LIGO-G1500318 – March 2015 LVC Meeting – Pasadena, CA
4.4.3 Feedforward decoupling of loops
53
difference in the two plants by placing a zero at the coupled cavity pole frequency in the path of the PDH error signal atBREFLjust before the summing node. Figure4.13shows the two individual loop gains, as well as the combined total loop gain, partway through the transition.
During lock acquisition, we control the power recycled Michelson vertex with the 3f2 (165MHz) IR PDH signals.
Initially, the common (CARM) and differential (DARM) arm lengths are controlled by the common and differential green laser beatnote signals (ALS). (Due to problems with vertex sensing, we currently operate with the signal recycling cavity misaligned.)
The ALS length noise is two or three times larger than the CARM line-width, so it is not possible to keep the arm cavities on IR resonance. However, the velocity is small enough that it is possible to blend in the PDH CARM error signal at zero CARM offset, without breaking the lock.
We initially blend the IR PDH signal with a crossover around 1Hz, pre-filtering it with an integrator at 20Hz. The integrator provides improved DC stability, which keeps the arms on resonance.
Both the common (CARM) and differential (DARM) arm lengths are blended in this way, before
increasing the RF error signal crossover frequencies.
This crossover is unconditionally stable as we
increase the crossover frequency, to transition to full RF control.
This scheme does not require slow reduction
Figure 4.14: Time series of transition from ALS to IR PDH control. The error signals are taken from the “total error” point for each system, after the blend. The arm cavities begin with an offset of approximately 3 nm, so there is no resonance of the infrared laser beam in the arm cavities (the power recycling gain is representative of this value). At∼10 s, the CARM offset is removed. At
∼60 s, the integrators for both CARM and DARM are engaged, and the CARM degree of freedom is almost fully transitioned to the PDH error signal, although the transition is not yet completed. At
∼170 s the DARM degree of freedom is transitioned completely to its PDH signal and the CARM transition is finished.
maintains full control, while allowing the PDH loop to engage.
Figure4.15shows a block diagram of the proposed control system. Green blocks represent the ALS loop and red blocks represent the PDH loop, also referred to here as the “Refl” loop. Initially, GREFL=0, so we only have a simple single loop system. Once the arm cavities are close to resonance and we have a valid PDH error signal, we begin increasing the gain of the secondary loop. Without any decoupling, the two loops can act against one another, and push the system into instability.
The purple decoupler block feeds the control signal of the Refl loop through an appropriate transfer function to the error point of the ALS loop. The transfer function of the decoupler is chosen so that any signal that arrives at the ALS loop’s error point due to the Refl loop is entirely cancelled out. If we calculate using the nodal matrix technique described in AppendixAthe transfer function
ALS Fool
Sensor ALS
Phase tracker Beat
box Actuator
ALS
Actuator REFL
Servo REFL Servo ALS
CARM
disturbance CARM
stabilized
A
ALSPlant ALS
P
ALSG
ALSS
ALSSensor REFL
S
REFL PlantREFL
P
REFLG
REFLA
REFLDecoupler
D
cplREFL error ALS error
Figure 4.15: Block diagram of feedforward decoupling of loops.
of the system shown in Figure4.15from the output ofGREFLto the input ofGALS, we find GoutREFL→GinALS
= Dcpl+AREFLPALSSALS
1−AALSGALSPALSSALS−AREFLGREFLPREFLSREFL−DcplAALSGALSGREFLPREFLSREFL
. (4.21) Setting this to zero implies thatDcplshould be set to
Dcpl=−AREFLPALSSALS. (4.22)
If we use this value ofDcplto examine the stability of the full system’s transfer function, we find that the closed loop is
CLG= 1
1−HALS−HREFL+HALSHREFL
(4.23) if
HALS=AALSGALSPALSSALS and HREFL=AREFLGREFLPREFLSREFL. (4.24) The open loop gain of the full system is then
OLG=HALS−HREFL+HALSHREFL. (4.25) For a single arm Fabry-P´erot cavity, the bode plot of the open loop gain in Equation4.25is shown in Figure4.16.
Figure 4.16: Open loop gain of feedforward decoupled loops. Shown are the individual loops, as well as the combined (“fool”) loop from Equation4.25.
While this technique has not yet been implemented for the full PRFPMI at the 40 m Lab, we have tested it on a single arm cavity. Figure4.17shows the measured transfer function between the PDH control signal and the ALS error point. We see in the blue trace that the two paths to the summing node (throughAREFLPALSSALSor throughDcpl) are well matched since the magnitude of the ratio is very close to 0 dB. The green trace shows that we get more than 30 dB isolation almost everywhere below 100 Hz [70].
We find that it is possible to increase the gain of the Refl loop, and the system stays stable.
Conveniently, if one of the mirrors is given an impulse large enough that the PDH signal is no longer valid, the ALS loop maintains control of the cavity. The cavity comes back to resonance and is once again controlled by the PDH loop within about 40 ms, as seen in Figure4.18[71].
Frequency (Hz)
102 103
Magnitude (dB)
-60 -50 -40 -30 -20 -10 0 10 Transfer function
T0=17/02/2015 07:50:51.100036 Avg=7 S_ALS out / D_cpl out S_REFL out / S_ALS out Transfer function
Frequency (Hz)
102 103
Coherence
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Coherence function
T0=17/02/2015 07:50:51.100036 Avg=7 S_ALS out / D_cpl out S_REFL out / S_ALS out Coherence function
Frequency (Hz)
102 103
Phase (deg)
-150 -100 -50 0 50 100 150 Transfer function
T0=17/02/2015 07:50:51.100036 Avg=7
S_ALS out / D_cpl out S_REFL out / S_ALS out
Transfer function
Figure 4.17: Measured loop decoupling. We see in the blue trace that the 2 paths to the summing node (throughAREFLPALSSALSor throughDcpl) are well matched since the magnitude of the ratio is very close to 0 dB. The green trace shows that we get more than 30 dB isolation almost everywhere below 100 Hz.
Figure 4.18: Impulse response of decoupled loop system, for single Fabry-P´erot cavity. ETMY is given an impulse that would normally unlock the PDH-only system; however, the ALS loop maintains control, and the cavity is relocked within about 40 ms [71].