Col Loch

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Control Electronics Optimization in FM

Summer Internship (May 13

Undergraduate Student (Senior Yr)

B. Tech

Training Report

Control Electronics Optimization in FM-Atomic Force

Microscopes

Summer Internship (May 13th to July 19

th 2013)

Submitted by

Gowtham Ragavendar G

Undergraduate Student (Senior Yr)

B. Tech – Engineering Physics

IIT Delhi

Under the guidance of

Dr. Ashwin Lal

Chief Technology Officer

i2n Technologies Pvt. Ltd.

Bangalore - 560012

Atomic Force

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Contents Page

1. Introduction – Force Microscopy 3

2. Literature Survey 3

2.1 AFM Basics

2.1.1 Dynamic Mode of Operation 3

2.2 Components of FM-AFM 5

2.2.1 AFM Probe – Cantilever & Tip 5

2.2.2 Quartz Tuning Fork 5

2.2.3 Piezo Actuators 6

2.2.4 Control Electronics 6

3. FM-AFM system at i2n Technologies 8

4. Experimental Data Collection – Primary Data 10

5. Problem Definition 17

6. Modeling the Problem 17

6.1 Control Engineering – Mathematical Modelling 17

6.1.1 Open Loop and Closed Loop Control 18

6.1.2 Transfer Functions and Block Representation 18

6.2 PID Controller 19

7. Solution Methodology 20

7.1 Control Loop Simulation 20

7.2 Optimization Criteria 21

8. Conclusions 23

8.1 Simulink plot 23

9. Future Outlook 23

10. Appendix 24

10.1 Transfer Functions 24

10.2 Bode Plot & Nyquist Plot – MATLAB Simulink Tool 25

11. References 28

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1. INTRODUCTION – Force Microscopy

Atomic Force Microscopy started in 1986 when the Atomic Force Microscope (AFM), was invented

by G. Binnig at Stanford University. Since then, a considerable number of improvements and

developments had been made, and many results had been achieved by simple contact

measurements. Nevertheless, contact AFM cannot generate true atomic resolution in a stable

operation. In 1995, non-contact AFM achieved true atomic resolution under attractive regime at

room temperature.

2. LITERATURE REVIEW

2.1 AFM Basics

The AFM consists of a cantilever with a sharp tip (probe) at its end that is used to scan the specimen

surface. The cantilever is typically silicon or silicon nitride with a tip radius of curvature on the

order of nanometers. When the tip is brought into proximity of a sample surface, forces between

the tip and the sample lead to a deflection of the cantilever according to Hooke's law. Typically, the

deflection is measured using a laser spot reflected from the top surface of the cantilever into an

array of photodiodes. Other methods that are used include optical interferometry, capacitive

sensing or piezoresistive AFM cantilevers. These cantilevers are fabricated with piezoresistive

elements that act as a strain gauge.

If the tip was scanned at a constant height, a risk would exist that the tip collides with the surface,

causing damage. Hence, in most cases a feedback mechanism is employed to adjust the tip-to-

sample distance to maintain a constant force between the tip and the sample. Traditionally, the

sample is mounted on a piezoelectric tube, which can move the sample in the z direction for

maintaining a constant force, and the x and y directions for scanning the sample. Alternatively a

'tripod' configuration of three piezo crystals may be employed, with each responsible for scanning in

the x,y and z directions. This eliminates some of the distortion effects seen with a tube scanner. In

newer designs, the tip is mounted on a vertical piezo scanner while the sample is being scanned in X

and Y using another piezo block. The resulting map of the area z = f(x,y) represents

the topography of the sample.

Noncontact AFM operates in static mode or dynamic mode, i.e., static AFM or dynamic AFM,

respectively. In static AFM the force Fts interacting between tip and sample translates into a

deflection of the cantilever, and the image is a map z(x, y, Fts) with Fts = constant.

2.2 Dynamic Mode of Operation

On the other hand, in dynamic AFM, the cantilever is deliberately vibrated. Dynamic mode imaging

is done using cantilevers of various frequency ranges Often as the cantilever resonant frequency

goes up, they get stiffer and have a higher Q. The higher Q provides greater amplitude amplification

of the drive signal and better frequency discrimination for small shifts due to surface interaction.

However, the extra stiffness of the cantilever might damage some materials, so there is a tradeoff to

be made on increasing the cantilever resonance.

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The two basic methods are Amplitude Modulation (AM

AFM). In AM-AFM the cantilever is vibrated at a fixed frequency near the cantilever eigenfrequency.

The tip-sample interactions cause changes in both amplitude and phase, these changes are

measured and are used as the feedback signal. The speed of AM

factor of the cantilever, which slows the detection of surface features thr

effect in which a the tip goes off a cliff on the surface but doesn’t detect it for a while.

In FM-AFM the cantilever is also driven to oscillate at

In this mode, the change in the oscil

FM-AFM typically requires extremely high

The control signal is used to generate dissipation images. Another control loop is used to k

constant the cantilever’s oscillation frequency by adjustin

control signal is used to generate topographic images. Using FM

dramatically and for atomic studies in vacuum the FM is now t

FIG1: Block diagram of atomic force microscope using beam deflection detection

(All images referenced in Section 11)

The two basic methods are Amplitude Modulation (AM-AFM) and Frequency Modulation (FM

AFM the cantilever is vibrated at a fixed frequency near the cantilever eigenfrequency.

sample interactions cause changes in both amplitude and phase, these changes are

measured and are used as the feedback signal. The speed of AM-AFM is often limited by the high Q

factor of the cantilever, which slows the detection of surface features through the Wile E. Coyote


AFM the cantilever is also driven to oscillate at fixed amplitude by an automatic control loop.

change in the oscillation frequency of the returned deflection signal is detected.

extremely high-Q cantilevers so that the frequency shift

The control signal is used to generate dissipation images. Another control loop is used to k

ation frequency by adjusting the tip-sample distance

control signal is used to generate topographic images. Using FM-AFM improved the resolution

dramatically and for atomic studies in vacuum the FM is now the preferred AFM technique.

: Block diagram of atomic force microscope using beam deflection detection

(All images referenced in Section 11)

AFM) and Frequency Modulation (FM-

AFM the cantilever is vibrated at a fixed frequency near the cantilever eigenfrequency.

sample interactions cause changes in both amplitude and phase, these changes are

AFM is often limited by the high Q-

ough the Wile E. Coyote


amplitude by an automatic control loop.

signal is detected.

Q cantilevers so that the frequency shift can be detected.

The control signal is used to generate dissipation images. Another control loop is used to keep

sample distance. The distance

AFM improved the resolution

he preferred AFM technique.

: Block diagram of atomic force microscope using beam deflection detection

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2.3 Components of an AFM

2.3.1 AFM Probe – Cantilever & Tip

The AFM probe is a consumable and measuring device with a sharp tip on the free swinging end of

a cantilever which is protruding from a holder plate used in Atomic force microscopes (AFM). The

dimensions of the cantilever are in the scale of micrometers. The radius of the tip is in the scale of a

few nanometers. The holder plate, also called holder chip, - often 1.6 mm by 3.4 mm in size - allows

the operator to hold the AFM probe with tweezers and fit it into the corresponding holder clips on

the scanning head of the Atomic force microscope.

The tip material is generally chosen for specific properties of the surface it will interact

with. There are two basic designs of cantilevers. The most common is the thin rectangular

bar “diving board” shape, used in contact and AC mode operation.

However, there are other methods to excite vibrations in the FM-AFM which do not include

cantilevers with bending and torsional properties. The configuration currently in operation at i2n

does not feature rectangular diving board or triangular cantilevers, instead uses a tuning fork with

tip attached, which act as basis for both the actuator and sensor system.

2.3.2 Quartz Tuning Fork

One major approach for implementation of vibration in high resolution FM-AFM imaging, and

currently used in-house at i2n, is the use of quartz tuning-fork sensors (QTF).

In this method, the piezoelectric quartz tuning-fork sensor/actuator is used to drive tip vibrations.

A sharpened 0.001 in. wire is fixed to one tine of a quartz tuning fork. In this configuration, the

probe tip oscillates perpendicular to the sample surface. The motion is actuated by driving one

electrode of the tuning fork by an AC voltage near the tuning-fork resonant frequency. The tuning-

fork oscillations are sensed by monitoring the current from the other electrode.

FIG3: Configuration of the piezoelectric quartz tuning-fork sensor/actuator.

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The AFM tip is mounted perpendicular to the tuning-fork tine so that the tip oscillates normally to

the sample surface, minimizing tip wear. Furthermore, the high spring constant of the tuning-fork

allows the use of small oscillation amplitudes as small 0.1 nm, offering potentially improved spatial

resolution.

2.3.3 Piezo Actuators

A piezoelectric tube is a ceramic tube in which the many molecular dipoles are polarized at an

elevated temperature by applying a positive voltage to an outer electrode, causing the molecules to

partially align themselves with the positive directed outward. This results in + outer and - inner

radial polarization, which is made permanent by cooling. When subject to small voltages, the

element experiences a temporary expansion & contraction, leading to tube bending. This bending

action allows for piezo tubes to be used for XY-scanning in AFMs. However, this XY-direction

control is effected via the supporting software for the AFM usage, and not as a part of the PID

control apparatus, and thus it will remain outside the scope of further analysis, in this report.

2.3.4 Control Electronics

As seen from the figure below, there are two main control loops visible in the AFM control system:

• Loop controlling position of AFM head in z axis - Maintaining position of tip along the

vertical.

• QTF driving loop (excitation) - Directly maintaining tip vibration amplitude and

frequency

FIG4: Block level diagram of the AFM setup

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Due to the high mechanical Q of quartz tuning forks, there is a delay of tens to hundreds of ms

before the tuning-fork oscillations reach their steady-state condition. As a result, traditional, ‘‘open-

loop,’’ methods of AFM are not appropriate for tuning-fork feedback. Since an AFM could not track a

surface at a scan rate faster than 0.1 Hz with a 100 ms lag in its sensor, we use a phase-locked-loop

(PLL) circuit to overcome this limitation by actively tracking the tuning-fork resonant frequency.

A Phase Locked Loop (PLL) present inside the VCO (Voltage Controlled Oscillator) performs the

function of frequency control. A phase-locked loop or phase lock loop (PLL) is a control system that

tries to generate an output signal whose phase is related to the phase of the input "reference"

signal. This circuit compares the phase of the input signal with the phase of the signal derived from

its output oscillator and adjusts the frequency of its oscillator to keep the phases matched. The

signal from the phase detector is used to control the oscillator in a feedback loop.

A VCO produces a repetitive output waveform at a frequency which is controlled by the DC level of

an input signal. As shown in Fig. below, the VCO includes a DSP module input to a digital frequency

synthesizer. An analog input on the DSP samples the output of the PSD; to maintain a fixed input

level, the DSP program raises or lowers the frequency it provides to the synthesizer. The DSP also

uses one of its analog outputs to produce a voltage (‘‘Freq Shift’’) proportional to the excursions of

the drive frequency from the tuning-fork free-oscillation resonant frequency. Since a PLL is formed,

this drive frequency tracks the resonant frequency of the tuning fork.

FIG5: Schematic of PLL used to track the tuning-fork resonant frequency. Left: VCO drives one electrode

of the tuning fork to excite resonance. Current from other electrode is monitored to sense mechanical

response of tuning fork. Analog multiplier and low-pass filter detect the phase difference between drive

and sense signals, which in turn controls VCO frequency. Right: VCO comprises a DSP chip controlling the

frequency output of a digital frequency synthesizer.

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3. FM-AFM SYSTEM AT i2n TECHNOLOGIES

The two control loops in the diagram FIG4 are independent loops which have output values which

together contribute towards the input of the QTF probe. The Frequency controller and Gain

controller in FIG4 are essentially PI controller blocks with set points provided manually. The VCO

function is obtained by using SR844 Lock-In amplifier with manually adjustable time constant

values (see figure below) The control systems were linked together using a LABView program

which allowed for user-end manipulation of PI coefficients and operating frequency range.

The Z-control loop is operated separately via the Open-Source GXSM (Gnome X Scanning

Microscopy) module, and thus was removed from the feedback loop considerations. The PI

controller blocks are accorded set points manually using a program developed in LABView

framework currently in use at i2n, while Z-control is performed via the GXSM tool.

FIG6: The FM-AFM system at i2n, including the SR844 RF Lock-In Amplifier (below) (top left). Also visible

are (top right) probe setup, and the computers required to operate the GXSM tool and the PID calibration

program

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4. EXPERIMENTAL DATA COLLECTION: Operating the AFM

In the second phase of the internship, the learnings from the literature review was put to

experiment by operating the FM-AFM in the premises of i2n. This allowed for a weeklong

immersion in the functioning controls of the running software used, and the best-practices and

safety precautions required to be adopted by user while operating the AFM.

In order to identify all the components of the control loop and to obtain data, the FM-AFM was

operated without an operating sample, using the LABView Program to input set points for

amplitude and frequency control loops. The control systems were subjected to short impulse

inputs, in order to obtain data that could be compared with theoretical expectations from impulse

response functions applied on the computer simulations. The Gain vs Frequency plot (Bode Plot) of

the QTF was obtained as shown below.

Screen1: Bode plot

The said LABView program was the user-interface system for changing the parameters required to

operate the AFM. The code for this implementation was devised in-house at i2n, and they are

beyond the scope of this report. A sample grab of the user-interface for modification of control

parameters is as shown below:

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Screen2: LABView program user-interface

The FM-AFM was operated using the LABView Program to input set points for amplitude and

frequency control loops. The control systems were subjected to short impulse inputs, in order to

obtain data that could be compared with theoretical expectations from impulse response functions

applied on the computer simulations. Presented below are graphs for various impulse responses

obtained on the digital Cathode Ray Oscilloscope monitor, followed by analysis of the graphs at a

later section of the report. (In each graph, the signal represented in blue, which is vertically lower,

is the output signal whose transitions are being considered)

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I. PHASE SHIFTS

a) P = 2, I = 0.010:

i) Phase change from -1.7 to -2.3

ii) Phase change from -2.3 to -1.7

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b) P = 2, I = 0.003:

i) Phase change from -1.7 to 2.3


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c) P = 10, I = 0.003

i) Phase change from -1.7 to 2.3


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II. AMPLITUDE SHIFTS

a) P = 0.2, I = 0.001

i) Amplitude shift 3.5 to 3

ii) Amplitude shift 3 to 3.5

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b) P = 0.2, I = 0.002:



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c) P = 0.05, I = 0.001:



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5. PROBLEM DEFINITION

For operating the FM-AFM to obtain high-resolution images at the nano level, the question at hand

is the selection of proper values as PI controller coefficients. The values of these coefficients are

to be selected optimally in order to avoid control loop instability, contact between tip and sample,

wear-and-tear of the tip and improper scanning of the surface. At i2n Technologies, the prevalent

procedure for arriving at optimum PI constants was via trial-and-error. This would require

considerable time and effort as preparation and calibration, and the set-up the FM-AFM would

become a tedious process. A need was felt to understand the basics of optimum constants in the

control loop in order to automate their selection. Thus, the project was aimed at identifying the

constraints responsible for selection of optimum PI coefficients and achieving critical damping

conditions inside the FM-AFM control loop.

6. MODELLING THE PROBLEM

Optimizing the PI controller to identify suitable constants required a thorough understanding the

basics of control engineering. Provided below is a brief review of the basic concepts on control

systems, which form the basis for understanding in subsequent sections of this report.

6.1 Control Engineering: Mathematical Modeling

6.1.1 Open-Loop and Closed-Loop Controller

An open-loop controller is a type of controller that computes its input into a system using only the

current state and its model of the system.

A characteristic of the open-loop controller is that it does not use any signal from its output to

determine if its output has achieved the desired goal of the input. This means that the system does

not observe the output of the processes that it is controlling. Consequently, a true open-loop system

can not engage in machine learning and also cannot correct any errors that it could make. It also

may not compensate for disturbances in the system.

To overcome these limitations of the open-loop controller, control theory introduces feedback. A

closed-loop controller uses feedback to control states or outputs of a dynamical system. Its name

comes from the information path in the system: process inputs (e.g., voltage applied to an electric

motor) have an effect on the process outputs (e.g., speed or torque of the motor), which is

measured with sensors and processed by the controller; the result (the control signal) is "fed back"

as input to the process, closing the loop.

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Closed-loop controllers have the following advantages over

rejection, guaranteed performance even with

variations and improved reference tracking performance

FIG7: Typical, single-input,single

Mathematically modeling open-loop and closed

these control systems in everyday use. Feedback and non

again after discussing certain tools that allow us to model them. One such

Function.

6.1.2 Transfer function

In classical control theory, the transfer function is the primary tool of operation, and is derived

from Laplace transforms.

The Transfer Function fully describes a control system. The Order, Type and Frequ

can all be taken from this specific function. Nyquist and Bode plots can be drawn

loop Transfer Function. These plots show the stability of the system when

the denominator of the transfer function,

can be derived.

A Transfer Function, by definition, i

temporal frequency, of the relation between the input and output of a

It is given by the ratio of the output of a system to the input of a system, in

considering its initial conditions and equilibrium point to be zero. If we

X(s), and an output function Y(s), we define

Transfer Function:

loop controllers have the following advantages over open-loop controllers:

guaranteed performance even with model uncertainties, reduced sensitivity to parameter

improved reference tracking performance.

input,single-output feedback loop with various parts described

loop and closed-loop systems allows us to predict the functioning of

these control systems in everyday use. Feedback and non-feedback circuits shall be discussed once

again after discussing certain tools that allow us to model them. One such tool is the Transfer

Transfer function

theory, the transfer function is the primary tool of operation, and is derived

The Transfer Function fully describes a control system. The Order, Type and Frequ

can all be taken from this specific function. Nyquist and Bode plots can be drawn

loop Transfer Function. These plots show the stability of the system when the loop is closed. Using

the denominator of the transfer function, called the characteristic equation the roots of the system

, by definition, is a mathematical representation, in terms of spatial or

temporal frequency, of the relation between the input and output of a time invariant lin

of the output of a system to the input of a system, in the Laplace domain

considering its initial conditions and equilibrium point to be zero. If we have an input function of

, we define the transfer function H(s) to be:

H(s) � ��

loop controllers: disturbance

reduced sensitivity to parameter

output feedback loop with various parts described

loop systems allows us to predict the functioning of

feedback circuits shall be discussed once

tool is the Transfer

theory, the transfer function is the primary tool of operation, and is derived

The Transfer Function fully describes a control system. The Order, Type and Frequency response

can all be taken from this specific function. Nyquist and Bode plots can be drawn from the open

the loop is closed. Using

equation the roots of the system

s a mathematical representation, in terms of spatial or

time invariant linear system.

the Laplace domain

have an input function of

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6.1.3 PID Controller

PID controllers are combinations of the proportional, derivative, and integral controllers:

Proportional controllers are simply gain values. These are essentially multiplicative coefficients,

usually denoted with a K.

Derivative controllers can be shown using Laplace calculus in simplified forms. In the Laplace

domain, we can show the derivative of a signal using the following notation: D(s) = L{ f’(t) } = sF(s)

The derivative controllers are implemented to account for future values, by taking the derivative,

and controlling based on where the signal is going to be in the future. Thus, even small amount of

high-frequency noise can cause very large derivatives, which appear like amplified noise.

To implement an Integral in a Laplace domain transfer function, we use the relation that Laplace

transform of a definite integral of f(t) = F(s)*1/s. Integral controllers of this type add up the area

under the curve for past time.

FIG 16. Schematic of PID controller

PID controllers are parallel combination of these 3 features. Because of this, PID controllers have

large amounts of flexibility. We will see below that there are definite limits on PID control.

The transfer function for a standard PID controller is an addition of the Proportional, the Integral,

and the Differential controller transfer functions (hence the name, PID). Also, we give each term a

gain constant, to control the weight that each factor has on the final output:

This flexibility is provided by the parallel nature of the PID combination. One or two of the

components may be removed to form an entirely new controller. These combinations are denoted

as PI and PD controllers. As we have seen already, i2n uses PI controllers in its AFM control system.

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7. SOLUTION METHODOLOGY

7.1 Modelling the Control Loops

As mentioned in the above section, MATLAB software created by Mathworks Inc. provides a tool

along with its software bundle called SIMULINK which allowed us to simulate the conditions of a

control loop by providing a platform to create equivalent block diagrams. Using SIMULINK models,

the FM-AFM control loop was reduced to control blocks representing various components of the

loop. As mentioned above, the frequency and gain control loops are treated independently. The

simplified versions of the two control loops can be seen from the block diagram below:

The transfer function of the PI controller is given by (Ps + I)/s, where P & I are the controller

coefficients. Constant ‘g’ denotes the gain; in the frequency control loop, the value of g is obtained

from the slope of the gain of the Bode plot. The constant ‘a’ denotes the time constant of the Lock-In

amplifier which is manually set on the SR844 RF Lock-In system. The Lock-In Amplifier acts as a 1st

order Low-pass filter, thus performing the function of the Voltage Controlled Oscillator. The output

from the PI controller is taken and subjected to the gain and low-pass filter, which is then used as

feedback to the PI controller.

7.2 Optimization Criteria

The Closed Loop transfer function of the system is obtained as ��

�� --{a}

The corresponding Open Loop transfer function is given by �� --{b}

Optimizing the control loop characteristics was performed via a 4-pronged analysis of the transfer

functions.

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1. First, the error function that goes in as input to the PI controller was kept to a minimum, as

per the requirements of the PI controller.

� � � −� �� + ��

� � � −� �� + �� + ��

When this is subjected to the condition that e<<0, we get optimization constraint #1 :

�� ≫ 0 --{#1}

2. The second condition to be checked is the stability of the system, as given by the Nyquist

Stability Criterion. The Nyquist plot that can be obtained for the standard Open Loop

Transfer {#b} shows that the plot originates at ω = 0 occurs on the real axis and ends at

origin without once circling (-1,0) point on the Polar plane. This is significant because of the

Nyquist stability criterion, which determines closed loop stability using open-loop

frequency response and open-loop polar plots.

The criterion states that:

For when the open-loop transfer function [G(s)H(s)] has no pole or zero on the jω

axis, if the open-loop transfer function has k poles in the right half-s plane and lim�→∞ "��#�� = constant, then for stability, the G(j ω)H(j ω) locus must encircle

the -1+j.0 point k times in the counterclockwise direction as ω varies from -∞ to +∞

Thus, since the open loop transfer function of our control loop satisfies this condition and

will be stable irrespective of the values of g,I and P.

3. The third condition to be checked is the Gain Margin and Phase Margin value.

We find from the Nyquist plot that the Gain Margin is ∞, but the equation for phase margin

throws up an interesting expression.

It is calculated by taking |"��#��| � 1, since we need to find the frequency values which

satisfy this condition. Those points lie on the radius = 1 circle surrounding (-1,0) point on

the polar plot. Solving we get those values of ω which I call &' as,

&'( + �)&')�1 − �)�)� − �)�)�) � 0

Thus, &') � *�� +�,��±.�/��+�� (�� )

By analyzing the Discriminant, we can say that out of ±, only the ‘+’ solution is acceptable,

because of positive squares underneath the root, and because of enforcing {#1}.

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Thus, the value of &' that we get here is to be used in the overall transfer function

0�� "��1 + "��#��

The Phase margin is obtained as P.M. = 180° + Phas0��e(. -----

{#2}

4. The final criterion is the condition of critical damping. This is performed by taking the value

of the denominator of the Closed Loop Transfer Function {a} = 0, since it ensures the case of

purely damped loop. Applying the said criterion, we get,

1�� + ��2) − 4�� 0

Thus, � � �� ) �)

----{#3}

Using these 4 criteria, we can arrive at a specific expression for &' , which is an equation in P and g,

given by

&') � �)�� + 1�2 5�� − 1� + .5�)�) − 6�� + 52 8 When this &' value is used in {#2}, we can obtain the Phase margin equation. A plot of this

equation for Phase Margin vs P for various values of g, is obtained on MATLAB and is attached

(below).

From this plot, we can obtain those values of g & P for which Phase Margin is less than 60°, which is

the acceptable range for optimum phase separation from (-1,0). Those values of P & g, and

consequently, I, are optimum for FM-AFM usage.

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8. CONCLUSION

The developed algorithm predicts to a satisfying degree the coefficients required to optimally tune

the PID controller. The algorithm has been translated to a code which is currently under testing to

reduce errors and include further parameters into the methodology.

8.1 Simulink plot

A projection of the plot created is provided here for reference:

From this plot, we obtain those values of g & P for which Phase Margin is less than 60° {that is, less

than 1.047), which is the acceptable range for optimum phase separation from (-1,0). Those values

of P & g, and consequently, I, are optimum for FM-AFM usage.

9. FUTURE OUTLOOK:

The future outlook of this approach to arriving at optimum PI coefficients is to include various

expressions for gain in the equation and to include system noise into consideration. The non-linear

effects of digital communication have not been considered and these will definitely play a role in

the functioning of the FM-AFM. The way to take this process forward is to automatise the obtaining

of the database of optimum P & I values for given a & g vales from the system at hand. This can be

pursued by creating a program in LABView (the software currently in use at i2n) using the above

algorithmic approach and testing the algorithm for consistency and accuracy.

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10. APPENDIX

10.1 Transfer Function – Impulse

For comparison, we will consider the time

relationship. In the time domain, we generally denote the input to a system as

of the system as y(t). The relationship between the input and

response, h(t).

As we know, Impulse Function δ(

that, for continuous-time input signal

linear mapping of the Laplace transform of the input,

transform of the output

A deeper understanding of the applications of Transfer Functions in control theory can be obtained

by taking the cases of the Impulse Response Func

From a Laplace transform table, we know that the Laplace transform of the impulse function,

is: L[δ(t)] = 1

So, when we plug this relationship into the input, output and transfer function, we get

In other words, the "impulse response" is the output of the system when we input an

function. From the Laplace Transform table, we can also see that the transform of the unit step

function, u(t) is given by: L[u(t)] =

Plugging that result into our relation for the transfer function gives us:

And we can see that the step response is simply the impulse response divided by

Impulse Response

For comparison, we will consider the time-domain equivalent to the above input/output

relationship. In the time domain, we generally denote the input to a system as x(t)

. The relationship between the input and the output is denoted

δ(t) is 0 at all values and undefined at t = 0. Simple calculus tells us

time input signal and output , the transfer function

of the Laplace transform of the input,

. That is,

Y(s) = H(s)X(s)


Impulse Response Function and Step Response Function

From a Laplace transform table, we know that the Laplace transform of the impulse function,


Y(s) = H(s)X(s)

Y(s) = (1) H(s)

H(s) = Y(s)

the "impulse response" is the output of the system when we input an

From the Laplace Transform table, we can also see that the transform of the unit step

)] = ��

Plugging that result into our relation for the transfer function gives us:

Y(s) = H(s)X(s)

Y(s) = ��H(s)

Y(s) = 9��

And we can see that the step response is simply the impulse response divided by s.

domain equivalent to the above input/output

x(t), and the output

the output is denoted as the impulse

is 0 at all values and undefined at t = 0. Simple calculus tells us

, the transfer function is the

, to the Laplace


Step Response Function:

From a Laplace transform table, we know that the Laplace transform of the impulse function, L(t)


the "impulse response" is the output of the system when we input an impulse

From the Laplace Transform table, we can also see that the transform of the unit step

.

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Mathematical modeling of input and output signals and transfer functions of the systems and

components of the control circuit required a simpler method of representation than conventional

linear equations. Thus the Block Diagram system was developed as a tool for easy representation of

classical control loops.

10.2 BODE PLOT, NYQUIST PLOT : MATLAB Simulink TOOL

A Bode Plot is a useful tool in control engineering that shows the gain and phase response of a given

LTI system for different frequencies. This Frequency Domain Analysis diagram provides us many

useful information regarding bandwidth of operation, number of poles and zeros etc

The frequency of bode plots are plotted against a logarithmic frequency axis. Every tick-mark on

the frequency axis represents a power of 10 times the previous value. Because each tick-mark is a

power of 10, they are referred to as a decade.

FIG. 17: An example of a Bode magnitude and phase plot set. The Magnitude plot is typically on the top,

and the Phase plot is typically on the bottom of the set.

The bode Magnitude plot measures the system Input/Output ratio in special units called decibels.

A Decibel is a ratio between two numbers on a logarithmic scale. To express a ratio between two

numbers (A and B) as a decibel we apply the following formula:

If we have a system transfer function T(s), we can separate it into a numerator polynomial N(s) and

a denominator polynomial D(s). We can write this as follows: T(s) = N(s)/D(s). To get the magnitude

gain plot, we must first transition the transfer function into the frequency response by using the

change of variables: s = jω.

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From here, we can say that our frequency response is a composite of two parts, a real part

imaginary part X: T(jω) = R(ω)+jX

The Bode magnitude and phase plots can be quickly and easily appro

straight lines. Once the straight-line graph is determined, the actual Bode plot

that follows the straight lines, and travels through the

Bode plots on MATLAB - This operation can be performed usi

MATLAB also offers a number of tools for examining the frequency response characteristics

system, both using Bode plots, and using Nyquist charts. To construct a Bode plot

function, we use the following command:

[mag, phase, omega] = bode(NUM, DEN, omega);

Or:

[mag, phase, omega] = bode(A, B, C, D, u, omega);

Where "omega" is the frequency vector where the magnitude and phase response points are

analyzed. If we want to convert the magnitude data into

conversion: magdb = 20 * log10(mag);

When talking about Bode plots in decibels, it makes the most sense (and is the most

occurrence) to also use a logarithmic frequency scale. To create such a logarithmic

omega, we use the logspace command, as such:

Nyquist Plot

A Nyquist plot is another parametric plot of a frequency response

theory. The most common use of Nyquist plots is for assessing the sta

with feedback.

From here, we can say that our frequency response is a composite of two parts, a real part

jX(ω)

The Bode magnitude and phase plots can be quickly and easily approximated by using a

line graph is determined, the actual Bode plot is a smooth curve

that follows the straight lines, and travels through the breakpoints.

This operation can be performed using this MATLAB command

MATLAB also offers a number of tools for examining the frequency response characteristics

system, both using Bode plots, and using Nyquist charts. To construct a Bode plot

command:

[mag, phase, omega] = bode(NUM, DEN, omega);

[mag, phase, omega] = bode(A, B, C, D, u, omega);


analyzed. If we want to convert the magnitude data into decibels, we can use the following

magdb = 20 * log10(mag);

When talking about Bode plots in decibels, it makes the most sense (and is the most

occurrence) to also use a logarithmic frequency scale. To create such a logarithmic

command, as such: omega = logspace(a, b, n);

parametric plot of a frequency response, of significant use in control

The most common use of Nyquist plots is for assessing the stability of a system

FIG17. Sample Nyquist plot

From here, we can say that our frequency response is a composite of two parts, a real part R and an

ximated by using a series of

is a smooth curve

is MATLAB command: bode

MATLAB also offers a number of tools for examining the frequency response characteristics of a

system, both using Bode plots, and using Nyquist charts. To construct a Bode plot from a transfer


decibels, we can use the following

When talking about Bode plots in decibels, it makes the most sense (and is the most common

occurrence) to also use a logarithmic frequency scale. To create such a logarithmic sequence in

, of significant use in control

bility of a system

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In Cartesian coordinates, the real part of the transfer function is plotted on the X axis. The

imaginary part is plotted on the Y axis. The frequency is swept as a parameter, resulting in a plot

per frequency.

Thus, this careful inspection of the Nyquist plot reveals a surprising relationship to the Bode plots

of the system. If we use the Bode phase plot as the angle ϴ, and the Bode magnitude plot as the

distance r, then it becomes apparent that the Nyquist plot of a system is simply the polar

representation of the Bode plots. To obtain the Nyquist plot from the Bode plots, we take the phase

angle and the magnitude value at each frequency ω. We convert the magnitude value from decibels

back into gain ratios. Then, we plot the ordered pairs (r, ϴ) on a polar graph.

The major application of Nyquist plots : the assessment of the stability of a closed-loop negative

feedback system is done by applying Nyquist stability criterion to the Nyquist plot of the open-loop

system. This method is easily applicable even for systems with delays and other non-rational

transfer functions, which may appear difficult to analyze by means of other methods. Stability is

determined by looking at the number of encirclements of the point at (-1,0). Range of gains over

which the system will be stable can be determined by looking at crossing of the real axis.

When drawn by hand, a cartoon version of the Nyquist plot is sometimes used, which shows the

shape of the curve, but where coordinates are distorted to show more detail in regions of interest.

Nyquist plots on MATLAB - This operation can be performed using the MATLAB command nyquist

In addition to the bode plots, we can create nyquist charts by using the nyquist command. The

nyquist command operates in a similar manner to the bode command that we have used so far):

[real, imag, omega] = nyquist(NUM, DEN, omega);

Or:

[real, imag, omega] = nyquist(A, B, C, D, u, omega);

MATLAB TOOL: SIMULINK

Simulink is a data flow graphical programming language tool for modeling, simulating and

analyzing multi-domain dynamic systems. Its primary interface is a graphical block diagramming

tool and a customizable set of block libraries. It offers tight integration with the rest of

the MATLAB environment and can either drive MATLAB or be scripted from it.

Learning this graphical interface language was the final crucial step before embarking upon the

solution to our problem at hand. Simulink allowed one to model the PI controller components and

easily generate Bode, Nyquist and Impulse response simulation plots for quick access.

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11. REFERENCES

1. Katsuhiko Ogata, Modern Control Engineering, Edition 5, 2011

2. Zhang, Miao, Zheng, Dong, Xi, Feedback Control Implementation for AFM Contact-Mode

Scanner, 2008, 3rd IEEE Int. Conf. on Nano/Micro Engineered and Molecular Systems, IEEE

3. Abramovitch, Andersson, Pao, Schitter, A Tutorial on the Mechanisms, Dynamics, and Control

of Atomic Force Microscopes, 2007, American Control Conference Journal, IEEE

4. Freidt and Carry, Introduction to the quartz tuning fork, 2007, American Journal of Physics,

AAPT

5. Kilpatrick, Gannepalli, Cleveland, Jarvis, Frequency modulation atomic force microscopy in

ambient environments utilizing robust feedback tuning, 2009, RSI, Am. Inst. Of Physics

6. Hrouzek, Feedback Control in an Atomic Force Microscope Used as a Nano-Manipulator,

2005, Acta Polytechnica

7. Bueno, Balthazar, Piquiera, Phase-Locked Loop Application to Frequency Modulation - Atomic

Force Microscope, 2010, Bra. Conf. on Dynamics, Control and Applications

8. Edwards, Taylor, Duncan, Fast, high-resolution atomic force microscopy using a quartz tuning

fork as actuator and sensor, 1997, OJPS, American Institute Of Physics

9. Singh, Kumar, Electronics and Software for Scanning Tunneling Microscope, Inter-

Universoty Accelerator Centre

10. Control Systems, Wikibooks Creative Commons Attribution-ShareAlike 3.0 Unported license

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