PID Tab
The PID tab is only available if the HF2PID Quad PID Controller option is installed on the HF2 Series Instrument (the installed options are displayed in the Device tab).
Some settings in the PID tab are interdependent with settings that are accessible from other tabs. If the PID output controls a certain variable, e.g. Signal Output Offset, this variable will be shown as readonly where it appears in other tabs (i.e. in the Lockin tab for this case). 
Features

Four fully programmable proportional, integral, derivative (PID) controllers

PID Advisor with multiple DUT models, transfer function, and step function modeling

More than 5 kHz regulation bandwidth

Input parameters: demodulator data, auxiliary inputs, oscillator frequency

Output parameters: output amplitudes, oscillator frequencies, auxiliary outputs and DIO
Description
The PID tab is the main control center for the feedback loop controllers in the instrument. Whenever the tab is closed or an additional one of the same type is needed, clicking the following icon will open a new instance of the tab.
Control/Tool  Option/Range  Description 

PID 
Features all control, analysis, and simulation capabilities of the PID controllers. 
The PID tab (see Figure 1) consists of four identical sidetabs, each of them providing access to the functionality of one of the four PID controllers and the associated PID Advisor.
With their variety of different input and output connections, the LabOne PID controllers are extremely versatile and can be used in a wide range of different applications including laser locking or highspeed SPM. Figure 2 shows a block diagram of all PID controller components, their interconnections and the variables to be specified by the user.
Setting up a Control Loop
Depending on the application there are a number of ways to set up a control loop. Let’s consider a few different approaches and see how the Advisor can help to reduce the effort and improve on the result and understanding of the setup.
Manual Setup
In cases where the transfer function of the device under test (DUT) is unknown and only little noise couples into the system from the environment, a manual approach is often the quickest way to get going. For manual configuration of a new control loop it is recommended to start with a small value for P and set the other parameters (I, D) to zero. By enabling the controller one will then immediately see if the sign of P is correct and if the feedback is acting on the correct output parameter for instance by checking the numbers (Error, Shift, Out) displayed in the PID tab. A stepwise increase of the integral gain I will then help to zero the PID error signal completely. Enabling the derivative gain D can increase the speed of the feedback loop, but it can also cause an instable feedback loop behavior . Monitoring the PID error in the Plotter Tab in parallel can be a great help at this stage. The math tools offered by the Plotter allow us to display the standard deviation and the average value of the error. These values should be minimized by tweaking the PID parameters and the associated histogram should have a symmetric (ideally Gaussian) envelope.
In order to characterize the feedback loop quantitatively, you can measure the step response in the Data Acquisition Tab . To do that measurement, change the PID setpoint manually after you have configured the DAQ Trigger level half way in between the old and new setpoint. DAQ Delay and Duration are chosen to roughly match the expected bandwidth. For a step response curve with fine time resolution, the PID data rate should be high enough.
PID Advisor
For many experimental situations the external device or DUT can be well approximated by a simple model. The LabOne PID Advisor allows you to simulate the behavior of a number of different DUT types in a feedback loop and choose feedback gain parameters based on the simulation. The DUTs are characterized by a model function with a number of parameters found on the Advisor subtab. All models include a setting for the delay that occurs outside the instrument. Depending on the targeted servo bandwidth, the external delay can often be the limiting factor and should be sensibly chosen.
The delay specified for each model is the earliest possible response to a stepwise change of the instrument output to be seen on the instrument input. It describes the causality of the system and does not affect the shape of the DUT transfer function. Standard coaxial cables cause a signal delay of about 5 ns/m. 
The most simple approach to modeling is to assume a DUT with a unity transfer function by using All Pass. The lowpass filters allow for limiting the bandwidth, to set an overall gain and a damping for the second order filter. With a Gain set to 1 and a Delay set to 0, All Pass can be used to model the PID controller independent of the external device. Resonator Frequency is a model that applies well in situations with a passive external component, e.g. a AFM cantilever or a quartz resonator, whose frequency should be tracked by a PLL over time. In cases where the amplitude of the resonator signal needs to be stabilized with a second control loop (automatic gain control), the Resonator Amplitude model is the right choice. Setting the resonance frequency and the Q factor, both can be obtained before by a frequency scan over the resonance using the Sweeper Tab , allows the Advisor to estimate the gain and lowpass behavior of the resonator. Internal PLL is used whenever an external oscillating signal is provided that shall be followed by one of the internal oscillators. The VCO setting describes a situation where the input variable of the DUT is a voltage and the output is a frequency. The gain parameter specifies how much voltage change on the input causes how much frequency shift on the VCO output. In case the frequency of the VCO can be tracked by using the external reference mode, one can easily measure this gain with the Sweeper Tab by scanning the Auxiliary Output voltage and displaying the resulting oscillator frequency. The gain is given by the slope of the resulting line at the frequency of interest.
With a model and parameters set to best describe the actual measurement situation, one can now continue by defining a target bandwidth for the entire control loop and the Advise Mode, i.e. the feedback gain parameters that shall be used for the control operation. Whenever the input signal is derived from one of the demodulators it is convenient to activate the box next to target bandwidth. With that in place the Advise algorithm will automatically adjust the demodulator bandwidth to a value about 5 times higher than the target bandwidth in order to avoid to be limited by demodulation speed. The Advisor algorithm will now calculate a target step response function that it will try to achieve by adjusting the feedback gain parameters in the next step. Before doing so in case of a newly set up DUT model, the algorithm will first try to estimate the PID parameters by using the ZieglerNichols method. When there has been a previous run, the user can also change the parameters in the model manually which will the be used as new start parameters of the next Advise run. Starting from the initial parameters, the Advisor will then perform a numerical optimization in order to achieve a leastsquares fit of the calculated step response to a target step response determined from the Target Bandwidth. The result is numerically characterized by an achieved bandwidth (BW) and a phase margin (PM). Moreover, the large plot area on the right can be used to characterize the result by displaying transfer functions, magnitude and phase, and step responses between different signal nodes inside the loop. Once the modeling is completed one can copy the resulting parameters to the physical PID by clicking on .
Name  Function  Parameters 

All pass 
\(H(s)=g\) 

Lowpass 1st 
\(H(s)=g\frac{1}{t_c s + 1} = g\frac{\omega_n}{s +\omega_n} \) 

Lowpass 2nd 
\(H(s)=g\frac{\omega^2_n}{s^2+2\omega_n\zeta s+\omega^2_n}\) 

Resonator frequency 
\(H(s)= 360^{\circ} \frac{t_c}{t_c s+1}\) with \(t_c=\frac{1}{2\pi BW}=\frac{2Q}{2\pi f_{res}}\) 

Resonator amplitude 
\(H(s)=g\frac{\omega / (2Q)}{s+\omega /(2Q)}\) with \(\omega=2\pi f_{res}\) 

Internal PLL 
\(H8s)=\frac{360^\circ}{s}\) 

VCO 
\(H(s)=g\frac{360^\circ}{s(t_c s+1)}\) with \(t_c=\frac{1}{2\pi f_{res}}\) 

It is recommended to use the Advisor in a stepwise approach where one increases the free parameters from P to PI, to PID . This can save time because it prevents optimizing into local minima. Also it can be quite illustrative to see which of the feedback parameters leads to which effect in the feedback behavior. 
The lowpass filter in the differential part is implemented as an exponential moving average filter described by \(y_t=(1\alpha)\cdot y_{t1}+\alpha x_t\) with \(\alpha = 2^{dshift}\), \(x_t\) the filter input, and \(y_t\) the filter output. The default value for dshift is 0 which corresponds to a disabled filter. On the UI the filter properties can be changed in units of bandwidth or a time constant. 
In case the feedback output is a voltage applied to sensitive external equipment it is recommended to make use of the center value and the upper and lower limit values. This will guarantee that the output stays in the defined range even when the lock fails and the integrator goes into saturation.
Functional Elements
Control/Tool  Option/Range  Description 

Setpoint Mode 
Defines the source of the PID setpoint value. 

Fixed 
Setpoint is manually set. 

Aux Input 1 
Setpoint is supplied by Auxiliary Input 1. 

Aux Input 2 
Setpoint is supplied by Auxiliary Input 2. 

PID Output 4 
Setpoint is supplied by the output of another PID. 

Center 
numeric value 
After adding the Center value to the PID output, the signal is clamped to Center + Range and Center  Range. 
Range 
numeric value 
Set the range of the PID controller output relative to the center 
Default Out 
numeric value 
Set the value for the default output if the PID is disabled. 
Default Out Enable 
ON / OFF 
Enable the default value when PID is off. 
Enable 
ON / OFF 
Enable the PID controller 
Input 
Select input source of PID controller 

Modulation Index 
Modulation depth 

Dual Frequency Tracking Z(i+1)Z(i) 
Used in dual frequency tracking applications 

Demod X(i+1)X(i) 
Used in dual frequency tracking applications 

Demod Z(i+1)Z(i) 
Used in dual frequency tracking applications 

Oscillator Frequency 
Oscillator frequency 

Demodulator X 
Demodulator cartesian X component 

Demodulator Y 
Demodulator cartesian Y component 

Demodulator R 
Demodulator magnitude component 

Demodulator Theta 
Demodulator phase 

Aux Input 
Auxiliary Input 

Aux Output 
Internal value of Auxiliary Output 

Input Channel 
index 
Select input channel of PID controller. 
Setpoint 
numeric value 
PID controller setpoint 
Filter BW 
numeric value 
Bandwidth of the demodulator filter used as an input. 
Filter Order 
Selects the filter roll off between 6 dB/oct and 48 dB/oct of the current demodulator. 

1 
1st order filter 6 dB/oct 

2 
2nd order filter 12 dB/oct 

3 
3rd order filter 18 dB/oct 

4 
4th order filter 24 dB/oct 

5 
5th order filter 30 dB/oct 

6 
6th order filter 36 dB/oct 

7 
7th order filter 42 dB/oct 

8 
8th order filter 48 dB/oct 

Harmonic 
1 to 1023 
Multiplier of the for the reference frequency of the current demodulator. 
Output 
Select output of the PID controller 

Output 1 Amplitude 
Feedback to the main signal output amplitude 1 

Output 2 Amplitude 
Feedback to the main signal output amplitude 2 

Oscillator Frequency 
Feedback to any of the internal oscillator frequencies 

Aux Output Offset 
Feedback to any of the 4 Auxiliary Output’s Offset 

DIO (int16) 
Feedback to the DIO as a 16 bit word 

Output Channel 
index 
Select output channel of PID controller. 
P (Hz/deg) 
numeric value 
PID proportional gain P 
I (Hz/deg/s) 
numeric value 
PID integral gain I 
D (Hz/deg*s) 
numeric value 
PID derivative gain D 
Rate 
RT load dependent 
PID sampling rate and update rate of PID outputs. Needs to be set substantially higher than the targeted loop filter bandwidth. The numerical precision of the controller is influenced by the loop filter sampling rate. If the target bandwidth is below 1 kHz is starts to make sense to adjust this rate to a value of about 100 to 500 times the target bandwidth. If the rate is set to high for low bandwidth applications, integration inaccuracies can lead to non linear behavior. 
Error 
numeric value 
Error = Set point  PID Input 
Shift 
numeric value 
Difference between the current output value Out and the Center. Shift = P*Error + I*Int(Error, dt) + D*dError/dt 
To Advisor 
Copy the current PID settings to the PID Advisor. 
Control/Tool  Option/Range  Description 

Advise 
Calculate the PID coefficients based on the used DUT model and the given target bandwidth. If optimized values can be found the coefficients are updated and the response curve is updated on the plot. Only PID coefficients specified with the advise mode are optimized. The Advise mode can be used incremental, means current coefficients are used as starting point for the optimization unless other model parameters are changed inbetween. 

Target BW (Hz) 
numeric value 
Target bandwidth for the closed loop feedback system which is used for the advising of the PID parameters. This bandwidth defines the tradeoff between PID speed and noise. 
Advise Mode 
Select the PID coefficients that are optimized. The other PID coefficients remain unchanged but are used during optimization. This enables keeping selected coefficients at a fixed value while optimizing the rest. The advise time will increase significantly with the number of parameters to be optimized. 

P 
Only optimize the proportional gain. 

I 
Only optimize the integral gain. 

PI 
Only optimize the proportional and the integral gain. 

PID 
Optimize the proportional, integral, and derivative gains. 

Filter BW 
numeric Value 
Defines the lowpass filter characteristic of the selected demodulator input. 
Auto Bandwidth 
ON / OFF 
Adjusts the demodulator bandwidth to fit best to the specified target bandwidth of the full system. If disabled, a demodulator bandwidth too close to the target bandwidth may cause overshoot and instability. In special cases the demodulator bandwidth can also be selected smaller than the target bandwidth. 
Filter Order 
Selects the filter roll off between 6 dB/oct and 48 dB/oct of the modelled demodulator. 

1 
1st order filter 6 dB/oct 

2 
2nd order filter 12 dB/oct 

3 
3rd order filter 18 dB/oct 

4 
4th order filter 24 dB/oct 

5 
5th order filter 30 dB/oct 

6 
6th order filter 36 dB/oct 

7 
7th order filter 42 dB/oct 

8 
8th order filter 48 dB/oct 

Harmonic 
1 to 1023 
Multiplier of the for the reference frequency of the modelled demodulator. 
DUT Model 
Type of model used for the external device to be controlled by the PID. A detailed description of the transfer function for each model is found in the previous section. 

All Pass 
The external device is modelled by an all pass filter. Parameters to be configured are delay and gain. 

LP 1st 
The external device is modelled by a firstorder lowpass filter. Parameters to be configured are delay, gain and filter bandwidth. 

LP 2nd 
The external device is modelled by a secondorder lowpass filter. Parameters to be configured are delay, gain, resonance frequency and damping ratio. 

Resonator Frequency 
The external device is modelled by a resonator. Parameters to be configured are delay, center frequency and quality factor. 

Internal PLL 
The DUT is the internal oscillator locked to an external signal through a phaselocked loop. The parameter to be configured is the delay. 

VCO 
The external device is modelled by a voltage controlled oscillator. Parameters to be configured are delay, gain and bandwidth. 

Resonator Amplitude 
The external device is modelled by a resonator. Parameters to be configured are delay, gain, center frequency and quality factor. 

Delay 
numeric value 
Parameter that determines the earliest response for a step change. This parameter does not affect the shape of the DUT transfer function. 
Gain 
numeric value 
Parameter that determines the gain of the DUT transfer function. 
BW (Hz) 
numeric value 
Parameter that determines the bandwidth of the firstorder lowpass filter respectively the bandwidth of the VCO. 
Damping Ratio 
numeric value 
Parameter that determines the damping ratio of the secondorder lowpass filter. 
Res Freq 
numeric value 
Parameter that determines the resonance frequency of the of the modelled resonator. 
Q 
numeric value 
Parameter that determines the quality factor of the modelled resonator. 
P (Hz/deg) 
numeric value 
Proportional gain P coefficient used for calculation of the response of the PID model. The parameter can be optimized with PID advise or changed manually. The parameter only gets active on the PID after pressing the button To PLL. 
I (Hz/deg/s) 
numeric value 
Integral gain I coefficient used for calculation of the response of the PID model. The parameter can be optimized with PID advise or changed manually. The parameter only gets active on the PID after pressing the button To PLL. 
D (Hz/deg*s) 
numeric value 
Derivative gain D coefficient used for calculation of the response of the PID model. The parameter can be optimized with PID advise or changed manually. The parameter only gets active on the PID after pressing the button To PLL. 
BW (Hz) 
numeric value 
Simulated bandwidth of the full close loop model with the current PID settings. This value should be larger than the target bandwidth. 
Target BW LED 
green/red 
Green indicates that the target bandwidth can be achieved. For very high PID bandwidth the target bandwidth might be only achieved using marginal stable PID settings. In this case, try to lower the bandwidth or optimize the loop delays of the PID system. 
PM (deg) 
numeric value 
Simulated phase margin of the PID with the current settings. The phase margin should be greater than 45 deg for internal PLL and 60 deg for all other DUT for stable conditions. An Infinite value is shown if no unity gain crossing is available to determine a phase margin. 
Stable LED 
green/red 
Green indicates that the phase margin is fulfilled and the PID system should be stable. 
To PID 
Copy the PID Advisor settings to the PID. 
Control/Tool  Option/Range  Description 

Advanced Mode 
ON / OFF 
Enables manual selection of display and advice properties. If disabled the display and advise settings are automatically with optimized default values. 
Display 
Select the display mode used for rendering the system frequency or time response. 

Bode Magnitude 
Display the Bode magnitude plot. 

Bode Phase 
Display the Bode phase plot. 

Step Resp 
Display the step response plot. 

Start (Hz) 
numeric value 
Start frequency for Bode plot display. For disabled advanced mode the start value is automatically derived from the system properties and the input field is readonly. 
Stop (Hz) 
numeric value 
Stop frequency for Bode plot display. For disabled advanced mode the stop value is automatically derived from the system properties and the input field is readonly. 
Start (s) 
numeric value 
Start time for step response display. For disabled advanced mode the start value is zero and the field is readonly. 
Stop (s) 
numeric value 
Stop time for step response display. For disabled advanced mode the stop value is automatically derived from the system properties and the input field is readonly. 
Transfer Function Selector 
Selection of the displayed transfer function of the loop. 2 presets and a manual selection are possible. In closed loop configuration all elements from output to input will be included as feedback elements. 

System 
From Setpoint to System Output. 

PID 
From Setpoint to PID Output. 

Manual 
Any transfer function in the open or closed loop can be visualized. 

Response In 
Start point for the plant response simulation for open or closed loops. In closed loop configuration all elements from output to input will be included as feedback elements. 

Demod Input 
Start point is at the demodulator input. 

Setpoint 
Start point is at the setpoint in front of the PID. 

PID Output 
Start point is at PID output. 

Instrument Output 
Start point is at the instrument output. 

DUT Output 
Start point is at the DUT output and instrument input. 

Response Out 
End point for the plant response simulation for open or closed loops. In closed loop configuration all elements from output to input will be included as feedback elements. 

PID Output 
End point is at PID output. 

Instrument Output 
End point is at the instrument output. 

DUT Output 
End point is at the DUT output and instrument input. 

Demod Input 
End point is at the demodulator input. 

System Output 
End point is at the output of the controlled system. 

ClosedLoop 
ON / OFF 
Switch the display of the system response between closed or open loop. 
TC Mode 
ON / OFF 
Enables time constant representation of PID parameters. 
Set Limits 
ON / OFF 
Switch the writing of PID limits when 'To PID' is pressed. Only applies in case of internal PLL. 
Advisor Link 
Automatically copy cursor values displayed below to the PID advisor. To enable cursor helpers, switch Advanced Mode on and set Display to Bode Magnitude with PID Transfer Function. Cursors will be displayed in Log and dB axis scale combinations. 

P 
Cursor value representing PID proportional gain P. Drag the plot cursor with the mouse pointer or directly insert numerical value here. 

I 
Cursor value representing PID integral gain I. Drag the plot cursor with the mouse pointer or directly insert numerical value here. 

D 
Cursor value representing PID derivative gain D. Drag the plot cursor with the mouse pointer or directly insert numerical value here. 