Abstract

The main goal of this research project is to enhance decision making of a controller using fuzzy

logic to adapt to various dynamic changes in an environment. More specifically, it will allow the

controller system to emulate human reasoning. This topic is related to the NAE Grand Challenge of

Reverse Engineering the Brain. To accomplish this goal, an inverted pendulum model will be

developed as a benchmark control platform. The model will be used to test and compare a Fuzzy

Logic based PID controller with a conventional PID controller. The Fuzzy PID controller for the

pendulum model will then be adapted to stabilize a multi-rotor aircraft. The particular advantage of

using a Fuzzy system is that it streamlines the design process by simplifying if-then rules for action-

reaction scenarios to a more general field. This would allow for a high level of controllability with

minimal effort on the part of the human pilot, due to the inherit autonomy gained through enhanced

performance in the dynamic environmental changes.

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Table of Contents

Abstract…………………………………………………………………………………………………..2

1. Introduction…………………………………………………………………………………………...4

1.1 Introduction to Multi-Rotor Control………………………………………………………..4

1.2 Introduction to Fuzzy Logic………………………………………………………………...4

1.3 Objective Goals……………………………………………………………………………..

62. Inverted Pendulum Problem…………………………………………………………………………..

62.1 Inverted Pendulum Model (IPM) Development…………………………………………….

62.2 Creating a PID Controller…………………………………………………………………...

72.3 Creating a Fuzzy PID Controller……………………………………………………………

9 2.3.1 Pendulum Angle Fuzzy PID Inference System………………………………………

11 2.3.2 Cart Position Fuzzy PID Inference System………………………………………….

132.4 Performance Envelope Testing Constraints……………………………………………….

142.5 Dual PID Performance Envelope…………………………………………………………..

152.6 Dual Fuzzy PID Performance Envelope…………………………………………………...

172.7 PID vs. Fuzzy PID ………………………………………………………………………….

193. Dynamic Quad-Rotor Model…………………………………………………………………………

203.1 Dynamic Quad-Rotor Model Overview……………………………………………………

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3.2 Dual Fuzzy PID Structure…………………………………………………………………..

213.3 Single Fuzzy PID Structure…………………………………………………………………

233.4 PID vs. Single Fuzzy PID Step Response………………………………………………….

263.5 PID vs. Single Fuzzy PID Flight Data……………………………………………………..

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4. Discussion…………………………………………………………………………………………….

295. Conclusions…………………………………………………………………………………………...

326. Acknowledgements…………………………………………………………………………………...

337. References…………………………………………………………………………………………….

33 8. Nomenclature…………………………………………………………………………………………

349. Appendicies...........................................................................................................................................

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1. Introduction

1.1 Introduction to Multi-Rotor Control

The multi-rotor copter is a versatile flight platform growing ever more popular in many fields due

to its low cost and utility. However, the multi-rotor aircraft is inherently unstable and extremely

hard to control without an onboard stabilization process. Using a previously developed quad-

rotor simulation, a Fuzzy PID controller was designed and tested. The flight mechanics of a

quad-rotor are straightforward and relate directly to the principle rotational axes of pitch, role,

and yaw of which can be considered independently. This will allow for a simple yet robust

controller that can be easily adapted to more complex platforms, such as a hex-copter or octo-

copter.

1.2 Introduction to Fuzzy Logic

Conventional computer based intelligence systems typically entail very specific boolean logic

principles in order to intelligently control a system. This boolean logic operates under the

principle of highs or lows, through logic gates to create a reaction, thus they contain rigid if-then

rule bases. Fuzzy logic is a different take on this type of control or intelligence system, which is

based on a more human description of the world. Fuzzy logic uses if-then statements that use

what are known as linguistic variables to describe system states. (Mendel 1995) These linguistic

variables a are human descriptors of a situation such as, the room is very hot, or somewhat cold,

versus the room is 75 or 10 degrees. This interpretation allows for a more generalized

development of logical systems, utilizing a ramped mapping of inputs to outputs according to

different functions, weights, and interpolation. The entire process of mapping the inputs to

outputs occurs in what is known as a Fuzzy Inference System (FIS), which works by utilizing

four basic components: fuzzification, the fuzzy rule base, fuzzy inference, and defuzzification, as

depicted below. (Bih 2006)

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Figure 1. Fuzzy Process Diagram.

i. Fuzzification

This is the process of collecting crisp input data and evaluating it in relation to

membership functions, where the amount of truth (or degree of membership) to each

membership is analyzed (i.e. 1.77m might be 75% tall, and 25% not tall) – this is known

as a fuzzy set. (Bih 2006)

ii. Fuzzy Rule Base

This process takes the newly developed fuzzy set and analyzes it across if-then rules

where logical operators (AND, OR, NOT, etc.) can be used. All rules that have any

membership (regardless of amount) that is greater than zero will be evaluated. (Bih 2006)

iii. Fuzzy Inference

Once the rules are evaluated, the resulting output is then mapped to the appropriate

defuzzification membership functions, where it will be converted into crisp data (i.e. if

temperature is hot, then cooling is activated). (Bih 2006)

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iv. Defuzzification

Defuzzification takes the fuzzy output data from the inference portion, and weights the

defuzzification membership functions according to the initial degree of membership for

that type of response. There, the output can be chosen as a crisp value by evaluating the

new weighted data in a variety of ways. Typically the centroid of the resulting, weighted

defuzzification membership functions is used. (Bih 2006)

2.

1.3 Objective Goals

In order to accomplish the autonomous flight and stabilization of a multi-rotor aircraft, the

following goals have been laid out:

1. Perform a Literature review. (PID/ Fuzzy logic control systems, MATLAB Fuzzy Logic

Toolbox)

2. Model an inverted pendulum.

3. Obtain or Create a PID controller to be used for the benchmark pendulum model.

4. Create a Fuzzy PID controller for the pendulum model.

5. Use the Fuzzy PID pendulum controller to develop a Fuzzy PID controller for a multi-

rotor aircraft.

6. Learn how to effectively compile a technical paper, poster and presentation.

7. Develop team collaboration.

8. Learn the basics behind technical research.

2. Inverted Pendulum Problem

2.1 Inverted Pendulum Model (IPM) Development

The first step in modeling most dynamic systems is to begin with the equations of motion that

govern their behavior. From these equations, a model is developed using the Simulink toolbox in

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MATLAB. Equations (1) and (2) below describe the nonlinear system for an IPM with θ being

the pendulum angle and x being the horizontal displacement of the cart. The equations describe a

single input multiple outputs (SIMO) system that is normally unstable about a reference angle, θ

= π radians counterclockwise from the negative y-axis. A force input will create an angle

movement and a cart displacement as the reaction outputs.

Figure 2. Inverted Pendulum Diagram

( I+ml2 ) θ+mglsin (θ )=−ml xcos (θ) (1)

(M +m ) x+b x+ml θ cos (θ )−ml θ2sin (θ )=F (2)

2.2 Creating a PID Controller

A PID controller works by evaluating the error present in the feedback from a system. More

specifically, the controller evaluates three specific error terms; proportional, integral, and

derivative. The proportional term is the current system error, the integral is the summation of the

errors up to that point, and the derivative term refers to the rate of change of error. These three

terms are then weighted based on the system constraints in order to provide the optimal control

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input. (Araki 2006) Multiplying each of the three terms by a gain results in the desired weights.

This type of controller can be powerful in providing accurate and quick action responses when

well-tuned. For this reason, the PID controller is the accepted convention for many feedback

control systems and is the chosen benchmark for comparison with the Fuzzy PID controller.

The two primary structures for a PID controller are the parallel PID and the ideal PID. The

parallel PID configuration splits the error signal into its proportional, integral, and derivative

terms then simultaneously applies the gains to the three terms before summing them into a single

output signal. On the other hand, the ideal PID applies the integral and derivative gains and sums

them with the original error term. Once these three terms are summed together, the entire signal

is multiplied by a proportional gain before feeding back into the system.

The graph in Figure 3 demonstrates the system’s response to a single step input using a single

custom built ideal PID controller. The controller is accounting for the error in θ and tracking the

cart velocity to match the needed force magnitude. However, the cart never returns to the

reference position once the system reaches a steady state. Two reasons for this type of response

are, the PID controller is only correcting the error between θ and its reference, and there is no

consideration taken for the cart’s reference position. The cart will simply move to any position

that makes θ steady. The addition of a second PID controller, that takes into account the error

between current position and cart origin, remedied the condition.

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Figure 3. Single PID Impulse Response

The introduction of a cart position PID controller in series after the pendulum PID controller

improved the cart position response without significantly diminishing θ response. While this

setup was a tremendous improvement over the single PID, it did not solve the problem of the cart

settling position. Analysis of signal data throughout the dual PID system revealed that once θ

settled to steady state the signal entering the cart position PID diminished to zero. With a near

zero error signal entering the position PID, the controller could not output an error correction

because there was no signal to amplify. It was being tricked into thinking the position was steady

because the error input was close to zero. By relocating the position PID before the angle PID, the

system would work to keep the pendulum stable while returning the cart to the reference position.

This setup did not allow for the signal to diminish until both the cart and angle were stable. This

became the final configuration for the PID control of the inverted pendulum model.

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Figure 4. Dual PID Impulse Response

2.3 Creating a Fuzzy PID Controller

Upon completing the PID controller, a Fuzzy Inference System (FIS) was developed to allow for

the dynamic modification of the gains in a PID controller. The conventional PID is a set in stone

method that is limited to constant gains that may or may not be ideal for each system condition.

Dynamically modifying the PID gains allows for a greater performance envelope and additional

resilience to noise and/or small system modifications. To create a Fuzzy PID controller, the

generalized effects of each gain was researched, and thus fuzzy logic rules were developed to

tailor these gains to their best value for a particular error range (or rate of change of error range).

The table used for the generalized fuzzy rule base is listed below:

Table 1. Effects of the Independent Tuning of Each Gain Coefficient. (Ang,Chong,&Li,2005)

Closed Loop Response Rise Time Overshoot Settling

TimeSteady-State

Error Stability

Increasing P Decrease Increase Small Increase Decrease Degrade

Increasing I Small Decrease Increase Increase Large

Decrease Degrade

Increasing D Small Decrease Decrease Decrease Minor

Change Improve

Implementing fuzzy logic into the IPM controller system required two separate FISs. Each FIS

was developed with different rule bases and membership functions that represented the

appropriate reaction needed for the cart position and the pendulum angle. The goal was to allow

for the necessary generality in the systems to modify gains (Proportional, Integral, and

Derivative) based on current system conditions. This was done by taking the values of the tuned

PID controller and allowing them to vary by ±10% according to the fuzzy rule base. The fuzzy

rule base was optimized to obtain the best result given the current system state based on gain

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reactions in Table 1. Once the fuzzy rule base was completed the FIS was placed into the ideal

PID controller prior to the gains. The architecture of the Fuzzy PID controller can be seen below

in Figure 5.

Figure 5. Fuzzy PID (Ideal) Controller Architecture

2.3.1 Pendulum Angle Fuzzy PID Inference System

The pendulum angle (θ) was the sole input for this particular FIS, and allowed for a

total range of input (universe of discourse) of -1.57 to 1.57 radians. This range was

split between the five possible membership functions seen in Figure 6 which included

overlap. The overlap between the memberships allowed a maximum of two function

weights to be applied at any given point based on which rules were firing at the time

of the calculation.

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Figure 6. Pendulum Angle Input Fuzzy Logic Membership Function

Upon evaluation of the signal input through the FIS, the deffuzzification process

entailed three output fuzzy sets to be converted to crisp values (the gains). Figure 7

depicts the deffuzzification membership functions used in this process.

Figure 7. Pendulum Angle Output Proportional, Integral and Derivative Gain Membership Functions

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The actual mapping of the input and output of any FIS is based on the fuzzy rule base.

Large numbers of trials were run in conjunction with the general trends and rules from

Table 1 to yield the final rule base. These results are shown in Tables 2 & 3. It should

also be noted that when creating the rule base in the inference process, all rules carried

the exact same weight.

Table 2. Pendulum Angle Fuzzy Rule Base

Pendulum Angle Fuzzy Rule Base

Pendulum Angle DeviationLarge

PositiveSmall

Positive Stable Small Negative

Large Negative

Proportional Gain Large Moderate Moderate Moderate Large

Integral Gain Small Small Moderate Small Small

Derivative Gain Small Small Moderate Small Small

Table 3. Cart Position Fuzzy Rule Base

Cart Position Fuzzy Rule Base

Cart Position DeviationSmall Positive Stable Small Negative

Proportional Gain Moderate Moderate Moderate

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Integral Gain Small Large Small

Derivative Gain Small Large Small

2.3.2 Cart Position Fuzzy PID Inference System

Similar to the Fuzzy PID controller used for θ, the cart position FIS was configured in

the exact same manner; taking a single input for cart position x, and setting the

proportional, integral, and derivative gains according to a fuzzy rule base. The cart

position input membership functions that were used can be seen below in Figure 8.

Figure 8. Cart Position Input Fuzzy Logic Membership Function

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Figure 9. Cart Position Output Proportional, Integral and Derivative Gain Membership Functions

2.4 Performance Envelope Testing Constraints

The performance envelope tests contained very specific constraints for which a MATLAB script

was written to analyze the results of each simulation run. The simulations were set to run for a

maximum of 50 seconds. A “stabilization” was defined as the time it takes the system to get

within the determined “bounds” and stay within those “bounds” for the remainder of the

simulation such that the time staying within the “bounds” was at least 20% of the simulation time.

The “bounds” for the test were a half degree deviation and a half meter deviation (from the

centered values). Additionally, the controller systems used were bounded for a maximum output

correction to the IPM of ±5 units. It should be noted that, the constraints place on the position

deviation were deliberately enlarged to allow for an emphasis on the pendulum angle

stabilization.

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2.5 Dual PID Performance Envelope

Once completed, extensive testing was conducted on the performance of the PID controller with

the pendulum model. To form a performance envelope, it was decided to vary the initial

conditions for both pendulum angle (θ) and cart position. Upon changing the initial conditions,

the pendulum/cart system was evaluated for its performance in terms of settling time and whether

or not the system settled within a desired threshold. To extensively test small increments to form

a high resolution image of the envelope, a script was created to automatically run 50 data trials

varying the initial position from -10 meters to 10 meters and 50 trials varying the pendulum’s

initial angle deviation from –pi/2 radians to pi/2 radians, yielding a total of 2,500 trials. The

results of the performance envelopes from this script are displayed below as a series of color

maps denoting the actual stabilization time of each degree of freedom within the system.

Figure 10. PID Performance Envelope (No Noise)

After the initial envelope test of the PID control (Figure 10), an additional test was conducted to

simulate harsh sensor noise in a real world system (Figure 11). The chosen value for the sensor

noise was a ±10% deviation from the input signal for both the position and the angle

measurements coming out of the pendulum model itself and going into the respective PID

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controllers. This is a drastic case of noise, but it describes how the system behaves (and breaks

down) given large amounts of noise.

Figure 11. PID Performance Envelope (Sensor Noise)

Finally, another test was conducted with no sensor noise but instead included an eternal force

noise (Figure 12). That is, the system simulated a ±10% deviation from the output of the PID

controller and then fed this noisy data into the force input of the system. These deviations were

held constant for two seconds, giving the behavior of external force inputs, such as wind.

Figure 12. PID Performance Envelope (Force Noise)

2.6 Dual Fuzzy PID Performance Envelope

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Similar to the PID controller, a comprehensive test was also conducted on the Fuzzy PID

controller varying the same initial conditions of the angle from –pi/2 radians to pi/2 radians and

the position from -10 meters to 10 meters, all in increments of 50 trials, yielding the 2,500 tests.

The resulting performance envelope is shown below.

Figure 13. Fuzzy PID Performance Envelope (No Noise)

The Fuzzy PID controller was also subjected to the ±10% deviation of the input signal to simulate

harsh sensor noise in the system from the cart position and pendulum angle. It should be noted

that these random signals were actually pseudo-random and used the same seed and generator as

in the PID test case, and thus resulted in the same noisy data changes. The results are depicted

below in Figure 14.

Figure 14. Fuzzy PID Performance Envelope (Sensor Noise)

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Lastly, the Fuzzy PID was also subjected to a ±10% force noise input between the Fuzzy PID and

the actual pendulum model itself to simulate external force influences. This system also utilized

the same seed and pseudo-random number generation methods as in the PID version of the test,

causing the deviation signal to be multiplied by the actual PID output to be the same. The

resulting contour plots can be seen below in Figure 15.

Figure 15. Fuzzy PID Performance Envelope (Force Noise)

2.7 PID vs Fuzzy PID

With the above performance envelopes established, data was collected to observe the overall

difference in their performance from a numerical standpoint. Below is a table that depicts the

average stabilization times from each of the envelope tests. In the far right column, the difference

between each noisy case and the no noise case was taken. They were then averaged to obtain the

resulting average change in stabilization time from the no noise test. The final row in the table

depicts the percent change when going from the PID to the Fuzzy PID in average stabilization

time. It should also be noted that even though some responses from the envelope tests are quite

long, all situations where the system “stabilized” within the established constraints were kept.

Table 4: Settling Time Comparison

Settling Time

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No Noise Sensor Noise Force NoiseAverage change in settling time

with Noise

PID 2.91 s 8.07 s 3.08 s 2.66 s

Fuzzy PID 4.79 s 4.82 s 4.81 s 0.03 s

Fuzzy PID compared to PID +64% -40% +56%

In addition to evaluating stabilization times, the actual sizes of the performance envelopes were

also compared in the following table. The value used to represent the size of an envelope was a

summation of the number of trials where the system stabilized within the established constraints

out of the 2,500 possible trials. The final row then depicts the percent change in the data point

representative area when going from the PID system to the Fuzzy PID system.

Table 5: Performance Envelope Comparison

Performance Envelope

No Noise Sensor Noise Force Noise

PID 204 pts 172 pts 232 pts

Fuzzy PID 312 pts 274 pts 334 pts

Fuzzy PID compared to

PID+53% +59% +44%

3. Dynamic Quad-Rotor Model (QRM)

3.1 Dynamic Quad-Rotor Model Overview

The Quad-Rotor Model (QRM) used in this project was a result of previous REU research teams

who developed and validated it using flight data recording, Cipher, and Conduit. This model

utilizes a dual PID control system similar to the inverted pendulum model. However, unlike the

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IPM, the Quad-Rotor Model (QRM) responds to the error in the angular velocity (P) and the

angular displacement or attitude (φ). Initial ventures into implementing a Fuzzy PID into the

QRM consisted of simply using the pendulum anglefuzzy PID to control φ for the QRM. The

resulting response was less than desirable. Further inspection of the original PID controllers

implemented in the QRM revealed a difference in architecture between the two controllers. The

IPM controller is an ideal PID, while the original QRM utilizes a parallel PID. The difference in

these two structures is the magnitude of influence that the proportional gain has on the response

signal. In the ideal PID, the integral and derivative signals are processed and summed prior to the

proportional gain. This provides the proportional gain an overwhelming influence over the

control signal. In the parallel PID controller, the signals are all processed and summed in parallel

resulting in smaller influence of the proportional gain. The FIS for the pendulum angle controller

provided a response that bore a slight resemblance of the desired signal pattern once restructured

into a parallel architecture.

3.2 Dual Fuzzy PID Structure

As a starting point for design of a new FIS, the input membership functions were built to increase

precision of the response as φ of the system approaches the region deemed as stable, while

providing broad ranges around the edges of the input range. It can be shown below in Figure 16

that the input membership function for φ, has a specific set of ranges from large, to small, to

stable. This allows for a smaller increment of change in response creating a smoother reaction as

the φ of the model approaches a stable state. On the other hand, the controller accounting for the

error in P has a very specific response around its stable region with large broad regions outside of

stable. These broad ranges around the edge of the input allow prolonged steady responses at

extreme φ to return the system to a stable state.

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Figure 16. Input Membership Functions for the Dual Fuzzy PID in QRM

The output range of the membership functions were initially centered on the tuned PID gain

values for both the P and 𝛗 control. The membership functions of each output variable were

designed to match the input membership functions with five functions for the 𝛗 controller and

three sets for the P controller. The rule base required for this system was found to be very

cumbersome to build and not overly effective at controlling the model. This FIS was quickly

replaced with one of a far less complex nature. The resulting output membership functions are

relatively simplistic involving only three fuzzy sets for each variable for the 𝛗 and P controllers

Figure 17 and 18. It was found that by maintaining simple output membership functions, a

simplistic rule base could be used without degradation in response.

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Figure 17. Output Membership Functions for the Attitude (φ) Controller in QRM

Figure 18. Output Membership Functions for Angular Velocity (P) Controller in QRM

3.3 Single Fuzzy PID Structure

The second Fuzzy PID structure implemented into the QRM was simplified into a single FIS

which was a multiple input multiple output (MIMO) system. This was employed to cut out the

calculations that go along with having two different fuzzy controllers. Additionally,

consolidating both inputs into a single FIS allows for better system awareness because the rules

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can be tailored to react to both the attitude and the angular velocity at the same time. Figure 19

depicts the input membership functions used for this FIS.

Figure 19. Input Membership Functions for the Single Fuzzy PID

The FIS structure that was created also added a 4 th output to the system which determined if there

was a need for extra proportional gain for quick responses. The FIS took in information about φ

and the magnitude of P. Say for example φ is negative and P is making the angle more negative,

the proportional gain will be high and the 4th output will scale up the φ error to create a stronger

response. Shown below in Figure 20 is the output membership function in which this FIS used,

including the 4th output mentioned.

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Figure 20 Output Membership Functions for the Single Fuzzy PID Controller

This setup is incredibly effective for rapid responses to larger error signals and gives the

controller more adaptability to any situation that falls in between extreme and stable conditions.

Figure 21 shows the diagram for the entire controller.

Figure 21. Single Fuzzy PID Controller Diagram

Since this new controller has such a different architecture to that of the previous Fuzzy PID

outlined in this document, the fuzzy rule base had to be adapted to take full advantage of both

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inputs as well as the additional output in order to maximize the performance. Thus the following

rule base was created and tested for a good response which will be outlined in the following

section.

Table 6. Single Fuzzy PID Rule Base (MIMO)

Single Fuzzy PID Rule BaseInput Output

Angle Angular Velocity P I D Angular P

- Negative Large Off Small Moderate- Stable Moderate Off Large Moderate- Positive Large Off Small Moderate

Negative - - On - ModeratePositive - - On - ModerateNegative Negative - - - HighNegative Positive - - - LowPositive Positive - - - HighPositive Negative - - - LowStable - - Off - Low

3.4 PID vs. Single Fuzzy PID Step Response

Step response data is the most common way to test controllers and their respective response

parameters. Parameters such as rise time, settling time, overshoot, peak and steady state final

value are important system characteristics that determine the effectiveness of the controller. In

order to confirm the performance of the QRM Single Fuzzy PID vs. the PID, it had to be run

through an array of response comparisons. Originally the model was being run strictly with

recorded flight data, which represented the percent stick deviation input by the pilot (this data

will be analyzed in section 3.5). In order to first quantify the performance difference of the

Single Fuzzy PID vs. the PID before worrying about the QRM response, the step data was

analyzed. Figure 22 and Figure 23 show that the Single Fuzzy PID has the ability to diminish

oscillations and produce smoother response curves compared to the PID for a simulated 20% user

step input. The settling time of the Single Fuzzy φ response was slightly reduced from 8.13

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seconds to 8.04 seconds while oscillations were damped around steady state. A very similar

reaction was also seen in P (Figure 23). As seen in Table 7, the rise time was tightened up by

0.08% compared to the PID which is a marginal amount, but still an improvement. The

overshoot and peak values were also increased which is indicative of a more responsive

controller. The last parameter to observe is the steady state final value. The fuzzy controller’s

final attitude value was 8.38% larger than the PID which is expected around the stable range

since the rule base optimizes the gain behaviors in this region.

The results seen below using the Single Fuzzy PID setup are very promising for its real world

implementation. The controller has an overall smoother response curve to the 20% step input

while still achieving similar if not better settling times which was the ultimate goal of the Fuzzy

PID setup. These key characteristics are all products of the active gain updating feature included

in the FIS.

Figure 22. Attitude Response to 20% User Inputted Step Function

(left: Dual PID Right: Single Fuzzy PID)

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Figure 23. Angular Velocity Response to 20% User Inputted Step Function

(left: Dual PID Right: Single Fuzzy PID)

Table 7. Step Response Comparison Data for the Attitude Step Response

Quad Model Roll (Phi) Step Response ComparisonRise Time Settling Time Settling Min Overshoot Peak Peak Time

Dual PID 0.084286 s 8.130 s 6.570 deg 170.469% 18.898 deg 1.440 sSingle Fuzzy PID 0.084216 s 8.036 s 7.121 deg 184.355% 20.248 deg 1.480 s

Change from PID to Fuzzy PID -0.08% -1.15% 8.38% 8.15% 7.14% 2.78%

3.5 PID vs. Single Fuzzy PID Flight Data

The Single Fuzzy PID proved itself when subjected to the common step response test, but how

does it perform when subjected to real time flight data? This is the real test that will prove or

disprove the smooth response claims made in previous sections. The data used was a 10 second

excerpt that was provided by past REU participants during the development the QRM simulation.

The section of data was validated in both the frequency and time domain to ensure that it was

“good” data. It consisted of three different user inputted peaks that represent a roll right, left, and

right again. These peaks range from 60% to -60% stick deviations. Figure 24 shows the

comparison of the Single Fuzzy PID vs. the Dual PID controller’s φ response to the flight data.

Three magnified pictures provide a better visual of the differences between the Fuzzy PID and

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PID responses. The trends are very consistent with the results from the step response in section

3.4. Parameters to note are, decreased settling time, fewer oscillations, and quicker overall

response time. These results confirm that the Single Fuzzy PID is able to handle noise anomalies

as well as large user inputs in a manner that would be suitable for implementation on an actual

quad-rotor system.

Figure 24. Step Response Comparison Data for the Attitude Step Response

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4. Discussion

In order to evaluate the impact of the addition of a Fuzzy Inference System to a PID controller,

both models outlined in this document must be looked at. First, the results and findings of the

Dual Fuzzy PID vs. PID implemented into the pendulum model will be discussed. This data will

then be compared to the Single Fuzzy PID vs. PID in the Quad model. This should provide a

more complete understanding of the capabilities of the Fuzzy PID in comparison to the PID.

When considering the results of the IPM envelope tests conducted on the PID and the Fuzzy PID,

a few different aspects must be observed. One prominent aspect in a control system is the settling

time in any given situation. In the previously listed performance envelopes, it can be observed

that in what would be considered ideal conditions, the PID proved to have a faster overall settling

time than the Fuzzy PID. This held true for force noise testing as well, however, didn’t hold up

when subject to sensor noise; the Fuzzy PID had a substantially shorter average settling time than

the PID in this test. The other facet of these tests though, is the actual range that the averages

span over. For instance, the PID lacked the overall consistency in settling time when subjected to

the different external factors, such as the sensor noise and the force inputs, resulting in the 2.66

second average settling time change across the tests, whereas the Fuzzy PID was extremely

consistent across all the tests, having only a 0.03 second average settling time change. Another

key aspect to consider is the actual size of the performance envelopes, which were conducted on

the basis of the number of counted trials that were considered settled. In the envelope tests, the

Fuzzy PID yielded a consistently larger (at least 44%) performance envelope over the PID. This

is a direct result of the Fuzzy PID controller’s ability to actively change the Proportional, Integral,

and Derivative gains depending on the current state of the system. In these tests, it is interesting

to also note that the Fuzzy PID was varying these gains by up to 10% from that of the original

tuned PID, yet it yielded an average of a 52% larger performance envelope from the three

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envelope tests conducted. Together, these results suggest that while the Fuzzy PID can have a

slower overall response in ideal conditions, it’s much more resilient and consistent in different

circumstances while allowing for a larger performance envelope. This would be expected, as the

tuned PID is optimized for a small solution space for ideal conditions while the Fuzzy PID strives

on small amounts of generalization around this solution space.

After obtaining good results with the Inverted Pendulum Model (IPM) tests, similar ideologies

were used in the development of the QRM Fuzzy PID controller. However, as noted in section 3

of this document, the finalized Fuzzy PID for comparison with the original PID consisted of a

unique structure containing a single Fuzzy Inference System (FIS) and an additional proportional

gain, differing drastically in overall architecture (see Figure 21). The results show that this new

Single Fuzzy PID controller (in a step response) was indeed able to settle 1.15% quicker, with

less overall oscillations than that of the tuned Dual PID system. This is again highly attributed to

ability of the FIS to adapt the gains with regards to the current system state. However, why was

the settling time quicker than the Dual PID, which is different than that of the IPM version? This

is likely attributed to the increase in contextual awareness that this new FIS has, as it takes in both

the roll attitude (𝛗) and roll rate (P). This is also why it can be shown that the Single Fuzzy PID

actually has a 0.08% faster rise time, and that the overall peak of the response is smoother, as

indicated by the 2.78% longer peak time. Lastly, it must be noted that the Single Fuzzy PID was

able to achieve an 8.38% higher settling minimum value, indicating a larger range of operating

motion when placed in an actual Quad-Rotor. The most significant fall back of the Single Fuzzy

PID lies in the overall increased overshoot (8.15% more frequent/present). These results also

carry over into the QRM with flight data, as shown in Figure 24, where the results are actually a

bit more noticeable than in the step responses. This behavior is highly consistent with that of the

IPM results, which concluded that a PID breaks down in performance once heavy, high frequency

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noise is introduced, whereas the Fuzzy PID was extremely resilient to this type of situation. This

is why the QRM with real flight data reflects the exaggerated difference in performance. Thus if

this Single Fuzzy PID were to be placed on an actual Quad-Rotor, the performance difference

between it and the PID would be expected to increase, with the Single Fuzzy PID staying

consistent, and the PID degrading. This will likely result in a much more stable aircraft in harsher

conditions, and an overall more consistent performance. It is also speculated that the Single

Fuzzy PID may be able to handle small system changes, such as in an increase in mass, due to the

active gain updating, however, this is not yet confirmed.

With all the aforementioned results in mind, a series of follow-up tests would be highly beneficial

in verifying/understanding the full extent of the Fuzzy PID’s capabilities and performance. The

first test would be in modifying the capabilities of the system to be more or less rigorous such as

more powerful motors, efficient propellers or quicker controller processing. These various

attributes would be able to verify a consistent performance of similar system types, but with

different mechanics or capabilities. The next test would be to modify the system’s state, such as

the mass or the moments of inertia, in order to investigate whether or not the Fuzzy PID is able to

handle these changes better than the PID could handle without modification. If this were the

case, the Fuzzy PID has an extremely bright future in an adaptive/autonomous system, which is

the end-goal of such a controller. This is also a speculated result that would be expected to be

observed, but again, would need verification.

5. Conclusions

When it comes to stabilizing a multi-rotor aircraft, the PID controller has remained a very popular

and proven solution. However, the tests outlined in this document show that the PID controller

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can be greatly improved upon by implementing a Fuzzy Inference System (FIS) to adaptively

update/modify the Proportional, Integral, and Derivative gains. The results indicate that a system

can become more stable overall with a consistent, faster response and a larger performance

envelope over a similar, PID controller. This is extremely beneficial in virtually any application

of such a multi-rotor aircraft, particularly when it comes to autonomous control, as it implements

some human reasoning into the stabilization controller to further abstract any autonomous

controller from the physics or changing environment at hand, as well as noise or other external

forces, while providing the best performance.

6. Acknowledgements

This material is based upon work supported by the National Science Foundation under Grant No.

DUE 0756921. Any opinions, findings, and conclusions or recommendations expressed in this

material are those of the author(s) and do not necessarily reflect the views of the National Science

Foundation.

7. References

Ang, K. H., Chong, G. C., & Li, Y. (2005). PID control system. IEEETransactionsonControl

Systems, 559-576.

Araki, M. (2006). PID Control in Control Systems, Robotics and Automation, vol II. In Encyclopedia

ofLifeSupportSystems(EOLSS).

Bih, J. (2006). Paradigm shift - An introduction of fuzzy logic. Potentials,IEEE,25(1), 6-21.

Mendel, J. (1995, Mar). Fuzzy logic systems for engineering: a tutorial. ProceedingsoftheIEEE,

83(3), 345-377.

34

Michigan, R. o. (Ed.). (96, 8 30). ModelingofanInvertedPendulum. Retrieved 6 19, 2014, from

Control Tutorials for Matlab:

http://www.ee.usyd.edu.au/tutorials_online/matlab/examples/pend/invpen.html

Razzaghi, K. (2011, 11 01). A New Approach on Stabilization Control of an Inverted Pendulum,

Using PID Controller. Advancedmaterialsresearch, pp. 403-408.

8. Nomenclature

b = Friction forceF = Input force

g = Gravity constantI = Moment of InertiaL = Pendulum rod lengthm = Pendulum mass

M = Cart massx = Cart displacementθ = Pendulum angleφ = Quad-rotor Attitude anglep = Quad-rotor Angular velocity

9. Appendices

Attached are appendices A-F, where the time schedule, benchmarking PID controllers for the

Inverted Pendulum Model and Dynamic Quad Model, and personal pages by each team member are

all outlined.

35

Appendix A: Time Schedule

During the first couple of days of the research program, the following timeline was adapted

in order to effectively meet the goals outlined in this document. Specifically, this was

planned as an eight week, tentative schedule.

Appendix B: Inverted Pendulum Model Ideal PID Controllers

Ideal PID Controller architecture used for both the position and the angle.

Tuned PID Gain Values

Angle PID Position PID

Proportional (P) 46.9986307007982 0.00985559843681718

Integral (I) 0.234602953123809 0.00983601972259795

Derivative (D) 0.125301282844126 11.2944188731606

Filter Coefficient (N) 100.839854415248 2.73635948094535

Appendix C: Dynamic Quad Model Parallel PID Controllers

Attitude (Phi) Parallel PID architecture used for the Dynamic Quad Model

Angular Velocity (P) Parallel PID architecture used for the Dynamic Quad Model

It should be noted that this system worked by feeding the angle (phi) minus the desired value (controller

input) into the “phi parallel PID” controller, then the corrected value would be fed into the “P parallel

PID” controller as the “phi PID out” where the angular velocity is also fed into. The result of this “P

parallel PID” controller is then used as the correction.

Appendix D: Adam Katterheinrich – Personal Page

Throughout the course of the 8 week REU program I have been exposed to a variety of

new experiences. First of all, the program has provided a professional research platform that has

given me the opportunity to take a mostly foreign topic and formulate a project outline to

complete before a strict deadline. Each Biweekly presentation has been a joy and a struggle to

buckle down and compile relevant information to show consistent progress. The process of

conducting a fuzzy logic and PID control literature review, collaborating with mentors to obtain

goals and objectives, compiling research data and spending time with hands-on work has opened

my eyes to the world of scientific research.

Working with Dr. Cohen and Wei has provided me with an experience that is unmatched

by any other program. My team was been able to pull together ideas from across the board to

develop a Fuzzy Logic PID Controller for both an inverted pendulum model as well as a quad-

rotor model. Using Simulink and MATLAB as the main programming platform provided ample

experience with modeling, simulation and dynamic feedback control. All of these skills are

marketable attributes that will stick with me into further engineering work.

Along with the computer simulations, we were able to spend time assembling a

hexacopter and conducting real flight tests. The hands-on portion of the research project gave the

program a solid edge on the competition because it allowed for implementation of theoretical

topics such as fuzzy logic into a real world platform. Working with the technology first hand has

motivated me to further pursue UAV technology or control systems in the future as career

possibilities. The University of Cincinnati has been pushing to increase its presence in the world

of “smart” drone technology and it is incredibly exciting to be a part of that movement. Our

research will be used in further development of an autonomous controller for rotorcraft and I plan

to continue pursuing the opportunities available in this field.

Finally, I would like to thank the NSF for funding this project and allowing me to

participate in such a well-organized research environment. Learning from experienced educators

and researchers has been a privilege and I plan to use all of the skills acquired in future

engineering work.

Appendix E: Nicholas Nielsen – Personal Page

First off, I would like to thank the NSF for supplying the funding for this wonderful

learning experience. Next, I would like to thank everyone involved in making this program a

reality. I have learned so much in the past 8 weeks that will be beneficial to me as I move on in

my career. Here I would like to make a short accounting of all the things that I have learned this

semester.

We started by conduction literature research that provided us with a solid background on

fuzzy logic. The readings assigned by Dr. Cohen prior to and during the first couple of weeks of

the program allowed us to understand the concepts of fuzzy logic enough to work with the fuzzy

logic toolbox in Matlab Simulink. While this only gave me a rudimentary understanding of

everything that is, involved with fuzzy logic it has given me just enough of a taste of this

wonderful way of thinking to make me want more. My ambitions lie in the field of drones and

drone research. I am very excited about the implications of our research and the potentials for

future research along these lines. I see fuzzy logic as a very solid stepping-stone toward more

intelligent systems. While we did not use it in this research, I learned a little bit about the science

of fuzzy neural networks. I am truly interested in the possibilities of creating a fuzzy neural

network to allow a control system such as the one in our research to “learn” as time goes on to

improve its abilities of adaptation. A system such as this could quite possibly learn to tune a

flight controller itself without any outside influence.

Next, we were face with understanding what a PID controller was and how they operate.

Considering that the PID controller was the benchmark with which we were going to be testing

the fuzzy logic controller, we had to understand what a PID controller was before we could build

or use one. This was difficult at first, not because the concept was that hard to understand but

because the methods of tuning were very complex with complicated calculation that must be done

to find the proper responses. After some exploring, we found a PID block with an auto tuning

function built in to it. This was much less intensive than learning how to calculate the proper

gains by hand and was much more efficient considering our time constraints

During this experience, we were also exposed to the hardware side of this project,

through the opportunity to assemble a hex-rotor aircraft for the lab. This allowed us hands on

experience with all of the components of a multi-rotor aircraft from the motors to the speed

controllers and power distribution boards to the flight controller and radio system. Once

assembled, we flashed the system software onto the flight controller and ran through the

calibration of the flight controller sensors. Finally, we accompanied Wei Wei to conduct the

maiden flight. Wei also showed us how to go about flight data acquisition using telemetry data

and a flight-data recording program built in Matlab. Wei then demonstrated how he uses Cipher

along with Conduit for controller verification and validating.

Throughout this process, I learned many other research and presentation skills to

accompany all the technical skills and knowledge I have gathered. The workshops were very

helpful and have given me a much greater appreciation for the research process and how to

effectively portray my research so that others can quickly and concisely learn about what I did

and how I did it. I learned more about many things from how to document my research

photographically, how to compose a proper technical paper, and how to present the material.

While all of the above listed lessons are going to be useful as I continue my career here at UC and

beyond, I have discovered something far more important to me. I have learned that the field of drones

and control systems particularly those of a fuzzy logic design are what I want to study. Again, I would

like to thank the NSF, Dr. Kukerti, Dr. Aure, Dr. Cohen, Dr. Kastner, and Wei Wei for all of their

guidance and input on this project.

Appendix F: Tyler Parcell – Personal Page

I have always been deeply fascinated with science and technology, as well as its impact

on the world around us. It has always intrigued me to see the true impacts the application of

technology can have in the advancement of society as a whole, and I desire to be a part of it.

More specifically, I desire to be a part of research and development of robotics and/or the

artificial intelligence behind controlling them. As a result, I decided to apply for this research

experience, and was selected for this project. My goal was to gain a better understanding behind

the fundamentals of research and development, as well as to advance both my knowledge and the

scientific field in which the project lies through the research conducted.

I entered the project with essentially no real controls experience at all. Additionally, I

had a minimal working knowledge of more advanced artificial intelligence systems, working only

with fundamental topics on the most basic of scale. This research project was an opportunity to

explore those two areas more in depth through the designing of an intelligent controller system.

The intelligent controller system that our team designed familiarized me with the concept of

feedback control systems, fuzzy logic, and PID controllers. Since my ultimate desire is in the

research and development of robotics and the artificial intelligence behind them, this acquired

skill set will prove to be extremely valuable.

In this project, I was introduced to the idea of non-linear systems, and the unique

challenges that such a system presents in a controls problem. I assisted in developing a simple

non-linear inverted pendulum model and controller, putting some of the aforementioned skills to

use with MATLAB’s Simulink. Lastly, I was also given the opportunity to understand more

regarding the movements/dynamics of controlling a multi-rotor aircraft on a more fundamental

scale, as well as create, along with our team, a controller for this more complex system.

I would like to thank the National Science Foundation for funding this project as well as

the University of Cincinnati for giving me the opportunity to be a part of it.