SAMPLE AND HOLD INPUTS THEORY AND APPLICATIONS, AND FAULT

DETECTION FOR SSCDS

By

Yingxu Wang

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Mechanical Engineering – Doctor of Philosophy

2019

ABSTRACT

SAMPLE AND HOLD INPUTS THEORY AND APPLICATIONS, AND FAULT

DETECTION FOR SSCDS

By

Yingxu Wang

The disadvantages of non-minimum-phase (NMP) systems for control applications are well

known. Prior research has shown that a NMP system can be discretized to an minimum-

phase (MP) system using either a zero-order hold or square-pulse sample and hold input.

First the research focused on the MP characteristics of the discrete-time system that

obtained from a continuous-time single-input-single-output NMP system by using different

sample and hold inputs. Two new sample and hold inputs (forward and backward triangular)

are studied in addition to the square pulse. Numerical simulations were adopted for studying

the MP property of the resulting discrete-time system as a function of sample and hold

parameters. The simulation results show that it is possible to find a smaller sampling

period that results in an MP discrete-time system using the proposed sample and hold

inputs compared to zero-order hold. The q-Markov Cover system identification with pseudo-

random binary signal (PRBS) was then used for a hardware-in-the-loop (HIL) simulation

study. A resistor-capacitor filter was used to represent the implementation error of the

sample and hold input due to unmodeled actuator dynamics. HIL simulation results show

that the proposed sample and hold input scheme is robust to actuator modeling error. The

MP properties of the discrete-time systems with three sample and hold inputs are compared.

The results show that the forward triangle sample and hold input has the best performance

due to its robustness to unmodeled actuator dynamics and the capability of retaining MP

property of the discretized system at small sampling periods.

Based on the analytical result, the advantages of using the SHIs for the single inverted

pendulum through simulations and experiments were studied.

In particular, it is shown

that the performance of the stabilized closed-loop system can be improved by designing an

dual-loop controller based on the MP discrete system obtained using the SHIs. Simulation

results in the presence of Coulomb friction show that the additional controller reduces cart

oscillations significantly; for a particular SHI, the steady state amplitude of oscillation was

reduced by up to 68.95%. Experiments confirmed the results obtained in simulations.

The dual-loop control technique is also demonstrated for performance improvement on

a mini Segway (MS), a robot developed for an undergraduate mechatronics class which

equipped with an extremely low cost microcontroller (Arduino).

It is important to show

that the special SHIs can be implemented in a low cost micro-controller. A dual-loop control

tuning method was also developed to optimize the overall closed-loop system performance.

Experimental result shows that the low cost microcontroller can be used for the dual-loop

SHI control scheme and the MS cart displacement oscillation is significantly reduced by more

than 60% over the baseline controller.

A solid-set canopy delivery system (SSCDS) is developed to deliver water and chemical

solutions to high-density fruit trees with the support from USDA-SCRI (US Department of

Agriculture Specialty Crop Research Initiative). For the purpose of detecting SSCDS faults,

two methods were proposed and they are flow and/or pressure sensors based in-line detection

method and thermal image based detection method. A FLIR thermal camera was mounted

on a modified commercial Unmanned Aerial Vehicle (UAV) so that any plug, partial plug,

and gusher faults can be captured by the thermal camera during the water and chemical

spray process. With the help of a image processing software, the thermal video is stitched

into one panorama image. By comparing it with a baseline panorama image, the SSCDS

faults can be detected. The main advantage of the proposed fault detection method is the

ability of distinguishing all three types of faults over a huge area within a fairly short amount

of time, utilizing only a UAV equipped with both regular and thermal video cameras. The

future work is to fly the UAV automatically based on a pre-planned route, collect the thermal

video, process the panorama image and diagnose the SSCDS faults.

Copyright by
YINGXU WANG
2019

ACKNOWLEDGMENTS

This thesis can not be completed without the selfless support from my advisor, Dr. Zhu.

He guided me through the years of my master and Ph.D career. I would like to express my

deepest sense of gratitude to him.

I’m very much thankful to Dr. Mukherjee, my co-advisor, for his valuable guidance. He

inspired me with his working attitude – paying attention to every detail.

My sincere thanks to Dr. Grieshop. His sense of humor makes my research fun and full

of joy. His expertise also guided me through a lot of hard times.

Also Dr. Li, who is so thorough in his work and set an excellent example for me

I’m deeply grateful to my fianc´e, Stephanie Sun. She supports me with all her heart with

no reason. She gives me the strength and hope in the hard times.

I also want to express my deepest gratitude to my parents. They supports me financially

and mentally.

I acknowledge with thanks the support from USDA-SCRI, who is the sponsor for my

agriculture project.

Last but not least, many thanks to my lab friends: Keith Koonter, Jie Yang, Jie Jerry

Yang, Shupeng Zhang, Dafa Yan, Tao Zeng, Mengyang Gu, Xuefei Chen, Xiaojian Yang,

Hailiang Zheng, Yifan Men, Ruixue Li, Tianyi He, Shen Qu, Ali Alhajjar, Anuj Pal, Wenpeng

Wei, Jian Tang, Aqeel Salim, Ruitao Song, Ali Al-jiboory, Andrew White, Zhen Ren, Yu He,

Dagmara Wehr, Ping Mi, Donghao Hao, Dawei Hu, Huan Li, Chengsheng Miao, Shupeng

Zhao, Ju Zhang, Hai-ou Liu, Xiaohong Tai.

v

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

CHAPTER 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

CHAPTER 2 SAMPLE AND HOLD INPUTS FOR MINIMUM-PHASE BEHAV-

2.2.1 Numerical Example 1.
2.2.2 Numerical Example 2.

IOR OF NON-MINIMUM-PHASE SYSTEM . . . . . . . . . . . . .
2.1 SHIS, DE and DT systems . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Background DE System . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Two Additional Switched SHIs
. . . . . . . . . . . . . . . . . . . . .
2.1.3 Discrete Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Simulation case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 A framework for DT system identification . . . . . . . . . . . . . . . . . . .
System Identification . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Experimental validation using HIL simulations . . . . . . . . . . . . . . . . .
2.4.1 HIL Simulation Experiment Setup . . . . . . . . . . . . . . . . . . . .
2.4.2 HIL Simulation Procedure . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 HIL Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3.1
2.3.2 Method and Results

CHAPTER 3 DEMONSTRATION OF PERFORMANCE IMPROVEMENT I

. . .
3.1 Single inverted pendulum (SIP) system . . . . . . . . . . . . . . . . . . . . .
3.2 Dynamic Model with Coulomb Friction Term . . . . . . . . . . . . . . . . .
3.3 Dual-loop controller design . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Inner-loop Stabilizing Controller . . . . . . . . . . . . . . . . . . . . .
3.3.1
SHI Parameter Selection For MP Characteristics . . . . . . . . . . . .
3.3.2
3.3.3 Outer-Loop Controller Design . . . . . . . . . . . . . . . . . . . . . .
3.4 Simulation and experimental validation . . . . . . . . . . . . . . . . . . . . .
3.4.1 Experimental Set Up . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simulation and Experiment Results . . . . . . . . . . . . . . . . . . .
3.4.2

CHAPTER 4 DEMONSTRATION OF PERFORMANCE IMPROVEMENT II . . .
4.1 Segway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Motor Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Equations of the Wheels . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Equations of the Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Dual-loop controller design . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Inner-loop Stabilizing Controller . . . . . . . . . . . . . . . . . . . . .

4.5.1

vi

7
7
7
9
10
12
12
15
16
17
17
19
20
20
22

24
24
27
31
31
33
35
35
35
36

45
45
45
47
48
51
54
54

4.6 Experimental Hardware

4.5.2
SHI Parameter Selection . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.3 Outer-Loop Design . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.1
4.6.2 Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.3 Microcontroller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.4 Experimental Set Up . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Experimental Implementation . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7.1
Inner-loop LQR Tuning . . . . . . . . . . . . . . . . . . . . . . . . .
4.7.2 Outer-Loop Controller Tuning . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 5 AGRICULTURE PROJECT: FAULT DETECTION ON SOLID-

SET CANOPY DELIVERY SYSTEM FOR APPLE FARM . . . . .
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Solid-Set Canopy Delivery System . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Emitters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Irrigation Systems
5.2.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.3
SSCDS Spray Pattern . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.4
. . . . . . . . . . . . . . . . . . . . . . . . . . .
In-Line Flow Sensor
5.2.5 Thermal Fault Detection . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.6 Drone Hardware Selection . . . . . . . . . . . . . . . . . . . . . . . .
5.2.6.1 Drone: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.6.2 Thermal Camera . . . . . . . . . . . . . . . . . . . . . . . .
5.2.6.3 Gimbal:
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.6.4 Flight Controller: . . . . . . . . . . . . . . . . . . . . . . . .
5.2.6.5 Video Transmitter: . . . . . . . . . . . . . . . . . . . . . . .
5.3 Experiment Results and Discussion . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Experimental Procedure and Results
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4 Discussion and Future Works

55
56
57
57
58
60
60
62
62
64

71
71
73
73
75
76
77
80
81
81
81
81
82
83
85
85
86
90
90

CHAPTER 6 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

vii

LIST OF TABLES

Table 2.1: Definition of the SPSHI, FTSHI and BTSHI

. . . . . . . . . . . . . . . .

10

Table 2.2: Pole-Zero Location of the DT, ID, and HIL-ID Systems

. . . . . . . . . .

19

Table 2.3: The Third Zero Locations of the HIL ID Systems With δ = 0.5s, P = 0.6

and 0.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

Table 3.1: QUANSER inverted pendulum system parameters [1]

. . . . . . . . . . .

30

Table 3.2: Experimental results for inner-loop LQR and dual-loop control with SP-

SHI and FTSHI

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

Table 4.1: Segway system parameters (To be determined in the future work) . . . . .

53

Table 4.2: The Mean Absolute Values of Cart Displacement and Control Voltage

For Inner-loop LQRc Tuning . . . . . . . . . . . . . . . . . . . . . . . . .

62

Table 4.3: The Mean Absolute Values of Outer-loop LQRd Tuning Data with dif-

ferent GRc Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

Table 4.4: The Mean Absolute Values of Outer-loop LQRd Tuning Data with GRc

= 0.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

Table 5.1: Flow sensor specification . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

Table 5.2: Transmitter and receiver specifications . . . . . . . . . . . . . . . . . . . .

79

Table 5.3: FLIR VUE Pro R Specifications

. . . . . . . . . . . . . . . . . . . . . . .

82

Table 5.4: Experiment #1 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . .

86

Table 5.5: Fault Details in the Orchard that with Hadar 7110 Emitters . . . . . . . .

87

Table 5.6: Experiment #2 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . .

88

Table 5.7: Fault Details in the Orchard that with Spinner Emitters . . . . . . . . . .

89

viii

LIST OF FIGURES

Figure 1.1: Flowchart of obtaining a DT system . . . . . . . . . . . . . . . . . . . .

Figure 1.2: Closed-loop system architecture of a continuous-time system with a

discrete-time controller . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

4

Figure 2.1: (a) continuous input, (b) its ZOSHI, (c) SPSHI, (d) FTSHI, and (e) BTSHT 8

Figure 2.2: Shaded regions showing sample and hold parameters that result in MP
characteristics of (a) SPSHI-DT, (b) FTSHI-DT, and (c) BTSHI-DT
systems. The plot in (d) is an overlay of those in (a), (b), and (c) . . . .

13

Figure 2.3: Pole-zero locations of all SHIs . . . . . . . . . . . . . . . . . . . . . . . .

14

Figure 2.4: Feasible regions of all SHIs overlaid for the second example . . . . . . . .

16

Figure 2.5: Q-Markov Cover system ID process with PRBS excitation . . . . . . . .

17

Figure 2.6: PRBS sampled and held by ZOSHI, SPSHI, FTSHI, and BTSHI . . . . .

18

Figure 2.7: System responses associated with ZOSHI, SPSHI, FTSHI, and BTSHI

.

18

Figure 2.8: HIL simulation setup for studying SHI robustness . . . . . . . . . . . . .

20

Figure 2.9: Filtered PRBS sampled and held by (a) SPSHI, (b) FTSHI, (c) BTSHI .

21

Figure 2.10: Zero movements of the HIL ID systems with two set of feasible sample

and hold parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

Figure 3.1: The Single Inverted Pendulum System . . . . . . . . . . . . . . . . . . .

24

Figure 3.2:

Inverted Pendulum Free Body Diagram . . . . . . . . . . . . . . . . . . .

25

Figure 3.3: Coulomb Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

Figure 3.4: The dual-loop controlled SIP system . . . . . . . . . . . . . . . . . . . .

32

Figure 3.5: The inverted pendulum simulation layout . . . . . . . . . . . . . . . . . .

33

Figure 3.6: The SIP p-δ feasible regions of BTSHI (Region 1), SPSHI (Region 1+2)

and FTSHI (Region 1+2+3).

. . . . . . . . . . . . . . . . . . . . . . . .

34

ix

Figure 3.7: The inverted pendulum dual-loop simulation layout . . . . . . . . . . . .

35

Figure 3.8: The SIP experiment hardware connection . . . . . . . . . . . . . . . . .

36

Figure 3.9: The SIP experiment set up in the lab . . . . . . . . . . . . . . . . . . . .

37

Figure 3.10: The simulated and experimental results of the SPSHI-DT with δ = 0.3s

for the SIP system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

Figure 3.11: The simulated and experimental results of the FTSHI-DT with δ = 0.3s

for the SIP system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

Figure 3.12: The simulated and experimental results of the SPSHI-DT with δ = 0.4s

for the SIP system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

Figure 3.13: The simulated and experimental results of the FTSHI-DT with δ = 0.4s

for the SIP system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

Figure 3.14: The best experimental results of dual-loop with SPSHI and FTSHI com-

pare with inner-loop controller . . . . . . . . . . . . . . . . . . . . . . . .

43

Figure 3.15: The trend of dual-loop control response . . . . . . . . . . . . . . . . . . .

44

Figure 4.1: Free body diagram of the mini segway robot

. . . . . . . . . . . . . . .

45

Figure 4.2: Motor diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

Figure 4.3: Free Body Diagram of the Wheel

. . . . . . . . . . . . . . . . . . . . . .

47

Figure 4.4: Free Body Diagram of the Pendulum . . . . . . . . . . . . . . . . . . . .

49

Figure 4.5: Cart turning angle definition . . . . . . . . . . . . . . . . . . . . . . . . .

50

Figure 4.6: Dual-loop control architecture for the MS system . . . . . . . . . . . . .

54

Figure 4.7: The feasible sampling period of ZOH that result in MP DT system and

the p-δ feasible regions of SPSHI.

. . . . . . . . . . . . . . . . . . . . . .

55

Figure 4.8: The MPU 6050 Chip . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

Figure 4.9: The Complementary Filter . . . . . . . . . . . . . . . . . . . . . . . . . .

58

Figure 4.10: 12V Motor and Hall Effect Encoder . . . . . . . . . . . . . . . . . . . . .

59

Figure 4.11: Arduino MKR Motor Carrier

. . . . . . . . . . . . . . . . . . . . . . . .

59

x

Figure 4.12: Arduino MKR1000 WiFi . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

Figure 4.13: The MS system hardware configuration . . . . . . . . . . . . . . . . . . .

61

Figure 4.14: The MS Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

Figure 4.15: The experimental result for inner-loop LQRc optimization . . . . . . . .

Figure 4.16: LQRc tuning data plot . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

64

Figure 4.17: The feasible region of the closed-loop MS system with SPSHI

. . . . . .

65

Figure 4.18: The Experimental result for Outer-loop Controller Optimization with

SPSHI and GRc = 0.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

Figure 4.19: The Experimental result for Outer-loop Controller Optimization with

SPSHI and GRc = 0.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

Figure 4.20: The Experimental result for Outer-loop Controller Optimization with

SPSHI and GRc = 1.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.21: LQRd tuning data plot . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

70

Figure 5.1: SSCDS Layouts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

Figure 5.2: The Hadar 7110 Emitter . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

Figure 5.3: The Spinner Emitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

Figure 5.4: SSCDS Spray Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

Figure 5.5: Flow sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

Figure 5.6: Transmitter ZF-4 and Receiver ZW21-J

. . . . . . . . . . . . . . . . . .

79

Figure 5.7: The transmitter is mounted inside the generator . . . . . . . . . . . . . .

79

Figure 5.8: The in-line fault sensing system prototype 1 . . . . . . . . . . . . . . . .

80

Figure 5.9: The in-line fault sensing system prototype 2 . . . . . . . . . . . . . . . .

80

Figure 5.10: Controller Tray . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

Figure 5.11: Modified Drone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

xi

Figure 5.12: Drone System Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

Figure 5.13: Thermal and Regular Footage of the Orchard with Hadar 7110 Emitters

87

Figure 5.14: Thermal and Regular Footage of the Orchard with Spinner Emitters . . .

89

xii

CHAPTER 1

INTRODUCTION

A continuous-time system is non-minimum phase (unstable) if there exists at least one zero

(pole) in the right-half plane. Similarly, a discrete-time system is non-minimum phase (un-

stable) if there is at least one zero (pole) outside the unit circle. If a stable continuous-time

system is discretized using a zero-order hold (ZOH), each pole p of the continuous-time

system is mapped into the pole epT for the discrete-time system, where T is the sampling

interval. Therefore, discretization preserves the stability characteristics of poles. The same

is not true for zeros.

The control design for non-minimum-phase (NMP) systems is challenging. For instance,

from root locus analysis that the closed-loop poles move towards the open-loop zeros when

the system proportional gain increases. This results in unstable closed-loop dynamics and

prevents the use of high-gain control to achieve the desired closed-loop performance. It is

also well known that asymptotic tracking is difficult to achieve when the system is NMP.

Control methods can only achieve approximate output tracking for NMP systems [2–8].

NMP systems are difficult to control, but discretization with ZOH may alleviate the

problem; it is shown in [8–21] that a NMP continuous-time system may map into an MP

discrete-time system. This happens, for example, when a ZOH input is used with a suffi-

ciently large sampling period [9]. Unfortunately, a precise mathematical formula for calcu-

lating the smallest feasible sampling period was not found. For an NMP continuous-time

system, a feasible sampling period is the one that results in an MP discrete-time system.

Based on [9], the range of feasible sampling periods was determined by using a symbolic

approach in [10] and genetic algorithms in [11]. The methods for finding feasible sampling

periods with a ZOH input when the continuous-time system is minimum-phase (MP) can be

found in [12–14]. For stable MP systems, the Nyquist algorithm was used in [12]; the results

were extended in [13] to unstable systems with no poles or zeros on the imaginary axis.

1

A first-order approximation, based on Taylor series, was used in [14] to find the smallest

feasible sampling period for single-input-single-output (SISO) systems.

The results cited above find feasible sampling periods with ZOH input, but it all have

restrictions on the system and the feasible sample period may not be small enough for closed-

loop control design purpose. To find a feasible sampling period smaller than that obtained

with the ZOH, other sample and hold inputs (SHI) were explored in [15–21]. The impulse

generator function (IG) was used in [15], but the reduction of the smallest feasible sampling

period was minor. The first-order hold was studied in [16] and [17], but it provided no

improvement over ZOH. In [18–21], fractional-order hold (FROH) schemes were studied.

They were found to reduce the lower bound of feasible sampling periods obtained with ZOH.

However, the FROH schemes have restrictions, such as the continuous-time system has to

be low-pass [18], [19]; they are only applicable to second-order or first-order plants with time

delay [20], and the resulting discrete-time system is MP only if the continuous-time system

is MP and when the transfer function relative degree is 1 or 2 [21].

An alternative approach that reduces the lower boundary of feasible sampling periods was

developed in [8]; it is based on the characteristics of discrete equivalent (DE) system obtained

with pulse SHI. It was shown that a continuous-time NMP system and its corresponding DE

system can have the same state and output variables at regular sampling intervals for pulse

SHI and ZOH input, respectively. A proper choice of the sampling period and the pulse duty

cycle can result in MP characteristics of the DE system; this implies that the NMP system

can exhibit MP behavior at sampled data points. In this paper, two additional SHIs are

investigated: forward and backward triangle SHIs. The feasible sampling periods for these

two triangular SHIs and the pulse SHI are compared with that of the conventional ZOH

input.

Since any microprocessor-based controller and its associated control design are in discrete-

time (DT) domain, it is important to study the MP property in discrete-time domain. This

is the main focus of this thesis. The flowchart in Fig. 1 shows the process of obtaining a

2

DT system from a continuous-time system using a specific SHI. The top figure shows the

response of a continuous-time system with an SHI. The middle figure shows the response

of the corresponding DE system with only the ZOH input. The output of the DE system,

as expected, matches the output of the continuous-time system with SHI shown in the top

figure. The bottom figure shows the discrete-time representation of the DE system, its input

and output. It will be discussed later how the resulting DT system can be MP even when

the continuous-time system is NMP through a proper choice of the sampling period and the

duty cycle of the SHIs.

Figure 1.1: Flowchart of obtaining a DT system

In Fig. 1.

the systems shown in the middle and bottom figures are virtual systems.

However, the DT system model can be used to design a discrete-time controller for the

original continuous-time system with SHI shown in Fig. 2. This is true because the output

of the DT system and the original continuous-time system are identical at each sampling

instant.

The main contribution of the first part of this thesis is to find the smallest feasible

3

Continuous-time DomainZOHDiscreteInput00.511.522.533.544.55-0.8-0.6-0.4-0.2000.511.522.533.544.55Discrete-time DomainContinuous-Time SystemDE SystemDT SystemContinuousInputContinuousInput00.511.522.533.544.55SHIsContinuous-time Domain00.511.522.533.544.55-1-0.8-0.6-0.4-0.2000.511.522.533.544.55-1-0.8-0.6-0.4-0.2000.511.522.533.544.55-1-0.8-0.6-0.4-0.20sampling period for a continuous-time NMP system using an SHI through case studies and

to show the potential of using the SHIs to improve the feasible sampling periods. This is

important since the smallest feasible sampling period using the conventional ZOH is often too

large to be used for practical applications. Through proper choice of duty cycle and sampling

period of the SHI, the DT system can be MP even though the original continuous-time system

is NMP. This allows us to design a discrete-time controller base on the MP DT model for

the continuous-time NMP system with the selected SHI. In this paper, it is demonstrated

through two examples that the smallest feasible sampling period of an NMP system can be

reduced significantly. Also, SHI implementation error was introduced in hardware-in-the-

loop (HIL) simulations to demonstrate the feasibility for practical applications.

Figure 1.2: Closed-loop system architecture of a continuous-time system with a
discrete-time controller

Based on the developed theory, an experimental validation was performed on a single

inverted pendulum (SIP). since it has been used as a benchmark station for validating and

demonstrating advanced control algorithms [22, 23]. This is because it is an under actuated

NMP system with a high degree of nonlinearity. Different control approaches have been

implemented over the past few decades for the SIP system such as PID control [24–26], LQR

[25–28], and LQG control [29]; the performance of the closed-loop systems is highly dependent

on the selection of the PID gains or LQR and LQG weighting matrices. Without advanced

tuning methods [24, 27, 28, 30], the implementation of the above-mentioned controllers is

challenging and time consuming. Fuzzy logic control algorithms [31, 32] have been used but

4

Continuous-Time SystemDiscrete-TimeControllerZOHSHIsZOH00.511.522.533.544.55-1-0.8-0.6-0.4-0.20the performance is not ideal. Fractional order PIDs [33,34] are also used but the performance

improvements over LQR are minor.

The first experimental validation focus on converting the stabilized continuous-time SIP

system into an MP discrete-time system for the purpose of improving the closed-loop system

performance. Two linear quadratic regulation (LQRs) loops were used: the inner loop

(designed by the manufacturer) optimizes the continuous-time NMP system for guaranteed

stability; the outer loop, sampled at 3.3 Hz and 2.5 Hz for two different SHIs, is used to

further improve the closed-loop system performance exploiting MP characteristics of the

DE system. Due to the presence of Coulomb friction, the positions of the cart and the

pendulum oscillate around the equilibrium configuration; the cart displacement oscillation

is significantly reduced by the outer-loop discrete-time controller.

The first experimental validation lies in experimental demonstration of performance im-

provement of an NMP SIP system using SHIs. For the studied SIP, the feasible sampling

rate is increased from 1.12Hz (ZOH) to 5.78Hz by using the SHIs proposed in [35]. As a

result, the steady-state amplitude of oscillation was reduced by 78.87% in simulations and

68.95% in experiments. The outer-loop controller was implemented using the same control

system hardware provided by the SIP manufacturer.

Since the controller was implemented into to a very sophistic microcontroller in the SIP

validation, the feasibility of implementing the proposed SHIs into an extremely low cost

microprocessor remains an open question, which is the top of this paper. To be specific,

this paper studies the performance improvement of a mini Segway (MS) with a low cost

microprocessor (Arduino) with a dual-loop control scheme. The MS is a two-wheel self-

balanced robot and is a multi-input-multi-output (MIMO) nonlinear system.

It is NMP

after linearization about the upright equilibrium point. Due to its under-actuated nature,

this platform is widely used for controller design and implementation studies, including PID

controllers [36] [37] [38] [39] [40] [41], Fuzzy logic controllers [42] [43] [44], Fuzzy combined

with PID [45], Adaptive control [46] and LQR control [47] [48] [41] [49].

5

In this study, the dual-loop control architecture that proposed in [50] was adopted. The

continuous-time inner-loop LQR was implemented at 100 Hz and a discrete-time outer-loop

LQR with the selected SHI was implemented at 5 Hz. The idea is to utilize the inner-loop

controller to stabilize the MS system and to convert the resulting NMP MS system into a

discrete-time MP system with the help of proposed SHIs. Consequently a high gain outer-

loop can be used to further improve dual-loop closed-loop system performance. It is obvious

that the overall system performance is affected by both inner- and outer-loop controllers.

To study the interaction between two controllers, multiple inner-loop LQR controllers were

designed with gains from low to high and the outer-loop controller was optimized for each

designed inner-loop controller. It was found that the inner-loop control gain shall be properly

chosen to have a decent closed-loop system performance without the outer-loop controller.

When the inner-loop control gain is relatively low, the nonlinear Coulomb friction dominates

the closed-loop system dynamics, leading to unstable system responses; and while the inner-

loop control gain is relatively high, the system nonlinarity could result in unwanted high

frequency oscillations.

The rest of the paper is organized as follows. In Chapter II, the SHIs and DE system

are introduced and implemented in dSPACE for HIL analysis. Analytical results are pre-

sented. The SIP implementation for performance improvement is showed in Chapter III.

The dual-loop implementation onto a MS system along with controller tuning within a low

cost microcontroller is presented in Chapter IV; Chapter V is another area that relates to

my study – fault detection of an apple orchard with IR camera that carried by a drone.

Concluding remarks are provided in Section VI.

6

CHAPTER 2

SAMPLE AND HOLD INPUTS FOR MINIMUM-PHASE BEHAVIOR OF

NON-MINIMUM-PHASE SYSTEM

2.1 SHIS, DE and DT systems

2.1.1 Background DE System

Consider the continuous-time input u(t) shown in Fig. 2.1 (a) with solid line.

This input can be discretized using a zero-order sample and hold function (ZOSHF),

shown in Fig. 2.1 (a) with dotted lines. Mathematically, the zero-order sample and hold

input (ZOSHI) can be defined as follows.

uz(t) = u(tk), if tk < t ≤ tk + δ, k = 0, 1, 2, ...

(2.1)

where tk, k = 0, 1, 2, ..., are the sampling time instants; δ is the sampling period; and u(tk)

is the zero-order hold (ZOH) value of u(t) over the interval (tk, tk + δ]. uz(t) is shown in

Fig. 2.1 (b).

For the linear system

˙x(t) = Ax(t) + B ¯u(t), x(t0) = x0

y(t) = Cx(t)

with the switched input ¯u(t) shown in Fig. 2.1 (c), and defined as

uz(t),

0,

¯u(t) =

if tk < t ≤ tk + pδ
if tk + pδ < t ≤ tk + δ

(2.2)

(2.3)

where k = 0, 1, 2, ... and p ∈ (0, 1] is the duty cycle of the switched input, the concept of DE

system was introduced as follows [21]:

Definition 1 (DE system): The time-invariant linear system with the ZOSHI

7

Figure 2.1: (a) continuous input, (b) its ZOSHI, (c) SPSHI, (d) FTSHI, and (e) BTSHT

˙¯x(t) = ¯A¯x(t) + ¯Buz(t), ¯x(t0) = x0

¯y(t) = C ¯x(t)

(2.4)

is a DE system of (2.2) if the state variables of (2.2) and (2.4) match at sampling times

tk, k = 0, 1, 2, ....

The state variables of (2.2) and (2.4) at time tk+1 = tk + δ are given below by the

8

()utktkt0t1t2tktkt0t1t2tkt0t1t2tkt0t1t2tktkt0t1t2t()Zutktp()ut()ut()ut(a)(b)(c)(d)(e)following equations.

x(tk + pδ) = eApδx(tk) +

eA(pδ−τ )B ¯u(tk + τ )dτ

(cid:90) pδ

0

= eApδx(tk) + A−1(eApδ − I)Bu(tk)

x(tk+1) = eA(δ−pδ)x(tk + pδ)

= eAδx(tk) + A−1(eAδ − eA(1−p)δ)Bu(tk)

(cid:90) δ
¯Aδ ¯x(tk) +
¯Aδ ¯x(tk) + ¯A−1(e

e

0

¯x(tk+1) = e

= e

¯A(δ−τ ) ¯BuZ (tk + τ )dτ
¯Aδ − I) ¯Bu(tk)

(2.5)

(2.6)

Comparing (2.5) and (2.6), it is clear that (2.2) and (2.4) will be DE if the following

conditions are met:

¯A = A, ¯B = [I − e−Aδ]−1[I − e−Apδ]B

(2.7)

When p = 1, the switched input in (2.3) is identical to the ZOSHI. In this case, the DE

conditions in (2.7) simply reduce to

¯A = A, ¯B = B

(2.8)

2.1.2 Two Additional Switched SHIs

The switched input ¯u(t) in (2.3), which is shown in Fig. 2.1 (c), will henceforth be referred to

the square pulse sample and hold input (SPSHI). Two new SHIs are introduced in this paper:

the forward triangle sample and hold input (FTSHI) and the backward triangle sample and

hold input (BTSHI). These are shown in Fig. 2.1 (d) and Fig. 2.1 (e), respectively. The

SPSHI, FTSHI, and BTSHI are defined in the following table:

9

Table 2.1: Definition of the SPSHI, FTSHI and BTSHI

tk < t ≤ tk + pδ

tk + pδ < t ≤ tk + δ

Input

SPSHI

BTSHI

FTSHI

¯u(t) = 2
p

pδ

+ uz(t)

¯u(t) = 1

p[uz(t)]

(cid:104) uz(t)(tk−t)
(cid:104) uz(t)(t−tk)

(cid:105)

¯u(t) = 2
p

pδ

(cid:105)

0

0

0

Similar to the SPSHI, the conditions for DE system with the FTSHI can be obtained

below.

¯A = A
¯B = (I − e−Aδ)−1[

A−1(e−Apδ − I) + I]B

1
pδ

And the BTSHI conditions are

¯A = A
¯B = (I − e−Aδ)−1[

1
pδ

A−1(I − e−Apδ) − e−Apδ]B

(2.9)

(2.10)

Note that when system matrices A, B, and C, and SHI parameters p and δ are given, (2.7),

(2.9), and (2.10) can be used to construct DE systems with SPSHI, FTSHI, and BTSHI.

Since is different from B, the zero locations of the DE system could be different from those

of the original system.

2.1.3 Discrete Time Systems

Consider the standard discrete-time (DT) system with the same initial conditions as the DE

system in (2.4)

x(tk+1) = Adx(tk) + Bdu(tk), x(t0) = x0

y(tk) = Cdx(tk)

(2.11)

10

The DT system will have the same state and output variables as the DE system with

SPSHI, described by (2.4) and (2.7), at sampling time tk(k = 0, 1, 2, ...) if Ad = eAδ, Cd = C

and following condition is met

Bd = A−1(eAδ − I)(I − e−Aδ)−1(I − e−Apδ)B

(2.12)

Similarly, the conditions for FTSHI are and

Bd = A−1(eAδ − I)(I − e−Aδ)−1[A−1 e−Apδ − I

pδ

+ I]B

(2.13)

The conditions for the BTSHI are and

Bd = A−1(eAδ − I)(I − e−Aδ)−1[A−1 I − e−Apδ

pδ

− e−Apδ]B

(2.14)

It should be noted that the continuous-time system in (2.2) with the SPSHI, its DE

system (2.7), and the corresponding DT system (2.12) have the same state and output

variables at the sampling time tk (k = 0, 1, 2, ...). The same is true for system (2.2) with

the FTSHI, its DE system (2.9), and its DT system (2.13); and for system (2.2) with the

BTSHI, its DE system (2.10), and its DT system (2.14).

The DT system corresponding to system (2.2) with p = 1 has the following system

matrices

Ad = eAδ, Bd = A−1(eAδ − I)B, Cd = C

(2.15)

A comparison of the system matrices in (2.12), (2.13), and (2.14) with those in (2.15) indi-

cates that although all four systems have identical ZOSHI, the Bd matrices are all different.

This suggests that the location of the zeros of the DT systems in (2.12), (2.13), and (2.14)

could be different from those in (2.15).

In the sequel, the system of (2.15) will be referred as the ZOSHI-DT system. Similarly,

the systems of (2.12), (2.13), and (2.14) will be referred as SPSHI-DT, FTSHI-DT, and

BTSHI-DT systems, respectively.

11

2.2 Simulation case studies

In this chapter, numerical simulation studies were conducted to illustrate the possibility

of shifting zeros of a DT system for a given continuous-time system by using different SHI

functions.

2.2.1 Numerical Example 1.

Consider the transfer function below

G(s) =

(s − 5)(s2 + 1)

(s + 2)(s + 1)(s2 + 2s + 5)

(2.16)

This represents a stable NMP system with zeros locate at 5, i and −i. Using (2.12),

(2.13), and (2.14), the zero locations of the SPSHI-DT, FTSHI-DT, and BTSHI-DT systems

were calculated for a range of p and δ values. The regions where all the zeros lie inside the unit

circle are defined as feasible regions and shown in Fig. 2.2(a), (b), and (c) for the SPSHI-DT,

FTSHI-DT, and BTSHI-DT systems, respectively. For the purpose of comparison, Figs. 2.2

(a), (b), and (c) are overlaid in Fig. 2.2 (d). The zero locations of the ZOSHI-DT system can

be observed from the top boundary of Fig. 2.2 (a) since it is a special case of the SPSHI-DT

with p = 1.

For very small values of p, the minimum δ value that results in MP characteristic is

the same for all three DT systems shown in Fig.2.2. This is because when the duty cycle

is extremely small, all three SHIs converge to a narrow pulse. As mentioned earlier, the

ZOSHI-DT system is a special case of the SPSHI-DT system for p = 1. Therefore, the

feasible sampling period for the ZOSHI-DT system is between 0.69s and 1.76s; see Fig. 2.2

(a). The same result can be obtained using the procedure outlined in [8]. Compare to the

ZOSHI-DT system, the smallest feasible sampling period of FTSHI-DT system is as low as

0.38s. This indicates that the system in (2.16) with FTSHI can have a much smaller feasible

sampling period than that with the conventional ZOH method.

12

Figure 2.2: Shaded regions showing sample and hold parameters that result in MP
characteristics of (a) SPSHI-DT, (b) FTSHI-DT, and (c) BTSHI-DT systems. The plot in
(d) is an overlay of those in (a), (b), and (c)

For feedback control, it is beneficial to have a small sampling period, i.e., a small δ.

Furthermore, it is desirable to have a large value of p, since a small value of p will increase

the maximum norm of the input over each sampling period and reduce the effectiveness of

the control. This implies that the region in the top left hand corner of the shaded region in

Fig. 2.2 (d) is of maximum relevance. An investigation of this corner region indicates that

for p = 0.8 (duty cycle of 80%), for example, the feasible sampling period can be as small

as 0.32s with FTSHI. For the SPSHI the smallest feasible sampling period is slightly higher,

and is equal to 0.42s; for BTSHI it is even higher, equal to 0.55s. When a sampling period

of δ = 0.3s is given, the FTSHI-DT system provides MP characteristics with the largest

duty cycle of 67.9%, the SPSHI-DT system has a smaller duty cycle of 48.5%, and for the

BTSHI-DT system, it reduces further down to 35.4%. In summary, for system (2.16), the

13

FTSHI-DT system provides MP characteristics with the largest duty cycle and the smallest

sampling period among all four DT systems.

For the purpose of studying the MP property of the associated DT systems, the zeros

of the SPSHI-DT, the FTSHI-DT, and the BTSHI-DT systems are calculated for system

(2.16) with sample and hold parameters of δ = 0.5s and p = 0.8 and the ZOSHI-DT system

is calculated with δ = 0.5s. For comparison purposes, the pole-zero plot is shown in Fig. 5:

Figure 2.3: Pole-zero locations of all SHIs

From Fig. 2.3, the poles of the DT systems, represented by cross marks, are identical.

Circles, triangles, squares, and pentagrams are used to represent the zero locations of the

SPSHI-DT, the FTSHI-DT, the BTSHI-DT, and the ZOSHI-DT systems, respectively. For

system (2.16), when the SHIs are used, the zeros of the corresponding DT systems are moved

from the locations associated with the conventional ZOH DT system toward the unite circle.

Note that sampling period of 0.5s is smaller than the smallest feasible sampling period of

ZOH method. According to Fig. 2.2, the choice of δ = 0.5, p = 0.8 is within the feasible

region of SPSHI-DT and FTSHI-DT but not of BTSHI-DT. As a result, the zeros of the

14

SPSHI-DT and FTSHI-DT systems are shifted inside the unit circle, while the zeros of the

BTSHI-DT system were also moved toward the unit circle but still is outside. It is interesting

to observe that for all three DT systems with different SHIs, the movement of the real zero

is much larger than the complex zeros.

2.2.2 Numerical Example 2.

Consider the transfer function from [9] as the second example

G(s) =

6(1 − s)

(s + 2)(s + 3)

(2.17)

This stable NMP system has an unstable zero located at 1. The feasible regions of the

SPSFI-DT, the FTSHI-DT, and the BTSHI-DT systems are overlaid in Fig. 2.4. For system

(2.17) the shapes of the feasible regions are quite different from those in Fig. 2.2, where Fig.

2.2 shows discontinuous feasible regions and the regions in Fig. 2.4 are continuous. This

implies that for different systems, their feasible regions could be quite different. According

to [9], the ZOSHI-DT system is MP with a sampling period not smaller than 1.2485s and Fig.

2.4 confirms it; see the top boundary of the SPSHI-DT feasible region. Note that when p = 1,

the SPSHI-DT system is equivalent to the ZOSHI-DT system. With the FTSHI, the feasible

sampling period can be as small as 0.69s when the duty cycle is 1. If a smaller duty cycle is

allowed, the feasible sampling period can be reduced further for all three SHIs. For system

(2.17) the lower feasible sampling period boundary is 0.4s (a three-time reduction). Note that

the bandwidth of system (2.17) is 1.227 Hz and the lower bound of the SHI feasible sampling

frequency is 2.5Hz, which is about twice of the system bandwidth. Although the SHI feasible

sampling frequency boundary is relatively low, compared to the system bandwidth, it is still

more than two times of the system frequency; and in addition, it is more than three times

higher than the feasible sampling frequency of conventional ZOH (0.8Hz). Further study

will be conducted to observe the closed-loop system performance using SHI.

15

Figure 2.4: Feasible regions of all SHIs overlaid for the second example

From the aspect of selecting the feasible sample and hold parameters, the FTSHI has the

outstanding advantage compared with the SPSHI and the BTSHI.

2.3 A framework for DT system identification

The results from the previous section show that it is possible to use a special SHI to

obtain a discrete-time MP system even though the original continuous-time system is NMP.

The SHIs are designed to be implemented into the physical control system. Note that

a wave-generating circuit can be used to generate pulse, forward or backward triangular

signals and a variable-gain amplifier, utilizing the generated signal as input, can be used to

generate the sample and held signals. The amplifier gain is the control input. Finally, an

ASIC (application specific integrated circuit) can be designed and used as the SHIs. The

next two sections investigate the SHI robustness when the SHI is not implemented precisely.

In practice it is impossible to implement the required SHI precisely, which could make the

resulting DT system not always be MP even though in theory it should be. To study the

SHI robustness, HIL simulation approach was used to simulate the system responses with an

inexactly implemented SHI input; based on the simulation results, system identification was

used to obtain the correspondent DT system model, and its MP property is correlated to

the implementation error. This section presents the system identification (ID) approach and

the next section discusses the SHI robustness using a system ID base on the HIL simulation

results.

16

2.3.1 System Identification

The system ID based upon the q-Markov COVariance Equivalent Realization (q-Markov

Cover) [51], [52] with pseudo-random binary signal (PRBS) as system excitation was used

to obtain the DT system model. The q-Markov Cover system ID is capable of identifying

discrete-time system model with multiple sampling rates. However, a single sampling rate

q-Markov Cover was used in this study and the PRBS excitation was sampled and held by

SPSHI, FTSHI, or BTSHI. The associated system output, along with the PRBS input, were

used for system ID to obtain the corresponding DT system model. The system ID process

is depicted in Fig. 2.5.

Figure 2.5: Q-Markov Cover system ID process with PRBS excitation

2.3.2 Method and Results

The PRBS sampling period is defined as δ to be consistent with the notation for SHI. In

principal, the order of PRBS shall be small to have a short period. However PRBS order is

inversely proportional to its lowest frequency component contained in the signal. Therefore

the order selection is to cover the lowest frequency of the dynamic system to be identified

with the lowest order possible. For this study the PRBS order was selected to be 11. To

generate the input to the continuous-time NMP system defined in (2.16), PRBS was sampled

and held by the SHI functions defined in Table 1 and the resulting PRBS are shown in Fig.

2.6 for δ = 0.5s and p = 0.3.

17

NMP SystemSystem ResponseQ-Markov Cover System ID DT System PRBSSHIFigure 2.6: PRBS sampled and held by ZOSHI, SPSHI, FTSHI, and BTSHI

The system responses of the continuous-time NMP system in (2.16) using the signals

shown in Fig. 2.6 are shown in Fig. 2.7.

Figure 2.7: System responses associated with ZOSHI, SPSHI, FTSHI, and BTSHI

Note that system (2.16) has a bandwidth of 0.1125Hz. The DT system sample rate is

2.0Hz, which is more than 10 times faster than the system bandwidth. According to [53],

the DT system should represent the continuous-time dynamics well. To accurately simulate

the SHI shape, the simulation step size of 0.001s was selected in Simulink.

Based upon the discrete-time PRBS and the associated sampled system response, the

system ID was conducted to obtain the associated DT model denoted as ID DT model.

From Fig. 2.2 (d), to ensure the ID DT models to be MP with exact SHIs, the sampling

period and duty cycle for all SHIs were chosen to be 0.5s and 30%, respectively. The system

poles and zeros are calculated using the DT system analytical equations (2.12), (2.13), and

(2.14) directly based on the ID DT models. The absolute values of poles and zeros are shown

in Table 2.2, where rows denoted by “AN” are the absolute values of poles and zeros from

the analytical equations, the rows of “ID” are associated with the poles and zeros of ID DT

18

SPSHIZOSHIFTSHIBTSHISPSHIZOSHIFTSHIBTSHITable 2.2: Pole-Zero Location of the DT, ID, and HIL-ID Systems

p

Z

ZOSHI

AN
ID

0.61
0.61

0.61
0.67

0.61
0.61

0.37
0.36

1.00
1.00

1.00
1.00

1.51
1.52

SPSHI

AN
ID

0.61
0.61
HIL-ID 0.61

0.61
0.61
0.61

0.61
0.62
0.62

0.37
0.32
0.34

0.95
0.95
0.95

0.95
0.95
0.95

0.23
0.25
0.25

FTSHI

AN
ID

0.61
0.61
HIL-ID 0.61

0.61
0.61
0.61

0.61
0.62
0.62

0.37
0.33
0.33

0.95
0.95
0.95

0.95
0.95
0.95

0.19
0.19
0.21

BTSHI

AN
ID

0.61
0.61
HIL-ID 0.61

0.61
0.61
0.61

0.61
0.62
0.62

0.37
0.34
0.34

0.94
0.94
0.94

0.94
0.94
0.94

0.28
0.31
0.29

models, and rows of “HIL-ID” will be discussed in the next section. From Table 2.2 since the

SHIs were implemented precisely, the pole and zero locations of the ID DT models and the

corresponding DT systems are very close in the sense of their absolute values. This implies

that the q-Markov Cover system identification using PRBS is able to identify the DT system

model accurately. It is concluded that it can be used for studying the MP characteristics

with SHI implementation error through HIL simulations.

2.4 Experimental validation using HIL simulations

In this section, HIL simulations were used to study the robustness of the SHIs under

different levels of unmodeled actuator dynamics to emulate the practical implementation of

SHIs. Q-Markov Cover system ID was conducted based on the PRBS input and the response

of dSPACE-simulated system (2.16). The robustness property of all the SHIs is observed by

19

comparing the zero locations of the DT systems with and without actuator dynamics.

2.4.1 HIL Simulation Experiment Setup

Figure 2.8: HIL simulation setup for studying SHI robustness

A dSPACE auto-box and a resistance-capacitor (RC) filter circuit were used for the HIL

simulations; see Fig. 2.8. The dSPACE auto-box was used to simulate system (2.16); it also

converts the discrete-time control signal (PRBS input) into sample and hold signal based on

the SHI and sends the signal to its D/A output channel. The analog signal is an RC filter

and sampled by the dSPACE A/D converter, where the sampling frequency is 1000Hz, which

is significantly higher than the SHI sampling rate (1/δ). The RC filter circuit was used to

simulate the implementation error of the sample and held input with a 1kΩ resistor and a

selection of capacitor of 1µF, 3.3µF, 10µF, 47µF, or 100µF. Note that the 1µF capacitor

provides the smallest unmodeled actuator dynamics and the 100µF capacitor provides the

largest one.

2.4.2 HIL Simulation Procedure

The PRBS was sampled and held by SPSHI, FTSHI, and BTSHI through real-time simula-

tions in dSPACE with the feasible sample and hold parameters of δ = 0.5s and p = 0.3. Note

that the time constant of the RC filter is much smaller than that of the simulated system.

20

NMP SystemQ-Markov Cover System ID DT System PRBSdSPACEAUTOBOXRC FilterSHIZOHTherefore, the RC filter dynamics was not significant enough to be included in the ID DT

model.

Figure 2.9: Filtered PRBS sampled and held by (a) SPSHI, (b) FTSHI, (c) BTSHI

Before simulating using different RC filters, the HIL simulation was proceeded with no

actuator dynamics, that is, R=C=0. In this case, the poles and zeros of the ID DT system

based on the HIL simulation results matched with those of the theoretical DT system, as

expected. The magnitudes of the poles and zeros of the analytical (AN) and HIL-ID DT

systems are shown in Table 2.2. When the RC circuit is used in the HIL simulations, the

input to the system generated by the SHIs is distorted. The level of distortion increases as

the RC circuit time constant gets large. The input signals filtered by the RC circuit (with

different C values) are shown in Fig. 2.9.

Note that for the 1µF and 3.3µF cases, the actuator dynamics effects are not significant

enough to alter the input signal. For the 10µF, 47µF, and 100µF cases, as the capacitance

increases, the input signals become fairly smooth and their magnitudes decrease dramatically.

21

0 µF1 µF3.3 µF10 µF47 µF100 µF00.10.20.30.40.50.60.70.80.9100.5100.10.20.30.40.50.60.70.80.9101200.10.20.30.40.50.60.70.80.91012(a)SPSHI(b)FTSHI(c)BTSHI2.4.3 HIL Simulation Results

The ID DT system of (2.16) has three zeros. The HIL simulation indicates that for the

selected values of

p = 0.3, δ = 0.5, C = 1, 3.3, 10, 47, 100µF

(2.18)

two of the zeros are MP complex conjugate pairs; they are marginally affected by the

unmodeled actuator dynamics. The third zero is inside the unit circle for small unmodeled

actuator dynamics; and it moves outside the unit circle when the unmodeled actuator dy-

namics is increased to a certain level. Another HIL simulation was conducted with δ = 0.5s

and p = 0.6. The HIL simulation results of SPSHI, FTSHI, and BTSHI with two p values

are shown in Table 2.3. It is clear from Table 2.3 that the real zero is less sensitive to the

unmodeled actuator dynamics when the duty cycle is smaller.

The locations of the real zeros of the ID systems with 30% and 60% duty cycles are

plotted in Fig. 2.10 as functions of the capacitance (unmodeled actuator dynamics).

It

confirms that the FTSHI and BTSHI has the best and worst robustness, respectively.

Table 2.3: The Third Zero Locations of the HIL ID Systems With δ = 0.5s, P = 0.6 and 0.3

Capacitance (µ F)

SPSHI

FTSHI

BTSHI

1
3.3
10
47
100

p = 0.6

p = 0.3

p = 0.6

p = 0.3

p = 0.6

p = 0.3

-0.45
-0.46
-0.49
-0.95
-1.05

-0.25
-0.25
-0.27
-0.35
-0.55

-0.32
-0.32
-0.34
-0.58
-0.74

-0.21
-0.21
-0.22
-0.30
-0.46

-0.62
-0.64
-0.67
-0.99
-1.70

-0.29
-0.30
-0.32
-0.42
-0.65

As a summary, based on the HIL simulation study, the MP characteristics of SHIs are

affected by both the level of unmodeled actuator dynamics (SHI implementation error) and

the selection of feasible SHI parameters. For a given sampling period, small duty cycle

improves the SHI robustness to unmodeled actuator dynamics; see Fig. 2.10. Among the

22

Figure 2.10: Zero movements of the HIL ID systems with two set of feasible sample and
hold parameters

three SHIs, the BTSHI has the worst robustness property, while the FTSHI provides the

best robustness and smallest feasible sampling period with a given duty cycle.

23

0102030405060708090100-1.8-1.6-1.4-1.2-1-0.8-0.6-0.4-0.20  SPSHI with p = 0.3SPSHI with p = 0.6FTSHI with p = 0.3FTSHI with p = 0.6BTSHI with p = 0.3BTSHI with p = 0.6Zero ValueCapacitanceCHAPTER 3

DEMONSTRATION OF PERFORMANCE IMPROVEMENT I

3.1 Single inverted pendulum (SIP) system

A Quanser SIP system is used for the purpose of demonstration - see Fig.3.1. The system

is comprised of a pendulum hinged on a cart that moves freely on a horizontal track, where

xc is the cart displacement, α is the pendulum angle, u is the control input, and f is the

overall friction.

Figure 3.1: The Single Inverted Pendulum System

The SIP system parameters are defined in Table II. The free body diagram of the SIP is

shown in the following figure.

The EOMs of the SIP system is derived as follows: Summing the horizontal force of the

24

Figure 3.2: Inverted Pendulum Free Body Diagram

cart

F = Mc ¨x + Beq ˙x + N + Fa

(3.1)

where N is the reaction force between cart and pendulum. Fa is the inertia force due to the

armature motor rotation. For the reaction force N , it is represented as

N = Mp ¨xp

where xp is the pendulum horizontal displacement. It can be specified as

xp = x − lp sin α
˙xp = ˙x − lp ˙α cos α
¨xp = ¨x − lp ¨α cos α + lp ˙α2 sin α

(3.2)

(3.3)

where x is the cart position. Substitute the ¨xp into (3.2), the reaction force N becomes

N = Mp ¨x = Mp(¨x − lp ¨α cos α + lp ˙α2 sin α)

the motor running armature inertia force Fa is specified as

Fa =

ηgKgTa

rm

(3.4)

(3.5)

where ηg is the gearbox efficiency Kg is the gearbox ratio. Ta is the motor torque, which

can be written as

Ta = Jm ¨α =

JmKg ¨x

rm

25

(3.6)

Jm is the motor moment of inertia, and rmp is the motor pinion radius. The resultant inertia

force becomes

Fa =

ηgK2
g Jm ¨x
r2
m

(3.7)

So, the first equation of motion becomes

F = (Mc + Mp +

g Jm

ηgK2
r2
mp

)¨x + Beq ˙x − Mplp ¨α cos α + Mplp ˙α2 sin α

(3.8)

In this EOM, F is the driving force from the motor.

F =

ηgKgTm

rmp

in where Tm is the motor torque specified as

Tm = ηmKt

(3.9)

(3.10)

ηm is the motor efficiency and Kt is the motor torque constant. As a result, the driving

force is as

F =

ηgηmKgKt(rmV − KgKm ˙x)

Rmr2
m

(3.11)

Rm is the motor armature resistance. For the second EOM, sum the moments about the

centre of the pendulum,

P lp sin α + N lp cos α − Bp ˙α = Jp ¨α

(3.12)

where Bp is the viscous damping coefficient at the pendulum pivot. Jp is the pendulum

moment of inertia. Since the EOM is about the center, Jp is calculated as

Jp =

1
12

MpL2

p =

1
3

Mpl2
p

To solve this EOM, the vertical displacement need to be considered

yp = lp cos α
˙yp = −lp ˙α sin α
¨yp = −lp ¨α sin α − lp ˙α2 cos α

26

(3.13)

(3.14)

As a result, the vertical force applied on the pendulum is as

P = Mp(¨yp + g) = −Mplp ˙α2 cos α − Mplp ¨α sin α + Mpg

(3.15)

substitute (3.14) & (3.15) into (3.12), which leads to the second EOM:

Jp ¨α = −Mpl2

p ¨α + Mpglp sin α + Mplp ¨x cos α − Bp ˙α

(3.16)

3.2 Dynamic Model with Coulomb Friction Term

Figure 3.3: Coulomb Friction

The coulomb friction can be written as

v(t) (cid:54)= 0 : Ff (t) = µFN sgn(v(t))

(3.17)

v(t) is the cart velocity ( ˙x) and FN is the gravity force, which is the cart and pendulum

weight in force:

FN = (Mc + Mp)g

(3.18)

Summing the horizontal force of the cart

F = Mc ¨x + Beq ˙x + µ(Mc + Mp)g ˙x + N + Fa

(3.19)

27

where N is the reaction force between cart and pendulum. Fa is the inertia force due to the

armature motor rotation. For the reaction force N , it is represented as

N = Mp ¨xp

(3.20)

where xp is the pendulum horizontal displacement. It can be specified as

xp = x − lp sin α
˙xp = ˙x − lp ˙α cos α
¨xp = ¨x − lp ¨α cos α + lp ˙α2 sin α

(3.21)

where x is the cart position. Substitute the ¨xp into (3.20), the reaction force N becomes

N = Mp ¨x = Mp(¨x − lp ¨α cos α + lp ˙α2 sin α)

the motor running armature inertia force Fa is specified as

Fa =

ηgKgTa

rm

(3.22)

(3.23)

where ηg is the gearbox efficiency Kg is the gearbox ratio. Ta is the motor torque, which

can be written as

Ta = Jm ¨α =

JmKg ¨x

rm

(3.24)

Jm is the motor moment of inertia, and rmp is the motor pinion radius. The resultant inertia

force becomes

Fa =

ηgK2
g Jm ¨x
r2
m

(3.25)

So, the first equation of motion becomes

F = (Mc + Mp +

g Jm

ηgK2
r2
mp

)¨x + Beq ˙x + µ(Mc + Mp)g ˙x − Mplp ¨α cos α + Mplp ˙α2 sin α (3.26)

where µ is the Coulomb friction coefficient and F is the motor driving force that given by

the expression of

F =

ηgKgTm

rmp

28

(3.27)

in where Tm is the motor torque specified as

Tm = ηmKt

(3.28)

ηm is the motor efficiency and Kt is the motor torque constant. As a result, the driving

force is as

F =

ηgηmKgKt(rmu − KgKm ˙x)

Rmr2
m

(3.29)

Rm is the motor armature resistance. For the second EOM, sum the moments about the

center of the pendulum,

P lp sin α + N lp cos α − Bp ˙α = Jp ¨α

(3.30)

where Bp is the viscous damping coefficient at the pendulum pivot. Jp is the pendulum

moment of inertia. Since the EOM is about the center, Jp is calculated as

Jp =

1
12

MpL2

p =

1
3

Mpl2
p

(3.31)

To solve this EOM, the vertical displacement need to be considered. As a result, the vertical

force applied on the pendulum is as

P = Mp(¨yp + g) = −Mplp ˙α2 cos α − Mplp ¨α sin α + Mpg

(3.32)

substitute (3.14) & (3.32) into (3.30)

Jp ¨α = −Mpl2

p ¨α + Mpglp sin α + Mplp ¨x cos α − Bp ˙α

which leads to the second EOM of

0 = (Jp + Mpl2

p)¨α − Mpglp sin α − Mplp ¨x cos α + Bpα

(3.33)

(3.34)

where the motor inductance is ignored. The values of the parameters that used in the

derivation process are provided in TABLE 3.1. The equilibrium point considered for the SIP

system is the pendulum upright position with the cart at the origin: α = 0 and xc = 0. For

29

Table 3.1: QUANSER inverted pendulum system parameters [1]

Symbol

Description

Value/Unit

R
Kt
ηm
Km
Jm
Kg
ηg
rm
rp

Beq

Bp

lp

Jp
Mp
Mc
Vm
µ

Motor Armature Resistance

2.6 Ω

Motor Torque Constant

0.00767 Nm/A

Motor Efficiency

100%

Motor EMF Constant

Rotor Moment of Inertia

0.00767 Ns/rad
3.9 × 10−7 kgm2

Gearbox Ratio

Gearbox Efficiency
Motor Pinion Radius
Position Pinion Radius

Equivalent Viscous Damping

Coefficient at Motor

Viscous Damping Coefficient

at Pendulum Pivot

Pendulum Length from
Pivot to Center of Mass

3.71
100%

6.35 × 10−3 m
1.48 × 10−2 m
5.4 Nms/rad

5.4 Nms/rad

0.3302 m

Pendulum Moment of Inertia

0.00788 kgm2

Pendulum Mass

Cart Mass

Motor Nominal Input Voltage

Coulomb friction coefficient

0.23 kg
0.94 kg

5 V
1.45

control design, the friction is assumed to zero. The linearized state space model with motor

driving force as input is given by (2.2), where x = [ xc α ˙xc



˙α ]T and

0

1

MplpBp

q

(Mc+Mp)Bp

q

(3.35)

0

0

0

0




A =

B =

0

0

q

q

 ,

0

0
Mpl2
p+Jp
q

Mplp

q

(Mplp)2

(Mc+Mp)Mpglp

1

0

q

−Beq(Mpl2

p+Jp)

MplpBeq

q

1 0 0 0


0 1 0 0

C =

30

In (3.35), q is defined as

q (cid:44) (Mc +

ηgK2
g Jm
r2
m

+ Mp)Jp + (Mc +

ηgK2
g Jm
r2
m

)Mpl2
p

For the purpose of state feedback, ˙xc and ˙α are obtained by low-pass filtering the derivatives

of xc and α.

For the given system parameters listed in TABLE 3.1, the system model matrices in

(3.35) are

A =



0

0

1

0

0

0

1

0
0 1.522 −4.787 −0.005
0 26.110 −11.029 −0.084

 , B =





0

0

0.886

2.043

(3.36)

The linearized model in (3.36) has an unstable pole at s = 4.90. Since the SHIs can only be

implemented for stable systems [35], a stabilizing controller will be designed first.

3.3 Dual-loop controller design

3.3.1

Inner-loop Stabilizing Controller

To stabilize the SIP and improve its performance using SHIs, a dual-loop control system

configuration is proposed - see Fig. 3.4. The inner loop (shown in the dashed region) uses a

continuous-time LQR implemented at 500 Hz sampling rate; it provides the control input v

to stabilize the SIP about its equilibrium configuration.

The cost function of the LQR is

(cid:90) ∞

0

J =

1
2

(cid:104)

x(t)T Qx(t) + v(t)T Rv(t)

(cid:105)

dt

(3.37)

where Q ≥ 0 and R > 0 are the weighting matrices. The state feedback controller is given
by u = −Kx, where the control gain K = R−1BT P and P is positive semidefinite solution

of the Ricatti equation

P A + AT P + Q − P BR−1BT P = 0

(3.38)

31

 =





˙xc

˙α

¨xc

¨α

0

0

0

0

1

0

0

1

64.019 −289.00
61.574 −42.999
147.507 −643.282 141.874 −99.147

(cid:21)



and the stabilized state



xc

α

˙xc

˙α



(3.39)

Figure 3.4: The dual-loop controlled SIP system

We used the LQR designed by Quanser with Q = diag[35, 350, 0.1, 0.1] and R = 0.02. The

resultant control gain was K =

41.83 −189.84 47.85 −28.09

equation of the inner-loop is:

(cid:20)



0

0

1.530

3.526

 w

+

Note that the inner-loop controller is discretized and implemented in the Quanser control

hardware. The inner-loop simulation layout is shown in Fig. 3.5

32

++Figure 3.5: The inverted pendulum simulation layout

3.3.2 SHI Parameter Selection For MP Characteristics

We begin by converting the system in (3.39) first into the DE system in (2.4) and then into

the DT system in (2.11). The Bd matrix in (2.11) will be chosen from (2.12), (2.13) and

(2.14) depending on the choice of SHI: SPSHI-DT, FTSHI-DT, and BTSHI-DT. The SHI
parameters p ∈ (0, 1] and δ ∈ (0, 1] will be chosen such that the DT system is MP. Parameters

p and δ that result in MP characteristics are defined as feasible duty cycle (FDC) and feasible

sampling period (FSP), respectively; the set of FSP and FDC is defined as the p-δ feasible

region. Since SIP is a two-output system, the SHI parameter selection should be based on

both transfer functions. The p-δ feasible regions of SPSHI-DT, FTSHI-DT and BTSHI-DT

systems for both transfer functions are shown in Fig. 3.6.

The ZOH is a special case of the SPSHI (p = 1). The p-δ feasible region for the ZOH,

shown by the thick horizontal line on the top of Fig. 3.6, indicates that the smallest FSP

is 0.89s (1.12 Hz). Fig. 3.6 also shows that the FSP can be reduced to 0.173s (5.78 Hz)

by using the SHIs with extremely small duty cycles; this will, however, result in large peak

magnitudes - see TABLE 2.1.

The Bode plots of the system in (3.39) indicate that the bandwidths of the stabilized

system are 0.29Hz (cart displacement) and 2.53Hz (pendulum angle). For implementation,

the outer-loop sampling rate should be at least twice the bandwidth [53] of the stabilized

33

Figure 3.6: The SIP p-δ feasible regions of BTSHI (Region 1), SPSHI (Region 1+2) and
FTSHI (Region 1+2+3).

SIP, implying that the sampling rate should be at least 5Hz. From practical considerations,

the sampling rate should be at least five times the bandwidth; this corresponds to a sampling

rate of 12.5Hz (0.08s sampling period). According to Fig. 3.6, the minimum FSP is 0.173s

(5.78 Hz) - this implies that it will not be possible to adequately control the pendulum.

Since the cart dynamics is considerably slower (0.29Hz), FSPs of 0.3s (3.3Hz) and 0.4s (2.5

Hz) can improve the performance of the cart displacement. Although the minimum FSP is

0.173s, higher values were chosen to avoid large peak input that may exceed the actuator

limit.

From Fig. 3.6, it can be seen that for the SPSHI, the maximum duty cycles for sample

periods of 0.3s or 0.4s are 0.656 or 0.798, respectively. The FTSHI system has a larger p-δ

feasible region and the corresponding feasible duty cycles are 0.886 and 1.0 for 0.3s and

0.4s sampling periods, respectively. The selected SHI parameters are marked with dotted

lines in Fig. 3.6. Based on the results in [35] and the p-δ feasible regions in Fig. 3.6, the

BTSHI-DT is not considered. The circle and square marks along the δ = 0.3s dashed-line

will be discussed later.

34

3.3.3 Outer-Loop Controller Design

A discrete-time LQR is designed for the outer-loop to provide additional control effort to

improve the performance of the stabilized DT system. Consider the cost function:

xT (k)Qdx(k) + wT

SHI (k)RdwSHI (k)

(3.40)

(cid:105)

∞(cid:88)

(cid:104)

k=0

J =

1
2

(cid:21)

where Qd = diag[10, 1000, 1, 1] and Rd = 0.02. These values were chosen through

trial and error to provide adequate control authority and avoid actuator saturation. For

the choice of SPSHI with p = 0.2 and δ = 0.3s, the LQR gains were obtained as Kd =
−8.63 12.80 −7.99 5.45

. The dual-loop simulation layout is shown in Fig. 3.7

(cid:20)

Figure 3.7: The inverted pendulum dual-loop simulation layout

3.4 Simulation and experimental validation

3.4.1 Experimental Set Up

To evaluate the effectiveness of the dual-loop control design, simulations were first conducted

and then validated through experiments using a Quanser SIP system [54]. The experimental

35

Figure 3.8: The SIP experiment hardware connection

hardware configuration is shown in Fig. 3.8.

In this system, the cart is driven by a motor with a planetary gear head. The cart displace-

ment is measured by a US Digital E2 encoder with a resolution of 0.023 mm. The pendulum

angle is measured by another similar encoder with a resolution of 0.018 deg (an upgrade

from the Quanser provided encoder with 0.086 deg resolution). The Quanser Q8-USB DAQ

system is used for data acquisition and control; the control signal to the motor is amplified

by the Quanser VoltPAQ. It should be mentioned that both control loops are implemented

in the same host computer and no additional hardware is therefore required.

3.4.2 Simulation and Experiment Results

The simulation and experimental results of cart displacement and pendulum angle for SPSHI-

DT systems are shown in Figs. 3.10 (a) and (c) for δ = 0.3s and in Figs. 3.10 (b) and (d)

for δ = 0.4s. The results for FTSHI-DT systems are similarly shown in Fig. 3.12.

In

all the figures, dotted lines are used to show simulation results and solid lines are used

for experimental results. The results are shown for three time segments: the first segment

36

Figure 3.9: The SIP experiment set up in the lab

(0−20s) corresponds to the response of the system in the absence of the outer-loop controller,
the second segment (20 − 40s) corresponds to dual-loop controller with p = 0.2 for SPSHI
and p = 0.4 for FTSHI, the third segment (40 − 60s) corresponds to p = 1.0 for both SHIs.

The choice of p = 1.0 is used to compare with ZOH.

The oscillations in the responses are mainly due to the Coulomb friction between the

cart and track; it was artificially introduced in the simulation model (see (3.26)), where the

value of µ = 1.45 was used to achieve a close match with experimental results.

The SIP system was studied in both simulations and experiments. For each SHI (SPSHI

or FTSHI) and sampling period (δ = 0.3s or δ = 0.4s), the total run time was 300s: started

with 50s of inner-loop only and followed by 50s of dual-loop control with p = 0.2. For

each additional 50s, the duty cycle is increased by another 0.2 until it reached 1.0. The

corresponding cart displacement, pendulum angle, and control voltage signals for δ = 0.3s

are shown in Figs. 3.10 (a), (b), and (c) for SPSHI-ST system and in Figs. 3.11 (a), (b),

and (c) for FTSHI-DT system, respectively.

37

Similarly, the simulation and experiment results for δ = 0.4s are shown in Figs. 3.12 (a),

(b), and (c) for the SPSHI-DT system and in Figs. 3.13 (a), (b), and (c) for the FTSHI-DT

system, respectively.

The simulation and experimental results are overlaid for each case using dotted and solid

lines, respectively. The vertical straight black-dashed lines are used to mark these 50s time

intervals. The oscillation responses in the experiments are mainly due to the coulomb friction

between the cart and track. The encoder noises are also contributing to the oscillation,

however, the contribution is minor due to its high resolution. These factors were simulated

by tuning the friction coefficient to match the simulated and experimental results.

Table 3.2: Experimental results for inner-loop LQR and dual-loop control with SPSHI and
FTSHI

p

|x|avg(mm) |x|avg(mm) |u|avg(V ) |u|avg(V )
δ = 0.4s
δ = 0.3s

δ = 0.3s

δ = 0.4s

Inner-loop N/A

16.440

SPSHI

FTSHI

0.2
0.4
0.6
0.8
1.0

0.2
0.4
0.6
0.8
1.0

5.105
5.371
8.125
14.477
22.233

8.481
7.348
7.792
9.286
12.427

9.307
9.322
10.384
13.559
21.533

12.115
10.970
10.615
11.237
12.711

0.806

0.531
0.578
0.712
0.771
0.859

0.677
0.644
0.698
0.758
0.826

0.717
0.771
0.840
0.866
1.018

0.812
0.777
0.843
0.890
0.889

The mean absolute values of cart displacement (|x|avg) and overall control voltage (|u|avg)
of experimental results are shown in TABLE 3.2 with δ = 0.3s and δ = 0.4s. The mean

38

Figure 3.10: The simulated and experimental results of the SPSHI-DT with δ = 0.3s for
the SIP system

absolute value for a variable is defined as the ratio of the L1-norm of the sampled data vector

and vector size. The best performances in terms of duty cycle p are highlighted with boxes

and marked with circle and square in Fig. 3.6 for SPSHI and FTSHI, respectively. The

experimental results in TABLE 3.2 show that with δ = 0.3s and p = 0.2, the mean absolute

value of cart displacement amplitude reduces significantly from 16.440 mm (inner-loop only)

39

Figure 3.11: The simulated and experimental results of the FTSHI-DT with δ = 0.3s for
the SIP system

down to 5.105 mm (68.95% reduction) for SPSHI and to 7.348 mm (55.33% reduction) for

FTSHI with δ = 0.3s and p = 0.4. The corresponding mean absolute values of the control

signals are also reduced from 0.806V (inner-loop only) to 0.531V (34.1% reduction) for the

SPSHI and to 0.644V for the FTSHI. The experimental control voltage signals show that

use of SHIs does increase control effort in terms of mean absolute value.

40

Figure 3.12: The simulated and experimental results of the SPSHI-DT with δ = 0.4s for
the SIP system

For the purpose of comparison, the best experimental results for SPSHI and FTSHI are

compared with the result obtained with the inner-loop controller only - see Fig. 3.14 for

cart displacement amplitudes and overall control signals, respectively. The SPSHI is seen to

outperform the FTSHI.

In order to understand the trend of system response under different SHIs, all the data

41

Figure 3.13: The simulated and experimental results of the FTSHI-DT with δ = 0.4s for
the SIP system

in TABLE 3.2 is plotted in Fig. 3.15. The |xc|avg for SPSHI and FTSHI are shown in Figs.
3.15 (a) and (b) for δ = 0.3s and 0.4s. Similarly, the |u|avg for SPSHI and FTSHI are shown
in Figs. 3.15 (c) and (d). Both the SPSHI and FTSHI outperform the baseline controller
in terms of |xc|avg. Also, the higher sampling rate δ = 0.3 requires less control effort |u|avg
while providing better performance in general.

42

Figure 3.14: The best experimental results of dual-loop with SPSHI and FTSHI compare
with inner-loop controller

43

Figure 3.15: The trend of dual-loop control response

44

CHAPTER 4

DEMONSTRATION OF PERFORMANCE IMPROVEMENT II

4.1 Segway

Mini segway (MS) is a two-wheel self-balanced robot. It utilizes a microcontroller (Ar-

duino), two motor-wheel sets and a cart body. The free body diagram of the MS system is

shown in Fig. 4.1, where two motors provide the torques (τL and τR) to prevent the cart

body from falling and also to move the cart forward, backward and make it turn; xcart is

the cart displacement in x direction; and θp and φ are the cart body roll angle and cart yaw

angle, respectively.

Figure 4.1: Free body diagram of the mini segway robot

4.1.1 Motor Modeling

Sine the left and right motor/wheel are identical, only the derivation for the left motor/wheel

is showed here. When the control current iL is applied, the motor torque is as

τmL = KmiL

(4.1)

45

Figure 4.2: Motor diagram

and there is back electro magnetic voltage VeL, defined as follows

VeL = KeωL

From Kirchhoff’s voltage law of the DC motor, we have

uL − RmiL − Lm

diL
dt

− VeL = 0

which results in the DC motor linear equation:

diL
dt

=

uL
Lm

− Rm
Lm

iL − VeL
Lm

The DC motor has the torque equation of

ΣM = τmL − Kf ωL − τaL = Iw

dωL
dt

as a result, the second DC motor linear equation is as

dωL
dt

=

Km
Iw

iL − Kf
Iw

ωL − τaL
Iw

So, the DC motor EOM for the left wheel is as

 diL

uL
dt =
Lm
dωL
dt = Km
Iw

− Rm
Lm
iL − Kf

iL − Ke
ωL
ωL − τaL

Lm

Iw

Iw

46

(4.2)

(4.3)

(4.4)

(4.5)

(4.6)

(4.7)

TmL + U£ TLSimplifying the motor dynamics by setting the inductance to zero

 iL =

ωL

− Ke
uL
Rm
Rm
iL − Kf

Iw

Iw

dωL
dt = Km
Iw

ωL − τaL
So the approximation of the motor angular accelerations are
ωL − Kf
Iw

uL − KmKe
IwRm

Km
IwRm

dωL
dt

=

ωL − τaL
Iw

combining equation (4.5) and (4.9), the motor generated torque is as

τmL =

−KmKe

Rm

ωL +

Km
Rm

uL

4.2 Equations of the Wheels

(4.8)

(4.9)

(4.10)

Figure 4.3: Free Body Diagram of the Wheel

In the cart model, the left motor angular speed is replaced by ˙θL and the right motor
angular speed is ˙θR. From the motor equation, the driving torques on the left and right

47

wheels are what’s left from the motor generated torque fights against motor internal friction.

 τL = −KmKe

τR = −KmKe

Rm

Rm

˙θL + Km
Rm
˙θR + Km
Rm

uL − Kf
uR − Kf

˙θL
˙θR

(4.11)

Summing the moments around the contacting point of the left wheel and ground:

ΣML = τL − FLr = τmL − Kf

˙θL − FLr = (Iw + r2mw)¨θL

(4.12)

where FL is the cart to left wheel force. Assuming no slip, the driving force on both left and

right wheels are as

FL =

−KmKe
Rmr
−KmKe
Rmr
After linear transformation of:

FR =

˙θL +

˙θL +

− Iw ¨θL
− mw ¨θLr
r
− Iw
¨θR − mw ¨θRr
r

Km
Rmr
Km
Rmr

˙θL
uL − Kf
r
˙θR
uR − Kf
r

˙φh
r +
r
r − ˙φh

 ˙θL = ˙x
 xL = x + φh

xR = x − φh

˙θR = ˙x

r

(4.13)

(4.14)

(4.15)

(4.16)

and

The driving forces become

FL =

FR =

−KmKe
Rmr2
−KmKe
Rmr2

˙xL +

˙xR +

Km
Rmr
Km
Rmr

uL − Kf ˙xL
uR − Kf ˙xR

r2 − Iw
r2 − Iw

r2 ¨xL − mw ¨xL
r2 ¨xR − mw ¨xR

4.3 Equations of the Pendulum

Summing the force in x-axis of the pendulum:

ΣFx = FL + FR = mp ¨xp

(4.17)

Since the pendulum acceleration in x-axis is as in (3.21), so equation (4.17) becomes

ΣFx = FL + FR = mp ¨xp = mp ¨x − mpl ˙θ2

psinθp + mpl ¨θpcosθp

(4.18)

48

Figure 4.4: Free Body Diagram of the Pendulum

Substitute equation (4.16) into equation (4.18) and we get the first segway EOM:

−2KmKe

Rmr2 ˙x +

Km
Rmr

(uL + uR) − (

2Iw

r2 + 2mw)¨x − 2Kf ˙x

r2 = mp ¨x − mpl ˙θ2

psinθp + mpl ¨θpcosθp

Summing the moments around the pendulum joint and we have the second EOM:

(Ip + mpl2)¨θp = −mpl¨xcosθp + mplgsinθp − (τL + τR)

(4.19)

(4.20)

When the cart is making a turn, the difference between the generated forces of the two

wheels result in the pendulum spinning about the Z-axis, which is defined in the following

figure.

The associated governing equation is as:

¨φIpz = (FL − FR)h

(4.21)

49

Figure 4.5: Cart turning angle definition

where Ipz is the pendulum moment of inertia about z-axis and it is calculated as:

(cid:90) ltot

mp
ltot

3

Ipz =

=

(ysinθp)2dy + 2Iw + 2mwl2

0
4mpl2sin2θp

+ 2Iw + 2mwl2

(4.22)

Substitute equation (4.16) into this we can get:

¨φIpz =

−KmKeh
Rmr2

( ˙xL− ˙xR)+

Kmh
Rmr

(uL−uR)− Iwh

r2 (¨xL− ¨xR)−mwh(¨xL− ¨xR)− Kf

r2 ( ˙xL− ˙xR)

According to equation (4.15), equation (4.23) becomes

¨φIpz =

−2KmKeh2

Rmr2

˙φ +

Kmh
Rmr

(uL − uR) − 2Iwh2
r2

¨φ − 2mwh2 ¨φ − 2Kf h
r2

˙φ

(4.23)

(4.24)

where r is the wheel radius, h is the half cart width, φ is the cart turning angle and x is the

cart position. Replace the θL, θR, xL and xR therms in equation (4.19), (4.20) and (4.24)

to get the EOM:

50

2Iw

r2 + 2mw)¨x = −2KmKe

Km
(mp +
Rmr
(Ip + mpl2)¨θp = −mpl¨xcosθp + mplgsinθp +

Rmr2 ˙x +

(uL + uR) − 2Kf ˙x
˙x − Km
2KmKe
Rm

r2 + mpl ˙θ2

psinθp − mpl ¨θpcosθp

(uL + uR) +

2Kf
r

˙x

Rmr
Kmh
Rmr

˙φ +

(uL − uR) − 2Kf h
r2

˙φ

(4.25)

(Ipz +

2Iwh2
r2 + 2mwh2) ¨φ =

−2KmKeh2

Rmr2

4.4 Linearization

The linearization is done with the following transformation

cosθp = 1, sinθp = θp, and

Then the EOM becomes:

= 0

(4.26)

(cid:18) dθp

(cid:19)2

dt

2Iw

r2 + 2mw)¨x = −2KmKe
Rmr2 ˙x +
(mp +
(Ip + mpl2)¨θp = −mpl¨x + mplgθp +

Km
Rmr

2KmKe

r2 − mpl ¨θp

(uL + uR) − 2Kf ˙x
2Kf
˙x − Km
(uL + uR) +
Rm
r
(uL − uR) − 2Kf h
r2

Kmh
Rmr

˙φ

Rmr
−2KmKeh2

˙φ +

Rmr2

(Ipz +

2Iwh2
r2 + 2mwh2) ¨φ =
For state space form purpose let

˙x

(4.27)

a = mp + 2mw +

2Iw
r2

, b = Ip + mpl2

and c = Ipz +

2Iw
r2 h2 + 2mwh2

(4.28)

we have

˙x +

Km
Rmra

¨x =

−2KmKe
Rmr2a
¨θp = −mpl
b

¨x +
−2KmKeh2

¨φ =

Rmr2c

mpgl

b

˙φ +

(uL + uR) − 2Kf ˙x
r2a
˙x − Km
2KmKe
Rmrb
Rmb
(uL − uR) − 2Kf h ˙x
r2c

θp +

Kmh
Rmrc

− mpl
a

¨θp

(uL + uR) +

2Kf ˙x

rb

(4.29)

51

In the state space form, the firction term Kf is assumed to be 0. To eliminate the double

dot terms on the right hand side of these equations, the final EOM is:

¨x =

¨θp =

¨φ =

−2KmKe(mplr + b)
Rmr2(ab − m2
pl2)
2KmKe(ra + mpl)
Rmr2(ab − m2
pl2)
−2KmKeh2

˙x − m2

pl2g
ab − m2
mpgla
ab − m2
(uL − uR)

˙x +

Rmr2c

˙φ +

Kmh
Rmrc

Km(b + mplr)
Rmr(ab − m2
pl2 θp +
pl2)
pl2 θp − Km(mpl + ar)
Rmr(ab − m2
pl2))

(uL + uR)

(uL + uR)

(4.30)

The state vector is chosen as [x ˙x θp

˙θp φ ˙φ]T , and the inputs are the applied terminal

voltages uL and uR. As a result, the state space form of the segway model is shown below:



(4.31)

0

0

0

0

0

0




A =

B =

1

−2KmKe(mplr+b)
Rmr2(ab−m2
pl2)

0

2KmKe(ra+mpl)
Rmr2(ab−m2
pl2)

0 0

0
−m2
pl2g
pl2 0 0
ab−m2
1 0

0

0

0

0

mpgla
ab−m2

0

pl2 0 0
0 0
0 0 −2KmKeh2

1

Rmr2c

0

0

0

0

0

0

Km(b+mplr)
Rmr(ab−m2
pl2)

Km(b+mplr)
Rmr(ab−m2
pl2)

0

−Km(mpl+ar)
Rmr(ab−m2
pl2))

0

−Km(mpl+ar)
Rmr(ab−m2
pl2))

0

Kmh
Rmrc

0

−Kmh
Rmrc



The linearized model in (4.31) is an unstable NMP system. Since the SHIs can only be

implemented for stable systems [35], a stabilizing controller will be designed first. The MS

system parameters used in this study are shown in Table 4.1.

With these parameter values, the MS state-space model becomes (4.32)

52

Description

Table 4.1: Segway system parameters (To be determined in the future work)
Value/Unit
0.027 V · s/rad
0.23 N · m/A
0.001 N · s/rad
0.0000409 kgm2
0.0012 kgm2 Ω

Motor EMF constant
Motor torque constant

Motor shaft friction coefficient

Cart body moment of inertia

Motor internal resistance

Wheel moment of inertia

4.8 Ω

Distance between cart body center of mass and y-axis

Wheel mass

Cart body mass

Wheel radius
Half cart width

0.02 m

0.0492 kg
0.7197 kg
0.034 m
0.086 m

Symbol
Kemf
Kτ
Rm
Kf
Iw
Ip
l
mw
mp
r
h



1

0 0
0 0 −26.4502

(4.32)

0

0

0




A =

[h!]

B =

1

0
0 0
0 −6.8776 −1.6517 0 0

0

0

0

1 0

0 156.8512 107.3635 0 0

0

0

0

0

0

0

0

0

0

0

2.4425

2.4425

0

0

−55.7028 −55.7028

0

0

19.3887 −19.3887



The MS system is open-loop unstable due to a positive pole located at s = 9.5715. A

stabilizing controller is mandatory prior to use SHI technique for performance improvement

since using SHI requires a stable plant.

53

4.5 Dual-loop controller design

4.5.1

Inner-loop Stabilizing Controller

A dual-loop control architecture proposed in [50] is used for the MS system, where the inner-

loop controller stabilizes the MS system and outer-loop controller with SHI further improves

MS system performance. For this study, as shown in the dashed region of Fig. 4.6, the

inner-loop utilizes a continuous-time LQR (referred as LQRc) implemented at 100Hz. The

Figure 4.6: Dual-loop control architecture for the MS system

cost function for LQRc is

Jc =

(cid:90) ∞

0

1
2

(x(t)T Qcx(t) + v(t)T Rcv(t))dt

(4.33)

where Qc ≥ 0 and Rc > 0 are the weighting matrices. The controller is in the form of
v = −Kcx, where Kc = R−1
c BT Pc and Pc is the stabilizing solution of the folowing Ricatti
equation

PcA + AT Pc + Qc − PcBR−1

c BT Pc = 0

(4.34)

For demonstration purpose, an initial set of LQRc weighting matrices were chosen as

Qci = diag[100 0.1 300 0.1 10 0.01] and Rci = diag[1 1] and the corresponding control gain

is:

Kci =

(4.35)

−7.07 −9.49 −14.64 −0.91 2.24

0.08
−7.07 −9.49 −14.64 −0.91 2.24 −0.08



54

With the initial LQRc, the MS system is stabilized and the resulting closed-loop system

remains NMP.

4.5.2 SHI Parameter Selection
For δ ∈ (0, 1] and p ∈ (0, 1], the zero locations of the Kci stabilizing MS system can be
calculated and the SHI feasible region is shown in Fig. 4.7, where the shaded area is the

feasible region that is the common area formed by the sampling period and duty cycle

resulting in a MP DT system for the closed-loop system with the inner-loop controller.

Figure 4.7: The feasible sampling period of ZOH that result in MP DT system and the p-δ
feasible regions of SPSHI.

The black horizontal line on the top of Fig. 4.7 represents the feasible region of ZOH, a

special case of SPSHI with p = 100%. The smallest sampling period for ZOH to achieve MP

DT system is as large as 0.75s. This sampling period can be significantly reduced down to

0.1184s by using the SPSHI with extremely small duty cycle. However for an implementable

sampling period (e.g.

less than 0.2s), the associated duty cycle is reasonable. Note that

achieving samll feasible sample period is very important for improving closed-loop system

55

performance using the outer-loop controller.

In order to have the outer-loop controller sample frequency to be adequate to cover system

dynamics for both outputs, the normalizes system bandwidths are calculated and they are

1.3Hz and 33.9Hz for cart displacement and cart pitch angle, respectively. For the outer-loop

controller, at least 67.8Hz sampling frequency is required [53]. In this case, it is impossible

to have the outer-loop controller to cover system dynamics for both outputs. Consequently,

only the cart displacement output is considered since it requires a controller with the lowest

sample frequency of 2.6Hz [53]. Therefore, small sampling period within the feasible region

shown in Fig. 4.7 is preferred. Due to the experimental validation result in [50], small duty

cycle leads to better performance. As a result, for the SPSHI, the outer-loop sampling period

is chosen as 0.2s, marked using the dashed-line in Fig. 4.7, and the implementation duty

cycle is chosen to be 20%.

4.5.3 Outer-Loop Design

Since the open-loop MS system is NMP, high gain control could be destabilizing. With the

SHI parameters selected in the previous section, it is able to convert the NMP MS system

from the continuous-time domain into discrete-time MP system. In order to improve the

stabilized MS system performance, the outer-loop controller, a discrete-time LQR (referred

as LQRd) with the SPSHI, was used (see Fig. 4.6), and LQRd provides additional control

effort for further performance improvement. The LQRd cost function is defined below.

∞(cid:88)

k=0

Jd =

1
2

(xT (k)Qdx(k) + ωT

SHI (k)RdωSHI (k))

(4.36)

Note that selection of LQRd weighting matrices (Qd and Rd) is critical for this study, where

an adequate control effort should be guaranteed while maintaning both actuators with their
saturation limit of ±12V .

56

4.6 Experimental Hardware

The target MS system is built with three main component groups: sensors, actuators

and microcontroller.

4.6.1 Sensors

In order to balance the MS system about its upright equilibrium point, the measurement

signals of cart body pitch angle, pitch angular velocity, each wheel displacement and velocity

are required.

1) Body pitch angle and velocity: MPU-6050 sensor that shown in Fig. 4.8 from IvenSense

Inc. is used for reading body angular information. It contains a MEMS (microelectrome-

chanical system) gyroscope and a MEMS accelerometer in a single chip. The gyroscope

is a sensor that measures the angular velocity in roll, pitch, and yaw directions and the

accelerometer measures gravitational and motion accelerations in X,Y and Z directions.

After disabling the sleep mode of the chip in Simulink, the gyroscope and accelerometer

readings are sent to the microcontroller via the Arduino I2C bus. The cart body pitch

angular velocity is obtained directly from the gyroscope raw data.

Figure 4.8: The MPU 6050 Chip

57

Theoretically, integrating the angular velocity yields the pitch angle, but the sensor noise

and bias cause the integrated angle signal drift. Note that the pitch angle can also be

obtained from the arc-tangent value of the X and Z direction gravitational accelerations

from the accelerometer signals, however the sensor noise and external forces could also

affects its accuracy easily. Hence, a complementary filter, developed in [55], is used to

have an accurate pitch angle signal; see the complementary filter in Fig. 4.9.

Figure 4.9: The Complementary Filter

To compensate the gyroscope drift with the accelerator reading, it applies a low-pass

filter and a high-pass filter to accelerometer and gyroscope readings, respectively. In the

MS that used in this paper, T s = 0.02 s was used for the filter.

2) Wheel displacement and velocity: The MS displacement is measured with two hall-effect

rotary encoders mounted on the two motors (As shown in Fig. 4.10). The motor encoders

output digital signals of both motor shaft positions with a resolution of 1320 samples per

revolution. The encoder readings can be converted into cart displacement and velocity

based on gear ratio and wheel radius.

4.6.2 Actuators

As shown in Fig. 4.10, two of these 12V DC motors with a 1 : 4 gear ratio are used, where the

motor peak speed is 333 RPM and peak output power is 4.8W. The motor torque parameter

58

Figure 4.10: 12V Motor and Hall Effect Encoder

Kτ is 0.23 N m/A. The MKR motor carrier that shown in Fig. 4.11 is used to drive both two

motors and read out both encode signals. It has an operating voltage of 5 to 12V and uses a

L298P motor driver. The maximum output current is 2A/channel, which is lower than the

motor maximum current of 6.5 A so that the motors are protected from excess current.

Figure 4.11: Arduino MKR Motor Carrier

59

4.6.3 Microcontroller

For the purpose of balancing the MS system, the microcontroller speed has to be adequate

for sampling sensor signals, communicating with the host computer via Wi-Fi, and generat-

ing control commands for both motors. The sample periond is limited by the feasible region

to 0.1s and any Arduino board is suitable for this task if the control algorithm is written

in Arduino language. However, for this study, the control algorithm was developed in Mat-

lab/Simulink, which requires additional throughput from the microcontroller to handle the

communication between the Arduino microcontroller and real-time Simullink via Wi-Fi; see

Fig. 4.13 below. Therefore, a more advanced Arduino board, Arduino MKR 1000 Wi-Fi

is selected for this study. It uses the microcontroller with a clock speed of 48 HMz that is

capable of handling the sampling frequency of 100Hz.

Figure 4.12: Arduino MKR1000 WiFi

4.6.4 Experimental Set Up

The MS experimental hardware configuration is shown in Fig. 4.13. The rectangular box

represents the MS cart. All the sensor signals are fed into the Arduino MKR 1000 and send

to the host computer via Wi-Fi. The host computer then generates control signals in the

60

Matlab/Simulink and sends back to the microcontroller to drive the motors using the MKR

motor shield.

Figure 4.13: The MS system hardware configuration

The MS built in the lab is shown in Fig. 4.14.

Figure 4.14: The MS Picture

61

4.7 Experimental Implementation

4.7.1

Inner-loop LQR Tuning

After the MS system platform was developed, the inner-loop LQRc is implemented to stabi-

lize the MS system. A ”baseline” system performance was established based on Qci and Rci.

Started from this ”baseline” and with Qci matrix fixed, the weighting matrix Rc was tuned

by introducing a gain GRc such that Rc = GRcRci. Experiments were conducted with GRc

from 0.6 to 1.2 with an increment of 0.1. The cart displacement and pitch angle are shown

in Fig. 4.15 (a) and (b). The corresponding control voltages are shown in Fig. 4.15 (c).

The mean absolute values of the cart oscillation magnitude (|xcart|avg), pitch angle
(|θp|avg) and the corresponding control voltage (|v|avg) for each GRc are shown in Table
4.2.

Table 4.2: The Mean Absolute Values of Cart Displacement and Control Voltage For
Inner-loop LQRc Tuning

N/A

0.3

GRc Value
0.7

0.5

19.35

15.32

18.63

12.17

4.31

2.56

1.08

0.85

3.96

2.44

1.16

1.00

0.9

1.1

16.58 ∞

0.89 ∞

1.10 ∞

|xcart|avg
(mm)
|θp|avg
(mm)
|v|avg
(V)

and the data is plotted in Fig. 4.16.

From Fig. 4.15, when GRc is 0.6, the resulting LQR control gain is high, the MS system

outputs including pitch angle and dispalcement oscillate with very high frequencies. When

GRc = 0.7 with a reduced control gain, the cart oscillation magnitude is reduced but is

still unstable. As a result, these two GRc values are not feasible since they do not result in

feasible LQRc controllers for stabilization purpose. For GRc = 0.8, the closed-loop system

62

Figure 4.15: The experimental result for inner-loop LQRc optimization

takes about 5s to stabilize, therefore, the GRc region between 0.7 and 0.8 is the transition

one from unstable to stable for the MS system (as shown in Fig. 4.16). The closed-loop

system performance is further improved with GRc = 0.9 and the cart is also stabilized with

GRc = 1.0 with an increased oscillation magnitude. When GRc reaches 1.1, the closed-loop

system has an unbounded oscillation magnitude for cart displacement, indicating an unstable

system. The data in Table 4.2 is plotted in Fig. 4.16. In spite of the undesired range and

unstable range, the feasible (stabilizing) GRc is between 0.8 and 1.0.

63

Figure 4.16: LQRc tuning data plot

4.7.2 Outer-Loop Controller Tuning

For each stabilizing GRc using for the inner-loop LQRc, the outer-loop LQRd controller was

tuned accordingly to achieve optimal performance. A ”baseline” LQRd was designed first

using weighting matrices Qdi = diag[100 0.1 80 0.1 10 0.01] and Rdi = diag[1 1]. With

the feasible LQRc controllers, the dual-loop feasible regions are shown in Fig. 4.17 . The

smallest feasible sampling periods are 0.1136s, 0.1162s and 0.1184s for GRc values of 0.8,

0.9, and 1.0, respectively. Since the feasible regions of the closed-loop systems with SPSHI

in Figs. 4.7 and 4.17 changes a little, the SHI parameters remain unchanged at p = 20%

and δ = 0.2s.

64

Figure 4.17: The feasible region of the closed-loop MS system with SPSHI

For the purpose of tuning, GRd was introduced so that Rd = GRdRdi, where GRd ranges

from 0.3 to 1.1 with a 0.2 increment while Qdi matrix is fixed. Smaller GRd values such as

0.1 and 0.2 were tested but the corresponding LQRd gain was too large for the MS system so

that the nonlinearity characteristics were excited and resulted in unstable system responses.

The control gain tuning is to achieve the best performance improvement of cart oscillation

magnitude. For GRc = 0.8, 0.9and1.0, the outer-loop tuning result is shown in Fig. 4.18,

Fig. 4.19 and Fig. 4.20,respectively.

The data in Fig. 4.18, Fig. 4.19 and Fig. 4.20 are shown in the table below.

The reason why the data in Table 4.3 is different with that in Table 4.2 for the inner-loop

controller case is due to the variation of the battery position and unmodeled actuator dynam-

ics of the MS system. However the data is adequate for observing the outer-loop controller

effectiveness. For a maximized performance improvement, GRc = 0.9 is apparently the best:

The inner-loop controller results in the smallest cart oscillation magnitude meanwhile the

outer-loop has the best performance improvement of 64.8% (It is 51.6% and 60.0% for GRc

values of 0.8 and 1.0, respectively). Therefor the criteria for tuning the inner-loop LQR is

65

Figure 4.18: The Experimental result for Outer-loop Controller Optimization with SPSHI
and GRc = 0.8

for the best local performance and only GRc = 0.9 is considered.

The experimental result in Fig. 4.19 is divided into six segments. Segment I corresponds

to the response of the inner-loop MS system with LQRc only, which is tuned in the previous

section and it is used as the baseline controller. Start from segment II to segment VI, the

GRd values increases from 0.3 to 1.1 with 0.2 increment.

The experimental results with SPSHI in Fig. 4.19 (a) show that the outer-loop LQRd

with GRd = 0.3 reduces the amplitude of the cart displacement significantly. However the

66

Figure 4.19: The Experimental result for Outer-loop Controller Optimization with SPSHI
and GRc = 0.9

cart body pitch angle that shown in Fig. 4.19 (b) is barely affected due to aforementioned

inadequate control frequency. From segment II to segment VI, with the increasing GRd

value, the cart displacement increases while the SHI control effort is decreasing.

The overall (u) and SPSHI (ω) control efforts and are shown in Fig. 4.19 (c) and Fig.

4.19(d), respectively. The SPSHI signal is 0 in segment I since only LQRRc is used. The

outer-loop is activated from segment II and it starts to generate additional control effort,

which reduced the cart displacement. As GRd increases, the SPSHI signal decreases and

67

Figure 4.20: The Experimental result for Outer-loop Controller Optimization with SPSHI
and GRc = 1.0

result the increase of cart displacement. The mean absolute values of the cart displacement,

overall control input voltage and SHI voltage in Fig. 4.19 for each GRd value are shown in

Table 4.4.

The data in Table 4.4 is plotted in the figure shown below.

The best cart displacement is boxed in the table. From the trend shown in Fig. 4.21 (a),

when GRd = 0.3, the cart oscillation magnitude reaches the local minimum, meanwhile the

cart pitch angle is slightly higher than the single loop case. The numerical results show that

68

Table 4.3: The Mean Absolute Values of Outer-loop LQRd Tuning Data with different GRc
Values

N/A

0.3

GRd Value
0.7
0.5

0.9

1.1

27.87 13.50 15.46 15.64 16.89 15.51

10.94

3.85

5.39

6.29

6.84

7.36

16.59

6.63

8.00

10.55 12.07 13.29

GRc = 0.8

|xcart|avg(mm)
|xcart|avg(mm)
|xcart|avg(mm)

GRc = 0.9

GRc = 1.0

Table 4.4: The Mean Absolute Values of Outer-loop LQRd Tuning Data with GRc = 0.9

N/A

0.3

GRd Value
0.5
0.7

0.9

1.1

10.94

3.85

5.39

6.29

6.84

7.36

0.95

0.98

1.00

0.99

1.08

1.11

0.69

0.47

0.52

0.55

0.56

0.57

0

0.047

0.04

0.03

0.027

0.023

|xcart|avg
(mm)
|θp|avg
(mm)
|u|avg
(V)
|ω|avg
(V)

the inner-loop mean absolute cart displacement amplitude is reduced from 10.94 mm to 3.85

mm at GRd = 0.3, which is 64.8% reduction. Note that even though the SPSHI signal in
Fig. 4.19 (d) ranges from −1 to 1V, it does not increase the overall control voltage in Fig.

4.19 (c). Also the trend in Fig. 4.21 (d) shows that higher SHI control signal can result in

low overall control voltage.

69

Figure 4.21: LQRd tuning data plot

70

CHAPTER 5

AGRICULTURE PROJECT: FAULT DETECTION ON SOLID-SET

CANOPY DELIVERY SYSTEM FOR APPLE FARM

A solid-set canopy delivery system (SSCDS) is developed to deliver water and chemical

solusions to high-density fruit trees with the support from USDA-SCRI (US Department

of Agriculture Specialty Crop Research Initiative). For the purpose of detecting SSCDS

faults, two methods were proposed and they are flow and/or pressure sensors based in-line

detection method and thermal image based detection method (main focus of this paper).

A FLIR thermal camera was mounted on a modified commercial Unmanned Aerial Vehicle

(UAV) so that any plug, partial plug, and gusher faults can be captured by the thermal

camera during the water and chemical spray process. With the help of a image processing

software, the thermal video is stitched into one panorama image. By comparing it with a

baseline panorama image, the SSCDS faults can be detected. The main advantage of the

proposed fault detection method is the ability of distinguishing all three types of faults over

a huge area within a fairly short amount of time, utilizing only a UAV equipped with both

regular and thermal video cameras. The future work is to fly the UAV automatically based

on a pre-planned route, collect the thermal video, process the panorama image and diagnose

the SSCDS faults.

5.1

Introduction

This project is funded by the USDA-SCRI to the research and extension team of Michi-

gan, Washington, and New York. A solid-set canopy delivery system (SSCDS) was built on

the apple orchard located at the Clarksville Research Center of the Michigan State University

in Clarksville, MI, USA. This project is to demonstrate high-density tree fruit production

and evaluate SSCDS spray coverage for additional perennial fruit crops. SSCDS consists of

a network of micro-sprayers positioned in the tree canopy/trellis and connected to a pump-

71

ing/mixing station. SSCDS application virtually eliminates applicator exposure common

to tractor-based sprayers, while increasing farmers’ ability to apply sprays during critical

weather periods, which will help to ensure that US growers remain competitive in an in-

creasingly globalized marketplace.

To ensure reliability and longevity, fault detection of the SSCDS is necessary. Due to the

nature of SSCDS, the fault detection should be low cost, precise and fast. The cost is the

most important concern for growers since the investment return period is too long to afford

a high cost system; also the diagnostic system needs to be precise and fast to locate the

faults in the irrigation system to minimize the fault impact. As a result, two fault detection

schemes were proposed and they are in-line and thermal fault detection.

Previously, in-line fault detection was conducted using flow meters, pressure sensors

and/or valves ( [56], [57] and [58]), however these schemes require to install sensors on each

emitter, which could result in unaffordable cost for larger orchards. A low cost scheme was

developed in [59], but it was unable to locate the fault precisely. For the SSCDS a low cost

and precise in-line fault detection scheme was developed, which consists of an in-line flow

sensor, a transmitter and a receiver. It was designed to detect faults in each section consisting

of three manifolds, where each manifold is formed by three emitters. The low-cost flow sensor

was used to report any abnormal flow within the corresponding section via a low-cost micro-

transmitter. The SSCDS faults can be identified as soon as spray starts. However, due

to device degradation, installation and device variations the in-line fault detection method

could lose its detection accuracy as the system ages.

The second fault detection scheme is based on thermal (IR) imaging technology. It is

widely used nowadays for fault detection of electrical assets ( [60]), ground water monitoring

( [61], [62]) and defense system applications ( [63]). This technology was also used in agricul-

ture area including food or crop health inspection ( [64], [65] ), plant stress monitoring ( [66]),

irrigation system fault detection ( [67] and [68]), etc. Based on these successful applications,

thermal imaging fault detection using UAVs were studied by [69], [70] and [71]. As a re-

72

sult, the aerial thermal technology was proposed for the SSCDS as the second fault detection

scheme. It utilizes a modified drone with an additional thermal (IR) camera mounted. Along

with the attached RGB camera, the drone takes both thermal and RGB videos at the same

time during the spray process. Since a commercial drone is used instead of a professional

drone, the fly time is limited by its battery size. With a video transmitter, an extra thermal

camera and a gimbal mounted, the fly time is reduced from 25 minutes (without payload) to

about 20 minutes. A fly pattern was designed to maximize thermal and RGB videos taken

using the modified drone so that it is able to monitor up to 5 rows (300 feet each) within

one run. With the help of image processing software, images from both thermal and RGB

videos can be stitched into panorama pictures, making it convenient to identify the faults

via image processing algorithms. The thermal and RGB panorama pictures provide detailed

information regarding system faults, comparing with the in-line fault detection method. It

is able to identify any single spray fault based on the stitched panorama pictures. Since the

thermal camera captures relative temperature, fault detection under multiple operational

temperature conditions were studied and the thermal image-based method is able to identify

all these faults. However, this method is limited by certain weather condition. For instance,

the drone is not operable under heavy wind (e.g., wind speed over 10m/s) or rain. Also the

thermal fault detection method requires relatively expensive devices such as the drone and

thermal camera (both over $2000), and it also requires a professional pilot license to operate

the drone. But it is possible, in the future, that thermal fault detection service can be done

by an independent professional company, which will significantly reduce the cost.

5.2 Solid-Set Canopy Delivery System

5.2.1 Emitters

A Solid-Set Canopy Delivery System (SSCDS) consists of two main parts: irrigation system

and sprayer. The irrigation system is designed based on the sprayer selection and orchard

layout. The apple trees are planted 3 feet apart in rows and separated in sections by wooden

73

poles every 7 to 10 trees; see Fig. 5.1. Two types of emitters were selected for SSCDS:

Figure 5.1: SSCDS Layouts

The first type of emitter is the Hadar 7110 shown in Fig. 5.2. It produces 8.7 gallon/hour

flow in the form of mist. Three emitters form a triangle manifold and manifolds were mounted

6 feet apart.

Figure 5.2: The Hadar 7110 Emitter

74

The second type of emitter, called Spinner shown in Fig. 5.3, has higher flow rate and

produces 26.4 gallon/hour flow in the form of droplets. There is one Spinner hanging over

each tree.

Figure 5.3: The Spinner Emitter

For both types of emitters, all manifolds were fed through a 32 psi stop drip device for

building up the pressure inside irrigation line and the system was pressurized to 60 psi at

run time.

5.2.2

Irrigation Systems

The irrigation layouts for Hadar 7110 and Spinner emitters are shown in Fig. 5.1 (a) and

(b), respectively.

Both irrigation layouts utilize a water pump and an air compressor. Chemical solution

will be pumped into the waterline and held within sections by the special designed valve

between every two sections. After the waterline is filled up, the airline is pressured by the

air compressor so that the solution trapped in the waterline is pushed out through the emitter

manifold section-by-section to complete the spray process.

75

5.2.3 SSCDS Spray Pattern

In order to deliver roughly the same amount of solution to the canopy in each section, a

special design of the spray system was developed. There are two pipe along the row as a

set: upper air line and lower water line. The spray manifold locates at a optimized location

in the lower water line. Between each segment, a valve box is used to connect the upper air

line and the lower water line. The spray pattern and the use of the valve box is shown in

Fig. 5.4

Figure 5.4: SSCDS Spray Pattern

The pattern can be summarized as:

76

1. Initial condition: no air or water pumped into the air and water lines.

2. The water line is filled with solutions. As the in-line pressure is building up, the

valve boxes are activated to segment the water line and keep the solutions within each

segmentation.

3. The air line is filled with 60 psi air flow that provided by a portable air compressor.

4. When the static air pressure in the air line reaches 35 psi, the valve between air

and water lines will be open to push the solution through the emitter manifold segment by

segment.

5. The valve box operates with pressure signal. when one segment finishes the spray

with the solution in the segmented lower water line, the valve box will close the air-water

valve so that the spray will perform in the next segment.

6. When all the solution are sprayed in all of the segments, the user stops the air pump

to finish the spray in this row and move onto the next row.

7. On each of the emitter manifold, there is a 25 psi stop drip mechanism that opens up

when there is no pressure to eliminate any residual solution in the waterline.

This is the SSCDC spray process. It requires a reliable design of the valve box and flow

sensor.

5.2.4

In-Line Flow Sensor

Due to the size of a farm, the in-line sensor has to be wireless so that the user will identify

which row is having an abnormal emitter. In this case, the in-line sensor design is divided

into two portion: Sensor and transmitter/receiver.

For the flow sensor, we decided to go with the DC Water Turbine Generator Water

(shown in Fig. 5.5), which is designed to be in the middle of the water pipe. When there

is water flowing through the pipe to push through the generator turbine, the generator is

activated. With a modification on the generator, the geneated voltage is not going to be

regulated so the flow intensity can be monitored. The generator we chose is called Yosoo

77

DC Water Turbine Generator Water 12V DC 10W Micro-hydro Water Charging Tool. It

has the specifications listed below:

Figure 5.5: Flow sensor

Table 5.1: Flow sensor specification

Condition

Value/Unit

12V

10M Ω

≥ 220mA(@12V )

Output voltage
Output current
Insulation resistance
Maximum static pressure
0.6MPa (87.02 Psi)
Maximum kinetic pressure 1.2MPa (174.05 Psi)
Stat pressure
0.05Mpa (7.25 Psi)
Mechanical noise
Generator life

55dB (Max)
≥ 3000h

For the wireless transmitter, it has to be small, low power consumption ut wide range

of working voltage, high tolerance of working temperature, long range and long life. With

these restrictions, we decided to go with the Transmitter ZF-4 and Receiver ZW21-J (shown

in the Fig. )that manufacture by a Chinese company. The specifications are listed below:

The selection of the transmitter and receiver fulled the task perfectly. The transmitter

will be mounted inside the generator (shown in Fig. 5.7), which solves the reliability issue.

Based on the concept, two in-line sensing systems were made for testing purpose and the

layout are shown in the Figs.5.8 and 5.9

78

Figure 5.6: Transmitter ZF-4 and Receiver ZW21-J

Table 5.2: Transmitter and receiver specifications

Receiving Frequency
Working Voltage
Working Current
Working Distance
Size
Working Temperature

Transmitter
315/433 MHZ
DC 5V - 12V

35 mA
2 miles

Receiver

300/440 MHZ
DC 3.6V – 5.5V
3.7 mA 4.7 mA

2 miles

0.77” x 0.73” x 0.26” 1.02”x 0.47”x 0.26”

-4 F to +140 F

-4 F to +140 F

Figure 5.7: The transmitter is mounted inside the generator

In Figs. 5.8 and 5.9, the flow generators are marked with red circles and the emitter

manifolds are marked with green circles. In Fig. 5.8, the solution deliver system was tested,

and in Fig. 5.9, different types of flow generators are tested.

79

Transmitter ZF-4 Receiver ZW21-JAir pipeWater pipeValve boxFigure 5.8: The in-line fault sensing system prototype 1

Figure 5.9: The in-line fault sensing system prototype 2

5.2.5 Thermal Fault Detection

Due to the nature of the SSCDS, the fault detection mechanism is necessary since the

emitters are easy to clog (plug) or fall off (gusher) over time.

In order to help growers

locate faults in the field without checking emitters one-by-one in person, the Michigan State

University (MSU) engineering team developed a fast and precise fault detection scheme –

the thermal fault detection mechanism, which uses a commercial drone with a thermal (IR)

camera mounted. The live thermal videos of the faulted sections were sent back to the user

with a video transmitter mounted on the drone. In this paper, the thermal fault detection

is introduced in details.

80

5.2.6 Drone Hardware Selection

5.2.6.1 Drone

The selection of commercial drone is critical due to multiple factors. The drone battery is

not only used for the drone, it also provides power to the thermal camera, the mounting

gimbal and the video transmitter. The battery capacity has to be sufficient for a long flight

time with payload. Extra space on the drone is required for mounting the thermal camera

without affecting the drone functions including steady hovering and route planing based on

GPS and distance sensors. Payload is the third factor since the drone can not take off or

perform the onboard functions with excessing payload. These factors require a large drone

but the size is also limited by its mobility and cost. After a long process of market research,

DJI inspire 1 V2 was selected. The battery supports up to 20 and 25 minutes of flying time

with and without payload (up to 550 g), respectively.

5.2.6.2 Thermal Camera

This project also requires a thermal camera that is light in weight, small in size, and low

operational power. As a result, the FLIR VUE Pro R, a thermal (IR) camera for professional

drone, was selected. The specifications of the camera are listed in Table 5.3.

This thermal camera meets the requirements for fault detection. It was used for record-

ing spray process. The thermal videos are broken into frames and stitched into panorama

pictures for post analysis, and the results are shown in the next section.

5.2.6.3 Gimbal

A GoPro direct mount for the camera was used at first, however the camera pointing direc-

tion changes as the drone moves, and high frequency vibrations with fairly large magnitude,

generated by the drone propellers, pass to the camera. All these factors result in unus-

able thermal footage. Therefore, the GoPro mount was replaced by a 3-axis gimbal. PID

81

Table 5.3: FLIR VUE Pro R Specifications

Size

Spectral Band

Thermal Imager

Weight

Zoom

Operating Temperature Range

Full Frame Rate

Lens Options

Sensor Resolution

Measurement Accuracy

Input Voltage

Power Dissipation (peak)

2.26” × 1.75” (including lens)

7.5 − 13.5 µm

Uncooled VOx Microbolometer

310g

Yes – Adjustable via PWM

−20◦C to +50◦C

30 Hz (NTSC); 25 Hz (PAL)

6.8mm; 45◦ × 35◦

336 × 256

±5◦C or 5% of reading

4.8 − 6.0 VDC
2.1 W (3.9 W)

(proportional-integral-derivative) controllers are used to control the three gimbal motors and

the PID gains are precisely tuned to minimize any unexpected motion or vibration so that

the thermal footage can be stabilized. Even though the camera movement range is lim-
ited by the gimbal frame, it still has ± 65 degree freedom in yaw, pitch and roll axises. A

small modification was made to the drone logic board so that the gimbal can be powered by

the drone battery. Even though the flight time is reduced due to the gimbal power usage,

reducing payload is more important for this project.

5.2.6.4 Flight Controller

Since the camera has a function of zoom in/out and the pilot needs to adjust the camera to

capture perfect videos at different altitudes due to plant height variations, an online flight

controller was used so that the pilot can adjust the camera settings and gimbal angle at the

same time during the flight. A RadioLink AT9S transmitter/controller and a R9DS receiver

82

are used. The receiver is placed on the drone and it supports both SBUS and PWM signals

with up to 10 output channels. It was connected to the gimbal and thermal camera so that

all gimbal and camera functions are accessible during the flight. A remote controller tray was

made to hold both of the flight controllers (DJI drone and R9DS flight controllers) so that

the pilot can operate the drone and control the camera at the same time using both hands.

The two controller assembly is shown in Fig. 5.10, where the lower white and upper dolden

controllers are for controlling the drone and adjusting the thermal camera, respectively.

the iPad mini between two controllers is connected to the drone controller for adjusting

parameters and displaying live RGB video. The black screen on the top is connected to a

video receiver to observe the live thermal video.

Figure 5.10: Controller Tray

5.2.6.5 Video Transmitter

For thermal video transmission, a set of DJI Downlink video transmitter and receiver is used.

The transmitter is mounted on the drone and powered by the drone battery as well. The

83

receiver was mounted on the back of the controller tray. The channel selection for the video

transmission was critical since it may interfere with the RGB video or drone control signal

transmissions, which could affect controlling the drone. The modified drone is shown in Fig.

5.11.

Figure 5.11: Modified Drone

A system diagram is shown in Fig. 5.12 to summarize the modified drone system archi-

tecture.

Figure 5.12: Drone System Layout

84

5.3 Experiment Results and Discussion

5.3.1 Experimental Setup

On a summer morning, the water temperature is lower than atmosphere. Due to transpi-

ration effect, the trees were slightly cooler than the atmosphere prior to spray. During the

spray, the sprayed area temperature could be even lower than the area that are not being
sprayed. When the outside temperature is low (e.g., below 50◦ F), the water is ”hotter” than

the atmosphere and the tree temperature situation is reversed prior to and during spray. The
thermal camera has a resolution of ± 0.2◦C in distinguishing the temperature differences

between objects. Based on theses concepts, the thermal camera is used to pick up the ”hot”

or ”cold” spots on trees and those spots will be identified as fault regions. The corresponding

thermal footage is referred as feasible thermal footage. The flight and spray patterns are the

key for a successful fault detection.

When flying the drone along orchard with 30 feet in height and 1.5 feet/second in speed,

the modified drone is able to cover up to four of 300 feet long rows in one run, including the

battery consumption during taking off and landing. Due to the fact that the drone position

can easily be disturbed by the gust wind, stable thermal video was not obtainable via manual

control when the wind speed exceeds 3 feet/second. Therefore the route following feature, a

drone built-in function, was used for obtaining thermal videos.

Prior to the experiment, all SSCDC emitters were in perfect working condition. For fault

detection experiments, faults were created before the spray, otherwise the thermal camera

would not be able to capture any faults since the canopy were all cooled down or heated

up by a perfect spray. The created faults includes plug, partial plug and gusher for both

emitters and these faults were placed randomly along the orchards.

85

5.3.2 Experimental Procedure and Results

The experiment was divided into two parts on two orchards for two types of emitters. Ex-

periment #1 was performed on the orchard that used the Hadar 7110 emitters. The weather

condition and the water temperature for this experiment are shown in Table 5.4

Table 5.4: Experiment #1 Parameters

Experiment Parameters

Outside temperature

Water temperature

Humidity

Wind Direction

Wind Speed

Value

76 F − 80 F
65 F − 67 F
77% − 85%

South to Southwest
10.2f t/s − 16.9f t/s

It was a hot windy day and the water was slightly colder than atmosphere. As shown in

Fig. 5.13 (a), no spray for the first run so that a clear thermal video (baseline) was obtained.

The faults shown in Table 5.5 were placed for the regular fault simulation footage in the

second run; see Fig. 5.13 (b). For the baseline of thermal fault detection, all the faults

were corrected and the thermal footage is shown in Fig. 5.13 (c). And Fig. 5.13 (d) is the

stitched panorama picture from the RGB camera. All of the panorama pictures are aligned

horizontally so that pictures (a), (b), (c) and (d) are showing the same orchard section. In

Fig. 5.13 (d), the white stripes are used for marking all the Hadar 7110 emitter positions,
which are also marked with A1 − A13 in Fig. 5.13 (c).

For the thermal footage in black and white pattern, temperature difference is indicated

by color: darker color means lower temperature and lighter color higher temperature. Com-

paring Fig. 5.13 (b) with Fig 5.13 (c), at A2 and A6 locations, the color of upper portion of

the trees in Fig. 5.13 (b) is obviously lighter than that in Fig. 5.13 (c). This means that at

these two locations, the solution was not sufficiently dense to cool down the trees; see Fig.

86

Figure 5.13: Thermal and Regular Footage of the Orchard with Hadar 7110 Emitters

Table 5.5: Fault Details in the Orchard that with Hadar 7110 Emitters

Fault Location Fault Type

A2

A6

A7

A9

Upper plug

Partial plug

Manifold plug

Lower gusher

A13

Upper gusher

5.13 (c). The situation got worse at location A7 in Fig. 5.13 (b): it has a fairly large ”hot”

spot, comparing to Fig. 5.13 (c), which means it is another fault. From these observations,

the thermal camera is capable of detecting plug faults. However, at A9 and A13 locations,

the gusher faults are not as obvious as plug faults. At location A9, there is a black line

marked with the red circle and the upper portion of the tree is ”hot”. Crossing reference

with Fig. 5.13 (d), it can be observed that the black line was not a white stripe for marking

emitter locations, which means that it could be a spout. So the gusher fault can be detected

by the thermal camera for this location. But the same fault happens at location A13. The

87

spout is not visible but the upper portion of the trees is darker, which means that the spout

was shooting at the leaves and spread out.

The second experiment was performed on the orchard with Spinner emitters on the other

day. The experimental parameters are shown in table 5.6.

Table 5.6: Experiment #2 Parameters

Experiment Parameters

Value

Outside temperature

Water temperature

Humidity

49 F

55 F

59% − 61%

Wind Direction

North - Northeast

Wind Speed

4.4f t/s

It was a cold day with almost no wind so water was hotter than atmosphere. In this

orchard, faults were also created on purpose and the locations are listed in Table 5.7. Due

to the fact that the emitters were too close to each other, only the faulted emitters were

marked with small pieces of white stripes on top of the irrigation line (see Fig. 5.14 (e)). The
faulted emitters are also marked with B1 − B4 at the corresponding locations in Fig. 5.14

(d). In this experiment, thermal plate ”fusion” was tested and are shown in Figs. 5.14 (a)

and (b), which are spray with and without created faults, respectively. Black-white thermal

plate was also used for fault and no fault sprays and they are shown in Fig. 5.14 (c) and

(d), respectively. The fusion video was taken first and the black-white video was taken 2

hours later after all the solution was evaporated and the canopy temperature recovered back

to normal. The experiment parameters were exactly the same for these two thermal plate

runs. By comparing Fig. 5.14 (a) with Fig. 5.14 (c), a huge difference in these two thermal

plates are observed. The black-white footage shows a better result with larger faulted area

than the thermal ”fusion” one, indicating that for this thermal camera, black-white plate

has a better sensitivity than the fusion plate.

88

Table 5.7: Fault Details in the Orchard that with Spinner Emitters

Fault Location Fault Type

B1

B2

B3

B4

plug

gusher

plug

gusher

Figure 5.14: Thermal and Regular Footage of the Orchard with Spinner Emitters

From Fig. 5.14, the gusher effects are easily detected using the thermal camera by

observing the ”hot” areas due to higher flow rate of ”hot” solution (water), however the

plug effects are not even visible in the thermal footage. Note that the Spinner emitter has

a much bigger droplets so it is fairly robust with high viscosity solutions. Even though the

thermal scheme was not able to detect the plug faults on a cold day, it is still feasible for

fault detection of Spinner emitters.

89

5.4 Discussion and Future Works

Due to the battery size limitation, the thermal fault detection is not able to cover an

orchard requiring more than 20 minutes of flight. For an applicable thermal fault detection, a

hybrid drone shall be a better choice. The hybrid drone utilizes a small internal combustion

engine to generate electric power for propellers and it has much longer flight duration. Due

to the limitations of season and weather, limited experiments were conducted in 2018. In

2019, similar experiments will be conducted to confirm the thermal fault detection capability

including the variations of thermal plates, orchards, weather conditions, video taking angle,

and emitters. For the future work, an adaptive parameter estimation algorithm and a PID

controller can be combined with online image processing algorithm to have the thermal

camera to track the trees so that the thermal camera angle control can be automated. Also,

a second online image processing algorithm can be developed so that the faults can be

automatically marked on the image in real-time.

5.4.1 Future Work

For the future work of the SSCDS project, more of the flow fault detection system need to be

made and calibrated for the in-line fault detection sensor, and more flights need to be made

in order to collect enough thermal image data including ”plug” and ”gusher” faults. Also,

an optimal fly hight and speed need to be determined for best fault detection. In addition,

a Matlab based software need to be developed or other image processing software need to

be found to process these thermal image data to achieve automatic fault detection.

90

CHAPTER 6

CONCLUSIONS

Two new sample and hold input (SHI) functions are proposed based on the square pulse SHI

that proposed previously in an other research. Numerical and simulation study show that

by selecting the proper sampling period and duty cycle the resulting discrete-time system

can be minimum phase (MP). Additionally, the forward triangle SHI provides a discrete-

time MP system with the smallest feasible sampling period for a given duty cycle. Also,

comparing with the conventional zero-order holder, the SHIs provides additional choice of

sampling periods to achieve MP system in discrete-time domain.

An experimental demonstration of performance improvement of a non-minimum phase

(NMP) system using special sample and hold inputs (SHIs) was performed. The SHIs are

used to convert the continuous-time NMP system into a discrete-time minimum-phase (MP)

system, thus allowing a discrete-time controller to be designed for improving the performance.

This concept is demonstrated with a single inverted pendulum (SIP) system through sim-

ulations and experiments using the square pulse sample and hold input (SPSHI) and the

forward triangle sample and hold input (FTSHI). The overall controller is comprised of an

inner-loop continuous-time stabilizing controller (provided by the SIP manufacturer) and an

outer-loop discrete-time performance enhancement controller. While the inner-loop stabiliz-

ing controller is required to convert the continuous-time NMP system into a discrete-time

MP system, the outer-loop discrete-time controller allows additional control effort to be

used for better performance. The inner-loop continuous-time controller was implemented at

500Hz whereas the outer-loop controller was implemented at less than 5Hz to ensure MP

characteristics of the discrete-time system. To achieve good match between simulation and

experiment results, Coulomb friction was added to the Simulink model. Both simulation

and experiment results show that the mean absolute value of the cart displacement can be

significantly reduced when the outer-loop controller is used. The effectiveness of both SHIs

91

is compared with the inner-loop controller experimentally; the improvement is 69% for the

SPSHI and 55% for the FTSHI. Two sampling rates were used for the discrete-time controller

and it was found that the controller with higher sampling rate provided better performance.

A MS platform was built for this paper to present the implementation of the previously

proposed dual-loop control architecture onto a non-minimum phase (NMP) mini segway

(MS) system with a low cost microcontroller. It is shown that the both inner- and outer-

loop controllers affect the overall MS system performance. In order to study this interaction,

a fine tuning was performed on the inner-loop continuous-time LQR while preserving the

MS stability. A GRc gain that ranges from 0.6 to 1.1 is introduced to the R matrix of the

inner-loop controller. The outer-loop LQR controller is optimized for the inner-loop LQR

with each GRc value.

It was found that GRc selection is the key factor to have a good

inner-loop closed-loop MS system performance. When GRc is too low, nonlinear Coulomb

friction affects the MS system and result in an undesired high frequency oscillation. When

GRc is too high, the system nonlinarity results in unstable performance. In the outer-loop,

a SPSHI was combined with a discrete-time LQR and implemented at 5Hz sampling rate.

The SPSHI was used to obtain a MP MS system from the NMP closed-loop system so that

the extra control effort that provided by the discrete-time LQR can be utilized to further

improve the MS performance. For the outer-loop LQR, a fine tuning was also performed to

obtain the optimized controller by introducing the GRd gain of the R matrix. Experiments

were also conducted with the varying GRd. For both tuning processes, the experimental

results are presented graphically and numerically.

For the solid-set canopy delivery system (SSCDS) project, based on the current design,

two fault detection schemes are developed by the engineering team of Michigan State Uni-

versity. The first one is the in-line fault detection with special valve and flow sensors; and

the second is thermal fault detection that utilized a DJI Inspire 1 V2 commercial drone and

FLIR VUE Pro R thermal (IR) camera. These two schemes will be used together, where

the in-line scheme detects the faulted sections first, followed by sending the drone to locate

92

the faulted emitter. Multiple experiments were conducted for different types of faults with

two types of emitters (mist Hadar 7110 and large-droplet Spinner emitters). Specific spray

and fly patterns are designed for thermal fault detection. From the results obtained so far,

the thermal (IR) camera is feasible for detecting these faults. To be more specific, it is able

to precisely detect the plug and gusher faults for Hadar 7110 emitter and gusher faults for

the Spinner emitters. As a result, fault detection feasibility of SSCDS is validated using the

thermal (IR) images captured by a drone.

93

BIBLIOGRAPHY

94

BIBLIOGRAPHY

[1] Quanser, “Ip02 base unit experiment set up and experiment,” Quanser User Manual,

2012.

[2] M. Tomizuka, “Zero phase error tracking algorithm for digital control,” Journal of

Dynamic Systems, Measurement, and Control, vol. 109, no. 1, pp. 65–68, 1987.

[3] E. Gross, M. Tomizuka, and W. Messner, “Cancellation of discrete time unstable ze-
ros by feedforward control,” Journal of dynamic systems, measurement, and control,
vol. 116, no. 1, pp. 33–38, 1994.

[4] B. Haack and M. Tomizuka, “The effect of adding zeroes to feedforward controllers,”
Journal of dynamic systems, measurement, and control, vol. 113, no. 1, pp. 6–10, 1991.

[5] D. Torfs, J. De Schutter, and J. Swevers, “Extended bandwidth zero phase error tracking
control of nonminimal phase systems,” Journal of dynamic systems, measurement, and
control, vol. 114, no. 3, pp. 347–351, 1992.

[6] H.-S. Park, P. H. Chang, and D. Y. Lee, “Continuous zero phase error tracking controller
with gain error compensation,” in American Control Conference, 1999. Proceedings of
the 1999, vol. 5, pp. 3554–3558, IEEE, 1999.

[7] J. A. Butterworth, L. Y. Pao, and D. Y. Abramovitch, “The effect of nonminimum-
phase zero locations on the performance of feedforward model-inverse control techniques
in discrete-time systems,” in American Control Conference, 2008, pp. 2696–2702, IEEE,
2008.

[8] R. Jafari and R. Mukherjee, “Intermittent output tracking for linear single-input single-
output non-minimum-phase systems,” in American Control Conference (ACC), 2012,
pp. 5942–5947, IEEE, 2012.

[9] K. J. ˚Astr¨om, P. Hagander, and J. Sternby, “Zeros of sampled systems,” Automatica,

vol. 20, no. 1, pp. 31–38, 1984.

[10] C.-H. Wang, W.-Y. Wang, and C.-C. Hsu, “Minimum-phase criteria for sampled systems
via symbolic approach,” International journal of systems science, vol. 28, no. 8, pp. 767–
775, 1997.

[11] C.-C. Hsu and T.-C. Lu, “Minimum-phase criterion on sampling time for sampled-
data interval systems using genetic algorithms,” Applied Soft Computing, vol. 8, no. 4,
pp. 1670–1679, 2008.

[12] Y. Fu and G. A. Dumont, “Choice of sampling to ensure minimum-phase behaviour,”

IEEE Transactions on automatic control, vol. 34, no. 5, pp. 560–563, 1989.

95

[13] M. Ishitobi, “Criteria for stability of zeros of sampled systems,” IEE Proceedings-Control

Theory and Applications, vol. 141, no. 6, pp. 396–402, 1994.

[14] T. Hagiwara, “Analytic study on the intrinsic zeros of sampled-data systems,” IEEE

Transactions on Automatic Control, vol. 41, no. 2, pp. 261–263, 1996.

[15] E.-W. Bai and Y.-Q. Wu, “Limiting zero distribution of sampled systems,” Automatica,

vol. 38, no. 5, pp. 843–851, 2002.

[16] T. Hagiwara, T. Yuasa, and M. Araki, “Limiting properties of the zeros of sampled-
data systems with zero-and first-order holds,” in IEEE CONFERENCE ON DECISION
AND CONTROL, vol. 2, pp. 1949–1949, INSTITUTE OF ELECTRICAL ENGINEERS
INC (IEE), 1992.

[17] T. Hagiwara, T. Yuasa, and M. Araki, “Stability of the limiting zeros of sampled-data
systems with zero-and first-order holds,” International Journal of Control, vol. 58, no. 6,
pp. 1325–1346, 1993.

[18] K. M. Passino and P. J. Antsaklis, “Inverse stable sampled low-pass systems,” Interna-

tional Journal of Control, vol. 47, no. 6, pp. 1905–1913, 1988.

[19] M. Ishitobi, “Stability of zeros of sampled system with fractional order hold,” IEE

Proceedings-Control Theory and Applications, vol. 143, no. 3, pp. 296–300, 1996.

[20] R. B´arcena and A. Etxebarria, “Sufficient conditions for the attainment of sampled sys-
tems without nonminimum-phase zeros,” International Journal of Control, Automation
and Systems, vol. 8, no. 2, pp. 244–249, 2010.

[21] M. Ishitobi and Q. Zhu, “Zeros of sampled systems with fractional order hold imple-
mented by zero order hold,” in Intelligent Processing Systems, 1997. ICIPS’97. 1997
IEEE International Conference on, vol. 1, pp. 698–702, IEEE, 1997.

[22] O. Boubaker, “The inverted pendulum: A fundamental benchmark in control theory
and robotics,” in Education and e-Learning Innovations (ICEELI), 2012 international
conference on, pp. 1–6, IEEE, 2012.

[23] O. Boubaker, “The inverted pendulum benchmark in nonlinear control theory: a sur-

vey,” International Journal of Advanced Robotic Systems, vol. 10, no. 5, p. 233, 2013.

[24] M. F. M. Salleh and M. A. Shamsudin, “Mathematical modelling and quadratic optimal
tuning based pid scheme for an inverted pendulum-cart system,” in Asian Simulation
Conference, pp. 191–202, Springer, 2017.

[25] A. Jose, C. Augustine, S. M. Malola, and K. Chacko, “Performance study of pid con-
troller and lqr technique for inverted pendulum,” World Journal of Engineering and
Technology, vol. 3, no. 02, p. 76, 2015.

96

[26] L. B. Prasad, B. Tyagi, and H. O. Gupta, “Optimal control of nonlinear inverted pen-
dulum system using pid controller and lqr: performance analysis without and with dis-
turbance input,” International Journal of Automation and Computing, vol. 11, no. 6,
pp. 661–670, 2014.

[27] C. Wongsathan and C. Sirima, “Application of ga to design lqr controller for an in-
verted pendulum system,” in Robotics and Biomimetics, 2008. ROBIO 2008. IEEE
International Conference on, pp. 951–954, IEEE, 2009.

[28] S. Trimpe, A. Millane, S. Doessegger, and R. D’Andrea, “A self-tuning lqr approach
demonstrated on an inverted pendulum,” IFAC Proceedings Volumes, vol. 47, no. 3,
pp. 11281–11287, 2014.

[29] R. Eide, P. M. Egelid, A. Stamsø, and H. R. Karimi, “Lqg control design for balancing

an inverted pendulum mobile robot,” 2011.

[30] M. J. Grimble, “Implicit and explicit lqg self-tuning controllers,” IFAC Proceedings

Volumes, vol. 17, no. 2, pp. 941–947, 1984.

[31] M. E. Magana and F. Holzapfel, “Fuzzy-logic control of an inverted pendulum with

vision feedback,” IEEE transactions on education, vol. 41, no. 2, pp. 165–170, 1998.

[32] L. B. Prasad, H. O. Gupta, and B. Tyagi, “Intelligent control of nonlinear inverted
pendulum dynamical system with disturbance input using fuzzy logic systems,” in Re-
cent Advancements in Electrical, Electronics and Control Engineering (ICONRAEeCE),
2011 International Conference on, pp. 136–141, IEEE, 2011.

[33] R. Or´ostica, M. A. Duarte-Mermoud, C. J¨auregui, and G. Lefranc, “Inverted pendulum
stabilization by means of fractional order pid controllers,” in Electrical, Electronics Engi-
neering, Information and Communication Technologies (CHILECON), 2017 CHILEAN
Conference on, pp. 1–7, IEEE, 2017.

[34] R. Or´ostica, M. A. Duarte-Mermoud, and C. J´auregui, “Stabilization of inverted pen-
dulum using lqr, pid and fractional order pid controllers: A simulated study,” in Auto-
matica (ICA-ACCA), IEEE International Conference on, pp. 1–7, IEEE, 2016.

[35] Y. Wang, R. Jafari, G. G. Zhu, and R. Mukherjee, “Sample-and-Hold Inputs for
Minimum-Phase Behavior of Nonminimum-Phase Systems,” IEEE Transactions on
Control Systems Technology, vol. 24, no. 6, pp. 2103–2111, 2016.

[36] H.-S. Juang and K.-Y. Lurrr, “Design and control of a two-wheel self-balancing robot
using the arduino microcontroller board,” in 2013 10th IEEE International Conference
on Control and Automation (ICCA), pp. 634–639, IEEE, 2013.

[37] N. G. M. Thao, D. H. Nghia, and N. H. Phuc, “A pid backstepping controller for two-
wheeled self-balancing robot,” in International Forum on Strategic Technology 2010,
pp. 76–81, IEEE, 2010.

97

[38] A. Unluturk, O. Aydogdu, and U. Guner, “Design and pid control of two wheeled
autonomous balance robot,” in 2013 International Conference on Electronics, Computer
and Computation (ICECCO), pp. 260–264, IEEE, 2013.

[39] T.-J. Ren, T.-C. Chen, and C.-J. Chen, “Motion control for a two-wheeled vehicle using
a self-tuning pid controller,” Control Engineering Practice, vol. 16, no. 3, pp. 365–375,
2008.

[40] K. Liu, M. Bai, and Y. Ni, “Two-wheel self-balanced car based on kalman filtering and
pid algorithm,” in 2011 IEEE 18th International Conference on Industrial Engineering
and Engineering Management, pp. 281–285, IEEE, 2011.

[41] W. An and Y. Li, “Simulation and control of a two-wheeled self-balancing robot,” in
2013 IEEE International Conference on Robotics and Biomimetics (ROBIO), pp. 456–
461, IEEE, 2013.

[42] Q. Yong, L. Yanlong, Z. Xizhe, and L. Ji, “Balance control of two-wheeled self-balancing
mobile robot based on ts fuzzy model,” in Proceedings of 2011 6th International Forum
on Strategic Technology, vol. 1, pp. 406–409, IEEE, 2011.

[43] X. Ruan and J. Cai, “Fuzzy backstepping controllers for two-wheeled self-balancing
robot,” in 2009 International Asia Conference on Informatics in Control, Automation
and Robotics, pp. 166–169, IEEE, 2009.

[44] X. Ruan, J. Cai, and J. Chen, “Learning to control two-wheeled self-balancing robot
using reinforcement learning rules and fuzzy neural networks,” in 2008 Fourth Interna-
tional Conference on Natural Computation, vol. 4, pp. 395–398, IEEE, 2008.

[45] W. Wei, M. Xiaoning, and W. Jijun, “Intelligent control in two-wheel self-balanced
robot,” in 2010 International Conference on Computer, Mechatronics, Control and Elec-
tronic Engineering, vol. 3, pp. 470–473, IEEE, 2010.

[46] A. Wasif, D. Raza, W. Rasheed, Z. Farooq, and S. Q. Ali, “Design and implementation
of a two wheel self balancing robot with a two level adaptive control,” in Eighth Inter-
national Conference on Digital Information Management (ICDIM 2013), pp. 187–193,
IEEE, 2013.

[47] C. Sun, T. Lu, and K. Yuan, “Balance control of two-wheeled self-balancing robot
based on linear quadratic regulator and neural network,” in 2013 Fourth International
Conference on Intelligent Control and Information Processing (ICICIP), pp. 862–867,
IEEE, 2013.

[48] W. Junfeng and Z. Wanying, “Research on control method of two-wheeled self-balancing
robot,” in 2011 Fourth International Conference on Intelligent Computation Technology
and Automation, vol. 1, pp. 476–479, IEEE, 2011.

[49] C. Xu, M. Li, and F. Pan, “The system design and lqr control of a two-wheels self-
balancing mobile robot,” in 2011 International Conference on Electrical and Control
Engineering, pp. 2786–2789, IEEE, 2011.

98

[50] Y. Wang, G. G. Zhu, and R. Mukherjee, “Experimental study of nmp sample and hold
input using an inverted pendulum,” in ASME 2018 Dynamic Systems and Control Con-
ference, pp. V001T01A004–V001T01A004, American Society of Mechanical Engineers,
2018.

[51] G. G. Zhu, R. E. Skelton, and P. Li, “Q-markov cover identification using pseudo-
random binary signals,” International Journal of Control, vol. 62, no. 6, pp. 1273–1290,
1995.

[52] G. G. Zhu, “Weighted multirate q-markov cover identification using prbs–an application
to engine systems,” Mathematical Problems in Engineering, vol. 6, no. 2-3, pp. 201–224,
2000.

[53] C. E. Shannon, “Communication in the presence of noise,” Proceedings of the IRE,

vol. 37, pp. 10–21, Jan 1949.

[54] Quanser, “Linear servo base unit with inverted pendulum,” May 2018.

[55] A.-J. Baerveldt and R. Klang, “A low-cost and low-weight attitude estimation system
for an autonomous helicopter,” in Proceedings of IEEE International Conference on
Intelligent Engineering Systems, pp. 391–395, IEEE, 1997.

[56] S. Barash, Y. Brandman, Y. Cohen, S. Shporn, J. Vishnipolsky, and E. Yanai,

“Computer-controlled irrigation system,” June 24 1980. US Patent 4,209,131.

[57] A. Povoa and D. Hills, “Sensitivity of microirrigation system pressure to emitter plug-
ging and lateral line perforations,” Transactions of the ASAE, vol. 37, no. 3, pp. 739–799,
1994.

[58] R. W. Coates, M. J. Delwiche, and P. H. Brown, “Control of individual microsprinklers

and fault detection strategies,” Precision Agriculture, vol. 7, no. 2, pp. 85–99, 2006.

[59] J. E. Barlow and E. S. Pattiani, “Remotely controlled irrigation timer with fault detec-

tion,” Aug. 24 2004. US Patent 6,782,311.

[60] Z. A. Jaffery and A. K. Dubey, “Design of early fault detection technique for electrical
assets using infrared thermograms,” International journal of electrical power & energy
systems, vol. 63, pp. 753–759, 2014.

[61] R. S. Deitchman and S. P. Loheide, “Ground-based thermal imaging of groundwater
flow processes at the seepage face,” Geophysical Research Letters, vol. 36, no. 14, 2009.

[62] R. R. Gillies and T. N. Carlson, “Thermal remote sensing of surface soil water content
with partial vegetation cover for incorporation into climate models,” Journal of Applied
Meteorology, vol. 34, no. 4, pp. 745–756, 1995.

[63] A. Akula, R. Ghosh, and H. Sardana, “Thermal imaging and its application in defence

systems,” in AIP conference proceedings, vol. 1391, pp. 333–335, AIP, 2011.

99

[64] G. Ginesu, D. D. Giusto, V. Margner, and P. Meinlschmidt, “Detection of foreign bodies
in food by thermal image processing,” IEEE Transactions on Industrial Electronics,
vol. 51, no. 2, pp. 480–490, 2004.

[65] V. Chelladurai, D. Jayas, and N. White, “Thermal imaging for detecting fungal infection
in stored wheat,” Journal of Stored Products Research, vol. 46, no. 3, pp. 174–179, 2010.

[66] M. Stoll and H. G. Jones, “Thermal imaging as a viable tool for monitoring plant stress,”

OENO One, vol. 41, no. 2, pp. 77–84, 2007.

[67] S. Thomson, T. Brooks, Y. Huang, J. Weick, D. K. Fisher, S. DeFauw, C. Ouellet-
Plamondon, and P. English, “Thermal imaging: from irrigation management to finding
leaks at the levee,” Resource Magazine, vol. 22, no. 3, pp. 10–13, 2015.

[68] M. M¨oller, V. Alchanatis, Y. Cohen, M. Meron, J. Tsipris, A. Naor, V. Ostrovsky,
M. Sprintsin, and S. Cohen, “Use of thermal and visible imagery for estimating crop
water status of irrigated grapevine,” Journal of experimental botany, vol. 58, no. 4,
pp. 827–838, 2006.

[69] V. Gonzalez-Dugo, P. Zarco-Tejada, E. Nicol´as, P. A. Nortes, J. Alarc´on, D. S. Intrigli-
olo, and E. Fereres, “Using high resolution uav thermal imagery to assess the variability
in the water status of five fruit tree species within a commercial orchard,” Precision
Agriculture, vol. 14, no. 6, pp. 660–678, 2013.

[70] Y. Pederi and H. Cheporniuk, “Unmanned aerial vehicles and new technological meth-
ods of monitoring and crop protection in precision agriculture,” in 2015 IEEE In-
ternational Conference Actual Problems of Unmanned Aerial Vehicles Developments
(APUAVD), pp. 298–301, IEEE, 2015.

[71] K. Ribeiro-Gomes, D. Hern´andez-L´opez, J. Ortega, R. Ballesteros, T. Poblete, and
M. Moreno, “Uncooled thermal camera calibration and optimization of the photogram-
metry process for uav applications in agriculture,” Sensors, vol. 17, no. 10, p. 2173,
2017.

100