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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
References
[1] A. Mathers, K. S. Moon, “A vibration-based PMN-PT energy harvester”, IEEE Sensor
Journal, Vol 9, No. 7, July, 2009.
[2] H. Fu, E. Yeatman, “Rotational energy harvesting using bi-stability and frequency up-
conversion for low-power sensing applications: theoretical modeling and experimental
validation”, Mechanical Systems and Signal Processing, May, 2018.
[3] Y. Choi, M. Lee, “Wearable biomechanical energy harvesting technologies”, Energies, Oct,
2017.
[4] C. Spelta, S. Savaresi, “Vibration reduction in a washing machine via damping control”,
The International Federation of Automatic Control, Seoul, Korea, July, 2008.
[5] Y. Liu, T.P. Waters, “A comparison of semi-active damping control strategies for vibration
isolation of harmonic disturbances”, Journal of Sound and Vibration, Vol 280 (2005), pp. 21-
39.
[6] J. Suhardjo, B.F. Spencer Jr, “Frequency domain optimal control of wind-excited buildings”,
Journal of Engineering Mechanics, Vol 118 (1992), pp. 2463-2481.
[7] J. Burl, “Linear optimal control H(2) and H(infinity) methods”, Addison-Wesley Longman
Publishing Co,. Inc., 1998.
[8] Z. T. Rakicevic, A. Bogdanovic, “Effectiveness of tune mass damper in the reduction of the
seismic response of the structure”, Bull Earthquake Eng, Vol 10 (2012), pp. 1049-1073.
[9] U. A. Korde, J. V. Ringwood, “Hydrodynamic control of wave energy devices”, Cambridge
University Press, 2016.
[10] R. K. Mehra, J. N. Amin, “Active suspension using preview information and model
predictive control”, IEEE International Conference on Control Application, Hanford, CT, Oct,
1997.
[11] A. Bemporad, M. Morari, “Control of systems integrating logic, dynamics and constraints”,
Automatica, Vol 35 (1999), pp. 407-427.
[12] D. Wagg, S. Neild, “Nonlinear vibration with control”, Springer, 2010.
[13] United States Department of Transportation, https://www.bts.gov/content/us-vehicle-
miles.
[14] J. Walls III, and M. R. Smith, “Life-cycle cost analysis in pavement design-interim
technical bulletin”, Federal Highway Administration, 1998.
[15] H. H. Refai, A. Othman, and H. Tafish, “Portable Weigh-in-motion for Pavement Design-
-phases 1 and 2”, Oklahoma Department of Transportation, Oklahoma City, OK, 2014.
[16] A. J. Weissmann, J. Weissmann, A. Papagiannakis, and J. L. Kunisetty, “Potential Impacts
of Longer and Heavier Vehicles on Texas Pavements”, Journal of Transportation Engineering,
vol. 139, no. 1, pp. 75-80, 2012.
[17] D. Cole, and D. Cebon, “Truck suspension design to minimize road damage”, Proceedings
of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, vol.
210, no. 2, pp. 95-107, 1996.
[18] B. Németh, and P. Gáspár, “Control design for road-friendly suspension systems using an
optimal weighting of LQ theorem”, Periodica Polytechnica. Transportation Engineering, vol.
38, no. 2, pp. 61, 2010.
[19] D. Cebon, “Vehicle-generated road damage: a review”, Vehicle system dynamics, vol. 18,
86
no. 1-3, pp. 107-150, 1989.
[20] L. Sun, “Optimum design of “road-friendly” vehicle suspension systems subjected to
rough pavement surfaces”, Applied Mathematical Modelling, vol. 26, no. 5, pp. 635-652, 2002.
[21] G. Tsampardoukas, C. W. Stammers, and E. Guglielmino, “Hybrid balance control of a
magnetorheological truck suspension”, Journal of Sound and Vibration, vol. 317, no. 3, pp.
514-536, 2008.
[22] H. Wang, and I. L. Al-Qadi, “Evaluation of surface-related pavement damage due to tire
braking”, Road Materials and Pavement Design, vol. 11, no. 1, pp. 101-121, 2010.
[23] P. Sweatman, “A study of dynamic wheel forces in axle group suspensions of heavy
vehicles”, Australian Road Research Board, Special Report SR27,1983.
[24] G. Tsampardoukas, C. W. Stammers, and E. Guglielmino, “Hybrid balance control of a
magnetorheological truck suspension”, Journal of Sound and Vibration, vol. 317, no. 3, pp.
514-536, 2008.
[25] M. Valášek, W. Kortüm, Z. Šika, L. Magdolen, and O. Vaculın, “Development of semi-
active road-friendly truck suspensions”, Control Engineering Practice, vol. 6, no. 6, pp. 735-
744, 1998.
[26] J. Eisenmann, “Dynamic wheel load fluctuations-road stress”, Strasse und Autobahn, vol.
4, no. 2, 1975.
[27] M. Agostinacchio, D. Ciampa, and S. Olita, “The vibrations induced by surface
irregularities in road pavements–a Matlab® approach”, European Transport Research Review,
vol. 6, no. 3, pp. 267-275, 2014.
[28] Karnopp. D., 1989. “Permanent magnet linear motors used as variable mechanical
dampers for vehicle suspensions”, Vehicle System Dynamics, 18(4), pp.187-200.
[29] Fodor, M.G. and Redfield, R., 1993. “The variable linear transmission for regenerative
damping in vehicle suspension control”, Vehicle System Dynamics, 22(1), pp.1- 20.
[30] Tang, X., Lin, T. and Zuo, L., 2013. “Design and optimization of a tubular linear
electromagnetic vibration energy harvester”, IEEE/ASME Transactions on Mechatronics, 19(2),
pp.615-622.
[31] Kawamoto, Y., Suda, Y., Inoue, H. and Kondo, T., 2008. “Electro-mechanical suspension
system considering energy consumption and vehicle manoeuvre”, Vehicle System Dynamics,
46(S1), pp.1053-1063.
[32] Cassidy, I.L., Scruggs, J.T., Behrens, S. and Gavin, H.P., 2011. “Design and experimental
characterization of an electromagnetic transducer for large-scale vibratory energy harvesting
applications”, Journal of Intelligent Material Systems and Structures, 22(17), pp.2009-2024.
[33] Li, Z., Zuo, L., Luhrs, G., Lin, L. and Qin, Y.X., 2012. “Electromagnetic energy-harvesting
shock absorbers: design, modeling, and road tests”, IEEE Transactions on vehicular technology,
62(3), pp.1065-1074.
[34] Maravandi, A. and Moallem, M., 2015. “Regenerative shock absorber using a two-leg
motion conversion mechanism”, IEEE/ASME Transactions on Mechatronics, 20(6), pp.2853-
2861.
[35] Salman, W., Qi, L., Zhu, X., Pan, H., Zhang, X., Bano, S., Zhang, Z. and Yuan, Y., 2018.
“A high-efficiency energy regenerative shock absorber using helical gears for powering low-
wattage electrical device of electric vehicles”, Energy, 159, pp.361-372.
87
[36] Graves, K.E., Iovenitti, P.G. and Toncich, D., 2000. “Electromagnetic regenerative
damping in vehicle suspension systems”, International Journal of Vehicle Design, 24(2-3),
pp.182-197.
[37] Li, Z., Zuo, L., Luhrs, G., Lin, L. and Qin, Y.X., 2012. “Electromagnetic energy-harvesting
shock absorbers: design, modeling, and road tests”, IEEE Transactions on Vehicular
Technology, 62(3), pp.1065-1074.
[38] Guo, S., Liu, Y., Xu, L., Guo, X. and Zuo, L., 2016. “Performance evaluation and
parameter sensitivity of energy-harvesting shock absorbers on different vehicles”, Vehicle
System Dynamics, 54(7), pp.918-942.
[39] Liu, Y., Xu, L. and Zuo, L., 2017. “Design, modeling, lab, and field tests of a mechanical-
motion-rectifier-based energy harvester using a ball-screw mechanism”, IEEE/ASME
Transactions on Mechatronics, 22(5), pp.1933-1943.
[40] Li, X., Chen, C., 2018. “Design and simulation of a novel mechanical power take-off for
two-body wave energy point absorber”, ASME, IDETC/CIE.
[41] Liu, Y., “Design, Modeling and Control of Vibration Systems with Electromagnetic
Energy Harvesters and their Application to Vehicle Suspension”, PhD Dissertation submitted
to Virginia Tech, Sep, 2016.
[42] Morselli, R., and Zanasi, R., “Control of Port Hamiltonian systems by dissipative devices
and its application to improve the semi-active suspension behavior”, Mechatronics, vol.18, no.
7, pp. 364-369, 2008.
[43] Poussot-Vassal, C., Savaresi, S. M., 2010. “A methodology for optimal semi-active
suspension systems performance evaluation”, Decision and Control (CDC), 49th IEEE
Conference, pp. 2892-2897.
[44] Mucka, P., 2017. “Simulated road profiles according to ISO 8608 in vibration analysis”,
Journal of Testing and Evaluation, 46(1), pp.405-418.
[45] F. Antonio, “Wave energy utilization: A review of the technologies”, Renewable and
Sustainable Energy Reviews, 14, pp. 899-918, 2010.
[46] T. Hirohisa, “Sea trial of a heaving buoy wave power absorber”. In: Berge H, editor.
Proceedings of 2nd International Symposium on Wave Energy Utilization, Trondheim, Norway,
pp. 403–17, 1982.
[47] C. G. Ryan, B. Giorgio, “A comparison of control strategies for wave energy converters”,
International Journal of Marine Energy, 20, pp. 45-63, 2017.
[48] W. David, B. Giorgio, “A comparison of WEC control strategies”, Sandia National Lab
Report, 2016.
[49] K. Budal, J. Falnes, “Interacting point absorbers with controlled motion”. In: B. Count
(Ed.), Power from Sea Waves, Academic Press London, Edinburgh, Scotland, pp. 381–399.
[50] Z. Qian, Y. W. Ronald, “An efficient convex formulation for model-predictive control on
wave-energy converter”, Journal of Offshore Mechanics and Arctic Engineering, 140, 2018.
[51] H. Eidsmoen, “Optimum control of a floating wave-energy converter with restricted
88
amplitude”, ASME J. Offshore Mech. Arct. Eng., 118(2), pp. 96–101, 1996.
[52] D. Evans, “Maximum wave-power absorption under motion constraints”, Appl. Ocean
Res., 3(4), pp. 200–203, 1981.
[53] V. Pedro, F. Antonio, J. Paulo, “Slack-chain mooring configuration analysis of a floating
wave energy converter”, 26th International Workshop on Water Waves and Floating Bodies,
2011.
[54] M. Kvasnica, P. Grieder, and M. Baotic, MPT home page: http://control.ee.ethz.ch/~mpt/.
[55] Y. Liu, “Regenerative vibration control of tall buildings using model predictive control”,
Dynamic Systems and Control Conference (DSCC), ASME Conference, 2013.
[56] A. Bemporad and M. Morari, 1999, “Control of systems integrating logic, dynamics, and
constraints”, Automatica, vol. 35, no. 3, pp. 407–427.
[57] J. Lofberg, YALMIP. [Online]. Available: http://users.isy.liu.se/johanl/yalmip/
[58] E. Rothberg, GUROBI home page: http://www.gurobi.com
[59] Liang. C., Ai. J., “ Design, fabrication, simulation and testing of an ocean wave energy
converter with mechanical motion rectifier”, Ocean Engineering, 136 (2017), pp. 190-200.
[60] Falnes J, Budal K. “Wave-power absorption by point absorbers”., Norwegian Maritime
Research, (1978), Vol 6, pp. 2-11.
[61] https://ocw.mit.edu/courses/mechanical engineering/2-22-design-principles-for-ocean-
vehicles-13-42-spring 2005/readings/lec6_wavespectra.pdf
[62] Ogura Industrial Corp, AMC Electromagnetic Clutch, https://ogura-
clutch.com/products.php?category=2&product=80.
[63] X. Xu, P. J. Antsaklis, “Optimal control of switched system new results and open problem”,
American Control Conference, Chicago, Illinois, Jun, 2000.
[64] F. Lewis, Optimal Control, Wiley Interscience, 1986. Chapter 3.
[65] M. Zefran, Continuous methods for motion planning, Ph.D. Thesis, University of
Pennsylvania, 1996.
[66] R. Markus, “Different model predictive control approaches for controlling point absorber
wave energy converters”, University Stuttgart, Institute for System Dynamics, 2011.
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