simulations and experiments. We showed that the proposed control system was able to properly generate a phase correction ϕc which tracks the total phase shift (Δϕ + ϕ0), as can be seen from the mathematical modeling. In addition, open and closed loop simulations and experiments were performed to demonstrate that the system works correctly and was able to suppress the induced phase shifts, even for a large disturbance imposed in one of the interferometers mirrors (i.e., very high robustness operation was achieved).

The closed loop system was designed to obey the reachability criterion and it was shown that even in the unlikely initial condition in an unstable point (where fϕ + ϕ0 + ϕc) = 0), the system leaves these points, and once fϕ + ϕ0 + ϕc) ≠ 0, the reachability criterion is satisfied.

The results demonstrated that the proposed control system provided to the interferometer the capability of measuring phase shift in a wide dynamic range, for example achieving in the extremes, 52.5 rad at 10 Hz and 5.8 rad in 500 Hz, which is a great advance compared with the literature [2, 4, 5, 9]. In addition, the gain of the sigmoid function (Csgn) can be easily increased and therefore, the system can be able to measure even larger phase shifts since the reachability criterion is guaranteed and the physical limits of the feedback actuator are respected. The phase shift compensation can alternatively be increased, by using a system with higher sampling frequency and feedback PZT actuators or optical modulators with higher phase displacement capabilities.

In terms of mechanical displacement, it was shown that the system was capable to measure up to 2.6 μm in 10 Hz. This feature allows, for example, the characterization of piezoelectric actuators of large excursion, as demonstrated in the results regarding the PK2 actuator (corroborated by the low-gain approach proposed in [13]).

It was shown that the control signal ϕc presents a straight line relationship with the total phase shift Δϕ + ϕ0 and, therefore, demodulation methods or a phase unwrapping algorithms are not necessary, which allows the system to work in real-time with wide dynamic range. In comparison with the fringe counting method, the proposed control system presented higher accuracy. Furthermore, the results show that the system is capable to detect arbitrary waveform phase shifts, not being restricted to sinusoidal signals.

The complexity of the control system was handled by the digital myRIO device, while the optical hardware could be kept simple and low cost, since the quadrature signals were generated using the spatial positioning between interference fringes, therefore without requiring polarizing optics (see [8, 16, 17], for example) which introduce further complications to the design and adds cost. Reset circuitry is not necessary, which avoids glitches and noise from its operation.

The presented system does not require to operate in low modulation depth to guarantee the accuracy of the system. On the other hand, the feedback actuator frequency response should be known in order to estimate the phase shift signal Δϕ, representing thus, a tradeoff of this approach. It is noteworthy that similarly equated systems can benefit from this control methodology, such as temperature sensors based on fiber optic interferometers, fiber optic gyroscopes and accelerometers, once their dynamic range can be increased keeping the sensitivity.

Funding

São Paulo Research Foundation (FAPESP) (#2017/17235-1); Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) (001); National Council for Scientific and Technological Development (CNPq) - Brazil (420673/2016-4, 309872/2018-9).

Acknowledgments

The authors would like to thank E. L. Moraes, G. A. Martinez, S. S. Lyra, G. F. S. Nunes, R. M. Giovanini and T. R. Pentiado.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

References

1. S. K. Sheem, T. G. Giallorenzi, and K. Koo, “Optical techniques to solve the signal fading problem in fiber interferometers,” Appl. Opt. 21(4), 689–693 (1982). [CrossRef]   [PubMed]  

2. K. Fritsch and G. Adamowsky, “Simple circuit for feedback stabilization of a single-mode optical fiber interferometer,” Rev. Sci. Instrum. 52(7), 996–1000 (1981). [CrossRef]  

3. D. A. Jackson, R. Priest, A. Dandridge, and A. B. Tveten, “Elimination of drift in a single-mode optical fiber interferometer using a piezoelectrically stretched coiled fiber,” Appl. Opt. 19(17), 2926–2929 (1980). [CrossRef]   [PubMed]  

4. A. D. Fisher and C. Warde, “Simple closed-loop system for real-time optical phase measurement,” Opt. Lett. 4(5), 131–133 (1979). [CrossRef]   [PubMed]  

5. A. Dandridge and A. B. Tveten, “Phase compensation in interferometric fiber-optic sensors,” Opt. Lett. 7(6), 279–281 (1982). [CrossRef]   [PubMed]  

6. E. Udd and W. B. J. Spillman, Fiber Optic Sensors: An Introduction for Engineers and Scientists (Wiley, 2011). [CrossRef]  

7. T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. H. Siegel JR., J. H. Cole, S. C. Rashleigh, and R. G. Priest, “Optical fiber sensor technology,” IEEE Trans. Microw. Theory Tech. 30(4), 472–511 (1982). [CrossRef]  

8. L. Shao and J. J. Gorman, “Pulsed laser interferometry with sub-picometer resolution using quadrature detection,” Opt. Express 25(6), 6335–6348 (2016).

9. F. Xie, J. Ren, Z. Chen, and Q. Feng, “Vibration-displacement measurements with a highly stabilised optical fiber Michelson interferometer system,” Opt. Laser Technol. 42(1), 208–213 (2010). [CrossRef]  

10. G. Basile, A. Bergamin, G. Cavagnero, and G. Mana, “Phase Modulation in High-resolution Optical Interferometry,” Metrologia 28(6), 455–461 (1991). [CrossRef]  

11. A. J. Fleming and B. S. Routley, “A closed-loop phase-locked interferometer for wide bandwidth position sensing,” Rev. Sci. Instrum. 86(11),115001 (2015). [CrossRef]   [PubMed]  

12. Z. Chao and F. Duan, “Closed-loop phase stabilizing and phase stepping methods for fiber-optic projected-fringe digital interferometry,” Rev. Sci. Instrum. 82(11), 113105 (2011). [CrossRef]   [PubMed]  

13. R. I. Martin, J. M. S. Sakamoto, M. C. M. Teixeira, G. A. Martinez, F. C. Pereira, and C. Kitano, “Nonlinear control system for optical interferometry based on variable structure control and sliding modes,” Opt. Express 25(6), 6335–6348 (2017). [CrossRef]   [PubMed]  

14. R. I. Martin, J. M. S. Sakamoto, M. C. M. Teixeira, and C. Kitano, “Variable structure and sliding modes nonlinear control system applied to a fiber optic interferometer,” Proc. SPIE 10453, 104531W (2017).

15. D. Wu, R. Zhu, L. Chei, and J. Li, “Transverse spatial phase-shifting method used in vibration-compensated interferometer,” Optik 115(8), 343–346 (2004). [CrossRef]  

16. J. Zhu, P. Hu, and J. Tan, “Homodyne laser vibrometer capable of detecting nanometer displacements accurately by using optical shutters,” Appl. Opt. 54(34), 10196–10199 (2015). [CrossRef]  

17. J. G. Park and K. Cho, “High-precision tilt sensor using a folded Mach-Zehnder geometry in-phase and quadrature interferometer,” Appl. Opt. 55(9), 2155–2159 (2016). [CrossRef]   [PubMed]  

18. V. I. Utkin, “Variable structure systems with sliding modes,” IEEE Trans. Autom. Control. 22(2), 212–222 (1977). [CrossRef]  

19. V. I. Utkin, Sliding Modes on Control and Optimization (Springer-Verlag, 1992). [CrossRef]  

20. V. I. Utkin, “Sliding Mode Control Design Principles and Applications to Electric Drives,” IEEE Trans. Ind. Electron. 40(1), 23–36 (1977). [CrossRef]  

21. E. A. Barbashin, Introduction to the theory of stability (Nauka, 1967).

22. J. Xu, Q. Xu, L. Chai, Y. Li, and H. Wang, “Direct phase extraction from interferograms with random phase shifts,” Opt. Express 18(20), 20620–20627 (2010). [CrossRef]   [PubMed]  

23. K. D. Young, V. I. Utkin, and U. Ozguner, “A control engineer’s guide to sliding mode control,” IEEE Trans. Control. Syst. Technol. 7(3), 328–342 (1999). [CrossRef]  

24. J. J. Slotine and S. S. Sastry, “Tracking control of non-linear systems using sliding surfaces, with application to robot manipulators,” Int. J. Control. 38(2), 465–492 (1983). [CrossRef]  

25. J. J. Slotine and W. Li, Applied Nonlinear Control (Prentice Hall, 1991).

26. J. C. Lozier, “A steady state approach to the theory of saturable servo systems,” IRE Trans. Autom. Control. 1(1), 19–39 (1956). [CrossRef]  

27. A. Berger and P. O. Gutman, “A new view of anti-windup design for uncertain linear systems in the frequency domain,” Int. J. Robust Nonlinear Control. 26(10), 2116–2135 (2015). [CrossRef]  

28. M. C. Turner, J. Sofrony, and G. Herrmann, “An alternative approach to anti-windup in anticipation of actuator saturation,” Int. J. Robust Nonlinear Control. 27(6), 963–980 (2016). [CrossRef]  

29. M. Ran, Q. Wang, and C. Dong, “Anti-windup design for uncertain nonlinear systems subject to actuator saturation and external disturbance,” Int. J. Robust Nonlinear Control. 26(15), 3421–3438 (2016). [CrossRef]  

30. International Standard ISO16063-41, “Methods for the calibration of vibration and shock transducers-Part 41: calibration of Laser vibrometers,” (International Organization for Standardization (ISO), 2011).

31. International Standard ISO 16063-11, “Methods for the calibration of vibration and shock transducers-Part 11: primary vibration calibration by laser interferometry,” (International Organization for Standardization (ISO), 1999), reviewed and confirmed (2009).

32. H. J. von Martens, “Expanded and improved traceability of vibration measurements by laser interferometry,” Rev. Sci. Instrum. 84(12), 121601 (2013). [CrossRef]  

References

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  • |

  1. S. K. Sheem, T. G. Giallorenzi, and K. Koo, “Optical techniques to solve the signal fading problem in fiber interferometers,” Appl. Opt. 21(4), 689–693 (1982).
    [Crossref] [PubMed]
  2. K. Fritsch and G. Adamowsky, “Simple circuit for feedback stabilization of a single-mode optical fiber interferometer,” Rev. Sci. Instrum. 52(7), 996–1000 (1981).
    [Crossref]
  3. D. A. Jackson, R. Priest, A. Dandridge, and A. B. Tveten, “Elimination of drift in a single-mode optical fiber interferometer using a piezoelectrically stretched coiled fiber,” Appl. Opt. 19(17), 2926–2929 (1980).
    [Crossref] [PubMed]
  4. A. D. Fisher and C. Warde, “Simple closed-loop system for real-time optical phase measurement,” Opt. Lett. 4(5), 131–133 (1979).
    [Crossref] [PubMed]
  5. A. Dandridge and A. B. Tveten, “Phase compensation in interferometric fiber-optic sensors,” Opt. Lett. 7(6), 279–281 (1982).
    [Crossref] [PubMed]
  6. E. Udd and W. B. J. Spillman, Fiber Optic Sensors: An Introduction for Engineers and Scientists (Wiley, 2011).
    [Crossref]
  7. T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. H. Siegel, J. H. Cole, S. C. Rashleigh, and R. G. Priest, “Optical fiber sensor technology,” IEEE Trans. Microw. Theory Tech. 30(4), 472–511 (1982).
    [Crossref]
  8. L. Shao and J. J. Gorman, “Pulsed laser interferometry with sub-picometer resolution using quadrature detection,” Opt. Express 25(6), 6335–6348 (2016).
  9. F. Xie, J. Ren, Z. Chen, and Q. Feng, “Vibration-displacement measurements with a highly stabilised optical fiber Michelson interferometer system,” Opt. Laser Technol. 42(1), 208–213 (2010).
    [Crossref]
  10. G. Basile, A. Bergamin, G. Cavagnero, and G. Mana, “Phase Modulation in High-resolution Optical Interferometry,” Metrologia 28(6), 455–461 (1991).
    [Crossref]
  11. A. J. Fleming and B. S. Routley, “A closed-loop phase-locked interferometer for wide bandwidth position sensing,” Rev. Sci. Instrum. 86(11),115001 (2015).
    [Crossref] [PubMed]
  12. Z. Chao and F. Duan, “Closed-loop phase stabilizing and phase stepping methods for fiber-optic projected-fringe digital interferometry,” Rev. Sci. Instrum. 82(11), 113105 (2011).
    [Crossref] [PubMed]
  13. R. I. Martin, J. M. S. Sakamoto, M. C. M. Teixeira, G. A. Martinez, F. C. Pereira, and C. Kitano, “Nonlinear control system for optical interferometry based on variable structure control and sliding modes,” Opt. Express 25(6), 6335–6348 (2017).
    [Crossref] [PubMed]
  14. R. I. Martin, J. M. S. Sakamoto, M. C. M. Teixeira, and C. Kitano, “Variable structure and sliding modes nonlinear control system applied to a fiber optic interferometer,” Proc. SPIE 10453, 104531W (2017).
  15. D. Wu, R. Zhu, L. Chei, and J. Li, “Transverse spatial phase-shifting method used in vibration-compensated interferometer,” Optik 115(8), 343–346 (2004).
    [Crossref]
  16. J. Zhu, P. Hu, and J. Tan, “Homodyne laser vibrometer capable of detecting nanometer displacements accurately by using optical shutters,” Appl. Opt. 54(34), 10196–10199 (2015).
    [Crossref]
  17. J. G. Park and K. Cho, “High-precision tilt sensor using a folded Mach-Zehnder geometry in-phase and quadrature interferometer,” Appl. Opt. 55(9), 2155–2159 (2016).
    [Crossref] [PubMed]
  18. V. I. Utkin, “Variable structure systems with sliding modes,” IEEE Trans. Autom. Control. 22(2), 212–222 (1977).
    [Crossref]
  19. V. I. Utkin, Sliding Modes on Control and Optimization (Springer-Verlag, 1992).
    [Crossref]
  20. V. I. Utkin, “Sliding Mode Control Design Principles and Applications to Electric Drives,” IEEE Trans. Ind. Electron. 40(1), 23–36 (1977).
    [Crossref]
  21. E. A. Barbashin, Introduction to the theory of stability (Nauka, 1967).
  22. J. Xu, Q. Xu, L. Chai, Y. Li, and H. Wang, “Direct phase extraction from interferograms with random phase shifts,” Opt. Express 18(20), 20620–20627 (2010).
    [Crossref] [PubMed]
  23. K. D. Young, V. I. Utkin, and U. Ozguner, “A control engineer’s guide to sliding mode control,” IEEE Trans. Control. Syst. Technol. 7(3), 328–342 (1999).
    [Crossref]
  24. J. J. Slotine and S. S. Sastry, “Tracking control of non-linear systems using sliding surfaces, with application to robot manipulators,” Int. J. Control. 38(2), 465–492 (1983).
    [Crossref]
  25. J. J. Slotine and W. Li, Applied Nonlinear Control (Prentice Hall, 1991).
  26. J. C. Lozier, “A steady state approach to the theory of saturable servo systems,” IRE Trans. Autom. Control. 1(1), 19–39 (1956).
    [Crossref]
  27. A. Berger and P. O. Gutman, “A new view of anti-windup design for uncertain linear systems in the frequency domain,” Int. J. Robust Nonlinear Control. 26(10), 2116–2135 (2015).
    [Crossref]
  28. M. C. Turner, J. Sofrony, and G. Herrmann, “An alternative approach to anti-windup in anticipation of actuator saturation,” Int. J. Robust Nonlinear Control. 27(6), 963–980 (2016).
    [Crossref]
  29. M. Ran, Q. Wang, and C. Dong, “Anti-windup design for uncertain nonlinear systems subject to actuator saturation and external disturbance,” Int. J. Robust Nonlinear Control. 26(15), 3421–3438 (2016).
    [Crossref]
  30. International Standard ISO16063-41, “Methods for the calibration of vibration and shock transducers-Part 41: calibration of Laser vibrometers,” (International Organization for Standardization (ISO), 2011).
  31. International Standard ISO 16063-11, “Methods for the calibration of vibration and shock transducers-Part 11: primary vibration calibration by laser interferometry,” (International Organization for Standardization (ISO), 1999), reviewed and confirmed (2009).
  32. H. J. von Martens, “Expanded and improved traceability of vibration measurements by laser interferometry,” Rev. Sci. Instrum. 84(12), 121601 (2013).
    [Crossref]

2017 (2)

R. I. Martin, J. M. S. Sakamoto, M. C. M. Teixeira, G. A. Martinez, F. C. Pereira, and C. Kitano, “Nonlinear control system for optical interferometry based on variable structure control and sliding modes,” Opt. Express 25(6), 6335–6348 (2017).
[Crossref] [PubMed]

R. I. Martin, J. M. S. Sakamoto, M. C. M. Teixeira, and C. Kitano, “Variable structure and sliding modes nonlinear control system applied to a fiber optic interferometer,” Proc. SPIE 10453, 104531W (2017).

2016 (4)

L. Shao and J. J. Gorman, “Pulsed laser interferometry with sub-picometer resolution using quadrature detection,” Opt. Express 25(6), 6335–6348 (2016).

J. G. Park and K. Cho, “High-precision tilt sensor using a folded Mach-Zehnder geometry in-phase and quadrature interferometer,” Appl. Opt. 55(9), 2155–2159 (2016).
[Crossref] [PubMed]

M. C. Turner, J. Sofrony, and G. Herrmann, “An alternative approach to anti-windup in anticipation of actuator saturation,” Int. J. Robust Nonlinear Control. 27(6), 963–980 (2016).
[Crossref]

M. Ran, Q. Wang, and C. Dong, “Anti-windup design for uncertain nonlinear systems subject to actuator saturation and external disturbance,” Int. J. Robust Nonlinear Control. 26(15), 3421–3438 (2016).
[Crossref]

2015 (3)

J. Zhu, P. Hu, and J. Tan, “Homodyne laser vibrometer capable of detecting nanometer displacements accurately by using optical shutters,” Appl. Opt. 54(34), 10196–10199 (2015).
[Crossref]

A. Berger and P. O. Gutman, “A new view of anti-windup design for uncertain linear systems in the frequency domain,” Int. J. Robust Nonlinear Control. 26(10), 2116–2135 (2015).
[Crossref]

A. J. Fleming and B. S. Routley, “A closed-loop phase-locked interferometer for wide bandwidth position sensing,” Rev. Sci. Instrum. 86(11),115001 (2015).
[Crossref] [PubMed]

2013 (1)

H. J. von Martens, “Expanded and improved traceability of vibration measurements by laser interferometry,” Rev. Sci. Instrum. 84(12), 121601 (2013).
[Crossref]

2011 (1)

Z. Chao and F. Duan, “Closed-loop phase stabilizing and phase stepping methods for fiber-optic projected-fringe digital interferometry,” Rev. Sci. Instrum. 82(11), 113105 (2011).
[Crossref] [PubMed]

2010 (2)

J. Xu, Q. Xu, L. Chai, Y. Li, and H. Wang, “Direct phase extraction from interferograms with random phase shifts,” Opt. Express 18(20), 20620–20627 (2010).
[Crossref] [PubMed]

F. Xie, J. Ren, Z. Chen, and Q. Feng, “Vibration-displacement measurements with a highly stabilised optical fiber Michelson interferometer system,” Opt. Laser Technol. 42(1), 208–213 (2010).
[Crossref]

2004 (1)

D. Wu, R. Zhu, L. Chei, and J. Li, “Transverse spatial phase-shifting method used in vibration-compensated interferometer,” Optik 115(8), 343–346 (2004).
[Crossref]

1999 (1)

K. D. Young, V. I. Utkin, and U. Ozguner, “A control engineer’s guide to sliding mode control,” IEEE Trans. Control. Syst. Technol. 7(3), 328–342 (1999).
[Crossref]

1991 (1)

G. Basile, A. Bergamin, G. Cavagnero, and G. Mana, “Phase Modulation in High-resolution Optical Interferometry,” Metrologia 28(6), 455–461 (1991).
[Crossref]

1983 (1)

J. J. Slotine and S. S. Sastry, “Tracking control of non-linear systems using sliding surfaces, with application to robot manipulators,” Int. J. Control. 38(2), 465–492 (1983).
[Crossref]

1982 (3)

1981 (1)

K. Fritsch and G. Adamowsky, “Simple circuit for feedback stabilization of a single-mode optical fiber interferometer,” Rev. Sci. Instrum. 52(7), 996–1000 (1981).
[Crossref]

1980 (1)

1979 (1)

1977 (2)

V. I. Utkin, “Variable structure systems with sliding modes,” IEEE Trans. Autom. Control. 22(2), 212–222 (1977).
[Crossref]

V. I. Utkin, “Sliding Mode Control Design Principles and Applications to Electric Drives,” IEEE Trans. Ind. Electron. 40(1), 23–36 (1977).
[Crossref]

1956 (1)

J. C. Lozier, “A steady state approach to the theory of saturable servo systems,” IRE Trans. Autom. Control. 1(1), 19–39 (1956).
[Crossref]

Adamowsky, G.

K. Fritsch and G. Adamowsky, “Simple circuit for feedback stabilization of a single-mode optical fiber interferometer,” Rev. Sci. Instrum. 52(7), 996–1000 (1981).
[Crossref]

Barbashin, E. A.

E. A. Barbashin, Introduction to the theory of stability (Nauka, 1967).

Basile, G.

G. Basile, A. Bergamin, G. Cavagnero, and G. Mana, “Phase Modulation in High-resolution Optical Interferometry,” Metrologia 28(6), 455–461 (1991).
[Crossref]

Bergamin, A.

G. Basile, A. Bergamin, G. Cavagnero, and G. Mana, “Phase Modulation in High-resolution Optical Interferometry,” Metrologia 28(6), 455–461 (1991).
[Crossref]

Berger, A.

A. Berger and P. O. Gutman, “A new view of anti-windup design for uncertain linear systems in the frequency domain,” Int. J. Robust Nonlinear Control. 26(10), 2116–2135 (2015).
[Crossref]

Bucaro, J. A.

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. H. Siegel, J. H. Cole, S. C. Rashleigh, and R. G. Priest, “Optical fiber sensor technology,” IEEE Trans. Microw. Theory Tech. 30(4), 472–511 (1982).
[Crossref]

Cavagnero, G.

G. Basile, A. Bergamin, G. Cavagnero, and G. Mana, “Phase Modulation in High-resolution Optical Interferometry,” Metrologia 28(6), 455–461 (1991).
[Crossref]

Chai, L.

Chao, Z.

Z. Chao and F. Duan, “Closed-loop phase stabilizing and phase stepping methods for fiber-optic projected-fringe digital interferometry,” Rev. Sci. Instrum. 82(11), 113105 (2011).
[Crossref] [PubMed]

Chei, L.

D. Wu, R. Zhu, L. Chei, and J. Li, “Transverse spatial phase-shifting method used in vibration-compensated interferometer,” Optik 115(8), 343–346 (2004).
[Crossref]

Chen, Z.

F. Xie, J. Ren, Z. Chen, and Q. Feng, “Vibration-displacement measurements with a highly stabilised optical fiber Michelson interferometer system,” Opt. Laser Technol. 42(1), 208–213 (2010).
[Crossref]

Cho, K.

Cole, J. H.

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. H. Siegel, J. H. Cole, S. C. Rashleigh, and R. G. Priest, “Optical fiber sensor technology,” IEEE Trans. Microw. Theory Tech. 30(4), 472–511 (1982).
[Crossref]

Dandridge, A.

Dong, C.

M. Ran, Q. Wang, and C. Dong, “Anti-windup design for uncertain nonlinear systems subject to actuator saturation and external disturbance,” Int. J. Robust Nonlinear Control. 26(15), 3421–3438 (2016).
[Crossref]

Duan, F.

Z. Chao and F. Duan, “Closed-loop phase stabilizing and phase stepping methods for fiber-optic projected-fringe digital interferometry,” Rev. Sci. Instrum. 82(11), 113105 (2011).
[Crossref] [PubMed]

Feng, Q.

F. Xie, J. Ren, Z. Chen, and Q. Feng, “Vibration-displacement measurements with a highly stabilised optical fiber Michelson interferometer system,” Opt. Laser Technol. 42(1), 208–213 (2010).
[Crossref]

Fisher, A. D.

Fleming, A. J.

A. J. Fleming and B. S. Routley, “A closed-loop phase-locked interferometer for wide bandwidth position sensing,” Rev. Sci. Instrum. 86(11),115001 (2015).
[Crossref] [PubMed]

Fritsch, K.

K. Fritsch and G. Adamowsky, “Simple circuit for feedback stabilization of a single-mode optical fiber interferometer,” Rev. Sci. Instrum. 52(7), 996–1000 (1981).
[Crossref]

Giallorenzi, T. G.

S. K. Sheem, T. G. Giallorenzi, and K. Koo, “Optical techniques to solve the signal fading problem in fiber interferometers,” Appl. Opt. 21(4), 689–693 (1982).
[Crossref] [PubMed]

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. H. Siegel, J. H. Cole, S. C. Rashleigh, and R. G. Priest, “Optical fiber sensor technology,” IEEE Trans. Microw. Theory Tech. 30(4), 472–511 (1982).
[Crossref]

Gorman, J. J.

Gutman, P. O.

A. Berger and P. O. Gutman, “A new view of anti-windup design for uncertain linear systems in the frequency domain,” Int. J. Robust Nonlinear Control. 26(10), 2116–2135 (2015).
[Crossref]

Herrmann, G.

M. C. Turner, J. Sofrony, and G. Herrmann, “An alternative approach to anti-windup in anticipation of actuator saturation,” Int. J. Robust Nonlinear Control. 27(6), 963–980 (2016).
[Crossref]

Hu, P.

Jackson, D. A.

Kitano, C.

R. I. Martin, J. M. S. Sakamoto, M. C. M. Teixeira, and C. Kitano, “Variable structure and sliding modes nonlinear control system applied to a fiber optic interferometer,” Proc. SPIE 10453, 104531W (2017).

R. I. Martin, J. M. S. Sakamoto, M. C. M. Teixeira, G. A. Martinez, F. C. Pereira, and C. Kitano, “Nonlinear control system for optical interferometry based on variable structure control and sliding modes,” Opt. Express 25(6), 6335–6348 (2017).
[Crossref] [PubMed]

Koo, K.

Li, J.

D. Wu, R. Zhu, L. Chei, and J. Li, “Transverse spatial phase-shifting method used in vibration-compensated interferometer,” Optik 115(8), 343–346 (2004).
[Crossref]

Li, W.

J. J. Slotine and W. Li, Applied Nonlinear Control (Prentice Hall, 1991).

Li, Y.

Lozier, J. C.

J. C. Lozier, “A steady state approach to the theory of saturable servo systems,” IRE Trans. Autom. Control. 1(1), 19–39 (1956).
[Crossref]

Mana, G.

G. Basile, A. Bergamin, G. Cavagnero, and G. Mana, “Phase Modulation in High-resolution Optical Interferometry,” Metrologia 28(6), 455–461 (1991).
[Crossref]

Martin, R. I.

R. I. Martin, J. M. S. Sakamoto, M. C. M. Teixeira, G. A. Martinez, F. C. Pereira, and C. Kitano, “Nonlinear control system for optical interferometry based on variable structure control and sliding modes,” Opt. Express 25(6), 6335–6348 (2017).
[Crossref] [PubMed]

R. I. Martin, J. M. S. Sakamoto, M. C. M. Teixeira, and C. Kitano, “Variable structure and sliding modes nonlinear control system applied to a fiber optic interferometer,” Proc. SPIE 10453, 104531W (2017).

Martinez, G. A.

Ozguner, U.

K. D. Young, V. I. Utkin, and U. Ozguner, “A control engineer’s guide to sliding mode control,” IEEE Trans. Control. Syst. Technol. 7(3), 328–342 (1999).
[Crossref]

Park, J. G.

Pereira, F. C.

Priest, R.

Priest, R. G.

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. H. Siegel, J. H. Cole, S. C. Rashleigh, and R. G. Priest, “Optical fiber sensor technology,” IEEE Trans. Microw. Theory Tech. 30(4), 472–511 (1982).
[Crossref]

Ran, M.

M. Ran, Q. Wang, and C. Dong, “Anti-windup design for uncertain nonlinear systems subject to actuator saturation and external disturbance,” Int. J. Robust Nonlinear Control. 26(15), 3421–3438 (2016).
[Crossref]

Rashleigh, S. C.

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. H. Siegel, J. H. Cole, S. C. Rashleigh, and R. G. Priest, “Optical fiber sensor technology,” IEEE Trans. Microw. Theory Tech. 30(4), 472–511 (1982).
[Crossref]

Ren, J.

F. Xie, J. Ren, Z. Chen, and Q. Feng, “Vibration-displacement measurements with a highly stabilised optical fiber Michelson interferometer system,” Opt. Laser Technol. 42(1), 208–213 (2010).
[Crossref]

Routley, B. S.

A. J. Fleming and B. S. Routley, “A closed-loop phase-locked interferometer for wide bandwidth position sensing,” Rev. Sci. Instrum. 86(11),115001 (2015).
[Crossref] [PubMed]

Sakamoto, J. M. S.

R. I. Martin, J. M. S. Sakamoto, M. C. M. Teixeira, and C. Kitano, “Variable structure and sliding modes nonlinear control system applied to a fiber optic interferometer,” Proc. SPIE 10453, 104531W (2017).

R. I. Martin, J. M. S. Sakamoto, M. C. M. Teixeira, G. A. Martinez, F. C. Pereira, and C. Kitano, “Nonlinear control system for optical interferometry based on variable structure control and sliding modes,” Opt. Express 25(6), 6335–6348 (2017).
[Crossref] [PubMed]

Sastry, S. S.

J. J. Slotine and S. S. Sastry, “Tracking control of non-linear systems using sliding surfaces, with application to robot manipulators,” Int. J. Control. 38(2), 465–492 (1983).
[Crossref]

Shao, L.

Sheem, S. K.

Siegel, G. H.

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. H. Siegel, J. H. Cole, S. C. Rashleigh, and R. G. Priest, “Optical fiber sensor technology,” IEEE Trans. Microw. Theory Tech. 30(4), 472–511 (1982).
[Crossref]

Slotine, J. J.

J. J. Slotine and S. S. Sastry, “Tracking control of non-linear systems using sliding surfaces, with application to robot manipulators,” Int. J. Control. 38(2), 465–492 (1983).
[Crossref]

J. J. Slotine and W. Li, Applied Nonlinear Control (Prentice Hall, 1991).

Sofrony, J.

M. C. Turner, J. Sofrony, and G. Herrmann, “An alternative approach to anti-windup in anticipation of actuator saturation,” Int. J. Robust Nonlinear Control. 27(6), 963–980 (2016).
[Crossref]

Spillman, W. B. J.

E. Udd and W. B. J. Spillman, Fiber Optic Sensors: An Introduction for Engineers and Scientists (Wiley, 2011).
[Crossref]

Tan, J.

Teixeira, M. C. M.

R. I. Martin, J. M. S. Sakamoto, M. C. M. Teixeira, and C. Kitano, “Variable structure and sliding modes nonlinear control system applied to a fiber optic interferometer,” Proc. SPIE 10453, 104531W (2017).

R. I. Martin, J. M. S. Sakamoto, M. C. M. Teixeira, G. A. Martinez, F. C. Pereira, and C. Kitano, “Nonlinear control system for optical interferometry based on variable structure control and sliding modes,” Opt. Express 25(6), 6335–6348 (2017).
[Crossref] [PubMed]

Turner, M. C.

M. C. Turner, J. Sofrony, and G. Herrmann, “An alternative approach to anti-windup in anticipation of actuator saturation,” Int. J. Robust Nonlinear Control. 27(6), 963–980 (2016).
[Crossref]

Tveten, A. B.

Udd, E.

E. Udd and W. B. J. Spillman, Fiber Optic Sensors: An Introduction for Engineers and Scientists (Wiley, 2011).
[Crossref]

Utkin, V. I.

K. D. Young, V. I. Utkin, and U. Ozguner, “A control engineer’s guide to sliding mode control,” IEEE Trans. Control. Syst. Technol. 7(3), 328–342 (1999).
[Crossref]

V. I. Utkin, “Sliding Mode Control Design Principles and Applications to Electric Drives,” IEEE Trans. Ind. Electron. 40(1), 23–36 (1977).
[Crossref]

V. I. Utkin, “Variable structure systems with sliding modes,” IEEE Trans. Autom. Control. 22(2), 212–222 (1977).
[Crossref]

V. I. Utkin, Sliding Modes on Control and Optimization (Springer-Verlag, 1992).
[Crossref]

von Martens, H. J.

H. J. von Martens, “Expanded and improved traceability of vibration measurements by laser interferometry,” Rev. Sci. Instrum. 84(12), 121601 (2013).
[Crossref]

Wang, H.

Wang, Q.

M. Ran, Q. Wang, and C. Dong, “Anti-windup design for uncertain nonlinear systems subject to actuator saturation and external disturbance,” Int. J. Robust Nonlinear Control. 26(15), 3421–3438 (2016).
[Crossref]

Warde, C.

Wu, D.

D. Wu, R. Zhu, L. Chei, and J. Li, “Transverse spatial phase-shifting method used in vibration-compensated interferometer,” Optik 115(8), 343–346 (2004).
[Crossref]

Xie, F.

F. Xie, J. Ren, Z. Chen, and Q. Feng, “Vibration-displacement measurements with a highly stabilised optical fiber Michelson interferometer system,” Opt. Laser Technol. 42(1), 208–213 (2010).
[Crossref]

Xu, J.

Xu, Q.

Young, K. D.

K. D. Young, V. I. Utkin, and U. Ozguner, “A control engineer’s guide to sliding mode control,” IEEE Trans. Control. Syst. Technol. 7(3), 328–342 (1999).
[Crossref]

Zhu, J.

Zhu, R.

D. Wu, R. Zhu, L. Chei, and J. Li, “Transverse spatial phase-shifting method used in vibration-compensated interferometer,” Optik 115(8), 343–346 (2004).
[Crossref]

Appl. Opt. (4)

IEEE Trans. Autom. Control. (1)

V. I. Utkin, “Variable structure systems with sliding modes,” IEEE Trans. Autom. Control. 22(2), 212–222 (1977).
[Crossref]

IEEE Trans. Control. Syst. Technol. (1)

K. D. Young, V. I. Utkin, and U. Ozguner, “A control engineer’s guide to sliding mode control,” IEEE Trans. Control. Syst. Technol. 7(3), 328–342 (1999).
[Crossref]

IEEE Trans. Ind. Electron. (1)

V. I. Utkin, “Sliding Mode Control Design Principles and Applications to Electric Drives,” IEEE Trans. Ind. Electron. 40(1), 23–36 (1977).
[Crossref]

IEEE Trans. Microw. Theory Tech. (1)

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. H. Siegel, J. H. Cole, S. C. Rashleigh, and R. G. Priest, “Optical fiber sensor technology,” IEEE Trans. Microw. Theory Tech. 30(4), 472–511 (1982).
[Crossref]

Int. J. Control. (1)

J. J. Slotine and S. S. Sastry, “Tracking control of non-linear systems using sliding surfaces, with application to robot manipulators,” Int. J. Control. 38(2), 465–492 (1983).
[Crossref]

Int. J. Robust Nonlinear Control. (3)

A. Berger and P. O. Gutman, “A new view of anti-windup design for uncertain linear systems in the frequency domain,” Int. J. Robust Nonlinear Control. 26(10), 2116–2135 (2015).
[Crossref]

M. C. Turner, J. Sofrony, and G. Herrmann, “An alternative approach to anti-windup in anticipation of actuator saturation,” Int. J. Robust Nonlinear Control. 27(6), 963–980 (2016).
[Crossref]

M. Ran, Q. Wang, and C. Dong, “Anti-windup design for uncertain nonlinear systems subject to actuator saturation and external disturbance,” Int. J. Robust Nonlinear Control. 26(15), 3421–3438 (2016).
[Crossref]

IRE Trans. Autom. Control. (1)

J. C. Lozier, “A steady state approach to the theory of saturable servo systems,” IRE Trans. Autom. Control. 1(1), 19–39 (1956).
[Crossref]

Metrologia (1)

G. Basile, A. Bergamin, G. Cavagnero, and G. Mana, “Phase Modulation in High-resolution Optical Interferometry,” Metrologia 28(6), 455–461 (1991).
[Crossref]

Opt. Express (3)

Opt. Laser Technol. (1)

F. Xie, J. Ren, Z. Chen, and Q. Feng, “Vibration-displacement measurements with a highly stabilised optical fiber Michelson interferometer system,” Opt. Laser Technol. 42(1), 208–213 (2010).
[Crossref]

Opt. Lett. (2)

Optik (1)

D. Wu, R. Zhu, L. Chei, and J. Li, “Transverse spatial phase-shifting method used in vibration-compensated interferometer,” Optik 115(8), 343–346 (2004).
[Crossref]

Proc. SPIE (1)

R. I. Martin, J. M. S. Sakamoto, M. C. M. Teixeira, and C. Kitano, “Variable structure and sliding modes nonlinear control system applied to a fiber optic interferometer,” Proc. SPIE 10453, 104531W (2017).

Rev. Sci. Instrum. (4)

A. J. Fleming and B. S. Routley, “A closed-loop phase-locked interferometer for wide bandwidth position sensing,” Rev. Sci. Instrum. 86(11),115001 (2015).
[Crossref] [PubMed]

Z. Chao and F. Duan, “Closed-loop phase stabilizing and phase stepping methods for fiber-optic projected-fringe digital interferometry,” Rev. Sci. Instrum. 82(11), 113105 (2011).
[Crossref] [PubMed]

K. Fritsch and G. Adamowsky, “Simple circuit for feedback stabilization of a single-mode optical fiber interferometer,” Rev. Sci. Instrum. 52(7), 996–1000 (1981).
[Crossref]

H. J. von Martens, “Expanded and improved traceability of vibration measurements by laser interferometry,” Rev. Sci. Instrum. 84(12), 121601 (2013).
[Crossref]

Other (6)

J. J. Slotine and W. Li, Applied Nonlinear Control (Prentice Hall, 1991).

E. A. Barbashin, Introduction to the theory of stability (Nauka, 1967).

International Standard ISO16063-41, “Methods for the calibration of vibration and shock transducers-Part 41: calibration of Laser vibrometers,” (International Organization for Standardization (ISO), 2011).

International Standard ISO 16063-11, “Methods for the calibration of vibration and shock transducers-Part 11: primary vibration calibration by laser interferometry,” (International Organization for Standardization (ISO), 1999), reviewed and confirmed (2009).

E. Udd and W. B. J. Spillman, Fiber Optic Sensors: An Introduction for Engineers and Scientists (Wiley, 2011).
[Crossref]

V. I. Utkin, Sliding Modes on Control and Optimization (Springer-Verlag, 1992).
[Crossref]

Supplementary Material (1)

NameDescription
» Visualization 1       This video shows firstly the output signals of the interferometer operating in open-loop, while a huge displacement is induced on the mirror by exerting pressure at the mirror placed on the sensor arm of the interferometer. When the control system is

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Figures (14)

Fig. 1
Fig. 1 Modified Michelson interferometer and fringe pattern with different phase shifts.
Fig. 2
Fig. 2 Block diagram of the interferometric dynamical system.
Fig. 3
Fig. 3 Equilibrium points in the interferometric characteristic curve.
Fig. 4
Fig. 4 Closed loop block diagram in Simulink software.
Fig. 5
Fig. 5 Open and closed loop simulations for: (a) x1 for t between 0 and 0.01 s and initial condition 0.7π. (b) x1 for t between 0 and 0.01 s initial condition 1.0001π. (c) f for t between 0 and 0.01 s and initial condition 0.7π. (d) f for t between 0 and 0.01 s and initial condition 1.0001π.
Fig. 6
Fig. 6 Open and closed loop input phases. (a) Simulation for t between 0 and 0.02 s for initial condition 0.7π rad. (b) Simulation for t between 0 and 0.02 s for initial condition 1.0001π rad.
Fig. 7
Fig. 7 Block diagram of the implemented control system.
Fig. 8
Fig. 8 Modified Michelson interferometer assemblies. (a) Assembled Modified Michelson interferometer 1. (b) Modified Michelson interferometer with PK2 actuator.
Fig. 9
Fig. 9 Modified Michelson interferometer output signals. (a) Lissajous figure of the quadrature interferometer. (b) Applied voltage to PZT and output signals of the interferometer. (c) Input (in yellow) and output quadrature signals (in purple and green) with low modulation depth.
Fig. 10
Fig. 10 Frequency response of the feedback actuator assembly.
Fig. 11
Fig. 11 Interferometer output signals x1 (yellow), f (green), input applied voltage to PK2 actuator (purple) and inverted control signal (pink) with high modulation depth at 10 Hz. (a) Open loop interferometer. (b) Closed loop interferometer.
Fig. 12
Fig. 12 Interferometer operating under deep phase-modulation. Output signals x1 (yellow), f (green), control signal (purple) and input applied voltage to PK2 actuator (pink). (a) Transition from open to closed loop. (b) Closed loop interferometer with arbitrary signal.
Fig. 13
Fig. 13 Supressing capability for different values of gain.
Fig. 14
Fig. 14 Comparison of frequency response of the PK2 actuator.

Equations (23)

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v ac 1 ( t ) = A 1 V 1 cos ( Δ ϕ ( t ) + ϕ 0 ( t ) + ϕ c ( t ) ) ,
v ac 2 ( t ) = A 2 V 2 sin ( Δ ϕ ( t ) + ϕ 0 ( t ) + ϕ c ( t ) ) .
x ˙ = f ( t , x , u ) ,
u = { u + ( x ) if s ( x ) > 0 u ( x ) if s ( x ) < 0 ,
lim s 0 + s ˙ < 0 and lim s 0 s ˙ > 0 .
d d t ( 1 2 s 2 ( x ) ) = s ( x ) s ˙ ( x ) η | s ( x ) | ,
s ( x ) = x 1 ( t ) = v ac 1 ( t ) ,
s ˙ ( x ) = x ˙ 1 = f ( Δ ϕ + ϕ 0 + ϕ c ) ( Δ ϕ ˙ + ϕ ˙ 0 + ϕ ˙ c ) ,
{ s ( x ) = x 1 = 0 s ˙ ( x ) = x ˙ 1 = 0 { ϕ c = Δ ϕ ϕ 0 + k π 2 ϕ ˙ c = Δ ϕ ˙ ϕ ˙ 0 where k is odd .
u = C sgn ( x 1 f ) ,
s s ˙ = ( x 1 f ) [ Δ ϕ ˙ + ϕ ˙ 0 C sgn ( x 1 f ) ] = ( x 1 f ) ( Δ ϕ ˙ + ϕ ˙ 0 ) C | x 1 f | .
C > C n + | Δ ϕ ˙ + ϕ ˙ 0 | ,
s s ˙ < ( x 1 f ) ( Δ ϕ ˙ + ϕ ˙ 0 ) C n | x 1 f | | Δ ϕ ˙ + ϕ ˙ 0 | | x 1 f | .
s s ˙ < C n | f | | s | .
f ˙ = A 1 V 1 cos ( Δ ϕ + ϕ 0 + ϕ c ) [ Δ ϕ ˙ + ϕ ˙ 0 C sgn ( x 1 f ) ] ,
f f ˙ = x 1 f [ Δ ϕ ˙ + ϕ ˙ 0 C sgn ( x 1 f ) ] ,
f f ˙ = x 1 f ( Δ ϕ ˙ + ϕ ˙ 0 ) + C | x 1 f | .
f f ˙ > x 1 f ( Δ ϕ ˙ + ϕ ˙ 0 ) + C n | x 1 f | + | x 1 f | | Δ ϕ ˙ + ϕ ˙ 0 | ,
f f ˙ C n | x 1 | | f | .
f ˙ = A 1 V 1 ( Δ ϕ ˙ + ϕ ˙ 0 C sgn ( x 1 f ) ) ,
u = C sgn [ ( A 1 V 1 ) 2 2 sin ( 2 ϕ t ) ] .
C = C sgn C LVA C PZT .
sgn ( x 1 f ) ( x 1 f ) ( | x 1 f | + ) ,

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