Control of Vibration Systems with Mechanical Motion Rectifier

and their Applications to Vehicle Suspension and Ocean Energy

Harvester

Qiuchi Xiong

Thesis submitted to the Faculty of the Virginia Polytechnic Institute and State University in

partial fulfillment of the requirements for the degree of

Master of Science

in

Mechanical Engineering

Lei Zuo, Chair

Steve Southward

Oumar Barry

March, 18th, 2020

Blacksburg, Virginia

Keyworks: Vibration Reduction, Vibration Amplification, Energy Harvesting, Vehicle

Suspension, Ocean Wave Energy Converter

Copyright 2020, Qiuchi Xiong

Control of Vibration Systems with Mechanical Motion Rectifier

and their Applications to Vehicle Suspension and Ocean Energy

Harvester

Qiuchi Xiong

SCHOLARLY ABSTRACT

Vibration control is a large branch in control research, because all moving systems may induce

desired or undesired vibration. Due to the limitation of passive system’s adaptability and

changing excitation input, vibration control brings the solution to change system dynamic with

desired behavior to fulfill control targets. According to preference, vibration control can be

separated into two categories: vibration reduction and vibration amplification. Lots of research

papers only examine one aspect in vibration control. The thesis investigates the control

development for both control targets with two different control applications: vehicle suspension

and ocean wave energy converter. It develops control methods for both systems with simplified

modeling setup, then followed by the application of a novel mechanical motion rectifier (MMR)

gearbox that uses mechanical one-way clutches in both systems. The flow is from the control

for common system to the control design for a specifically designed system. In the thesis, active

(model predictive control: MPC), semi-active (Skyhook, skyhook-power driven damper: SH-

PDD, hybrid model predictive control: HMPC), and passive control (Latching Control)

methods are developed for different applications or control performance comparison on single

system. The thesis also studies about new type of system with switching mechanism, in which

other papers do not talk too much and possible control research direction to deal with such

complicated system in vibration control. The state-space modeling for both systems are

provided in the thesis with detailed model of the MMR gearbox. From the simulation, it can be

shown that in the vehicle suspension application, the controlled MMR gearbox can be effective

in improving vehicle ride comfort by 29.2% compared to that of the traditional hydraulic

suspension. In the ocean wave energy converter, the controlled MMR WEC with simple

latching control can improve the power generation by 57% compared to the passive MMR

WEC. Besides, the passive MMR WEC also shows its advantage on the passive direct drive

WEC in power generation improvement. From the control development flow for the MMR

system, the limitation of the MMR gearbox is also identified, which introduces the future work

in developing active-MMR gearbox by using an electromagnetic clutch. Some possible control

development directions on the active-MMR is also mentioned at the end of the thesis to provide

reference for future works.

Control of Vibration Systems with Mechanical Motion Rectifier

and their Applications to Vehicle Suspension and Ocean Energy

Harvester

Qiuchi Xiong

GENERAL AUDIENCE ABSTRACT

Vibration happens in our daily life in almost all cases. It is a regular or irregular back and forth

motion of particles. For example, when we start a vehicle, the engine will do circular motion

to drive the wheel, which causes vibration and we feel wave pulses on our body when we sit in

the car. However, this kind of vibration is undesirable, since it makes us uncomfortable. The

car manufacture designs cushion seats to absorb vibration. This is a way to use hardware to

control vibration. However, this is not enough. When vehicle goes through bumps, we do have

suspension to absorb vibration transferred from road to our body. The car still experiences a

big shock that makes us feel dizzy. On the opposite direction, in some cases when vibration

becomes the motion source for energy harvesting, we would like to enhance it. Hardware can

be helpful, since by tuning some parameters of an energy harvesting device, it can match with

the vibration source to maximize vibration. However, it is still not enough due to low

adaptability of a fixed parameter system. To overcome the limitation of hardware, researches

begin to think about the way to control vibration, which is the method to change system

behavior by using real-time adjustable hardware. By introducing vibration control, the theory

behind that started to be investigated. This thesis investigates the vibration control theory

application in both cases: vibration reduction and vibration enhancement, which are mentioned

above due to opposite application preferences. There are two major applications of vibration

control: vehicle suspension control and ocean wave energy converter (WEC) control. The

thesis starts from the control development for both fields with general modeling criteria, then

followed by control development with specific hardware design-mechanical motion rectifier

(MMR) gearbox-applied on both systems. The MMR gearbox is the researcher designed

hardware that targets on vibration adjustment with hardware capability, which is similar as the

cushion seats mentioned at the beginning of the abstract. However, the MMR cannot have

capability to furtherly optimize system vibration, which introduces the necessity of control

development based on the existing hardware. In the suspension control application, the control

strategy introduced successfully improve the vehicle ride comfort by 29.2%, which means the

vehicle body acceleration has been reduced furtherly to let passenger feel less vibration. In the

WEC application, the power absorbed from wave has been improved by 57% by applying

suitable control strategy. The performance of improvement on vibration control has proved the

effect on further vibration optimization beyond hardware limitation.

v

Acknowledgements

I would like to express my sincere gratitude to everyone who helped me during my research.

I would like to thank my advisor, Prof. Lei Zuo. He has been giving me valuable instructions

and guidance, which greatly helped me with my research. From the opportunities he provided,

I could get access to explore the field I worked in and received knowledges about my

concentrated fields.

I would like to thank my committee members Prof. Steve Southward and Prof. Oumar Barry

who provided very valuable suggestions to my thesis.

I would like to thank the partial financial support from DOE Grant # EE0007174, NSF Grant

#1530122 and # 1530508, as well as CIT/CRCF Award # MF16-004-En during the research.

I would like to thank my lab mates, Xiaofan Li, Qiaofeng Li, Boxi Jiang, Shuo Chen, Bonan

Qin, Jia Mi, Feng Qian, Lisheng Yang, Hongjip Kim, Hyunjun Jung, Yuzhe Chen, Rui Lin,

Jianuo Huang, Kan Sun, Weihan Lin, Mingyi Liu, Shifeng Yu, Yongjia Wu, Sijing Guo, for the

help on my research work and technical discussions.

At last, I would like to thank my parents, Chunming Xiong and Jianshi Meng who supported

me in my daily life.

Sincerely,

Qiuchi Xiong

vi

Contents

1 Background and Literature Review……………………………………………………..1

1.1 Necessity of Vibration Control……………………………………………………….1

1.2 Vibration Control Methods Review………………………………………………….2

2 Vibration Suppression Control: Vehicle Suspension……………………………………5

2.1 Active Suspension Control Design for Vehicle Dynamic Tire Load Reduction at

Traffic Signal Light……………………………………………………………………………5

2.1.1 Introduction…………………………………………………………………..5

2.1.2 Class-8 Heavy Duty Truck Modeling…………………………………………7

2.1.3 LQR Output Feedback Control Development………………………………...9

2.1.4 MPC Feedforward Control Development…………………………………...10

2.1.5 Simulation Results and Conclusion…………………………………………11

2.2 Semi-active Suspension Control Design for Vehicle Ride Comfort Improvement for

MMR-based Energy Harvesting Suspension…………………………………………………15

2.2.1 Introduction…………………………………………………………………15

2.2.2 MMR-based Energy Harvesting Shock Absorber Modeling………………..17

2.2.3 MMR-based Shock Absorber Effect on Vibration Reduction in Bump Scenario

with Simple Control…………………………………………………………………………..20

2.2.4 Skyhook Control on Traditional Suspension and SH-PDD Control on MMR-

based Suspension……………………………………………………………………………..23

2.2.5 Rule-based Control on MMR-based Suspension……………………………26

2.2.6 Simulation Results and Conclusion…………………………………………29

3 Vibration Amplification Control: Ocean Wave Energy Converter (WEC)………….36

3.1 Active and Semi-active Control for Normal Two-body WEC……………………...36

3.1.1 Introduction…………………………………………………………………...36

3.1.2 Two-body WEC Modeling…………………………………………………..38

3.1.3 MPC Control Development (Active)………………………………………..41

vii

3.1.4 HMPC Control Development (Semi-active)………………………………...44

3.1.5 Simulation Results and Conclusion…………………………………………47

3.2 Latching Control on MMR-based Two-body WEC………………………………...53

3.2.1 Introduction…………………………………………………………………53

3.2.2 MMR-based PTO Modeling………………………………………………...57

3.2.3 Latching Control on MMR-based Two-body WEC………………………...62

3.2.4 Simulation Results and Conclusion…………………………………………65

3.2.5 Flywheel Effect on System Peak-to-average Ratio Reduction for Two-body

WEC………………………………………………………………………………………….69

4 Summary and Future Work…………………………………………………………….73

4.1 Vibration Suppression Control: Vehicle Suspension………………………………….73

4.2 Vibration Amplification Control: Ocean Wave Energy Converter……………………74

4.3 Future Work: The Concept of Active-MMR and Control Research Directions………75

4.3.1 Introduction…………………………………………………………………75

4.3.2 Active-MMR Concept Design and Modeling……………………………….76

4.3.3 Possible Research Direction in Control……………………………………..80

5 References……………………………………………………………………………….85

Appendix A: Vehicle Suspension Nomenclature…………………………………………..89

Appendix B: Ocean Wave Energy Converter Nomenclature…………………………….92

viii

List of Figures

1.1 BRIDGE DESTRUCTION DUE TO WIND………………………………...1

1.2 SMALL-SCALE ENERGY HARVESTERS………………………………..1

1.3 HARMONIC WAVE SUPERPOSITION……………………………………2

1.4 THE SIMPLEST 1-DOF SYSTEM………………………………………….3

2.1 AMERICAN HEAVY-DUTY TRUCK ANNUAL HIGHWAY MILAGE

DATA……………………………………………………………………………5

2.2 ROAD DAMAGE CAUSED BY VEHICLE BRAKING AT

INTERSECTION………………………………………………………………..6

2.3 HALF TRUCK MODEL…………………………………………………….7

2.4 LQR FEEDBACK CONTROL LOOP……………………………………..10

2.5 BRAKING PRESSURE AND VEHICLE VELOCITY DATA OF A

HUMAN-DRIVING VEHICLE……………………………………………….12

2.6 RELATIONSHIP BETWEEN BRAKING INTENSITY AND ROAD

STRESS FACTOR……………………………………………………………..12

2.7 COMAPRSION BETWEEN PASSIVE VEHICLE AND LQR-

CONTROLLED VEHICLE……………………………………………………13

2.8 COMPARISON BETWEEN PASSIVE VEHICLE AND MPC-

CONTROLLED VEHICLE……………………………………………………14

ix

2.9 (A) COMPARISON OF ROAD STRESS FACTOR BETWEEN PASSIVE

VEHICLE AND CONTROLLED VEHICLE (B) THE IMPROVEMENT OF

THE ROAD STRESS FACTORS BROUGHT BY CONTROLLED ACTIVE

SUSPENSION…………………………………………………………………15

2.10 MMR-BASED SHOCK ABSORBER DESIGN…………………………17

2.11 SCHEMATIC DIAGRAM OF THE MMR-BASED SHOCK

ABSORBER…………………………………………………………………...18

2.12 DYNAMIC MODELING OF THE MMR-BASED SHOCK

ABSORBER…………………………………………………………………...18

2.13 BUMP CONTROL CONCEPT OF MMR-BASED SHOCK

ABSORBER…………………………………………………………………...21

2.14 BUMP TIME-BASED PROFILE………………………………………...22

2.15 TRADITIONAL SUSPENSION BASELINE…………………………….22

2.16 RIDE COMFORT COMPARISON IN BUMP SCENARIO……………...23

2.17 SKYHOOK CONTROL CONCEPT……………………………………...24

2.18 RULE-BASED CONTROL LOGIC DIAGRAM………………………...28

2.19 ROAD PROFILE INPUT FOR B-CLASS+C-CLASS…………..……….31

2.20 3-D PLOT TO DETERMINE OPTIMAL br …………………………..31

2.21 CONTROL FORCE COMPARISON FOR ALL CONTROLLED

MODELS………………………………………………………………………32

x

2.22 RIDE COMFORT COMPARISON AMONG DIFFERENT MODELS….33

2.23 PASSIVE MMR VS PASSIVE TRADITIONAL…………………………33

2.24 POWER GENERATION FOR CONTROLLED MMR SHOCK

ABSORBER…………………………………………………………………...34

3.1 (A) OVERTOPPING SYSTEMS (B) OSCILLATING BODY (C)

OSCILLATING WATER COLUMN (OWC) ……………………….…………37

3.2 TWO-BODY WEC…………………………………………………………37

3.3 TWO-BODY WEC SCHEMATIC DIAGRAM……………………………38

3.4 WAVE EXCITATION FORCE INPUTS FOR THE BUOY AND THE

SUBMERGED BODY…………………………………………………………48

3.5 BUOY AND SUBMERGED BODY RELATIVE POSITIONS AND

VELOCITY……………………………………………………………………48

3.6 RELATIVE VELOCITY AND WAVE EXCITATION FORCE ON THE

WEC……………………………………………………………………………49

3.7 GENERATOR FORCE FOR THE WEC…………………………………..49

3.8 PTO DAMPING COMPARISON………………………………………….50

3.9 PTO DAMPING COMPARISON BETWEEN SEMI-ACTIVE & PASSIVE

MODELS………………………………………………………………………50

3.10 POWER GENERATION COMPARISON………………………………..51

3.11 RACK-PINION BASED PTO…………………………………………….54

xi

3.12 BALL-SCREW MECHANISM…………………………………………..54

3.13 MMR-BASED WEC SYSTEM…………………………………………..55

3.14 (A) MMR BENCH TEST SETUP (B) SINUSOID TEST RESULTS……55

3.15 TWO-BODY LATCHING CONTROL CONCEPT……………………....56

3.16 SCHEMATIC DIAGRAM OF SINGLE-BODY WEC SYSTEM………..60

3.17 SCHEMATIC DIAGRAM OF MMR PTO………………………………..61

3.18 WAVE INPUT FREQUENCY SPECTRUM……………………………...66

3.19 TIME DOMAIN WAVE INPUT…………………………………………..66

3.20 FINDING OPTIMAL EXTERNAL RESISTANCE FOR PASSIVE

MMR…………………………………………………………………………...67

3.21 POWER COMPARISON BETWEEN PASSIVE SINGLE-BODY AND

TWO-BODY MMR WECS……………………………………………………67

3.22 POWER COMPARISON AMONG DIFFERENT MODELS…………….68

3.23 (A) 0.005 kg-m2 (B) 0.01 kg-m2 (C) 0.05 kg-m2……………………...…...70

3.24 (A) 0.005 kg-m2 (B) 0.01 kg-m2………………………………………..….70

4.1 OGURA MAGNETIC CLUTCH AND CROSS-SECTION DRAWING….75

4.2 (A) MAGNETIC POWDER CLUTCH (B) MULTI-PLATE MAGNETIC

CLUTCH………………………………………………………………………76

4.3 ACTIVE-MMR GEARBOX DESIGN CONCEPT………………………..76

xii

4.4 ACTIVE-MMR OPERATION MODES…………………………………...77

4.5 SHORTEST PATH PROBLEM…………………………………………….82

4.6 DYNAMIC PROGRAMMING FLOW FOR STATE-SPACE MODEL…...84

xiii

List of Tables

TABLE I. LQR CONTROL PERFORMANCE………………………………. 13

TABLE II. MPC CONTROL PERFORMANCE………………………………14

TABLE III. ROAD ROUGHNESS LEVELS CLASSIFIED BY ISO 8608……30

TABLE IV. COMPARISON ABOUT VEHICLE RIDE COMFORT AND

AVERAGE POWER GENERATION………………………………………….34

TABLE V. AVERAGE PTO DAMPING COMPARISON……………...……...52

TABLE VI. AVERAGE POWER COMPARISON…………………………….53

TABLE VII. AVERAGE POWER COMPARISON…………………...……….69

TABLE VIII. 0.005 kg-m2 FLYWHEEL ATTACHED………………...…….....70

TABLE IX. 0.01 kg-m2 FLYWHEEL ATTACHED……………………...…......71

TABLE X. 0.05 kg-m2 FLYWHEEL ATTACHED………………………..……71

1

Chapter 1

Background and Literature Review

1.1 Necessity of Vibration Control

Vibration exists in almost all the motion of mechanical system during operation. It is due to

uneven disturbance input (excitation). The vibrating energy from excitation source will be

transferred to mechanical system to cause vibration. Most of the vibration is unwanted and

even destructive. For example, the wind induced vibration on bridge may cause bridge

destruction, precise equipment noise may cause undesired accuracy destruction. However, on

the opposite, sometimes, vibration may be useful. For instance, energy harvesting from

vibration has been investigated by many researches through different applications. Both small

scale vibration energy harvesters, such as beam vibrator [1], backpack energy harvester, and

large scale energy harvesters, such as ocean wave energy converter (WEC) require vibration

amplification.

Fig 1.1. Bridge Destruction due to Wind

(a) Beam Energy Harvester [2] (b) Backpack Energy Harvester [3]

Fig 1.2. Small-scale Energy Harvesters

A passive system has no capability to be adapted to excitation source, which causes problem

in undesired vibration or desired vibration. A simple wave superposition theory shows that

2

when two harmonic waves are out of phase, the resulting wave will be flat. However, when

two waves are in phase, wave amplitude will be added together, which is called resonance.

Natural frequency is one fundamental characteristic of a system. It reflects the frequency a

system will vibrate when there is no excitation. When the input excitation has same frequency

as the system, resonance will happen to enhance vibration. When the input excitation has large

frequency difference compared to that of the system, vibration will be reduced. For undesired

vibration, the target should be to reduce that; for desired vibration, the target should be to

enhance that. However, the system natural frequency is fixed due to fixed parameters. The

resulting vibration may not be desirable due to difference between system and excitation

frequency. Therefore, vibration control is necessary to search for some solutions to change

system characteristics to be adapted with external excitation to achieve desired motion.

(a) (b)

Fig 1.3. Harmonic Wave Superposition (a) Out-of-phase (b) In-phase

1.2 Vibration Control Methods Review

Although vibration control can have two different control targets: vibration reduction and

enhancement, the control methods can be similar. So many researchers have developed

different sub-optimal, optimal methods in literatures. Based on the analysis of a mechanical

system, the simplest model can be a system with mass, damper, and spring. The natural

frequency of the system depends on the mass and spring stiffness. The damped frequency

depends on mass, damping and spring stiffness. Hence, vibration control focuses on the control

of mass, damper, and stiffness. According to real application, system damping can be the easiest

way to be adjusted with more flexibility. For example, in [4], a commercialized friction damper

is applied. It changes system damping by adjusting friction force between a foam covering a

piston and external case of the component. The friction is controlled via magnetic fluid

generated magnetic field intensity saturated inside the foam. Damping control or damping force

control becomes popular in vibration control in many industry fields. Start from simplest input,

harmonic excitation with single wave frequency, frequency domain analysis is developed. A

set of control methods are developed in frequency domain. The major control concept is to

change controlled system frequency domain response to let it has low amplitude at excitation

frequency. For example, in [5], a skyhook control is introduced to be implemented on a 1-DOF

system with harmonic base excitation. The resulting frequency response shows that the

skyhook control reduces the amplitude of controlled system at low frequency range (0Hz-

10Hz). Skyhook is the simplest control strategy applied in vibration control field. Other

3

damping tuning control, like 𝐻2 [6] or 𝐻 [7] controls are also applied as another optimal

damping tuning method for vibration reduction. Besides damping tuning, another method, tune

mass damper (TMD) tunes attached mass, damper and stiffness system to affect the controlled

system to vary system frequency response. Such method is widely used in high building

vibration due to wind or earthquake as depicted in [8]. Above methods mentioned are semi-

active methods, which means damper will always absorb energy. However, to achieve better

system dynamic, maybe sometimes, it is preferred to let an actuator to drive controlled system

to achieve certain dynamic behavior. Such concept is called active control. In [9], complex-

conjugate control specifically applied on ocean wave energy converter (WEC) is introduced. It

uses power take-off (PTO) force to cancel out WEC inertia and spring forces and match device

impedance to achieve maximum power extraction under harmonic excitation. It is an active

control method, since PTO force may drive the system to match impedance.

Fig 1.4. The Simplest 1-DOF System

In the frequency domain, control methods design based on system frequency response can be

effective with harmonic excitation. However, multi-frequency excitation or random excitation

happens everywhere, which limits frequency tuned control methods’ performance in real

application. Time domain optimal control methods become popular to be dealt with random

excitations. The LQR, MPC methods that are based on state-space model has been widely used

in vibration control. They use excitation time domain input to optimize defined control

performance cost function in time domain. The cost function can be the integration over a

certain time period or the whole time frame to get overall minimum value to achieve optimal

solution. Both methods have been applied actively or semi-actively for linear systems. For

example, in [10], the MPC method is applied in active suspension to improve vehicle ride

comfort and road handling. In [11], MPC application with systems with logic operator is

introduced to make MPC semi-active damping control possible. Besides linear system optimal

control, other optimization methods are also applied in vibration control for nonlinear systems.

In [12], adaptive control design based on stability theory is introduced in vibration control on

nonlinear system.

Vibration control has applied a wide range of control theories. The major control targets for all

vibration system can be conclude as in section 1.1: vibration reduction and amplification. The

thesis will investigate the control development in both aspects with vehicle suspension and

ocean wave energy converter applications. These two applications are symbolic in vibration

reduction and amplification control. The thesis will discuss how control is developed with

4

system design constraints, due to the application of a novel mechanical motion rectifier (MMR)

gearbox. The thesis will be arranged as follow: Chapter 2 presents vibration control in a vehicle

suspension; Chapter 3 examines the vibration amplification control in WEC; Chapter 4

provides conclusion and future work.

5

Chapter 2

Vibration Suppression Control: Vehicle Suspension

This chapter introduces the control application on vehicle suspension on both suspension

control targets separately via two different projects. One project is about active control

development, another project is about semi-active control development. The chapter

investigates major control concepts applied on vehicle suspension for vibration reduction. Both

projects show effectiveness of control strategies on vehicle ride dynamics optimization.

2.1 Active Suspension Control Design for Vehicle Dynamic Tire

Load Reduction at Traffic Signal Light

2.1.1 Introduction

American ground logistic heavily depends on heavy-duty truck. Based on the statistical data

from the United States Department of Transportation [13], from Fig 2.1, the total annual

mileage driven by heavy-duty truck (2 axles, 6 tires or more) on highway has increased from

40000 million miles to nearly 120000 million miles within the last 45 years.

Fig 2.1. American Heavy-duty Truck Annal Highway Mileage Data

0

20,000

40,000

60,000

80,000

100,000

120,000

140,000

1970 1980 1990 1995 2000 2005 2010 2015

Tota

l An

nu

al M

ileag

e (M

illio

ns)

Year

Total Annual Mileage Driven by Heavy-duty Truck

6

More annual mileage driven also means more chances truck will enter city, due to the location

of vendors. The weight of the trucks creates great impacts on the premature failure of

pavements. [14]. Refai et al. [15] stated that heavy-duty vehicles account for 79% (or $60

million) of annual expenditures required for roadway repaving in the State of Oregon.

Weissmann et al. [16] pointed out that the annual overlay costs induced by overweight trucks

can reach 59.5 million dollars, by analyzing the relevant data of 5 major Texas truck corridors.

Per empirical data in Chattanooga, TN area, pavements at signalized intersections needed to be

repaved in three years where there was frequent heavy-duty truck traffic. Pavement resurfacing

at signalized intersections were normally scheduled every fifteen years with “regular” traffic

conditions, i.e. less or none heavy-duty trucks, in the same area. Hence, to protect pavements

in such areas is significant and economically beneficial.

From the studied mechanism of traffic-induced pavement failure, dynamic tire loads greatly

influence the pavement failure. Cole et al. [17] concluded that fatigue failure of pavements is

likely to be governed by peak dynamic forces, rather than average dynamic forces. A

performance index to access the road damage caused by peak dynamic tire loads is commonly

expressed as road stress factor [18]. Additionally, a literature review shows that the dynamic

tire loads can increase the theoretical road damage by 20%~400% of the damage induced by

static loads for typical vehicles and operating conditions, based on analysis and assumptions

[19].

Lots of researchers focus on vehicle dynamic tire load control with normal driving operation,

for example, [20] and [21]. In such driving conditions, braking will not be considered. However,

during heavy braking scenario, vehicle will have significant weight transfer from the rear to

the front and induce 8 to 32 times pavement surface cracking and 2.0 to 2.6 times

rutting/shoving potential compared to normal high-speed vehicle load [22]. The damage will

be increased with increased vehicle mass. Therefore, for heavy-duty truck with trailer, road

damage will be significantly increased during braking at signalized intersection, where braking

happens frequently due to red light. Hence, to extend the pavement life at signalized

intersection, a good solution is to propose suspension control strategies with the

consideration of braking motion, with a purpose to minimize vehicle dynamic tire loads,

especially on the steering wheels, which will cause more damage due to braking weight

transfer.

Fig 2.2. Road Damage Caused by Vehicle Braking at Intersection

The project is to investigate an active control strategy to reduce heavy-duty truck steering and

7

second axle wheels’ dynamic tire load during heavy brake. A class-8 heavy duty truck with a

trailer is introduced. In the model, the braking deceleration is treated as disturbances, along

with the road roughness excitations. Then, the road stress factor which represents the road

damage caused by dynamic tire loads is introduced, and its relation with the braking

deceleration is also explored. It turns out that the road stress factor at 0.5g braking intensity is

enlarged to 2.7 times, compared to the vehicle without braking. To reduce this type of road

damage, the LQR and MPC control algorithms are applied to the active suspensions and the

objective function is to minimize the road stress factor of the front wheels. Results show that

the active suspensions can improve the road stress factor from 1.164 to 1.082 for the steering

wheel and 1.429 to 1.056 for the tractor drive wheel, compared with the corresponding passive

wheels. The improvements are considered to be nontrivial, since the common range of a road

stress factor varies from 1.11 to 1.46 [23].

2.1.2 Class 8 Heavy Duty Truck Modeling

In this section, a heavy articulated class-8 truck with 3 axles is selected and modelled. The

truck is assumed to travel in a straight line near the intersection. Therefore, both lateral and

yaw motions are neglected. For simplification, the vehicle is modeled as a half truck, as shown

in Fig 2.3. The articulation joint (commonly called as the 5th wheel) is simplified as a high

stiffness spring and damper [24].

3um

8

Fig 2.3. Half Truck Model

The dynamic motion is governed by the follow equations:

1. Vertical Ride Motion

5 5c c sf df sr dr s dm z F F F F F F= + + + − − (2.1)

5 5t t st dt s dm z F F F F= + + + (2.2)

1u wf tf sf dfm z F F F= − − (2.3)

2u wr tr sr drm z F F F= − − (2.4)

3u wt tt st dtm z F F F= − − (2.5)

2. Pitch Motion

2 2 1 1 5 3 5 3 1c c sr dr sf df s d bf g br g xJ F l F l F l F l F l F l F h F h F D = + − − − − + + + (2.6)

5 4 5 4 5 5 2t t s d st dt x bt gtJ F l F l F l F l F D F h = − − + + + + (2.7)

3. Force Equations

2( )sf f wf c cF k z z l = − − (2.8)

2( )sr r wr c cF k z z l = − − (2.9)

5( )st t wt t tF k z z l = − − (2.10)

5 5 3 4( )s c c t tF k z l z l = + − + (2.11)

5 5 3 4( )d c c t tF c z l z l = + − + (2.12)

( )tf tf gf wfF k z z= − (2.13)

( )tr tr gr wrF k z z= − (2.14)

( )tt tt gt wtF k z z= − (2.15)

x bf br cF F F m a= + − (2.16)

9

The parameters from Equation (2.1) to Equation (2.16) are introduced in Appendix A, Table I.

Then, by combining equation (2.1)-(2.16) the system is converted to state-space model as:

1 2

,

X AX B u B w

Y CX Du D

= + +

= + = 0 (2.17)

where

[ , , , , , , , , , , , , , ]

[ , , ]

[ , , , , , , ]

[( ), ( )]

T

c c c c wf wf wr wr t t t t wt wt

T

df dr dt

T

bf br bt gf gr gt

T

gf wf gr wr

X z z z z z z z z z z

u F F F

w F F F a z z z

Y z z z z

=

=

=

= − −

1 2, , , ,A B B C D are coefficient matrices, w denotes the external disturbances. Y represents

the steering tire deflection and second axle tire deflection, and the corresponding tire dynamic

loads of the steering and second axles are 1tfk Y and 2trk Y . Since the target is to reduce the tire

dynamic loads of the steering and second axles, the tire deflection on these two axles are chosen

as outputs.

2.1.3 LQR Output Feedback Control Development

To reduce the dynamic tire loads of the tractor, the performance index is written as:

0 0

( ) ( )T T T T T

LQRJ Y QY u Ru dt X C QCX u Ru dt

= + = + (2.18)

where

1

1

2

23

0 00

, 0 00

0 0

T

RQ

Q C QC R RQ

R

= = =

Since the control target is to reduce tractor dynamic tire load, the LQR cost function is designed

to minimize tire deflection of tractor with the system output matrix C integrated into the cost

function.

10

Fig 2.4. LQR Feedback Control Loop

2.1.4 MPC Feedforward Control Development

The LQR method calculates the cost function based on infinite time horizon, which means

sudden change in system states cannot be accounted. Besides, LQR method will not consider

system constraints due to physical limitation. For instance, the active control force for the

suspension cannot be unlimited. Therefore, with the consideration of the limitations of LQR

methods, Model Prediction Control (MPC) is applied. The MPC is a receding horizon optimal

control method. It is based on discretized linear model, and can be written as

N k NX Mx GU= + , where kx is the state at the first step in each horizon, ,N NX U are

respectively the state vector and the control input vector in the receding horizon of N steps, and

M and G are obtained from equation (2.17) via state-space discretization. The matrices and

vectors are represented by

1

2

2 1

1 2

1

0 0

0, , ,

k k d d

k k d d d d

N N

N N N

k N k N d d d d d d

x u A B

x u A A B BX U M G

x u A A B A B B

+

+ +

− −

+ + −

= = = =

(2.19)

where 1 2

,

X AX B u B w

Y CX Du D

= + +

= + = 0

1 1 2

,

k d k d k d k

k d k d k d

x A x B u B w

y C x D u D

+ = + +

= + = 0 by zero order hold

The main idea is to formulate a finite-horizon control problem by solving

. .

T T

MPC N N N NJ X Q X U R U

s t U U U

= +

(2.20)

where ,N NQ R are the weight matrices of the states and control inputs; ,U U are the upper

bound and lower bound of the control inputs and the absolute values are 40000N. The weight

matrices are { , , }, { , , }T T

N d d d d NQ diag C QC C QC R diag R R= = . The reason to use T

d dC QC

11

in the NQ matrix is essential, since the output is the control target, 'T T T

k k k d d ky Q y x C QC x= .

The problem is solved at each time step k with a time horizon of N. In each step, a series of

control inputs can be calculated and only the first control input ku is selected as the current

control input. The most efficient way to solve MPC problem is to convert it into a quadratic

programming problem. Hence, the cost function is converted as

1

2

.

T T

MPCJ U HU f U

s t AU b

= +

(2.21)

where , , [ ; ], [ ]T T

N N N kH G Q G R f G Q Mx A I I b U U= + = = − =

2.1.5 Simulation Results and Conclusion

I. Performance Assessment

As mentioned in the chapter introduction, the road damage with vehicle vibration is mainly

related to peak dynamic tire load. The road friendliness of these peak dynamic tire loads is

assessed by road stress factor [18], [25], [26]. It can be represented by

2 41 6 3DLC DLC = + + (2.22)

where DLC is the dynamic load coefficient and is calculated as ( ) /dyn statRMS F F . The dynF is

the dynamic tire load and statF is the static tire load. For typical highway conditions of

roughness and speed, the dynamic road stress factor is between 1.11 and 1.46 [23].

The system is solved discretely in MATLAB by using the Tustin approximation. The

parameters employed in the simulation are listed in Appendix A, Table II.

II. Disturbance Inputs

In the numerical calculation, both the road roughness and the braking deceleration are regarded

as disturbance inputs. The road roughness is created as Class C road, based on ISO-8608 [27].

The human driver braking data is recorded from road tests, as shown in Fig 2.5. It’s assumed

that the truck is not in the emergency braking scenario and there is no tire slip. The braking

fluid pressure in the braking cylinder is recorded and then transformed to deceleration data

with no time delay. Since the target is to explore the pavement improvements of suspension

control at signalized intersections, only one segment of braking motion shown in the box in Fig

2.5 from 636s to 652s is selected.

12

Fig 2.5. Braking Pressure and Vehicle Velocity Data of a Human-driving Vehicle

To explore the relationship between the braking intensity and road stress factor, different

constant deceleration for passive vehicle and calculated corresponding road stress factor are

obtained. The braking intensity is set from 0.1g to 0.5g with the result shown in Fig 2.6. It can

be seen that the tractor drive axle is not sensitive to braking intensity, while the road stress

factor of the steering axle is very sensitive and the road stress factor at 0.5g braking intensity

becomes 2.7 times of that without braking. The reason can be the great weight transfer at large

braking intensity value. As the braking intensity increases, larger weight transfers from the

trailer axle to the tractor second axle and from the tractor second axle to the steering axle. For

the tractor second axle, the two types of weight transfer may have cancelled each other and

induced a stable road stress factor. For the steering axle, it suffers greatly from weight transfer

and increases significantly with braking intensity.

Fig 2.6 Relationship between Braking Intensity and Road Stress Factor

III. Control Weight Matrix Selection

For the LQR control, the weight matrix Q for output minimization is selected as [7.2×1014, 0;

0, 4×1012]; matrix R is selected as [5.74, 0, 0; 0, 2.8, 0; 0, 0, 0.2]. The corresponding weights

are set with large values, since the minimization of tractor tires’ deflection is the target.

13

For the MPC control, the simple time is 0.01s, which is based on the experiment data simple

time for brake force. The time receding horizon is selected as 10, since a horizon greater than

10 steps will not change optimal control inputs. The weight matrix Q for output minimization

is selected as [2×1014, 0; 0, 5×1012]; matrix R is selected as [470, 0, 0; 0, 2, 0; 0, 0, 0.1].

IV. Simulation Results

With the parameters from Table II, the simulation is done for both LQR and MPC methods

with same road roughness and braking data inputs. The LQR results are shown in Fig 2.7.

(a) (b) (c)

Fig 2.7. Comparison between Passive Vehicle and LQR-controlled Vehicle for (a) Dynamic

Tire Load of Steering Wheel; (b) Dynamic Tire Load of Tractor Drive Wheel; (c) Control

Input Forces

In Fig 2.7 (a) and (b), the dynamic tire load comparison between the passive truck and the

LQR-controlled truck is shown. The controlled peaks of steering tire dynamic load within the

braking period has been reduced at almost every load peak point. Since the LQR control

algorithm has been designed to minimize tire deflection root mean square value, both the

positive and negative tire dynamic load values can be reduced as expected. Based on the road

stress factor evaluation, the improvement of the steering axle is 2.9%, while the improvement

of the second axle reaches 25.5%. Although the improvement percentage in the steering axle

is much smaller than the second axle, the decrement of road damage is not small, since the road

stress factor on the second axle is larger than the steering axle for passive model as shown in

Table I. The actual decrement of the second axle is nontrivial. The decrement of the second

axle reaches 0.365. It is considerable for the road stress which commonly varies from 1.11 to

1.46.

The comparison of dynamic tire loads between the passive vehicle and MPC controlled vehicle

is shown in Fig 2.8. The controlled dynamic tire load of the steering axle is much better than

the LQR controlled vehicle. Based on the road stress factor evaluation, the steering axle

improvement is 7.1 %, which is more than twice of the LQR method. The drive axle

14

improvement is similar compared to that of the LQR method (26.1%). The better improvement

is due to MPC prediction capability.

(a) (b) (c)

Fig 2.8. Comparison between Passive Vehicle and MPC-controlled Vehicle for (a) Dynamic

Tire Load of Steering Wheel; (b) Dynamic Tire Load of Tractor Drive Wheel; (c) Control

Input Forces

The control effect with various human braking intensity has also been studied. To fairly justify

the controlled effect at various braking intensity, the human braking input is scaled up by timing

different factors which is called the “Braking Scale Factor” in Fig. 2.9. The factor range is from

0 to 5. The 0 represents no braking input. As shown in Fig 2.9, the road stress factors on the

second axle are not sensitive to the braking intensity and the improvement of the controlled

vehicle is stabilized at around 26%. However, the road stress factor on the steering axle is very

sensitive to the braking intensity and the road damage caused by the steering wheel can only

be reduced when the braking scale factor is smaller than 3.8 which approximately corresponds

to an emergency braking (the 0.4g deceleration is commonly considered as an emergency

braking and the maximum human braking deceleration in this paper is around 0.1g). The reason

is that the weight transfer at large braking intensity becomes too large and the control input

force at the second axle could serve as another excitation which furtherly increases the road

damage caused by the steering wheels. This also proves the difficulty in reducing the road stress

factor on the steering axle.

15

(a) (b)

Fig 2.9. (a) Comparison of Road Stress Factor between Passive Vehicle and Controlled

Vehicle; (b) the Improvement of the Road Stress Factors Brought by Controlled Active

Suspension

V. Conclusion

In this sub-chapter, an example of vehicle suspension control targeted on dynamic tire load

reduction is introduced. It proves effectiveness of linear optimal control methods, like LQR

and MPC, can be applied to reduce vehicle dynamic tire load. The simulation is done with class

8 half-truck model. Since the project is related to pavement protection at signalized intersection,

road damage assessment is done to show the effectiveness of tire load control in pavement life

extension. The relationship between braking intensity and road damage is explored as well.

Both LQR and MPC methods are applied. MPC shows better performance on the steering axle,

compared with that of the LQR. It needs to be noted that the improvement on the steering axle

is only 7.1%, and it decreases with increasing braking intensity. The possible reason is the great

weight transfer caused by braking.

2.2 A Semi-active Suspension Control Design for Vehicle Ride

Comfort Improvement for MMR-based Energy Harvesting

Suspension

2.2.1 Introduction

Subchapter 2.1 introduces how active control can be applied to reduce vehicle dynamic tire

load via active suspension system. However, active control requires suspension to provide

active force, which means energy needs to be fed back to the vehicle body. Such suspension

requires high energy input to provide effective control force. When vehicle passing through

rough road, it is a scenario of kinetic energy transfer from wheel to suspension, then to vehicle

body. Such undesired energy transfer affects both wheel dynamic tire load and vehicle body

ride comfort, since undesired vibration energy cannot be absorbed effectively and will directly

16

transfer to passengers and wheels. Traditional suspension uses hydraulic reservoir with oil as

energy dissipation solution. Such method directly converts vibration energy into heat waste,

which will be absorbed by oil. Traditional suspension is not controllable and cannot dissipate

energy effectively, which will cause undesired vibration that results in bad ride comfort.

However, such vibration energy can be useful if there is a solution to convert it into useful

energy. Such motivation rises the research in energy harvesting suspension. The earliest

research focused on the linear regenerative electromagnetic shock absorbers (LESAs) were

proposed by Karnopp, [28], Fodor and Redfield, [29] etc. The linear electromagnetic motor

was applied and produced a back electromotive force (EMF) attenuating the suspension

vibration. The energy-harvesting efficiency for LESAs is generally high, even up to 70%-78%

[30]. However, the power density is too small. It can only provide damping of 940 Ns/m under

a short circuit condition, which is not sufficient for even a compact-size passenger car.

To overcome such low-damping defects, the rotary regenerative shock absorbers (RESAs) have

been proposed by utilizing some mechanisms, such as ball-screw mechanism [31], [32], rack-

pinion mechanism [33] or some other motion conversion mechanisms [34], [35], to convert

linear suspension motion into rotation movements of generators. Graves et al. [36]

demonstrated that the rotary electromagnetic module can significantly amplify the damping

force and regeneration efficiency due to the transmission gear ratio. Zuo et al. [37] designed a

rack-pinion based shock absorber and established the bench and road field tests. The

investigated suspension has a good power density and damping range of 1800Ns/m-8000Ns/m,

but with a relatively low energy-harvesting efficiency, 33%-56%. Such low efficiency is mainly

caused by the reciprocating suspension vibration being converted into bidirectional rotation of

the generator [38].

Based on this issue, MMR-based regenerative electromagnetic shock absorbers (MMRSAs)

have been developed to convert reciprocating linear vibration into unidirectional rotation of

generator and produce stable voltage with small ripples. The MMR mechanism is connected

with a ball-screw transmission to furtherly improve energy conversion efficiency. However,

when the output shaft has a higher speed compared to the input shaft, system will be disengaged

into another dynamic model. Such piecewise linear feature brings challenge in control

development.

The control development of the MMR has been discussed in [41]. However, the methods

(Skyhook-Power Driven Damper, LQR clipped control) are only designed in engage mode.

To fully use the MMR engage/disengage feature in control, a rule-based control method

that includes MMR mode switching feature is compared with the MMR-based suspension

with SH-PDD (Skyhook-Power Driven Damper) damping control, a power flow skyhook

controlled traditional suspension and a passive traditional suspension. A quarter car

suspension model of a heavy-duty pickup truck is applied with random road excitation under

different road class levels. From the simulation, the controlled MMR suspension with rule-

based damping control shows better performance in ride comfort improvement compared to

the SH-PDD damping controlled MMR suspension, the skyhook controlled traditional

suspension and passive traditional suspension. The improvement can reach up to 29.2% in

vehicle body acceleration reduction compared to passive traditional suspension.

17

2.2.2 MMR-based Energy Harvesting Shock Absorber Modeling

In this section, the design and working principle of the MMR-based shock absorber is

introduced. Then, the modeling of the shock absorber will be provided. The major innovation

of the MMR design is to convert bi-directional motion of the suspension into unidirectional

rotation of the generator to greatly improve system energy harvesting capability. With the

MMR gearbox, an engageable equivalent inerter can also contribute to the ride comfort

improvement. In Fig 2.10, it shows the design of the MMR-based shock absorber. In the design,

the conventional hydraulic chamber is replaced by MMR gearbox driven by a ball-screw

mechanism. The generator is driven by the output shaft of the MMR gearbox on the side. When

suspension deflects, a nut inside the shock absorber will have bi-directional vibration, which

will drive the ball-screw to convert vertical motion into rotational motion. Then, the ball-screw

will drive the input shaft of the MMR gearbox at the bottom. Inside the gearbox, there are two

bevel gears connected with two one-way clutches installed in opposite directions. Another

bevel gear is connected with the generator shaft on the side of the gearbox. A one-way clutch

can be locked in one direction (engage) and freely rotate in the other direction (disengage).

Hence, if the input shaft rotates in clockwise direction from the top view, the one-way clutch

on the bottom will be engaged. The input shaft will drive the bevel gear on the side, then drive

the generator rotating in counterclockwise direction in the top view of the generator. If the input

shaft rotates in counterclockwise direction, the upper one-way clutch will be engaged. The

upper bevel gear will rotate in counterclockwise direction and drive side bevel gear to rotate in

counterclockwise direction as well. Hence, no matter which direction the input shaft rotates,

the generator will only rotate in one direction. The MMR-based shock absorber has been

designed and developed by the Center of Energy Harvesting Materials and Systems Lab

(CEHMS).

Fig 2.10. MMR-based Shock Absorber Design

The modeling of the MMR-based shock absorber is summarized based on [39] and [40]. In Fig

2.11, a schematic diagram of the MMR-based shock absorber is shown.

18

Fig 2.11. Schematic Diagram of the MMR-based Shock Absorber

In Fig 2.12, the MMR concept is integrated with simplified quarter car dynamic model with

both engagement and disengagement modes.

(a) (b)

Fig 2.12. Dynamic Modeling of the MMR-based Shock Absorber (a) Engage (b) Disengage

As shown in Fig 2.12, the engaged model will introduce a set of equivalent damping eC , and

equivalent inerter em resulted from the MMR gearbox components and the generator circuit.

Therefore, based on Newton’s 2nd law, the dynamic equation of the suspension system can be

modeled as

2 1 2 1 2 1 2( ) ( ) ( ),s s e e gen inM z K z z m z z C z z = − + − + − = (2.23)

1 0 1 1 2 1 2 1 2( ) ( ) ( ) ( ),us us s e e gen inM z K z z K z z m z z C z z = − − − − − − − = (2.24)

where sM is the sprung mass; usM is the unsprung mass; 1z is the unsprung mass

19

displacement; 2z is the sprung mass displacement; 0z is the road input; sK is the shock

absorber stiffness; usK is the tire stiffness; is the rotational speed of the output shaft; in

is the rotational speed of the input shaft. During the engage period, the generator will be driven

by the input shaft, therefore, results in same rotational speed. In the internal dynamic of the

MMR gearbox and generator system, an external resistor, eR is connected with the generator

in series. The equivalent damping can be described as a function of the external resistor as

shown below

2 2 2

lg2( 2 )( )

( )

b g m b sg bs m

e

m m

r r J r J J J d flm

l fd d l

+ + + +=

− (2.25)

2 22 ( ) 3( )

( ) 2( )

b g m t ee v

m m i e

r r d fl k kC c

l fd d l R R

+= +

− + (2.26)

where br is the gear ratio between the large bevel gear and the small bevel gear on the side of

the gearbox; gr is the generator gearhead ratio; lg, , ,m sg bsJ J J J are the inertia of the

generator, small bevel gear, large bevel gear and the ball-screw; md is the pitch diameter of

the ball-screw; f is the friction coefficient [39]; l is the screw lead; ,t ek k are the

generator torque constant and voltage constant; iR is the generator internal resistance; eR is

the external resistance; vc is the generator viscous damping.

When the output shaft has a higher speed than the input shaft, disengage will occur. In such

situation, the generator will be decoupled from the suspension system, which therefore

eliminates the equivalent damping and inerter. However, with the purpose to provide accurate

model, the inertia of the MMR gearbox is still included in the suspension model. Since the

inertia of the MMR gearbox is small compared to the engaged equivalent inerter, it is not shown

in Fig 2.12. Based on Newton’s 2nd law, the disengaged dynamic equations can be formulated

as

2 1 2 1 2( ) ( ),s s e dis gen inM z K z z m z z −= − + − (2.27)

1 0 1 1 2 1 2( ) ( ) ( ),us us s e dis gen inM z K z z K z z m z z −= − − − − − (2.28)

20

where

2

lg2( 2 )( )

( )

b sg bs m

e dis

m m

r J J J d flm

l fd d l

−

+ + +=

− (2.29)

e dism − is the inertia of the MMR gearbox.

Since the generator is decoupled from the suspension system, it forms a dynamic system itself

with the external resistor, eR as shown in equation (2.30).

0

m

m

ct

J

gen gen m gen genJ c e −

+ = = (2.30)

3

2( )

e tm v ele v

i e

k kc c c c

R R= + = +

+ (2.31)

where mc is the generator damping.

The generator damping is composed by electrical damping part elec and mechanical damping

part vc . The vc is the generator viscous friction damping. It is a constant value. It will

consume partial mechanical power generated by suspension deflection due to viscous friction

of the generator. It is tested based on MMR-based shock absorber open-loop circuit bench test

[39]. The damping caused by resistance and viscous friction will let the generator decay

exponentially during disengage period.

2.2.3 MMR-based Shock Absorber Effect on Vibration Reduction in Bump Scenario

with Simple Control

Bump is a sudden pulse excitation to vehicle body that will bring heavy shock to passengers.

When vehicle passes through bump, vibration energy will suddenly transfer to vehicle body

and cause undesired jerk. The MMR engagement/disengagement feature can introduce or

eliminate equivalent inerter and damping by controlling generator speed. When the system is

engaged, the suspension will be hard due to added equivalent damping. During disengagement,

suspension will only have spring force and will be soft. By switching hard and soft mode of

the MMR-suspension, ride comfort when vehicle goes through a bump can be improved. From

vehicle dynamic analysis, when vehicle is about to go through the bump, the suspension should

be set to soft mode, since the upward motion of the wheel will not transfer motion to the vehicle

body. When the vehicle just passes the bump, the suspension should be set to hard mode to

support vehicle body from bouncing due to bump excitation. The control concept of

engagement can be depicted in Fig 2.13.

21

Fig 2.13. Bump Control Concept of MMR-based Shock Absorber

Based on the control concept, a simulation is done with a heavy-duty pickup truck (ex. Ford

F250) going through a bump with 0.0518m height and 0.3093m width. Fig 2.14 shows the

time-based profile of the bump. Since bump is designed to let vehicle to reduce speed for safety,

the vehicle passing speed is set at a low value, 4m/s. The simulation parameters are displayed

in Table III under Appendix. A with quarter vehicle setup. The baseline is a traditional hydraulic

suspension system as what is used in normal vehicle. It is a passive system without control of

suspension damping or force. Fig 2.15 shows the schematic diagram of the traditional

suspension baseline. The dynamic model of the traditional suspension can be concluded as

2 1 2 1 2( ) ( )s s pM z K z z c z z= − + − (2.32)

1 0 1 1 2 1 2( ) ( ) ( )us us s pM z K z z K z z c z z= − − − − − (2.33)

where pc is the damping of the traditional suspension. Other parameters are same as the

MMR model.

The vehicle parameters are selected based on Ford F250 pickup truck. Some of the parameters

that are not available, e. g., , ,p s usc K K , are selected by searching for truck model in CarSim

software with similar vehicle weight as the Ford F250. Then, the MMR equivalent damping is

tuned to find good performance in vehicle body vibration reduction.

22

Fig 2.14. Bump Time-based Profile

Fig 2.15. Traditional Suspension Baseline

In Fig 2.16, the simulation result of the vehicle sprung mass vertical acceleration (vehicle body)

is shown for controlled MMR suspension, passive MMR suspension and traditional suspension.

The MMR control can only be a semi-active control, since generator will never drive the

suspension, because higher generator speed compared to input shaft will result in

disengagement. Hence, generator will not feed energy back into the suspension, and the control

is called as semi-active control. From the result, it is obvious to see that the controlled MMR

model has reduced the first peak of sprung mass acceleration by more than half compared to

that of the traditional suspension. Then, followed by the passive MMR model. The controlled

MMR also reduced the second peak by nearly 20%. The first two peaks are the most important

to ride comfort, since the bump is just passed by the vehicle, and passengers will experience

large shock at the beginning. The mode switching of the MMR engagement helped suspension

reducing shock at the beginning of the bump. However, since the MMR-based suspension has

smaller damping, it will cause more vibration after the first two biggest shocks. The preliminary

results show effectiveness if MMR can be controlled. It can help improving vehicle ride

comfort in bump scenario. The next part will extend the MMR control into random excitation

road profile with connected road with different road classes.

23

Fig 2.16. Ride Comfort Comparison in Bump Scenario

2.2.4 Skyhook Control on Traditional Suspension and SH-PDD Control on MMR-based

Suspension

In this section, two semi-active control methods from [41] are introduced: Power Driven

Skyhook and Skyhook-Power Driven Damper (SH-PDD). From the dissertation, the two

methods are applied with analysis in the frequency domain with harmonic excitation. The

section extends the two methods in random excitation. The MMR-based suspension system is

a piecewise linear system due to engagement mode transition. Therefore, the SH-PDD control

is analyzed only in engage mode, since frequency domain analysis can only be done with linear

system. In this section, SH-PDD method is extended with comparison of engage and disengage

modes’ power cost at each time step to consider possible mode of disengagement. The

simulation is done in discrete time space.

I. Skyhook Control

Skyhook (SH) was initially proposed to reduce sprung mass vibration. It is a widely applied

control strategy developed for semi-active suspension. Its main idea is to virtually create an

ideal suspension system in which the chassis is “hooked” to a virtual inertial frame called “sky”

by a passive damper skyc , then using the real suspension with an electromagnetic semi-active

damper to emulate the dynamics of this ideal suspension. Fig 2.17 shows the concept of

skyhook control.

24

Fig 2.17. Skyhook Control Concept

By applying the skyhook damping, the system dynamic can be modified as

2 1 2 1 2( ) ( )s s SHM z K z z c z z= − + − (2.34)

1 0 1 1 2 1 2( ) ( ) ( )us us s SHM z K z z K z z c z z= − − − − − (2.35)

where SHc is the skyhook control damping

From energy perspective, the power of the sprung mass absorbed by the suspension can be

expressed as

2 1 2 2 1 2 2 1 2 2 1 2( ) ( ) , ( ) , ( )sc ss SH s sc SH ss sP P c z z z K z z z P c z z z P K z z z+ = − + − = − = − (2.36)

where scP is the suspension damping power absorbed from sprung mass; ssP is the

suspension spring power absorbed from sprung mass.

To reduce vehicle body sprung mass vibration, the reasonable way is to transfer kinetic power

of sprung mass as much as possible to suspension damper at every moment. In the system, the

suspension spring energy is not controllable. Therefore, the control law on skyhook damping

can be concluded as

( )SHc t = max 2 1 2

min 2 1 2

, ( ) 0

, ( ) 0

c if z z z

c if z z z

−

− (2.37)

When 2 1 2( ) 0z z z− , the power flow is from vehicle body to suspension. In such case, it is

preferable to absorb as much power as possible from vehicle body. Therefore, the skyhook

damping is set to maximum value. When 2 1 2( ) 0z z z− , the power is transferred from

25

suspension to vehicle body, which is undesirable. Hence, the skyhook damping is set to

minimum value to reduce the power transferred to minimum. The simplest skyhook control

strategy only considers two damping stages max min,c c . This control method is simple and easy

to be implemented.

II. SH-PDD Control

The SH-PDD algorithm is a mixed control method that combines the skyhook control and

Power-Driven-Damper (PDD) proposed in [42] using port Hamiltonian techniques. The PDD

control law can be concluded as

( )PDDc t =

2

max 2 1 2 1 max 2 1

2

min 2 1 2 1 min 2 1

max min2 1 2 1

2 1

2 1

, ( )( ) ( ) 0

, ( )( ) ( ) 0

, ( ) 0 & ( ) 02

( ),

( )

s

s

s

c if K z z z z c z z

c if K z z z z c z z

c cif z z z z

K z zotherwise

z z

− − + −

− − + −

+− = −

− −

−

(2.38)

The skyhook control strategy only considers the energy flow from sprung mass to suspension.

By continuing the analysis of the suspension energy flow from skyhook control analysis, it can

be extended by adding energy flow analysis from suspension to unsprung mass. The power that

the suspension damper releases to the unsprung mass is

2 1 1( )( )ucP c t z z z= − (2.39)

The power that the suspension spring releases to the unprung mass is

2 1 1( )us sP K z z z= − (2.40)

By combining the energy flow equations from sprung mass to suspension and energy flow

equations from suspension to unsprung mass-(2.36), (2.39), (2.40), the net power flow into

suspension is

2 1 2 1 2 1 2 1( )( )( ) ( )( )net sc ss uc us sP P P P P c t z z z z K z z z z= + − − = − − + − − (2.40)

The power flow of each component represents its energy transfer ability, since sc ssP P+ can

represent how much power can be transferred from sprung mass to suspension or vice versa

and uc usP P+ can represent how much power can be transferred from suspension to unsprung

mass or vice versa. The netP can reflect the capability of suspension to decouple power flow

26

between sprung and unsprung mass. Since suspension spring is not controllable, only the

,sc ucP P can be adjusted to reflect power transfer abilities of suspension. If 0sc ucP P+ , the

suspension can transfer all energy absorbed from sprung mass to unsprung mass. During this

period, skyhook can be applied to dissipate energy away from sprung mass. Otherwise, if

0sc ucP P+ , skyhook and PDD behave oppositely since more energy is absorbed by

suspension, more energy remains in it. Therefore, SH-PDD method uses 0sc ucP P+ as

switching law and is formed as

( )SH PDDc t− =

2 2

2 1( ), 0

( ),

SH

PDD

c t if z z

c t else

− (2.41)

It means that when 2 2

2 1 0z z− , the skyhook control law is applied to let suspension dissipate

energy away from sprung mass as much as possible. When 2 2

2 1 0z z− , the PDD control law

is applied to try to balance energy flowing into suspension. By substituting explicit rules of

skyhook and PDD, the SH-PDD control law can be concluded as

( )SH PDDc t− =

2 2 2

max 2 1 2 1 2 1 max 2 1

2 2 2

min 2 1 2 1 2 1 min 2 1

2 1

2 1

, 0 ( )( ) ( ) 0

, 0 ( )( ) ( ) 0

( ),

( )

s

s

s

c if z z or K z z z z c z z

c if z z or K z z z z c z z

K z zotherwise

z z

− − − + −

− − − + −

− −

−

(2.42)

In the MMR suspension application, the max min,c c will be the maximum and minimum

available equivalent damping generated by the MMR and generator system.

It is obvious to see that the control law is designed in MMR engage mode, since MMR

suspension will not have equivalent damping during disengage mode. However, during

disengage mode, the soft suspension setup may result in lower vehicle body acceleration.

Hence, a comparison of vehicle body acceleration between engage and disengage modes needs

to be done to determine mode selection in each simulation time step. Therefore, the control

method is extended with the consideration of disengage effect on vehicle ride comfort. The

concept is that during each time step, the engage mode vehicle body acceleration will be

compared to that of the disengage mode. The mode that will result in lower vehicle body

acceleration will be selected as the mode that will be applied in current time step. Then, the

simulation continues to the next time step with same comparison.

2.2.5 Rule-based Control on MMR-based Suspension

From the introduction of SH-PDD method, MMR mode control idea has been initiated. From

27

the modeling of the MMR suspension, when the generator speed is higher than the input shaft

speed, the generator will be disengaged. Hence, by increasing generator speed to a higher value

than the input shaft speed, system will be disengaged. During the disengagement, by changing

system damping to a large value, the generator speed will decay faster to be engaged with the

suspension. Therefore, by controlling generator speed as well as damping, system engagement

can be controlled. With the mode control concept, a rule-based control strategy that considers

mode control is developed. For simplification, the dynamic of engagement/disengagement

control is ignored. In the control model, engagement control is assumed to occur instantly. The

MMR system is a piecewise linear system that will switch system dynamic according to the

speed comparison between the input shaft and generator speed. Such feature brings great

challenge to global optimization method formulation. Hence, an instant optimization method

that compares the instant power of the engage and disengage model at each time step in

discretized manner is formulated.

Based on the MMR suspension dynamic equations mentioned in section 2.2.2, the state-space

model during engage and disengage periods are formulated as

Engage model-

2 2 2 22 1

2

1 0

1

2

0 1 0 1

0 0 0 1

s e e e e us e es e e

us e us e us e us e

e e e es e s e s e s e

us e us e us e us e

s e e es e

s e s e

eus e us e

s e

K m m C m K m CK C C

M m M m M m M m

z z m m m mM m M m M m M m

M m M m M m M mz

z z

z K m m CK C

M m M m

mM m M m

M m

−

−− − −

− + + +

− + − + − + − + −

+ + + + = − − −

+ +

+ − ++

( )

2 1

2

0

1 0

1

2 2 2

0

0

1

0e ee

us s e

e e eus e us e

s e s e s e

z z

zz

z z

zm CC

K M m

m m mM m M m

M m M m M m

− + − − − − +

− + − + − + + +

(2.43)

Disengage model-

2 22 1

2

1 0

1

2

0 1 0 1

0 0

0 0 0 1

0

s e dis e dis uss

us e dis us e dis

e dis e diss e dis s e dis

us e dis us e dis

s e diss

s e dis us

e disus e dis us e

s e dis

K m m kK

M m M m

z z m mM m M m

M m M mz

z z

z K mK

M m K

mM m M m

M m

− −

− −

− −− −

− −

−

−

−− −

−

−

−−

− +

− + − + −

+ + = − −

+ −

+ − ++

( )

2 1

2

0

1 0

1

2

0

0

1

0

0e dis

dis

s e dis

z z

zz

z z

z

m

M m−

−

− + − −

− +

(2.43)

Then, the system is discretized by sample time sT with first order hold (FOH) for better

approximation.

At time k, the Fig 2.18 shows the logic flow of the rule-based control strategy

28

Fig 2.18. Rule-based Control Logic Diagram (a) 1st time step ride comfort comparison (b) 2nd

time step ride comfort comparison (c) Generator speed control

In the figure, gn is the combined gear ratio between the suspension deflection and generator

speed. The rp is a coefficient that ensures the generator speed will be controlled to be higher

than the input shaft speed to cause disengagement, 1rp .

At each time step, the control strategy will compare the instant vehicle body acceleration for

both engage and disengage model for two time steps in the future. The available equivalent

damping range will be divided with certain grid size. The control strategy will find the optimal

damping value that results in minimum vehicle body acceleration.

The vehicle ride comfort index can be expressed as [43]

2

2

1

2

0

1

1( )

, 1,2,...,1

( )

N

irms N

i

z iN

a i N

z iN

=

=

= =

(2.44)

From the equation, it is obvious to see that by reducing instant vehicle body acceleration,

vehicle ride comfort index can be reduced. Therefore, the control algorithm targets on reducing

instant vehicle body acceleration.

The vehicle road handling index is also considered as

29

2

1 0

1

1( ( ) ( ))

, 1,2,...,( )

N

us

i

rms

s us

K z i z iN

i NM M g

=

−

= =+

(2.45)

From the equation, it’s easy to see that by reducing instant tire deflection, the index can be

reduced for better road handling. By combining the ride comfort and road handling

optimization, a cost function is formulated as

( ) 2( 1) 2( ) 2 1( 1) 0( +1)/ / e k k k S norm k k tire normf q z z T z p z z z+ − + −= − + − (2.46)

where ,q p are the weighting values; the subscript 1k + means state value at 1t k= + ;

2 , norm tire normz z− −are the vehicle body acceleration and tire deflection for passive traditional

suspension at the same time step.

In the engaged model, since different damping values can be applied, the minimum value of

the cost function at each time step needs to be found. Then, the minimum cost function value

for engage model will be compared to the cost function value of disengage model to determine

which mode can result in smaller vehicle body acceleration and dynamic tire load. The

engagement at time 1t k= + will also affect the dynamics of engagement at 2t k= + .

Therefore, the control strategy will process comparison for two time steps in the future at each

time step.

2.2.6 Simulation Results and Conclusion

I. Vehicle Parameters

A heavy-duty pickup truck (ex. Ford F250) quarter car model is applied as the target vehicle

for simulation. The vehicle parameters are displayed in Appendix. A table IV. The gear ratio

between the large bevel and the small bevel gear on the side is optimized for minimum vehicle

body acceleration. The passive MMR model is simulated with gear ratio from 0.1 to 2 based

on design limitation. Then, the simulation chooses the optimal gear ratio in the controlled

model simulation. However, the optimized gear ratio will also affect MMR shock absorber

equivalent damping. For fair comparison, all controlled models will use same damping range

as optimized MMR shock absorber. The passive traditional model also uses maximum available

damping of MMR shock absorber as the constant damping of the system.

In the simulation, the equivalent damping caused by vc is 4.4kN-s/m. The external resistance

eR changes from 0 to 50 ohms, which is the range that the system damping will be sensitive

to external resistance change. When increase external resistance, system equivalent damping

will decrease. The corresponding damping range is from 7.9kN-s/m to 1kN-s/m. Hence, for

SH-PDD and skyhook methods, the maximum and minimum damping correspond to the

minimum and maximum external resistances.

30

II. Road Profile Input

To justify control performance, a random road profile with changing road grades is considered

as road input for simulation. To thoroughly show advantage of controllable suspension, various

road conditions with changing grades can be more appropriate, since traditional suspension

does not have capability to adapt damping according to road change. The road profile changes

from class B road to class C road.

The stochastic road excitation was established according to the road roughness grade classified

by ISO 8608 [44]. The road elevation PSD has a form

0

0

( ) ( )( ) w

d d

nG n G n

n

−=

(2.47)

where ( )dG n is unevenness index, w is waviness, 0n is reference spatial frequency and n is

spatial frequency. Variance of roughness is

2

1

2 ( )n

z dn

G n dn =

(2.48)

where 1 2,n n are lower and upper limits of spatial frequency. According to the harmonic

superposition method, the road elevation can be expressed as

_( ) 2 sin(2 )m

i mid i i

i

q x xn = +

(2.49)

Divided the interval 1n to 2n into m cells and _mid in is the intermediate frequency of each

cell (i=1,2,3,…,m). is a uniformly distributed random number on [0,2]. x is the

displacement in the vehicle’s forward direction.

TABLE III. ROAD ROUGHNESS LEVELS CLASSIFIED BY ISO 8608

Road Class

0( )dG n 6 3( 10 )m−

1

0 0.1n m−=

Geometric Mean

A 16

B 46

C 256

D 1024

E 4096

The generated road profile is

31

Fig 2.19. Road Profile Input for B-class+C-class

III. Simulation Results

The simulation results for choosing the best gear ratio between the large bevel gear and small

bevel gear, as well as the vehicle body acceleration for different models and power generation

of energy harvesting shock absorber are shown. Fig 2.20 shows a 3-D plot considers both the

gear ratio, br and external resistance, eR for passive MMR suspension. The simulation is

done in time domain with the consideration of the MMR feature for the road profile input. It

can be seen that the lowest point in the plot gives 0.9br = . Hence, the controlled models use

same damping range as optimized MMR suspension with this gear ratio.

Lowest point

32

Fig 2.20. 3-D Plot to Determine Optimal br

Fig 2.21 shows the control force comparison for skyhook, SH-PDD and rule-based controller.

With same damping boundary, the skyhook controller applies largest control force for most of

the time, since it can only select control force between minimum and maximum damping forces.

The SH-PDD controller has one more tuning value capability in damping compared to that of

the skyhook control, which reduces control effort sometimes. However, the rule-based

controller has the best damping tuning flexibility, which results in even lower control effort to

achieve better control performance.

Fig 2.21 Control Force Comparison for All Controlled Models

Fig 2.22 shows the ride comfort comparison among passive traditional suspension, skyhook

controlled traditional suspension, SH-PDD controlled MMR suspension, and the rule-based

MMR suspension. For fair comparison, the same logic to choose to engage or disengage at

each time step is also applied to the SH-PDD method. It can be shown that the passive

traditional shock absorber in black has the worst vehicle body acceleration under the same road

excitation compared to other controlled models. When the road class changes, the passive

traditional shock absorber has no capability to change damping, which will force the vehicle

body to vibrate with higher body acceleration. The skyhook performs better compared to the

passive model, however, due to limited damping tuning options, it is worse than the rule-based

MMR model. The SH-PDD method introduces another damping tuning value, however, it does

not show noticeable improvement compared to skyhook method. Different from skyhook and

SH-PDD methods, rule-based controller has various damping tuning capability that can change

damping at each time step for minimum instant vehicle body acceleration. Therefore, it

performs best among all control models.

33

Fig 2.22. Ride Comfort Comparison among Different Models

The passive MMR model is also compared with passive traditional model with optimized bevel

gear ratio, br , and external resistance, eR . The parameters are optimized based on Class B

road. The road profile is generated by white noise with fixed spectrum in frequency domain.

After several trials, it is noticed that optimize the two parameters based on one class road and

various classes road can let passive MMR model perform well in different road profiles.

Therefore, the passive MMR model is optimized based on one road profile. Fig 2.23 shows the

ride comfort comparison between the two models. It can be noticed easily that optimized MMR

shock absorber can have better ride comfort compared to that of the traditional shock absorber.

From the rms value comparison, the passive MMR model improves the ride comfort by 16.7%.

34

Fig 2.23. Passive MMR vs Passive Traditional

Fig 2.24 shows the power generation for the controlled MMR shock absorber with SH-PDD

and rule-based methods on generator electrical damping, which means the maximum

recoverable energy as electricity without considering the recovery efficiency. However, since

the power generation is not the control target, sometimes the controller will try to reduce

vehicle body acceleration to choose large damping, then cause really high power peaks.

Fig 2.24. Power Generation for Controlled MMR Shock Absorber

Table IV shows the rms value of vehicle body acceleration comparison for all models. The

passive traditional shock absorber will be the baseline. The passive MMR shock absorber

improves 16.7% compared to the baseline. The skyhook control has 18.9% of improvement

compared to the baseline. The SH-PDD model has 19% improvement and the rule-based model

has 29.2% improvement compared to the baseline.

( )0 , 0, ,2 ,...

t

k

avg s s

P dtP k T T

t= = (2.50)

TABLE IV. COMPARISON ABOUT VEHICLE RIDE COMFORT AND AVERAGE

POWER GENERATION

System Dynamic Power 2( / )s rmsz m s−

Improvement Avg. Power (W)

Traditional 2.528 0% ---

Passive MMR 2.106 16.7% 137

Traditional+skyhook 2.050 18.9% ---

SH-PDD+Engagement control 2.048 19.0% 189

Rule-based 1.789 29.2% 187

35

IV. Conclusion

In this sub-chapter, a rule-based control strategy is designed for a mechanical motion rectifier

(MMR) based energy harvesting suspension with the simulation on a quarter car model with

pickup truck parameters. Passive traditional suspension, skyhook controlled traditional

suspension, and SH-PDD controlled MMR suspension are compared with the rule-based

controller on ride comfort performance. The rule-based controller achieves the best

performance (29.2%) in ride comfort compared to the passive traditional suspension.

Chapter Summary:

In this chapter, vibration suppression control is investigated with the application on vehicle

suspension. There are two major targets for suspension vibration control: vehicle dynamic tire

load reduction and vehicle ride comfort improvement. For the control objective on vehicle

dynamic tire load reduction, two active control methods, LQR and MPC, are applied for active

suspension control. The two control methods show effectiveness on vehicle dynamic tire load

reduction when braking hardly for heavy duty truck. The LQR method achieve 2.9% on

steering axle tire load reduction and 25.5% on second axle tire load reduction. The MPC

method achieves better performance with prediction capability. For the control objective on

vehicle ride comfort improvement, a MMR-based energy harvesting suspension is introduced

for vehicle body vibration reduction. The capability of engage/disengage feature of MMR with

controllable equivalent inerter and adjustable equivalent damping show effectiveness of

vehicle vibration reduction when vehicle passes through a bump. By using simple engagement

control on MMR-based suspension, the first two peak of vehicle body acceleration has been

reduced by nearly 50% compared with traditional hydraulic suspension. Then, the MMR-based

suspension is extended to random road excitation. Three new methods: skyhook, SH-PDD, and

rule-based control are introduced. The skyhook method is applied with traditional suspension.

The SH-PDD and rule-based methods are applied with MMR-based suspension. The controlled

MMR-base suspension performs better compare with passive traditional suspension and

controlled traditional suspension under random road excitation with changing road classes. The

rule-based control method has the best performance by improving the ride comfort by 29.2%

compared to that of the passive traditional suspension.

36

Chapter 3

Vibration Amplification Control: Ocean Wave

Energy Converter (WEC)

Previous chapter talks about vibration reduction control in vehicle suspension to reduce

undesired vibration. However, in energy harvesting through vibration, it is obvious that the

control target will be in the other way: vibration amplification. This chapter talks about the

application of vibration control in a promising energy harvesting field: ocean wave energy

converter (WEC). The chapter talks about control of two-body point absorber control. The first

sub-chapter investigates model predictive control (MPC) and hybrid model predictive control

(HMPC) methods on direct drive power take-off (PTO-mechanism that convers wave vibration

into electricity) system. The second sub-chapter extends control development on MMR-based

power take-off system.

3.1 Active and Semi-active Control for Normal Two-body WEC

3.1.1 Introduction

The energy from ocean wave is the most conspicuous form of ocean energy. The possibility of

converting wave energy into usable energy has inspired numerous inventors. The major wave

energy converter can be categorized into three types: oscillating water column (OWC),

oscillating body, and overtopping systems [45] as shown in Fig 3.1. Point absorber, one type

of oscillating body, becomes popular due to its capability to absorb energy from waves in

different directions. The simplest point absorber is a heaving buoy reacting against a fixed

frame of reference (the sea level). The first development of such device can be dated back to

1980, which was tested in Tokyo Bay [46]. However, compared to the higher natural frequency

of the point absorber (due to size limitation), the majority of energy in waves exists at low

frequencies, which results in low energy harvesting efficiency. Therefore, two-body wave

energy converter system (a buoy attached with a submerged second body) that includes a

submerged second body has been developed to expand the power band of the device [47].

37

(a) (b) (c)

Fig 3.1. (a) Overtopping Systems (b) Oscillating Body (c) Oscillating Water Column (OWC)

Fig 3.2. Two-body WEC

Based on the vibration theory, the majority of power produced by WEC devices occurs during

resonant absorption when the wave excitation force is in-phase with the device velocity [48].

Ocean wave is a wave spectrum consists of multiple frequencies and amplitudes. Therefore,

control of WEC system becomes inevitable to let device velocity match with the input

excitation force in time to maximize energy extraction. Based on the velocity matching theory,

several sub-optimal and optimal control strategies have been developed. Starting from the

single frequency wave WEC control, Budai and Falnes [49] introduced reactive control in the

early 1970s according to the matching of the PTO damping to the impedance of the system

transfer function from excitation force to device velocity. Such control strategies can only have

good performance in regular wave case and cannot be applied with PTO load limitation, which

makes it a type of sub-optimal strategy. The application of reactive control or similar strategies

has not been applied to two body system due to lack of analytical optimal force solution.

With the consideration of energy maximization among a power band with frequency spectrum,

time domain optimal control can be a better way to maximize power absorption at each time

point. However, in practice, most WECs will be subjected to limitations placed on physical

motion of the absorber and the capabilities. Hence, it is necessary to develop constrained

optimal control. In such a category of approach, MPC becomes more and more attractive due

38

to its capability in handling hard constraints on states and inputs, which serves the objective to

maximize energy extraction and satisfies machinery requirements for safety and operations

[50], [51], [52].

The MPC considers control input as a force applied by generator. It is an active control method

that requires generator to feed back energy into the system to maximize power. However, for

some kinds of PTO systems, energy feedback may not be capable. Therefore, a constrained

optimal semi-active control method is required to be developed for the WEC systems to

set system damping to be positive all the time to avoid power feedback. Hybrid Model

Predictive Control (HMPC) is a method to convert active control into semi-active control

by considering additional constraints on generator damping with switch logic depending

on the sign of relative velocity between the buoy and the submerged body for a two-body

WEC system. The sub-chapter investigates the development of HMPC application on WEC

system with comparison with active MPC method. The final results show that the HMPC can

have the capability to choose positive system damping to avoid power feedback. The active

method will always have the highest average power generation. However, semi-active method

reduces damping control effort by 88% with only 14% of power generation reduction, which

is still effective in power enhancement.

3.1.2 Two-body WEC Modeling

The PTO system has several types: hydraulic, turbine or mechanical mechanism. Hydraulic

PTO will convert vibration into fluid flow to drive hydraulic motor. The OWC will use air flow

caused by housing air pressure change due to wave motion to drive turbine to generate

electricity. The overtopping system will directly use water flow to drive turbine to generate

electricity. The mechanical mechanism PTO can convert vibration motion into generator

rotation by using mechanical transmission. Such setup becomes popular due to high efficiency

compared to the other two types. For a direct drive PTO with linear generator, the generator

will directly convert bidirectional motion of buoy vibration into bidirectional rotation. System

will not have switching mechanism. Hence, a linear state-space model can be developed.

A two body WEC system can be simplified as Fig 3.3. The device will convert relative vibrating

motion between the buoy and the submerged body into generator rotation to generate electricity.

Fig 3.3. Two-body WEC Schematic Diagram

Based on the model, some assumptions to keep the state-space model as LTI system are

addressed

39

1. The formulation is based on Linear Wave Theory (LTW).

2. Frequency dependent parameters of the two-body WEC are assumed to be constant.

3. The radiation forces (generated by wave propagation) are assumed to be linear and no

convolution terms are used to calculate them.

4. Nonlinear viscous drag force is ignored for both buoy and submerged body.

In the schematic diagram, 1 2,x x represent the position of the buoy and the submerged body.

The reference level is the calm sea water surface level. According to Newton’s law, the dynamic

equation of the buoy can be written as

1 1 1 12 1 1gen e r h rF F F F F m x+ − − − = (3.1)

where 1x is the buoy acceleration, 1m is the buoy mass, genF is the force produced by the

PTO system, 1rF is buoy radiation damping force, 12rF is the radiation damping force on the

buoy generated by the submerged body motion, 1hF is the buoy hydrostatic force, and 1eF is

the wave excitation force encountered by the buoy. The radiation force is a combination of

damping and inerter forces generated by wave propagation. It will consume vibrating energy

of floater on the water surface. If there is no wave input, floater vibrating motion will decay

due to radiation damping force. In the linear wave theory, the radiation force is modeled as the

combination of buoy added mass as well as radiation damping force. The hydrostatic force is

a restoration force to let floater vibrate on the water surface. It is caused by the difference

between the floater’s changing buoyancy force (vibration) and weight. It is modeled as linear

force respect to the buoy position. The forces are

1 1 1 1 1rF A x b x= + (3.2)

12 12 2rF A x= (3.3)

2

1 1 1 1h buoyF g r x k x = = (3.4)

where 1A is the buoy added mass, 12A is the buoy added mass due to submerged body motion,

1b is the buoy radiation damping, 12b is the buoy radiation damping due to submerged body

motion. In the simulation, the distance between the buoy and submerged body is assumed to

be long enough to ignore the radiation damping due to relative motion. The added mass and

radiation damping are generated by WAMIT software according to designed buoy and tank

40

shapes. For simplification, they are assumed as constants at infinite frequency. Same

assumption is also valid for submerged body. is the water density, buoyr is the buoy cross-

section radius, g is the gravitational acceleration. 2

buoyg r can be formed as a single

variable 1k , the hydrostatic stiffness.

The same procedure can be applied to the submerged body. It follows that

2 2 21 2 2gen e r rF F F F m x− + − − = (3.5)

2 2 2 2 2rF A x b x= + (3.6)

21 21 1rF A x= (3.7)

The buoyancy force is constant due to fully submerge of the body. The buoyancy force of the

submerged body is equal to its weight, which means it can freely move vertically in the water.

The two-body WEC is a self-reacting WEC system, since the power is generated via the relative

motion between two bodies. It is found in [53] that the effect of slack mooring is negligible on

the absorption power of a self-reacting WEC. Therefore, the mooring force is ignored in the

system formulation.

With the purpose to control the system via MPC, a discrete-time state-space representation is

necessary. The formulation process is not straightforward, due to the coupling terms within the

coupled radiation calculation. This coupling occurs in the second derivative; hence, a state-

space model cannot be formed without reformulating first. Equation (3.1)-(3.7) can be rewritten

as

1 1 1 1 1 12 2

1

1 1

gen eF F b x k x A xx

m A

+ − − −=

+ (3.8)

2 2 2 21 1

2

2 2

gen eF F b x A xx

m A

− + − −=

+ (3.9)

By plugging (3.8) into (3.9) and vice versa to get rid of the coupling term 12 2A x and 21 1A x .

The equation can be restated as

12 21 12 12 12 21 1 1 1 1 1 1 1 2 2

2 2 2 2 2 2 2 2

( ) gen e gen e

A A A A A bm A x F F b x k x F F x

m A m A m A m A+ − = + − − + − +

+ + + + (3.10)

1em

41

21 12 21 21 21 1 21 12 2 2 2 2 2 1 1 1

1 1 1 1 1 1 1 1 1 1

( ) gen e gen e

A A A A A b A km A x F F b x F F x x

m A m A m A m A m A+ − = − + − − − + +

+ + + + + (3.11)

The state vector is ( )1 1 2 2

Tx x x x x= and the initial conditions are ( )0 0 0 0 0

Tx = .

The system state-space form is

1 1 2 2gen e ex Ax BF B F B F

Y Cx

= + + +

= (3.12)

where

1 1 12 2 12

1 1 1 2 2 1 1 2 2

21 1 21 1 2 21

2 1 1 2 1 1 2 2 2 1 1

0 1 0 0 0

10

( ) ( ),

0 0 0 1 0

10

( ) ( ) ( )

e e e e e

e e e e e

k b A b A

m m m m A m m m AA B

A k A b b A

m m A m m A m m m m A

− − + + +

= =

− − − + + +

(3.13)

12

1 1 2 2

1 2

21

2 1 1 2

0 0

1

( ) 1 0 1 0, ,

0 0 1 0 10

1

( )

e e

e e

A

m m m AB B C

A

m m A m

− + −

= = = −

−

+

(3.14)

In what follows, the inputs 1 2, ,gen e eF F F are denoted by , ,u v w respectively. Then, the state-

space model is discretized by “Zero Order Hold (ZOH)” with the sampling time sT to obtain

the following discrete-time model

4

1 1 2 0, k d k d k d k d k

k d k

x A x B u B v B w x

y C x

+ = + + +

= (3.15)

where

1 1 2 2

0 0 0

, , , , s s s

s

T T T

AT A A A

d d d d d d dA e B e d B B e d B B e d B C C = = = = = (3.16)

3.1.3 MPC Control Development (Active)

2em

42

For the single-body WEC, a reference buoy velocity can be calculated based on the transfer

function from the wave excitation force to the buoy velocity. However, for two-body system,

there is no information about an optimal velocity trajectory in time domain. Hence, the MPC

is formulated with the purpose to directly maximize power extraction.

The optimization problem for the two-body WEC can be defined as

2

1( ) 2( ) 1 1,

1

min ( , ), ( , ) [ ( ) ]k k

N

k k k k k k k kx u

k

J x u J x u q x x u ru− −

=

= − − − − (3.17)

subject to

1 1 2

min 1( ) 2( ) max

min 1( ) 2( ) max

min max

, 1,...,

, 1,...,

, 0,..., 1

k d k d k d k d k

k k

k k

k

x A x B u B v B w

x x x x k N

x x x x k N

u u u k N

+ = + + +

− =

− =

= −

1( ) 2( ),k kx x represent velocities of the buoy and the submerged body at time k.

The problem has constraints on relative position (stroke length) and velocity between the buoy

and the submerged body. There are also constraints on the generator force.

The vector of the predicted states, control input and excitation forces are formulated as

( ) ( )1 2 0 1 1,T TT T T

N NX x x x U u u u −= = (3.18)

( ) ( )0 1 1 0 1 1,T T

N NV v v v W w w w− −= = (3.19)

Solving system 1 1 2k d k d k d k d kx A x B u B v B w+ = + + + and substituting in itself yields

0x u v wX J x J U J V J W= + + + (3.20)

where

2

2

1 2 3

0 0 0

0 0

, 0

d

d

d d d

d

x u d d d d d

N

d N N N

d d d d d d d

BA

A B BA

J J A B A B B

AA B A B A B B− − −

= =

(3.21)

genP−

43

1(2)

1(2) 1(2)

2

1(2) 1(2) 1(2)( )

1 2 3

1(2) 1(2) 1(2) 1(2)

0 0 0

0 0

0

d

d d d

d d d d dv w

N N N

d d d d d d d

B

A B B

A B A B BJ

A B A B A B B− − −

=

(3.22)

and dim( , , ) (4 ),dim( ) (4 4)u v w xJ J J N N J N= =

The objective function (3.17) can be reformulated as

(2) (4) 1 2 1 2ˆ ˆ ˆ( , ) ( ) ( ) ( )T T T T T T T TJ X U q X X U U RU q S X S X U U RU qX S S U U RU= − + = − + = − + (3.23)

with the matrix ( )R diag r= with dim( ) ( )R N N= , and the matrices 1 2,S S with

1 2dim( , ) ( 4 )S S N N= . 1 2,S S extract the second state (buoy velocity) and fourth state

(submerged body velocity) from X .

( )

( )

( )

( )1 2

0 1 0 0 0 0 0 0 1 0

,

0 0 1 0 0 0 0 0 0 1

S S

= =

(3.24)

By using (3.20), the objective function can be formulated as a quadratic function only depends

on U .

0

1 ˆ( ) ( ) ( )2

T T T T T T T T

u x v wJ U U qJ S R U q x J V J W J SU= + + + + (3.25)

where H is the Hessian matrix. It is positive semi-definite (PSD) to ensure the QP problem to

be a convex problem with available solution.

The constraints can also be augmented respect to U. Therefore, the active MPC can be written

as

0

1 ˆmin ( ) ( ) ( ) ( )2

T T T T T T T T

u x v wU

J U J U U qJ S R U q x J V J W J SU= = + + + + (3.26)

subject to 0

D

D u D x D v D w

D dU

E J e E J x E J V E J W

− − −

S

H TF

44

3.1.4 HMPC Control Development (Semi-active)

Damping control means control system damping to create damping force to control the system

dynamics. In the HMPC method, the PTO damping is calculated as

1 2

gen

pto

FC

x x

−=

− (3.27)

where genF is the generator force and the 1 2x x− is the relative velocity between the buoy

and the submerged body. The negative sign is used to represent that the damping force is always

in the opposite direction compared to the relative velocity. If the sign of the generator force is

the same as that of the relative velocity, a negative PTO damping will occur. In the active

control problem, negative PTO damping may appear to cause negative generated power. It is

obvious to see that PTO damping is generated by a nonlinear equation because both relative

velocity and generator force are changing variables. To simplify the problem, a linear

representation of the ptoC is defined here. The Hybrid System Descriptive Language

(HYSDEL) software tool is used to define the HMPC problem [54]. The HYSDEL is a high-

level descriptive language used to describe a hybrid dynamic system. It can directly describe

the formulation of the hybrid MPC system for the semi-active damping control for two-body

WEC.

Linear representation of ptoC

The generator force can be re-written as

1 2 1 2 1 2( ) ( ) ( )( )gen pass pto pass uF C x x u C x x C C x x= − − + = − − = − − − (3.28)

where max min

2pass

C CC

+= , u is the damping force input, uC is the PTO damping for u,

max min,C C are the maximum and minimum PTO damping a system can achieve. The reason to

choose passC to be the average value of max min,C C is to determine a damping value for the

passive model. Then, the HMPC algorithm will search for optimal control force that results in

a damping variation around the passive system damping. Based on the modification, the

generator (damping) force limitation can be expressed as

max 1 2 1 2 min 1 2 1 2

min 1 2 1 2 max 1 2 1 2

( ) ( ) ( ), if 0

( ) ( ) ( ), if 0

pass

pass

C x x C x x u C x x x x

C x x C x x u C x x x x

− − − − + − − −

− − − − + − − − (3.29)

45

From equation (3.29), the constraints for u respect to PTO damping have been converted into

linear inequality equations. However, the constraints depend on the sign of the relative velocity,

which means a hybrid MPC [55] needs to be applied.

Since the generator force consists of both control force and damping force generated by passC ,

with the purpose to be compared to active control, the limitation on the generator force is same

for the semi-active control problem, which will result in a different force limitation on u. the

force constraint can be rewritten as

min max min 1 2 max( )gen passu F u u C x x u u − − + (3.30)

where max min,u u are the same parameters with same values in the active control problem

Hybrid MPC Formulation

I. Cost Function Modification

Since the control force u has been modified as part of the generator force due to linear

representation of PTO damping constraints, the extracted power can be written as

2

1 2 1 2 1 2 1 2 1 2( ) ( ( ))( ) ( ) ( )gen gen pass passP F x x u C x x x x C x x u x x= − − = − − − − = − − − (3.31)

Therefore, based on the genP , the cost function can be reformulated as

2 2

1( ) 2( ) 1 1( ) 2( ) 1

1

( , ) [ ( ( ) ( )) ]N

k k pass k k k k k k

k

J x u q C x x u x x ru− −

=

= − − − − + − − (3.32)

II. Constraints and MPC Problem Formulation

First, the polyhedron regions the problem will be solved should be defined. The added

constraints are on input force u, PTO damping. The state constraints will not change for semi-

active MPC.

Polyhedron 1:

1 2 min

min max1 2

min max1 2

0

( )

( ) 02

( ) 02

pass

u

u C x x u

C Cu x x

C Cu x x

− + − −

−− + −

−+ −

, if 1 2 0x x− (3.33)

genP−

46

Polyhedron 2:

1 2 max

max min1 2

max min1 2

0

( )

( ) 02

( ) 02

pass

u

u C x x u

C Cu x x

C Cu x x

−

− −

−− + −

−+ −

, if 1 2 0x x− (3.34)

We can consider a logic binary variable ( )0 1T

s = . 1 2 1 21 0, 0 0s sx x x x = → − = → − .

By defining variable , , 0s c boundz z z as:

sz = , 1

, 0

s

s

u

u

=

− =, cz =

min max1 2

min max1 2

max min1 2

max min1 2

( )2

, 1

( )2

( )2

, 0

( )2

s

s

C Cu x x

C Cu x x

C Cu x x

C Cu x x

− − + −

= − + −

− − + −

= − + −

(3.35)

boundz = 1 2 min

1 2 max

( ) , 1

( ) , 0

pass s

pass s

u C x x u

u C x x u

− + − + =

− − − = (3.36)

The input constraints combined with the original dynamic system can be formulated as a mixed

logical dynamic system (MLD) [56] for controller design.

1 1 2 1 2

2 3 1 4 5

k d k d k d k d k s k k

s k k k k

x A x B u B v B w B B z

E E z E u E x E

+ −

−

= + + + + +

+ + + (3.37)

where , , ,k k k kx u v w are state variables, semi-active control force, buoy excitation force and

submerged body excitation force at current time step k. s k − is s at time step k.

( )T

k s c boundz z z z= are the auxiliary binary and continuous variables at time step k.

1 2, , ,d d d dA B B B are discretized system matrices from the original state-space model in (3.15).

The sampling time is sT . The matrices 1 2 1 2 3 4 5, , , , , ,B B E E E E E are calculated automatically

by the Multi Parametric Toolbox (MPT) [54] according to the descriptive HYSDEL language.

47

Based on the hybrid logic constraints, the hybrid MPC can be formulated as

2 2

2( ) 4( ) 1 2( ) 4( ) 1, ,

1

min ( , ) min [ ( ( ) ( )) ]k k k k

N

k k pass k k k k k kx u x u

k

J x u q C x x u x x ru− −

=

= − − − − + − − (3.38)

subject to

1 2( ) 4( ) 1 2 1 2

2 3 1 4 5

min 1( ) 3( ) max

min 2( ) 4( ) max

[ ( )]k d k d k pass k k d k d k s k k

s k k k k

k k

k k

x A x B u C x x B v B w B B z

E E z E u E x E

x x x x

x x x x

+ −

−

= + − − + + + +

+ + +

−

−

(3.39)

where 1( ) 2( ) 3( ) 4( ), , ,k k k kx x x x represent the first to fourth state at time k.

The procedure is to write the HYSDEL code, then convert the code as an MPT structure system

into MATLAB. Since the system has measurable disturbances v, w, the YALMIP [57] toolbox

inside the MPT toolbox is applied to manually input disturbance prediction according to

simulation time step. The YALMIP toolbox has the flexibility to customize cost function.

Finally, using the external called solver GUROBI [58], that contains the capability to solve

MIQP problem, to optimize the problem based on user-defined cost function. The GUROBI

solver has the capability to directly solve nonlinear cost function defined as in the problem.

3.1.5 Simulation Results and Conclusion

I. Parameters and Inputs

The active (MPC) and semi-active (HMPC) control law uses same parameters with same

adjusted weights for states and control inputs. The active control simulation is implemented in

the Simulink environment. The semi-active control simulation is implemented with MATLAB

code. The simulation parameters for the WEC system are shown in Table I in Appendix B.

The wave input is a superposition of three regular waves with different amplitudes, frequencies

and phases. The excitation force amplitudes are obtained from the coefficients calculated from

WAMIT. Since the distance between the buoy and the submerged body is assumed to be long

enough, the surface wave motion cannot have large effect on the submerged body, which results

in a much smaller excitation force on the submerged body. The wave parameters are displayed

in Table II in Appendix B. The wave excitation forces for the buoy and the submerged body

are displayed in Fig 3.4. From the plot, it can be shown that the combined wave has a period

equal to 10s.

48

Fig 3.4. Wave Excitation Force Inputs for the Buoy and the Submerged Body

II. Results Comparison and Discussion

Since the toolbox solver will need a long time to solve the problem in a standard desktop, the

simulation time is set to 20s, which equals to two full periods of the input wave. The passive

WEC model uses the constant PTO damping equals to passC in the simulation. The results of

the relative position, relative velocity between the buoy and the submerged body for semi-

active, active and passive systems are displayed in Fig 3.5. The relative velocities of the three

cases are also plotted with the excitation force applied on the buoy in Fig 3.6. The generator

forces for semi-active, active, and passive systems are displayed in Fig 3.7. The resulting PTO

damping for the three cases are displayed in Fig 3.8 based on the calculation

1 2/ ( )PTO genC F x x= − − . Since the PTO damping for active case is much larger compared to

that of the other 2 cases, which makes the PTO damping for semi-active case hard to see in Fig

3.8, the PTO damping for semi-active control is displayed in Fig 3.9 with the comparison with

that of the passive case. The generator power for the three cases are displayed in Fig 3.10.

Fig 3.5. Buoy and Submerged Body Relative Position (Left) and Velocity (Right)

pT

49

Fig 3.6. Relative Velocity and Wave Excitation Force on the WEC

Fig 3.7. Generator Force for the WEC

50

Fig 3.8. PTO Damping Comparison

Fig 3.9. PTO Damping Comparison Between Semi-active & Passive Models

51

Fig 3.10. Power Generation Comparison

The relative position is set to be 0.75 0.75x− ; the relative velocity is set to be

1 1x− ; the generator force is set to be 50000 50000genF− for both semi-active and

active systems. From Fig 3.5 on the left side, the relative positions of the active and semi-active

cases are within the pre-defined boundary, 0.75 0.75x− . The relative velocities on the

right side are also within the boundary, 1 1x− . In Fig 3.7, the generator forces for the

active and semi-active cases are within the boundary, 50000 50000genF− . By comparing

Fig 3.8 and Fig 3.9, the PTO damping for the active system (peak value can be 1.5×106 Ns/m)

is much larger compared to that of semi-active (peak value is only 1.5×104 Ns/m) and passive

systems (1×104 Ns/m). It also worth to note that the HMPC in semi-active cases tries to find

damping around pre-defined passive damping used in the passive case. It is mentioned above

in sub-section 3.1.4 that the algorithm will try to find damping around passive damping set

based on max min,C C , since the control force will heavily depend on passC set in the generator

force equation. Hence, it will help the semi-active case to avoid large or small damping change

during control. Besides, the active PTO damping changes from positive value to negative value

frequently. However, the PTO damping for semi-active system is constrained within the pre-

defined damping constraint, 0 20000genC . Since the damping value for the semi-active

system is much smaller compared to that of the active system, the relative velocity becomes

larger for semi-active system, which can be verified in Fig 3.5 on the right side. However, since

the damping for semi-active system is much smaller compared to that of the active system and

the relative velocity does not vary significantly for the active and semi-active systems, the

generator forces for the semi-active system is much smaller compared to that of the active and

passive systems. The average damping for the three systems is analyzed as below

52

As for the PTO damping for the semi-active system, it is always positive. Therefore, the

average damping value is obtained by taking the rood mean square (RMS) value of the PTO

damping. As for the active system, the damping value changes between positive and negative

values. Therefore, the average damping is calculated separately for positive and negative values.

Then, the average PTO damping is calculated based on the summation average of the positive

and negative damping values, which can be expressed as the equation below

( ) ( )( )

2

pto positive pto negative

pto active

rms C rms Cavg C

− −

−

+= (3.40)

where ( ), ( )pto positive pto negativerms C rms C− − are the root mean square values for the positive and

negative PTO damping values.

The damping suppression percentage from the semi-active system compared to that of the

active system is calculated as

sup

( ) ( )100%

( )

pto active pto semi active

pto active

avg C avg CP

avg C

− − −

−

−= (3.41)

where ( ), ( )pto active pto semi activeavg C avg C− − − are the average PTO damping for active and semi-

active systems. The passive PTO damping is a constant. The resulting average PTO damping

for the active, semi-active and passive systems are displayed in Table IV

TABLE V. AVERAGE PTO DAMPING COMPARISON

System Average PTO Damping Suppressing Percentage

Respect to Active System (%)

Semi-active 9994 88

Active 84092 0

Passive 10000 N/A

Based on the theory, semi-active control will result in lower power extraction since system

damping is restricted. From Fig 3.10, it is obvious to see that the power generation for semi-

active system is less than that of the active system. From Fig 3.10, both semi-active and active

systems can have larger generated power compared to that of the passive system, which shows

the effectiveness of the controller. The average power for the three systems is calculated based

on the equation as

0

1T

avg genP P dtT

= (3.42)

where genP is the instant generator power, T is the simulation horizon (20s). The average

53

power is the time integration of the generator power over the simulation horizon. The

integration is calculated by the MATLAB command “trapz()”. The calculated average power

is displayed in Table V. The power loss percentage of the semi-active system compared to the

power of the active system is calculated directly based on the power improvement percentage

difference for those two systems.

TABLE VI. AVERAGE POWER COMPARISON

System Average Generated Power (W) Power Improvement Percentage (%)

Semi-active 1033 35

Active 1139 49

Passive 763 0

From the table, it can be shown that the power loss for semi-active system is around 14%

compared to that of the active system. The damping suppression effect for the semi-active

system can be around 88% compared to that of the active system. The semi-active controller

successfully considers both the generator force as well as the PTO damping which ensures safer

operation for a PTO system in a WEC without sacrificing large amount of power generation.

III. Conclusion

A hybrid model predictive control (HMPC) strategy is applied on a two-body WEC targeted at

power extraction maximization with the consideration of PTO damping (semi-active) as well

as PTO force limitations. The two-body WEC system has been modeled as a linear time

invariant state-space model by setting the frequency dependent radiation damping as constant.

A standard formulation of Quadratic Programming (QP) MPC has been applied with active

control force on the WEC system as compared model. The HMPC has been formulated as a

mixed integer quadratic programming (MIQP) problem and is solved by the MPT toolbox in

MATLAB. The simulation results with same input parameters are analyzed and discussed for

semi-active, active and passive cases. From the result, the semi-active controller has the ability

to limit both the generator force and the PTO damping. The PTO damping control effort has

been reduced by 88% compared to that of the active system with 14% of power loss. The

controller displays the advantage in providing PTO with a safer operation condition and also

provides the possibility to be applied to the PTO system with linear generator.

3.2 Latching Control on MMR-based Two-body WEC

3.2.1 Introduction

Previous sub-chapter investigates time domain optimal control for direct drive PTO based on

linear generator. Lots of designs use rack and pinion to convert vertical bidirectional motion

into bidirectional rotation as shown in Fig 3.11. The two racks will move with the buoy. Two

pinion gears will be driven by the racks. The generator shaft can be driven by the pinion gears

to rotate. The mechanism to convert up-down motion into rotational motion effectively reduce

the required generator damping to achieve same energy absorption capability, which is

mentioned in section 2.2 in suspension application. However, the rack and pinion mechanism

54

has one major issue, the backslash impact force will cause great system damage when the

generator changes rotation direction. Besides, the generator will always attach to the buoy.

Generator velocity will change between clockwise and counter clockwise directions, which

means generator velocity will need to pass zero line frequently. Such significant change of

velocity will greatly reduce generator efficiency.

Fig 3.11. Rack-pinion based PTO [59]

With the purpose to solve the issue caused by rack-pinion based direct drive PTO design, the

mechanical motion rectifier (MMR) with ball-screw converting mechanism is introduced. The

ball-screw mechanism is shown in Fig 3.12. It contains a nut with helical slots to let ball rotate

through. The screw will be inserted into the nut. Lots of steel balls will be inserted into the slot

between the nut and screw. When the nut moves up and down, the screw will rotate. Such

mechanism smoothly converts vertical motion into rotational motion, which greatly reduces

backslash impact force compared to that of the rack-pinion mechanism.

Fig 3.12. Ball-screw Mechanism

In Fig 3.13, the design of MMR-based PTO is shown. The nut of the ball-screw is connected

to the buoy. The ball-screw will be used to drive the input shaft of the MMR gearbox. Then,

the MMR gearbox will use two one-way clutches to convert bidirectional motion of the input

shaft to unidirectional motion of generator driven by the output shaft. The working principle is

same as what is mentioned in the MMR-based suspension design in section 2.2. The MMR

PTO successfully avoids generator to change velocity between positive and negative values to

improve generator efficiency. From the bench test of MMR PTO with sinusoid excitation, it

can be shown in Fig 3.14 that the generator will never drop to zero velocity and always has

positive value.

Rack-pinion

55

Fig 3.13. MMR-based WEC System

(a) (b)

Fig 3.14. (a) MMR Bench Test Setup (b) Sinusoid Test Results

The design of MMR PTO proves its advantage in power conversion. However, such mechanism

brings new challenge on the control aspect. As mentioned in section 2.2, MMR will have mode

switch between engage/disengage modes that will change system dynamic equations. The

switching law depends on the velocity difference between input shaft and output shaft. When

the output shaft rotates faster, the MMR will disengage. Such phenomenon brings the system

into a piecewise linear system. There are two ways to control the MMR PTO. The first one is

to develop linear control law for engage mode without considering disengage mode. If follow

this way, many optimal control methods can be use. For example, the MPC method mentioned

in sub-chapter 3.1 can be applied. The second way is to find global optimal solution with the

consideration of disengage mode. Such problem is extremely hard to solve due to the switching

mechanism and also will consume lots of computational power, because global optimal

56

methods, like Dynamic Programming (DP), will require heavy computational effort.

To avoid heavy computational effort in global optimal control development, sub-optimal

methods become necessary, since they are easier to be implemented and require much less

computational time. From the investigation of lots of literatures, the time domain optimal

control force for a two-body WEC system based on irregular wave input does not have

analytical solution, which means the optimal condition has not been known. Therefore,

feedback control theory is hard to implement. According to the author’s investigation through

many control methods applied on WEC system, latching control brings attention due to its

simplicity. The latching control is a passive control method that will not require energy

feedback from the generator to the WEC. It simply uses braking mechanism to lock the WEC

from moving some times to let it be in-phase with the input wave again to harvest energy. For

a single body WEC system, latching control will lock the buoy at a fixed position, which means

the generator will not rotate during locking period. Then, when the next wave peak comes, the

buoy will be released to let generator rotate to harvest energy. The control law will lock the

buoy periodically or by following certain rules according to input wave excitation. From single

body latching control implementation original proposed in [60], it locks the WEC motion when

its velocity vanishes and release it such that ideally the velocity becomes in-phase with the

excitation force. The results show that under certain conditions, a significant increase in the

energy extraction can be achieved, when compared with the non-controlled case. For a two-

body WEC system, the latching control will lock the relative motion between the buoy and the

submerged body, which means it will be converted to a single-body system. The generator will

not rotate due to the lock of the two bodies. Then, when the wave peak comes, the controller

will release the two bodies to harvest energy again. The control concept can be concluded in

Fig 3.15.

Fig 3.15. Two-body Latching Control Concept

The control development of two-body WEC system has not been investigated a lot. Currently,

just limited literatures talk about this topic. However, the MMR PTO engagement feature may

become an advantage with latching control implementation. As mentioned in previous

paragraph, during locking period, the generator will not harvest energy. However, during MMR

disengagement, the generator may still have rotation due to its inertia, then the velocity will

decay due to generator damping. Therefore, MMR PTO can have some time to continuously

harvesting energy even during locking period. This sub-chapter will investigate two-body

latching control on the MMR-based PTO. From the simulation, the latching control

significantly improves the power absorption. However, due to latching, system peak-to-

average power ratio has been increased significantly as well due to fast generator velocity

decay. Flywheel is added to the output shaft as a way to reduce generator velocity decay

rate. The effect of flywheel rotational inertia is also investigated.

57

3.2.2 MMR-based PTO Modeling

I. Two-body WEC Model Refine

From sub-chapter 3.1, two-body WEC system has been modeled without the consideration of

the radiation damping caused by buoy and submerged body interaction. The applied model

weights around 2600 kg, which means the model size is large. Therefore, the column or truss

used to connect the two bodies are long enough to ignore radiation damping caused by buoy

and submerged body interaction. In the MMR based WEC, only the 1/30th scale MMR PTO

parameters are available to the author. The 1/30th scale model has a buoy diameter equal to

0.75m. For such model, the distance between buoy and submerged body will be reduced

significantly, which results in the necessity to include radiation damping due to bodies’

interaction. Hence, the WEC model will be reformulated as

Dynamic equation during engagement: 1 2genx n x x −

Buoy dynamic:

1 11 1 12 2 1 2 11 1 12 2 1 2 1 2 1 1 1( ) ( ) ( ) ( ) s e

e pto ptom A x A x m x x b x b x c x x k x x k x f+ + + − + + + − + − + = (3.43)

where 1 2,x x are buoy and submerged body position, 12b is the interacted radiation damping

caused by submerged body on buoy, 11b is the buoy radiation damping, 11A is the buoy

added mass, 12A is the added mass caused by submerged body on buoy, 1

ef is the buoy

wave excitation force, , ,e pto ptom c k are the equivalent mass, equivalent damping and

equivalent stiffness introduced by MMR system. In the simulation, the ptok is set to 0. The

1

sk is the hydrostatic stiffness of the buoy. n is the combined gear ratio between the

generator shaft and the relative velocity between the buoy and the submerged body.

Submerged body dynamic:

2 22 2 21 1 2 1 21 1 22 2 2 1 2 1 2 2 2( ) ( ) ( ) ( ) s e

e pto ptom A x A x m x x b x b x c x x k x x k x f+ + + − + + + − + − + = (3.44)

where 21b are the interacted radiation damping caused by buoy on submerged body, 22b is

the submerged body radiation damping, 22A is the submerged body added mass, 21A is the

added mass caused by buoy on submerged body, 2

ef is the submerged body wave excitation

58

force. The 2

sk is the hydrostatic stiffness of the submerged body. 2

sk is set to zero, since

submerged body buoyancy force will not change due to fully submerge in the water.

After doing some re-arrangement, two equations for buoy and submerged body will be

reformulated as

12 21 21 12 22 12 121 11 1 11 1 12 2 1 2

2 22 2 22 2 22 2 22

12 12 21 2 1 1 2 1

2 22 2 22

( )( ) ( ) ( )[ ] [ ] [ ] (1 )( )

( )(1 )( )

e e e e ee pto

e e e e

ss ee e

pto

e e

m A m A b m A b m A m Am A m x b x b x c x x

m A m m A m m A m m A m

m A m A k mk x x k x x f

m A m m A m

− − − − −+ + − + + + + + − −

+ + + + + + + +

− −+ − − + + − −

+ + + +

122

2 22

0 ee

e

Af

m A m

−=

+ +

(3.45)

21 12 11 21 12 21 212 22 2 21 1 22 2 1 2

1 11 1 11 1 11 1 11

21 21 11 2 2 2 1 2

1 11 2 22

( )( ) ( ) ( )[ ] [ ] [ ] ( 1)( )

( )( 1)( )

e e e e ee pto

e e e e

ss ee e

pto

e e

m A m A b m A b m A m Am A m x b x b x c x x

m A m m A m m A m m A m

m A m A k mk x x k x x f

m A m m A m

− − − − −+ + − + + + + + − −

+ + + + + + + +

− −+ − − + + − −

+ + + +

211

1 11

0 ee

e

Af

m A m

−=

+ +

(3.46)

Then, a state-space model can be formulated for the two-body WEC engage model.

1 2

1 2e e e e eX A X B f B f= + + (3.47)

where

0 1 0 0

( )12 21 12 12[1 ] [ ] ( 1) (1111 2 22 2 22 2 22( )( ) ( )( ) ( )( )12 21 12 21 12 21

1 11 1 11 1 112 22 2 22 2 22

m A b m A m A ms e e e ek k b c kpto pto ptom A m m A m m A me e em A m A m A m A m A m Ae e e e e em A m m A m m A me e em A m m A m m A me e e

eA

− − −− − − − + − −

+ + + + + ++

− − − − − −+ + − + + − + + −

+ + + + + +=

( )12 ( )12 2 22 12 12) [ ] ( 1)122 22 2 22 2 22 2 22

( )( ) ( )( ) ( )( )12 21 12 21 12 211 11 1 11 1 11

2 22 2 22 2 220 0 0 1

(

sm A keA b m A m Ae eb cptom A m m A m m A m m A me e e em A m A m A m A m A m Ae e e e e em A m m A m m A me e em A m m A m m A me e e

me

−− − −− − + −

+ + + + + + + +−

− − − − − −+ + − + + − + + −

+ + + + + +

−−

)21 ( )1 21 11 21 21( 1) [ ] (1 )212 22 1 11 1 11 1 11

( )( ) ( )( ) ( )( )21 12 21 12 21 122 22 2 22 2 22

1 11 1 11 1 11

sA k m A b m A m Ae e ek b c kpto pto ptom A m m A m m A m m A me e e em A m A m A m A m A m Ae e e e e em A m m A m m A me e em A m m A m m A me e e

− − −− − − + −

+ + + + + + + ++

− − − − − −+ + − + + − + + −

+ + + + + +

( )21 12 21 21( 1) [ ] (1 )2221 11 1 11 1 11( )( ) ( )( ) ( )( )21 12 21 12 21 12

2 22 2 22 2 221 11 1 11 1 11

m A b m A m Ase e ek b cptom A m m A m m A me e em A m A m A m A m A m Ae e e e e em A m m A m m A me e em A m m A m m A me e e

− − −− − − + −

+ + + + + +−

− − − − − −+ + − + + − + + −

+ + + + + +

12

12 21 2 221 11

2 22 12 211 11

2 221 2

21

1 11

21 122 22 2 22

1 11

0 0

1

( )( )

( )( )

,0

0

1

( )( )

e

e e ee

e e ee

ee e

e

e

e ee

e

m A

m A m A m A mm A m

m A m m A m Am A m

m A mB B

m A

m A m

m A m Am A m m A

m A m

−

− − + + + + −

+ + − −+ + −

+ += = −

+ + − −

+ + − + + + +

( )1 1 2 2

21 12

1 11

,

( )( )

T

e ee

e

X x x x x

m A m Am

m A m

= − −

− + +

Dynamic equation during disengagement: 1 2genx n x x −

Buoy dynamic:

During disengagement, equivalent mass, damping and stiffness will be eliminated. Therefore,

the equation for buoy will be

59

1 11 1 12 2 1 2 11 1 12 2 1 1 1( ) ( ) s e

e dism A x A x m x x b x b x k x f−+ + + − + + + = (3.48)

where e dism − is the disengaged equivalent inertia. During disengagement, the MMR gearbox

and ball-screw mechanism is still connected with the WEC. Hence, the equivalent inertia

caused by MMR gearbox and ball-screw mechanism still exists.

Submerged body dynamic:

2 22 2 21 1 2 1 21 1 22 2 2 2 2( ) ( ) s e

e dism A x A x m x x b x b x k x f−+ + + − + + + = (3.49)

Then, the state-space model for disengage model is

1 2

1 2dis dis e dis eX A X B f B f= + + (3.50)

where

0 1 0 0

( )( ) 1221 12 2[ ]112 22 2 221

( )( ) ( )( ) (12 21 12 211 11 1 11 1 11

2 22 2 22

sm A kb m A e dise disbs

k m A m m A me dis e dism A m A m A m A me dis e dis e dis e dis e dism A m m A m m A me dis e dis e dis

m A m m A me dis e dis

disA

−− −−− + −− + + + +− −

− − − −− − − − −+ + − + + − + + −− − −+ + + +− −=

( )22 12[ ]122 22

)( ) ( )( )12 21 12 211 11

2 22 2 22

0 0 0 1

( )21 1

2 22 1( )( )21 12

2 221 11

b m Ae disbm A me dis

A m A m A m Ae dis e dis e dism A me dism A m m A me dis e dis

sm A ke dis

m A m e dism A m Ae dis e dism A me dis

m A me di

−−− ++ + −

− − − −− − −+ + −−+ + + +− −

−−−

+ + −− −− −+ + −−+ + −

( ) ( )11 21 12 21[ ] [ ]21 221 11 1 112

( )( ) ( )( ) (21 12 21 122 22 2 22 2 22

1 11 1 11

b m A b m Ae dis e disb bskm A m m A me dis e dis

m A m A m A m A m Ae dis e dis e dis e dis e dism A m m A m m A me dis e dis e dism A m m A ms e dis e dis

− −− −− + − +−+ + + +− −

− − − − −− − − − −+ + − + + − + + −− − −+ + + +− −

12 21

1 11

2 22

1

21

1 11

21 12

2 22

1 11

)( )21 12

1 11

0

1

( )( )

0

( )( )

e dis e dis

e dis

e dis

dis

e dis

e dis

e dis e dis

e dis

e dis

m Ae dis

m A me dis

m A m Am A m

m A m

B

m A

m A m

m A m Am A m

m A m

− −

−

−

−

−

− −

−

−

−− + + −

− −+ + −

+ +

=

−

+ +

− −+ + −

+ +

12

2 22

12 21

1 11

2 222

21 12

2 22

1 11

0

( )( )

,

0

1

( )( )

e dis

e dis

e dis e dis

e dis

e disdis

e dis e dis

e dis

e dis

m A

m A m

m A m Am A m

m A mB

m A m Am A m

m A m

−

−

− −

−

−

− −

−

−

−

+ +

− −+ + −

+ +=

− −+ + −

+ +

Notice: Single-body MMR-based WEC system modeling

To show the advantage of two-body WEC design, a single-body MMR-based WEC is also

modelled with same buoy and PTO parameters. The only difference is the elimination of the

second submerged body.

The schematic diagram is shown in Fig. 3.16:

60

Fig 3.16. Schematic Diagram of Single-body WEC system

The dynamic equation is based on the force balanced equation on buoy itself without a

connected second submerged body.

Dynamic equation during engagement: 1genx n x

Buoy dynamic:

1 11 1 1 11 1 1 1 1 1 1( ) s e

e pto ptom A x m x b x c x k x k x f+ + + + + + = (3.51)

In the equation, the terms related to submerged body has been eliminated to formulate buoy

only dynamic equation. Since the PTO system will be the same, the internal dynamic for PTO

equivalent mass em and equivalent damping ptoc will be the same for both systems. For

single-body system, ptok is also set to zero.

By reformulating equation (3.51), the state-space model for single-body system in engage

mode can be concluded as:

1 1

1111

1 1

1 111 11 1 11

0 1 0

1es

pto

ee e

x xfc bk

x xm A mm A m m A m

= +− −− + ++ + + +

(3.52)

Dynamic equation during disengagement: 1genx n x

Buoy dynamic:

1 11 1 1 11 1 1 1 1( ) s e

e dism A x m x b x k x f−+ + + + = (3.53)

By reformulating equation (3.53), the state-space model for single-body system in disengage

mode can be concluded as:

61

1 1

11 11

1 1

1 111 11 1 11

0 1 0

1es

e dise dis e dis

x xfk b

x xm A mm A m m A m

−− −

= +− − + ++ + + +

(3.54)

II. MMR PTO Modeling

Then, the expressions of the mentioned , ,e pto e dism c m − will be derived. The generator model

during disengage mode will also be introduced. A simplified MMR PTO schematic diagram is

shown in Fig 3.17.

Fig 3.17. Schematic Diagram of MMR PTO

Engage model:

From torque balance analysis starting from ball-screw nut to the generator, the equivalent mass,

equivalent damping during engage mode are

2 2 2

2 2

4 ( 2 ) 4 ( )2

bs cp is gb gen g

e push bn

J J J J J nm m m

l l

+ + += + + + (3.55)

2 2

2

4

( )

e t g

pto

e i

k k nc

R R l

=

+ (3.56)

where ,push bnm m are the push-tube and ball-screw nut mass, , , , ,bs cp is gb genJ J J J J are the ball-

screw, coupling, input shaft, bevel gear, generator inertia, l is the ball-screw lead, ,e tk k are

generator voltage and torque constants, ,e iR R are the generator circuit external resistance and

generator inner resistance, gn is the generator gearhead ratio originally installed by the

manufacturer.

62

The ptoc is based on DC generator. If an AC generator is applied, the

2 2

2

6

( )

e t g

pto

e i

k k nc

R R l

=

+.

The equation is similar as what is mentioned in section 2.2.2. However, in the ocean application,

the ball-screw friction coefficient f , generator viscous damping vc are ignored here, since

WEC system has a much larger MMR system, the inertia and damping are much larger. Based

on experiment, the two terms are ignorable. The MMR PTO uses 1:1 bevel gear inside the

gearbox. Therefore, br is set to 1. The equivalent mass and damping are simplified as in

equation (3.55), (3.56).

Disengage model:

In the disengage model, the equivalent inertia is

2

2

4 ( )2

bs cp is

e dis push bn

J J Jm m m

l

−

+ += + + (3.57)

During the disengage mode, the generator will form a system by itself. The dynamic equation

is

0 1

00

gen

gen

ct

gen gen m

gengen gen gen gen gen

gen gen

gen

x xcm x c x x e

x xm

−

−+ = = =

(3.58)

where

2

gen cp gb gen gm J J J n= + + (3.59)

2e tgen g

e i

k kc n

R R=

+ (3.60)

where ,gen genm c are the combined generator mass and generator damping. The generator is

still attached to the bottom bevel gear with a coupling. Therefore, the masses of the coupling

and bevel gear need to be included in the generator model. From the generator model, it can be

shown that the generator velocity will decay exponentially with decay rate equal to

gen

gen

ct

me−

.

3.2.3 Latching Control on MMR-based Two-body WEC

63

As mentioned in [60] Falnes locks the two-body WEC when the relative velocity vanishes. The

reason to lock at this time point is to avoid large impact force due to sudden brake on a moving

WEC system. Most of the latching control will lock this way to protect system. Hence, the

latching control development for the MMR-based WEC also follows this rule.

Then, the MMR dynamic will be analyzed when latching control happens. When the MMR is

engaged at the beginning of the locking period. The relative velocity will suddenly drop to zero.

Therefore, the generator will have faster velocity or same velocity compared with input shaft,

which will cause system disengagement. Hence, the system will jump into disengage mode

with zero relative velocity. When the MMR is disengaged at the beginning of the locking period,

relative velocity will suddenly drop to zero as well. However, since the system is already

disengaged, the generator velocity must be higher compared to the input shaft. The system will

continually disengage. The mode switching rule will be added into original MMR WEC

switching law to be adapted with latching control feature. The system switching rule can be

summarized as

Mode=

1 2

1 2

,

,

, engage before lock

, disenage before lock

gen

gen

engage n x x

disengage n x x

disengage

disengage

−

− (3.61)

The next step is to determine best latching period. The latching control in the current control

development uses constant generator damping for most of the time except for the time instant

when locking and releasing happen. The damping is chosen as the optimal damping for the

passive MMR WEC optimized for given input wave to ensure fair comparison. Then, the

optimal latching period is targeted on average power maximization for a certain time length

simulation. The latching time is pre-defined with a range with self-defined grid size. The

simulation will be done for each latching period to identify one latching time for maximum

average power over a period.

The MMR WEC has some capability for damping tuning, which gives more flexibility in

control. At the instant when the system locking starts, to avoid large impact force, system

equivalent damping can be set to maximum available value to generate maximum damping

force to reduce relative speed faster for minimum braking effort. Besides, during locking period,

power should be absorbed as much as possible. Therefore, system damping will be set to

maximum value to absorb power as much as possible. So, the control rule can be summarized

as

• Latch happens when relative velocity vanishes

• Use maximum damping when re-latch happens

• Use maximum damping during disengagement

Next, the dynamic model during latching period is derived.

Engagement equations-

64

1 11 1 12 2 1 2 11 1 12 2 1 2 1 2 1 1 1( ) ( ) ( ) ( ) s e

e pto ptom A x A x m x x b x b x c x x k x x k x f+ + + − + + + − + − + = (3.62)

2 22 2 21 1 1 2 21 1 22 2 1 2 1 2 2 2 2( ) ( ) ( ) ( ) s e

e pto ptom A x A x m x x b x b x c x x k x x k x f+ + − − + + − − − − + = (3.63)

2 2 2

2 2

4 ( 2 ) 4 ( )2

bs cp is gb gen g

e push bn

J J J J J nm m m

l l

+ + += + + + (3.64)

2 2

2

4

( )

e t g

pto

e i

k k nc

R R l

=

+ (3.65)

Add latching force 𝑢𝑒-

1 11 1 12 2 1 2 11 1 12 2 1 2 1 2 1 1 1( ) ( ) ( ) ( ) s e

e pto pto em A x A x m x x b x b x c x x k x x k x f u+ + + − + + + − + − + = + (3.66)

2 22 2 21 1 1 2 21 1 22 2 1 2 1 2 2 2 2( ) ( ) ( ) ( ) s e

e pto pto em A x A x m x x b x b x c x x k x x k x f u+ + − − + + − − − − + = − (3.67)

When the system is latched, 1 2 1 2 0x x x x− = − = , equation (3.66), (3.67) becomes

1 11 1 12 1 11 1 12 1 1 2 1 1 1( ) ( ) s e

pto em A x A x b x b x k x x k x f u+ + + + + − + = + (3.68)

2 22 1 21 1 21 1 22 1 1 2 2 2 2( ) ( ) s e

pto em A x A x b x b x k x x k x f u+ + + + − − + = − (3.69)

If set 1 2x x c− = , where c is the relative position when the WEC is locked, equation (3.68),

(3.69) becomes

1 11 1 12 1 11 1 12 1 1 1 1( ) s e

pto em A x A x b x b x k c k x f u+ + + + + + = + (3.70)

2 22 1 21 1 21 1 22 1 2 2 2( ) s e

pto em A x A x b x b x k c k x f u+ + + + − + = − (3.71)

Add (3.70) to (3.71)

1 11 2 22 12 21 1 11 12 21 22 1 1 1 2 2 1 2( ) ( ) s s e em A m A A A x b b b b x k x k x f f+ + + + + + + + + + + = + (3.72)

Set 2 0sk = , since the submerged body is fully submerged.

65

1 11 2 22 12 21 1 11 12 21 22 1 1 1 1 2( ) ( ) s e em A m A A A x b b b b x k x f f+ + + + + + + + + + = + (3.73)

State-space model: set 1x x=

1

1 11 12 21 22

21 11 2 22 12 21 1 11 2 22 12 211 11 2 22 12 21 1 11 2 22 12 21

0 1 0 0

1 1

e

s

e

x x fk b b b b

x x fm A m A A A m A m A A Am A m A A A m A m A A A

= +− − − − −

+ + + + + + + + + ++ + + + + + + + + +

(3.74)

Disengage equations-

1 11 1 12 2 1 2 11 1 12 2 1 1 1( ) ( ) s e

e dism A x A x m x x b x b x k x f−+ + + − + + + = (3.75)

2 22 2 21 1 1 2 21 1 22 2 2 2 2( ) ( ) s e

e dism A x A x m x x b x b x k x f−+ + − − + + + = (3.76)

2

2

4 ( )2

bs cp is

e dis push bn

J J Jm m m

l

−

+ += + + (3.77)

2 2, ,

gen

gen

ct

m e tgen gen cp gb gen g gen g

e i

k ke m J J J n c n

R R

−

= = + + =+

(3.78)

Apply latching force 𝑢𝑑𝑖𝑠 and do derivation as engage model-

1

1 11 12 21 22

21 11 2 22 12 21 1 11 2 22 12 211 11 2 22 12 21 1 11 2 22 12 21

0 1 0 0

1 1

e

s

e

x x fk b b b b

x x fm A m A A A m A m A A Am A m A A A m A m A A A

= +− − − − −

+ + + + + + + + + ++ + + + + + + + + +

(3.79)

Based on the derivation above combined with original MMR switching model, the latching

control model will have 3 different dynamic models.

3.2.4 Simulation Results and Conclusion

I. Wave Input and WEC Parameters

The input wave is a JONSWAP wave spectrum [61], which is a concentrated spectrum with

noticeable significant height and dominant frequency. The wave equation is

41

2 24( )4 2

1950( )2 52

4

320 ( )2( )

Tp

p

sT e

p

H

S eT

−−

−

−−

=

(3.80)

66

where is dominant wave frequency, pT is the wave period, sH is the significant wave

height, is a constant (it is set to 3.3), is the wave frequency variance.

The frequency domain spectrum is shown in Fig 3.18. The original wave input is incorporated

with random phases as well. The variance value equals to 0.07 for wave components with wave

period greater than pT ; it is set to 0.09 for wave components with wave period less than

pT .

Fig 3.18. Wave Input Frequency Spectrum

In time domain, JONSWAP is a random wave with variance value mentioned above.

Fig 3.19. Time Domain Wave Input

The WEC parameters are obtained directly from 1/30th scale MMR-based two-body WEC

developed by CEHMS prototype. The parameters are shown in Table III under Appendix B.

The buoy and submerged bodies’ added mass and radiation damping, as well as interacting

added mass and radiation damping are frequency dependent. For simplification, since the

JONSWAP spectrum has significant frequency, it is used as the frequency to find corresponding

added mass and radiation damping for both bodies.

II. Find Optimal External Resistance for Two-body Passive MMR WEC under JONSWAP

Wave

67

In subsection 2.2, the optimization of two MMR-based suspension components are mentioned-

bevel gear ratio and external resistance in generator circuit. There is one pair of optimal gear

ratio and external resistance for a type of input to have maximum average power in a time

period. In the ocean application, the bevel gear ratio is already designed to be 1:1 in the

prototype, which means only the external resistor will be optimized for the passive model. The

simulation is done by choosing external resistance range from 1 to 100 ohms with time domain

simulation including the MMR feature with JONSWAP spectrum mentioned above. Fig 3.20

shows the relationship between the external resistance and average power. It can be shown that

the maximum average power happens when external resistance equal to 17 ohms. This value

will be used for passive MMR WEC. Since the JONSWAP spectrum will randomly change in

time domain, the simulation for both controlled and passive models all use same time domain

wave input for fair comparison.

Fig 3.20. Finding Optimal External Resistance for Passive MMR

The single-body is also optimized with the target for maximum average power. The power

comparison between passive single-body and passive two-body MMR WEC is shown in Fig

3.21.

68

Fig 3.21. Power Comparison Between Passive Single-body and Two-body MMR WECs

From Fig 3.21, it is obvious to see the two-body system can absorb much more energy

compared to that of the single-body system. Based on the average power calculation equation

(3.42), the two-body WEC has 0.15W average power. While the single-body WEC has 0.07W

average power. The power difference in passive model confirms the advantage of two-body

WEC design.

III. Comparison between Controlled and Passive Two-body Models

In the simulation, another normal WEC is also introduced for comparison. The normal WEC

is just a system that directly connects generator to the input shaft without MMR feature.

Therefore, generator will rotate bidirectionally with the input shaft. The normal WEC also uses

same external resistance for fair comparison. It also contains controlled and passive models.

The whole simulation has time length equal to 100s to cover 50 periods of the input wave to

prove control performance. The resulting instant power comparison is shown in Fig 3.22.

Fig 3.22. Power Comparison among Different Models

From the figure, it is easy to see that the passive MMR WEC already has advantage over the

passive normal WEC with higher power generation, which proves the effectiveness of the

MMR-based design. When the wave amplitude is larger, the generator is easier to be driven by

the input shaft in the MMR WEC, which result in a higher peak power. When the power drops,

the MMR can have disengage feature to let generator decay by itself rather than directly let the

input shaft drive the generator as what happened in normal WEC. The power drop rate is slower

compared to that of the normal WEC.

With the latching control, both normal and MMR WECs show significantly improvement in

power absorption. While the MMR shows higher power peak compared to that of the normal

WEC system. The reason can be the engagement feature provides less restriction on generator

velocity, because direct drive normal WEC will always restrict generator velocity with the input

69

shaft. When buoy has small motion, the low velocity of input shaft will directly restrict the

velocity of the generator. From the MMR controlled model, it is obvious to see that the power

drops fast after a peak. It is due to large generator velocity decay rate. It means that even with

minimum available damping setting, the decay rate is still too fast, which will introduce the

way to add flywheel to reduce the decay rate in the next subchapter. Table VI provides the

average power calculation based on power integration over the simulation time period.

TABLE VII. AVERAGE POWER COMPARISON

Case Average Power Peak to Average Power Ratio

Controlled MMR 0.30 W 44.74

Passive MMR 0.19 W 7.02

Controlled Normal 0.35 W 23.24

Passive Normal 0.15 W 6.90

From the average power table, it can be shown that the controlled model can increase average

power by 57% in MMR WEC, which proves the control performance of latching control.

However, the latching control causes great impact on peak-to-average power ratio, which

means power fluctuates too much.

IV. Conclusion

To improve power absorption of two-body WEC system, MMR-based PTO is introduced. The

MMR PTO is modelled in detail with refined two-body WEC system model with the

consideration of interreference radiation damping. A latching control strategy is developed and

implemented on both MMR and normal WECs. The input wave is chosen to be a JONSWAP

irregular wave spectrum. From simulation result, the latching control can greatly improve

power generation on both MMR and Normal WEC. However, on the other hand, greatly

increase system peak-to-average power ratio. The latching control is easy to be implemented

in real time, which provides some research directions of the effective MMR control

development.

3.2.5 Flywheel Effect on System Peak-to-average Ratio Reduction for Two-body WEC

As mentioned in previous part, latching control greatly increase system peak-to-average ratio

combined with fast decay rate of generator velocity. Flywheel is a device to keep the motion

of a mass by using inertia effect. Hence, a flywheel is added to the generator shaft to lower the

decay rate when disengage happens. Since the system is small, large flywheel will be hard to

drive. Three different inertia of flywheels are simulated: 0.005 kg-𝑚2, 0.01 kg-𝑚2, and 0.05

kg-𝑚2. The simulation is done with finding optimal external resistance for passive MMR WEC

as well, since the flywheel will alter the power generation of the system. Fig 3.23 shows the

optimal external resistances for different flywheel inertia.

70

(a) (b) (c)

Fig 3.23. (a) 0.005 kg-𝑚2 (b) 0.01 kg-𝑚2 (c) 0.05 kg-𝑚2

In the simulation, only the MMR WEC is added with flywheel. So, the normal WEC result will

not change. The power results are shown in Fig 3.24.

(a) (b)

Fig 3.24. (a) 0.005 kg-𝑚2 (b) 0.01 kg-𝑚2

It can be seen that when flywheel inertia increases, the generator is harder to be driven.

Therefore, power has been reduced with increasing flywheel inertia. However, the generator

decay rate has been reduced, since the plot shows it drops slower compared to no flywheel case.

From the power plot for 0.01 kg-𝑚2case, the controlled MMR generated power has already

been reduced a lot. The case with 0.05 kg-𝑚2 has an unnoticeable power generation for the

controlled MMR WEC. So, it is not shown here. The summary table is also provided for three

different flywheel cases.

TABLE VIII. 0.005 kg-𝑚2 FLYWHEEL ATTACHED

Case Average Power Peak to Average Power Ratio

Controlled MMR 0.35 W 22.92

Passive MMR 0.11 W 12.63

Controlled Normal 0.345 W 23.24

Passive Normal 0.15 W 6.90

TABLE IX. 0.01 kg-𝑚2 FLYWHEEL ATTACHED

71

Case Average Power Peak to Average Power Ratio

Controlled MMR 0.16 W 21.8

Passive MMR 0.09 W 17.53

Controlled Normal 0.35 W 23.24

Passive Normal 0.15 W 6.90

TABLE X. 0.05 kg-𝑚2 FLYWHEEL ATTACHED

Case Average Power Peak to Average Power Ratio

Controlled MMR 0.06 W 20.05

Passive MMR 0.09 W 30.10

Controlled Normal 0.35 W 23.24

Passive Normal 0.15 W 6.90

The flywheel does have effect on peak-to-average power ratio reduction. However, it requires

power reduction as sacrifice. If flywheel inertia is too large, controlled model becomes worse

compared to passive model. A tradeoff between flywheel inertia and power performance

introduces an optimization of flywheel inertia to balance average power and peak-to-average

power ratio.

Chapter Summary:

The chapter investigates the vibration control applied on ocean wave energy converter. The

first project examines the time-domain active and semi-active control comparison on a two

body WEC system without considering PTO internal dynamics. Both MPC and HMPC

methods are developed based on system physical constrains. The MPC considers system

relative motion constraints with actuator force constraint. The HMPC method adds an actuator

damping constraint. The two body WEC dynamic model is provided in detail with an irregular

wave profile combined with 3 harmonic waves. Both controlled models are compared with a

passive system with fixed actuator damping. From the simulation result, it shows the MPC

method improves the average power by 49% compared to the passive model. The HMPC

method improves the average power by 35% compared to the passive model. As for the MPC

model, active actuator force causes extremely large damping and sometimes will feed power

back from actuator to the WEC. With the damping constraint, HMPC model eliminates power

feedback from actuator to the WEC and restricts the damping within defined damping

constraint. The HMPC reduced the average actuator damping requirement by 88% with only

14% of power loss. In the second project, a MMR based PTO is introduced to improve

efficiency by regulating bidirectional motion of WEC into unidirectional rotation of the

generator. The detailed dynamics of PTO is considered. With the engage/disengage feature of

the MMR PTO, a latching control method is applied on a two body WEC system developed by

CEHMS with 0.75 meters buoy. The simulation is done with JONSWAP irregular wave profile.

From simulation, the latching control can improve the average power by 57% compared to the

72

passive MMR PTO. However, the control method introduces large peak-to-average power ratio.

Therefore, flywheel is investigated to identify its effect on balancing system power generation

capability as well as peak-to-average power ratio reduction. The simulation shows flywheel is

effective in reducing system peak-to-average power ratio. However, when increasing flywheel

inertia, power generation will be reduced. Hence, an optimal flywheel inertia can be found to

balance both control performance targets.

73

Chapter 4

Summary and Future Work

4.1 Vibration Suppression Control: Vehicle Suspension

The thesis examined the vibration reduction control on vehicle suspension. Two major control

targets, ride comfort and dynamic tire load reduction, have been investigated separately. Lots

of researchers focus on dynamic tire load control to improve vehicle handling during

normal operation (cruising, etc.). The vehicle dynamic tire load effect during hard

braking on pavement life has not been investigated. Therefore, the first project starts the

research in such a field. A heavy-duty truck with trailer is used for analysis with active

suspension control via LQR and MPC methods. The control target is to reduce tractor dynamic

tire load during heavy braking scenario. A road damage index is introduced to quantify road

damage near signalized intersection. The relationship between braking intensity and road

damage index is also investigated. From the result, it proves both LQR and MPC methods are

effective on vehicle dynamic tire load reduction. The MPC can have better performance.

However, due to weight transfer when braking, which has larger effect with larger vehicle mass,

the dynamic tire load on steering axle is hard to be reduced.

The second project introduces energy harvesting suspension with MMR gearbox. The MMR

reduces vehicle vibrating energy by energy harvesting. Previous research developed control

methods for engage mode. The control method that considers MMR engagement has not

been developed. Hence, the second project introduced a control method that includes

MMR engagement control. The control target changes to vehicle ride comfort improvement.

The project investigates two scenarios: bump and random excitation road profile. A heavy-duty

pickup truck quarter model is applied with pre-designed MMR-based suspension prototype.

The MMR design introduces engage/disengage feature to convert input bidirectional rotation

into unidirectional motion on generator to improve system efficiency. The bump scenario

introduces engagement control to reduce initial peak of vehicle body vibration when it passes

the bump. The controlled MMR suspension successfully reduces the first vibration peak by

50%. In the random excitation scenario, a rule-based control strategy is compared with simple

skyhook control and SH-PDD control methods. The three semi-active methods can have better

performance compared to a passive traditional suspension. However, the MMR suspension

with SH-PDD and rule-based methods can have better performance compared to a controlled

traditional suspension.

The research focuses on ride comfort and dynamic tire load reduction separately. However, in

suspension control, these two targets behave oppositely. There is a tradeoff between two control

targets that has been struggled by researchers for so long time. The future work can be the

control development on MMR-based suspension with the consideration between dynamic tire

load and ride comfort. The MMR design also introduces new control challenge, since it is a

passively switching system depends on velocity difference between input and output shafts.

74

Such limitation will make control harder to achieve with optimal condition. The re-design of

the MMR gearbox is necessary to let switching mechanism be controlled flexibly, which will

introduce some research directions in active MMR design.

4.2 Vibration Amplification Control: Ocean Wave Energy

Converter

The second part of the thesis addressed the vibration amplification control on the other hand as

well. The application is on vibration energy harvesting: ocean wave energy harvesting. The

work starts from the control development on a normal two-body WEC system via both active

(MPC) and semi-active (HMPC) control methods. From the literature review, HMPC

method has not been applied on two-body WEC system before. Therefore, the project

investigated the applicability of such a method with damping constraints on WEC system.

Both methods show power generation improvement compared to passive model. The MPC

method has maximum power improvement among those three models. However, the damping

is not restricted, which causes negative damping and also introduces extremely large damping

values. The HMPC method adds extra constraints on system damping and converts the control

problem to a switching logic MPC problem. The method successfully limits the damping

around a pre-set passive damping value without violating existing constraints in the MPC

problem. The power has been reduced by 14%. However, the damping has been reduced by

88%.

In the normal WEC, generator is directly connected to the input shaft, which will result in

bidirectional rotation, since WEC will vibration up and down. To improve power capture

efficiency, MMR-based PTO is introduced. A simple latching control strategy is developed

for MMR-based PTO. Latching control is widely applied for single-body WEC. However,

the research for two-body application is very limited. For the MMR PTO, generator can

have more flexibility in energy harvesting due to less restriction from the WEC side in

the disengage mode. Hence, latching control is introduced in two-body MMR PTO for the

first time. The simulation is done with irregular JONSWAP wave input. A 1/30th scale two-

body MMR WEC prototype parameters are used in the problem with detailed modeling. The

simulation first optimizes the damping of the passive MMR WEC, then use that to be compared

with the controlled MMR WEC. The normal WEC is also introduced for comparison. The result

shows that passive MMR can have better power harvesting performance compared to that of

the passive normal WEC. The latching control greatly increases the power generation on both

MMR and normal WECs. The MMR WEC shows higher peak power. However, the fast

generator velocity decay rate and latching cause high peak-to-average power ratio. By adding

flywheel, it can help to reduce peak-to-average power ratio, but with the sacrifice of power loss.

Although MMR has advantage in energy harvesting, the control challenge still exists. The

piecewise linear system with unpredicted engagement is the major obstacle in control

development. Besides, the MMR feature will never allow generator to drive the WEC brings

impossibility of active control development. Both challenges show necessity in re-design of

75

MMR system with the capability to have flexibly controlled engagement and active control

feature.

4.3 Future Work: The Concept of Active-MMR and Control

Research Directions

4.3.1 Introduction

As mentioned in section 4.1 and 4.2, the control challenge in MMR based system has been

identified. The first one is the restricted passive engagement. The second one is the lack of

capability to apply active control for better control performance.

To solve the problems mentioned above, flexibly controlled clutch should be applied to replace

mechanical one-way clutch. Since clutch can be controlled to engage or disengage, there is no

limit on engagement control. Besides, if engage clutch when generator has higher velocity

compared that of the input shaft. The power can be transferred from generator to device.

Electromagnetic clutch addresses more and more attention in industry. The most common

application is the magnetic clutch used to engage A/C to the engine in vehicle powertrain. Fig

4.1 shows a compact electromagnetic clutch produced by OGURA company. It uses magnetic

coil to generate magnetic force to push the output plate to be contacted with a friction plate.

The input shaft is connected with the friction plate. Therefore, when electric current flows in

the clutch, it will engage the output side with the input shaft. When power is cut down, it will

instantly lose magnetic force and let the output side disengage with the input shaft. This is

friction plate magnetic clutch. There are other types of magnetic clutch that uses magnetic

powder to stick inner wall and outer wall of the clutch for engagement or use tooth to engage

for large torque load requirement.

Fig 4.1. OGURA Magnetic Clutch and Cross-section Drawing

76

(a) (b)

Fig 4.2. (a) Magnetic Powder Clutch (b) Multi-plate Magnetic Clutch

This kind of clutch are compact with fast response time (30~50ms) [62], which is fast enough

compared to wave period (6~10s). If the torque requirement is high, multiple friction plates

can be added to increase total friction for higher torque requirement.

4.3.2 Active-MMR Concept Design and Modeling

I. Design

By using two magnetic clutches, a design of active-MMR gearbox can be designed to fully

achieve passive MMR gearbox (mechanical one-way clutch) feature plus active control

capability and flexible engagement control. The concept design of the gearbox is shown in Fig

4.3.

(a) (b)

Fig 4.3. Active-MMR Gearbox Design Concept (a) CCW (b) CW

From the design, when the input shaft rotates in counter clockwise (CCW) direction, as in Fig

4.3 (a), the left side clutch will be engaged. The input shaft will drive the output shaft directly.

When the input shaft rotates in the clockwise (CW) direction, as in Fig 4.3 (b), the right side

clutch will be engaged. The input shaft will drive the spur gear on the right shaft and change

Input Shaft

Spur Gear

Magnetic Clutch

Spur Gear and Chain

Output Shaft

77

rotation direction to counter clockwise direction. Then, the spur gear and chain on the right

shaft will drive the output shaft to rotate in counter clockwise direction. Hence, the active-

MMR will fully achieve passive MMR feature to convert input shaft bidirectional rotation into

unidirectional rotation of the generator. When the output shaft rotates with higher velocity

compared to input shaft. The clutches can be controlled to generate torque in both directions

separately when necessary.

II. Modeling

The operation mode of the active MMR gearbox can be summarized in Fig 4.4. In the figure,

beside the two engage modes, there is also a disengage mode, which both clutches will be

disengaged. Such mode should be kept when wave input is small and generator still has velocity.

Disengage mode will release the generator to let it decay by itself to extract energy completely.

Fig 4.4. Active-MMR Operation Modes

First, the ball-screw mechanism dynamic can be concluded as

1 22 gen

bs

g

x x

l n

−= = (4.1)

( )2( )

2( )

pto m mbs mpto bs

m m m

F d l fdd flF

d l fd d fl

−+= =

− + (4.2)

where gen is generator velocity, bs is ball-screw velocity, bs is the ball-screw torque.

Other parameters have similar meaning as the MMR model developed in section 2.2 and 3.2.

78

If ignore friction coefficient f of the ball-screw and generator viscous damping vc . The

relationship between PTO force and ball-screw torque can be simplified as

20 , 0

2

ptobspto bs v

F lf F c

l

= = = = (4.3)

a. Left Side Engage

The torque balance equations for left side engage mode from ball-screw to generator are

bs bs clutch bs spur bs chain bs bs spur chain g genJ J J J n + + + = − − − (4.4)

clutch upper bs spur bs spurJ J − + = (4.5)

clutch lower bs chain bs chainJ J − + = (4.6)

gen gen gen gen genJ c + = (4.7)

3,

2( )

t e t egen AC gen DC

i e i e

k k k kc c

R R R R− −= =

+ + (4.8)

where , , , ,clutch spur chain clutch upper clutch lowerJ J J J J− − are clutch, spur gear, chain, clutch

upper part (input side), clutch lower part (output side) inertia, , , ,bs spur chain gen are ball-crew,

spur gear, chain, and generator torques. The generator model is same in section 2.2 and 3.2.

Both AC and DC generator models are provided.

By combining equation (4.3)-(4.8), if use AC generator, the dynamic equation for the PTO can

be

2 ( 2 2 2 ) 3

( )

bs clutch spur chain g gen t e g

gen gen pto

g i e

J J J J n J k k nF

n l R R l

+ + + ++ =

+ (4.9)

Next, find relationship between buoy and submerged body relative velocity and generator

velocity.

79

1 22

gen

g

lx x

n

− = (4.10)

Then, replace generator velocity with relative velocity between buoy and submerged body. In

the whole system dynamic equation, push rode and ball nut mass, ,pr bnm m are also included,

since current WEC design uses two push rods and one ball nut.

2 2 2 2

1 2 1 22 2

4 ( 2 2 2 ) 6[2 ]( ) ( )

( )

bs clutch spur chain g gen g t e

pr bn pto

i e

J J J J n J n k km m x x x x F

l R R l

+ + + ++ + − + − =

+

(4.11)

2 2

2

4 ( 2 2 2 )bs clutch spur chain g gen

e

J J J J n Jm

l

+ + + += (4.12)

2 2 2 2

2 2

6 4,

( ) ( )

g t e g t e

pto AC pto DC

i e i e

n k k n k kc c

R R l R R l

− −= =

+ + (4.13)

From equation (4.11), active-MMR system also introduces a pair of equivalent mass and

damping, which is similar as passive MMR.

b. Right Side Engagement

The right side engagement mode will follow similar torque balance derivation. The result

dynamic equation for the whole PTO is same as left side engagement case.

c. Both Disengage

In this mode, both clutches will be disengaged. The clutch upper part (input side) will rotate

with input shaft; the clutch lower part (output side) will rotate with generator. Therefore, the

PTO switches to two separated systems: the WEC side and the generator side systems. The

torque equations are

bs bs spur bs clutch upper bs bs spurJ J J −+ + = − (4.14)

spur bs clutch upper bs spurJ J −+ = (4.15)

gen gen

clutch lower chain chain

g g

J Jn n

− + = (4.16)

80

gen gen

chain clutch lower g gen chain

g g

J J nn n

−+ = − (4.17)

gen gen gen gen genJ c + = − (4.18)

By combining equation (4.3), (4.10), (4.14), (4.15), the WEC side dynamic equation is

2

2

4 ( 2 2 )[2 ]( )

bs spur clutch upper

pr bn b t pto

J J Jm m x x F

l

−+ ++ + − = (4.19)

2

2

4 ( 2 2 )2

bs spur clutch upper

e dis pr bn

J J Jm m m

l

−

−

+ += + + (4.20)

By combining equation (4.16)-(4.18), the generator side dynamic equation is

2 2

32 2( ) 0

2( )

t e gclutch lower chaingen gen gen

g g i e

k k nJ JJ

n n R R −+ + + =

+ (4.21)

3,

2( )

t e t egen AC gen DC

i e i e

k k k kc c

R R R R− −= =

+ + (4.22)

4.3.3 Possible Research Direction in Control

The control problem for active-MMR system is an optimal control problem with switching

dynamics. Such problem has changing LTI system. The system switching is flexible without

any restriction. If consider control input u as a force that can be generated by generator. The

target is to find an optimal control method that can include logic operator. In section 3.1, HMPC

is introduced. It is capable to consider hybrid system with logic switch operator. However, one

important case for HMPC is that the system has a switching rule. For example, in the damping

control problem, system constraints switch according to the sign of the velocity. In the active-

MMR control problem, such switch rule is not available, since it switches whenever the

operation needs it to switch. From a thorough investigation of literature, only two control

concepts talk about switching system control. The first one is two-stage optimization method,

while the second one is dynamic programming.

I. Two Stage Optimization Method

This method is introduced by Dr. Xuping Xu in [63]. In the paper, the control problem of

switching system has been defined and analyzed. A simple example of the method will be

provided here.

81

Consider a switched system ( , )S F D= , find a switching sequence from 0t to ft and a

control input u (piecewise continuous function), such that the cost function

0

( ( )) ( ( ), ( ))ft

ft

J x t L x t u t dt= + (4.23)

is minimized, where 0 0 0, , ( )ft t x t x= are given.

For example, the switching system ( , )S F D= has two dynamic systems

1 3{ , }F f x u f x u= = + = − + and ( , )D I E= with {1, 2}I = and {(1,2)}E = , where

{1, 2}I = is the set of indices of the subsystems (two subsystems in F ), {(1,2)}E = is the set

of events. ( {(1,2)}E = means system will switch from subsystem 1 to subsystem 2 once in a

time period starting from 0t to ft ).

The cost function is

22

0( )J u t dt= (4.24)

Find an optimal switching instant 1t and an optimal input u such that 0 0, 2ft t= = ,

(0) 1, (2) 1x x= = to minimize the cost function.

The first stage fixes the switching sequence. In the problem, it is fixed that system will switch

from subsystem 1 to 2 once, but the optimization problem is to find switching time 1t .

Then, convert cost function with 1t . For example, define the Hamiltonian functions

1 1( , , )H x u for 1[0, )t t and 2 2( , , )H x u for 1[ , 2]t t . By using the state, costate and

stationary conditions [64] along with the Weierstrass-Erdmann corner condition 1 1 2 2( ) ( )t t =

at 1t [65], solve the minimum cost as a function of 1t as

1 1 1 1

1 1

4 4 2 2 4 2 4

1 1 4 2 2 2 2 2

2( 2 1)( 2 )( )

( 2 )

t t t t

t t

e e e e eJ t

e e e

− −

− −

− + − +=

− + . When 1 1t = , 1J has minimum value 0.

82

Therefore, it is confirmed if only switch once from subsystem 1 to 2, the switching time should

be at 1 1t = .

The second stage of optimization is to consider other types of switching sequences. For

example, changing number of switching times, switching orders. Then, formulate cost function

according to switching instants. If multiple switches are done as a sequence, the cost function

should be formulated as 1 1 1 2( , ,..., )kJ J t t t= for k times of switches. Then, minimize 1J

based on all switching instants to determine a sequence of switching instants. The method is

obviously computationally heavy. User needs to define switching plan (switching times and

order) first. For a flexible switching system like active-MMR, such method will be hard to

formulate. However, such method brings idea about how to solve problem with switching

dynamics.

II. Dynamic Programming

Dynamic programming is the most flexible nonlinear global optimization method, which is

based on principle of optimality. The flexibility is expressed as the self-defined cost function

capability. Dynamic programming is a discrete time optimization method. User can self-define

cost-to-go function at each time step according to any specific target. The dynamic

programming algorithm will calculate system cost at final time point, then formulate cost-to-

go function for the previous time step as the summation between current cost with the past cost.

The calculation will be done recursively in backward manner to the first time step to find the

minimum control command sequence. Since dynamic programming can consider all possible

tracks from the states at first time step to final time step, it must be a global optimization method.

By using a shortest path example and a simple state-space model as an example, the dynamic

programming algorithm is introduced here.

Shortest path problem:

The problem is to find the shortest path from Honolulu to Heathrow London. The distance

between each pair of places are displayed in Fig 4.5. Distances are treated as cost from one

place to another place in the problem formulation.

Fig 4.5. Shortest Path Problem

83

4

3

3

3

2 2 3 2 3

2 2 3 2 3 2 3

2 2

Stage 4:

0

Stage 3:

88

76

Stage 2:

min{ , } min{45 88,56 65} 121

min{ , , }

min{71 88,48 65,63 76} 1 3

65

1

min{

H

B

N

A

C CB B CN N

S SB B SN N SA A

L LN

f

f

f

f

f f f f f

f f f f f f f

f f

−

−

−

−

− − − − −

− − − − − − −

− −

=

=

=

=

= + + = + + =

= + + +

= + + + =

= + 3 2 3, } min{ ,57 76} 10944 65N LA Af f f− − −+ + =+=

1 1 2 1 2 1 2

Stage 1:

min{ , , }

min{105 121,75 113, } 174

Shortest path: Honolulu Los Angeles New York Heathrow London

Shortest path length: 174

65 109

Ho HoC C HoS S HoL Lf f f f f f f− − − − − − −

→

+

= + + +

= + + =

→ →

By calculating cost-to-go function backward, the shortest path has been found. In this problem,

the cost function is just the summation of total distance from start point to final destination.

State-space problem:

Consider a discretized state-space model with two states and one input, the cost function can

be formulated as

1

0

( , ) ( )N

k k N N

k

J L x u G x−

=

= +

(4.25)

Subject to

1 1

min max

min max

( , ) , 0,1,..., 1k k k k k k

k

k

x f x u x Ax Bu k N

x x x

u u u

+ += = = + = −

where ( , )k kL x u is the transitional cost, ( )N NG x is the terminal cost at final time point

The dynamic programming algorithm will calculate in sequence

*

* *

1 1

Step : ( ) ( ) Terminal cost

Step , for 0

( ) min[ ( , ) ( )] Cost-to-go functionk

N N N N

k k k k k ku

N J x G x

k k N

J x L x u J x+ +

=

= +

84

Fig 4.6 shows the calculation flow of a dynamic programming problem for two states state-

space model with one control input. The user needs to define states and control input grid size

at each time step. More grid points mean more accurate solution. However, dynamic

programming will require heavy computational power. If grid size is too small, it will need

extremely long time to solve.

Fig 4.6. Dynamic Programming Flow for State-space Model

The dynamic programming is also capable to include system switching. It is an extension of

the number of costs need to calculate at each time step, since cost for both engage and disengage

modes should be calculated together to find optimal switching sequence as well.

The control development for active-MMR is challenging, since global optimal solution must

require high computational power. Due to the lack of analytical solution of optimal force in

time domain for two body WEC system, based on author’s knowledge, the methods mentioned

above are the possible research directions in active-MMR control problem.

85

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89

Appendix A: Vehicle Suspension Nomenclature

TABLE I. NOMENCLATURE FOR SECTION 2.1

Name Symbol Unit

Tractor CG Displacement 𝑧𝑐 m

Trailer CG Displacement 𝑧𝑡 m

Tractor CG Pitch Angle 𝑐 rad

Trailer CG Pitch Angle 𝑡 rad

Tractor Front Unspurng Mass Displacement 𝑧𝑤𝑓 m

Tractor Front Road Roughness 𝑧𝑔𝑓 m

Tractor Rear Unsprung Mass Displacement 𝑧𝑤𝑟 m

Tractor Rear Road Roughness 𝑧𝑔𝑟 m

Trailer Axle Unspurng Mass Displacement 𝑧𝑤𝑡 m

Trailer Axle Road Roughness 𝑧𝑔𝑡 m

Tractor Front Suspension Spring Force 𝐹𝑠𝑓 N

Tractor Front Suspension Damping Force 𝐹𝑑𝑓 N

Tractor Rear Suspension Spring Force 𝐹𝑠𝑟 N

Tractor Rear Suspension Damping Force 𝐹𝑑𝑟 N

Trailer Suspension Spring Force 𝐹𝑠𝑡 N

Trailer Suspension Damping Force 𝐹𝑑𝑡 N

Equivalent Hitch Spring Force 𝐹𝑠5 N

Equivalent Hitch Damping Force 𝐹𝑑5 N

Hitch Longitudinal Force 𝐹𝑥 N

TABLE II. SIMULATION PARAMETERS IN SECTION 2.1

Name Symbol Value Unit

Tractor Front Suspension Stiffness 𝑘𝑓 300000 N/m

Tractor Front Suspension Damping (Passive Vehicle) 𝑐𝑓 10000 N-s/m

Tractor Rear Suspension Stiffness 𝑘𝑟 967430 N/m

Tractor Rear Suspension Damping (Passive Vehicle) 𝑐𝑟 27627 N-s/m

Trailer Suspension Stiffness 𝑘𝑡 155800 N/m

Trailer Suspension Damping (Passive Vehicle) 𝑐𝑡 44506 N-s/m

Tractor Steering Tire Stiffness 𝑘𝑡𝑓 847000 N/m

Tractor Second Axle Tire Stiffness 𝑘𝑡𝑟 2000000 N/m

Trailer Tire Stiffness 𝑘𝑡𝑡 2000000 N/m

Tractor Hitch Stiffness 𝑘5 20000000 N/m

Tractor Hitch Damping 𝑐5 200000 N-s/m

Tractor Sprung Mass 𝑚𝑐 4400 kg

Trailer Sprung Mass 𝑚𝑡 12500 kg

Tractor Front Unsprung Mass 𝑚𝑢1 270 kg

Tractor Rear Unsprung Mass 𝑚𝑢2 520 kg

Trailer Unsprung Mass 𝑚𝑢3 340 kg

Distance between Hitch and Tractor CG 𝐷1 0.1 m

Distance between Hitch and Trailer CG 𝐷2 1.2 m

90

Distance between Steering Axle and Tractor CG 𝑙1 1.2 m

Distance between Tractor Drive Axle and Tractor CG 𝑙2 4.8 m

Distance between Tractor CG and Hitch 𝑙3 4.134 m

Distance between Trailer CG and Hitch 𝑙4 6.973 m

Distance between Trailer Axle and Trailer CG 𝑙5 4 m

Tractor CG Height ℎ𝑔 1.22 m

Trailer CG Height ℎ𝑔𝑡 2 m

Moment of Inertia of Tractor Body 𝐽𝑐 18311 kg-𝑚2

Moment of Inertia of Trailer Body 𝐽𝑡 251900 kg-𝑚2

Tractor Steering Tire Brake Force 𝐹𝑏𝑓 N/A N

Tractor Driving Tire Brake Force 𝐹𝑏𝑟 N/A N

Trailer Tire Brake Force 𝐹𝑏𝑡 N/A N

TABLE III. VEHICLE SUSPESNION PARAMETERS IN SECTION 2.2

Parameter Symbol Value Unit

Sprung Mass 𝑀𝑠 575 kg

Unsprung Mass 𝑀𝑢𝑠 265 kg

Sprung Stiffness 𝐾𝑠 125 kN/m

Unsprung Stiffness 𝐾𝑢𝑠 750 kN/m

Damping of Traditional Hydraulic

Shock Absorber (Passive baseline) 𝑐𝑝 15000 Ns/m

MMR-base Shock Absorber

Equivalent Damping 𝐶𝑒 4437 Ns/m

MMR-base Shock Absorber

Equivalent Inerter 𝑚𝑒 50 kg

TABLE IV. VEHICLE AND SUSPENSION PARAMETERS IN SECTION 2.2

Name Symbol Value Unit

Sprung Mass 𝑀𝑠 575 kg

Unsprung Mass 𝑀𝑢𝑠 265 kg

Suspension Stiffness 𝐾𝑠 125 kN/m

Tire Stiffness 𝐾𝑢𝑠 750 kN/m

Traditional Shock Absorber Damping 𝑐𝑝 7.9 kN-s/m

Vehicle Speed v 18 m/s

Ball-screw Pitch Diameter 𝑑𝑚 0.008 m

Ball-screw Lead l 0.006 m/rev

Ball-screw Friction Coefficient f 0.15

Generator Inertia 𝐽𝑚 41.21 10− kg-𝑚2

Large Bevel Gear Inertia 𝐽𝑙𝑔 510− kg-𝑚2

Small Bevel Gear Inertia 𝐽𝑠𝑔 76.5 10− kg-𝑚2

Ball-screw Inertia 𝐽𝑏𝑠 62 10− kg-𝑚2

Gear Ratio between Large Bevel Gear

and Small Bevel Gear 𝑟𝑏 0.9

Generator Gearhead Ratio 𝑟𝑔 1

Generator Voltage Constant 𝑘𝑒 0.114 V/rad

91

Generator Torque Constant 𝑘𝑡 0.114 Nm/A

Generator Internal Resistance 𝑅𝑖 1.1 ohm

Generator Viscous Damping 𝑐𝑣 0.0023 N-s/m

System Gear Ratio between Input and

Output Shafts 𝑛𝑔 150

External Resistance 𝑅𝑒 0:1:50 ohm

Simulation Sample Time 𝑇𝑠 0.0056 s

Ride Comfort Weight q 0.6

Road Handling Weight p 0.4

92

Appendix B: Ocean Wave Energy Converter Nomenclature

TABLE I. PARAMETERS OF THE SYSTEM IN SECTION 3.1 [66]

Parameters Symbol Value Unit

Buoy Mass 𝑚1 2625.3 kg

Submerged Body Mass 𝑚2 2650.4 kg

Buoy Added Mass 𝐴1 8866.7 kg

Submerged Body Added Mass 𝐴2 361.99 kg

Buoy Added Mass due to Submerged Body 𝐴12 361.99 kg

Submerged Body Added Mass due to Buoy 𝐴21 361.99 kg

Buoy Radiation Damping 𝑏1 5000 Ns/m

Submerged Body Radiation Damping 𝑏2 50000 Ns/m

Buoy Hydrostatic Stiffness 𝑘1 96743 N/m

Sampling Time 𝑇𝑠 0.1 s

Prediction Step N 100

Weight on Power Generation q 106

Control Input Weight r 10−6

Prediction Horizon T 10 s

Maximum Generator Force 𝑢𝑚𝑎𝑥 50000 N

Minimum Generator Force 𝑢𝑚𝑖𝑛 -50000 N

Maximum PTO Damping 𝐶𝑚𝑎𝑥 20000 Ns/m

Minimum PTO Damping 𝐶𝑚𝑖𝑛 0 Ns/m

Maximum Stroke Length ∆𝑥𝑚𝑎𝑥 0.75 m

Minimum Stroke Length ∆𝑥𝑚𝑖𝑛 -0.75 m

Maximum Relative Velocity ∆�̇�𝑚𝑎𝑥 1 m/s

Minimum Relative Velocity ∆�̇�𝑚𝑖𝑛 -1 m/s

Passive Model PTO Damping 𝐶𝑝𝑎𝑠𝑠 10000 Ns/m

TABLE II. WAVE PARAMETERS IN SECTION 3.1

Buoy Wave 1 Wave 2 Wave 3

Excitation Force Amplitude (N) 6074.8 19624 48121

Frequency (rad/s) 2.4 3 4

Phase (rad) -0.5 0 0.5

Submerged Body Wave 1 Wave 2 Wave 3

Excitation Force Amplitude (N) 100.5428 324.1803 324.1803

Frequency (rad/s) 2.4 3 4

Phase (rad) -0.5 0 0.5

TABLE III. 1/30TH SCALE MMR WEC PARAMETERS IN SECTION 3.2

Parameters Symbol Value Unit

Buoy Mass 𝑚1 60.1 kg

Submerged Body Mass 𝑚2 334.6 kg

Buoy Added Mass 𝐴11 88.5 kg

Submerged Body Added Mass 𝐴22 64.2 kg

93

Buoy Added Mass due to Submerged Body 𝐴12 9.4 kg

Submerged Body Added Mass due to Buoy 𝐴21 9.2 kg

Buoy Radiation Damping 𝑏11 43.5 Ns/m

Submerged Body Radiation Damping 𝑏22 0.4 Ns/m

Buoy Radiation Damping due to Submerged Body 𝑏12 4.4 Ns/m

Submerged Body Radiation Damping due to Buoy 𝑏21 4.4 Ns/m

Buoy Hydrostatic Stiffness 𝑘1𝑠 4338.5 N/m

Submerged Hydrostatic Stiffness 𝑘2𝑠 0 N/m

Equivalent Stiffness 𝑘𝑝𝑡𝑜 0 N/m

Push Rode Mass 𝑚𝑝𝑢𝑠ℎ 0.26 kg

Ball Nut Mass 𝑚𝑏𝑛 1.19 kg

Ball-screw Inertia 𝐽𝑏𝑠 4.95×10−5 kg-𝑚2

Coupling Inertia 𝐽𝑐𝑝 1.74×10−5 kg-𝑚2

Input Shaft Inertia 𝐽𝑖𝑠 1.13×10−6 kg-𝑚2

Bevel Gear Inertia 𝐽𝑔𝑏 2.03×10−5 kg-𝑚2

Generator Inertia 𝐽𝑔𝑒𝑛 5.4×10−5 kg-𝑚2

Generator Gearhead Ratio 𝑛𝑔 3.5

Generator Voltage Constant 𝑘𝑒 0.6044 V/rad

Generator Torque Constant 𝑘𝑡 0.605 Nm/A

Generator Inner Resistance 𝑅𝑖 3.9 ohm

Generator External Resistance 𝑅𝑒 0:1:100 ohm

Ball-screw Lead l 0.04 m/rev

Input Wave Significant Height 𝐻𝑠 0.045 m

Input Wave Period 𝑇𝑝 2 s