Dissertation Zhen Li
Dissertation Zhen Li
Dissertation Zhen Li
SF
=
1 e
(usin
v cos
) +r, (1.1)
e = usin
+v cos
, (1.2)
where e, dened as the distance between the origins of SF and B, and
:=
SF
, are referred to as the cross-track error and heading error, respectively; u,
6
Figure 1.2: Illustration of the coordinates in the earth frame (inertial frame) E, the
ship body-xed frame B and the Serret-Frenet frame SF.
v, r are the surge, sway and yaw velocity, respectively. is the heading angle of
the vessel,
SF
is the path tangential direction as shown in Figure 1.2 [65] and is
the curvature of the given path. In Figure 1.2, T and N are the tangent and normal
directions of the curve C at the origin of SF. The control objective of the path
following problem is to drive e and
to zero. When environmental disturbances
(such as wind, wave and current) exist, the path following errors e and
often can
not be eliminated simultaneously. In such circumstances, the primary objective is
to maintain a small or near-zero cross-track error e, while keeping certain necessary
heading error
to counteract disturbances.
For most path following problems for surface vessels in open sea, the path is often
a straight line or a way-point path [15], which consists of piecewise straight lines with
the curvature being zero. Even if the desired path has non-zero curvature, it is
often possible to approximate the curve by many piecewise straight lines. Therefore
7
the heading error dynamics (1.1) can often be simplied as:
= r. (1.3)
A challenge in the path following of marine surface vessels is the inherent non-
linearity, from either the ship dynamics or path following kinematics. Many dier-
ent nonlinear design methodologies have been attempted. For example, Lyapunovs
direct method is used in [8, 31] while the cascade control is employed in [37, 56].
Most papers published on this topic adopt the back-stepping as the design method-
ology [1416, 19, 25, 35, 57, 65, 66]. Instead of using linear approximations, they often
explore the inherit nonlinearities to achieve better performance. However, since the
controller attempts to cancel or compensate for high-order nonlinearities, it yields a
very complicated control law. Meanwhile, most of the control methodologies are ex-
plored with analytical and/or numerical investigation and no experimental eorts are
reported, with the exception of [25, 37] and [17], where the Cybership (I and II) with
infrared camera system and a model ship with Dierential Global Positioning Sys-
tem (DGPS) are used for experimental validation, respectively. Some path following
algorithms for marine surface vessels, such as the LQR approach [29] and PID-type
controllers [32], have already achieved industrial applications. All of these industrial
path following systems pay great attention to the robustness and easy implementa-
tion.
Motivated by these recent developments in path following of marine surface vessels,
this dissertation presents a novel back-stepping design for an integrated model of
the surface vessel dynamics and 2-DoF path following kinematics. The focus is on
developing the controller that lends itself for easy tuning and implementation, which
is one of the key considerations of industrial path following systems. The details
of this work, namely path following without roll constraints, will be presented in
8
Chapter 3 and also have been reported in [40, 41].
1.1.4 Path Following with Roll Constraints for Marine Sur-
face Vessels
Roll motion, no matter whether it is induced by maneuvering or environmental
disturbances, is normally considered to be detrimental to the operation and safety of
marine surface vessels [24]. In particular, roll motion aects ship performance in the
following ways [47, 53]:
Acceleration induced by roll can contribute to the development of seasickness in
the crew and passengers, resulting in reduced crew performance and passenger
comfort.
Roll acceleration may cause cargo damage, especially for high speed container
ships.
Large roll angles reduce the capability of equipment on board. For example,
the performance of weapon launching systems, net shing equipment and high
precision electrical devices such as sonar will be strongly inuenced.
Given that the roll motion produces high accelerations and is considered as the
principal villain for seasickness and cargo/device damage, roll reduction in a seaway
becomes an important consideration in hull design and vessel motion control. A
noticeable amount of work has been reported in roll reduction and summarized in
[64]. Although roll reduction in the course-keeping has been achieved by proper
implementation of bilge keels [30], anti-rolling tanks [10, 26], active n stabilizers
[64,68] and rudder roll stabilizations (RSS) [4,6,33,55,68,71], roll reduction in course-
changing, such as in path following and heading control, has not been thoroughly
studied. Recently, a progress has been made in [22] to achieve roll reduction in the
9
track keeping using the rudder control. Given the stringent safety and performance
requirements for both military and commercial vessels, achieving path following while
enforcing roll constraints in a seaway deserves attention and will be one of the research
foci of this dissertation.
Typical nonlinear control methodologies in path following such as those pursued
in [8, 17, 19, 25, 31, 37, 40, 56, 66] do not take roll constraints explicitly into account in
the design process. The constraint enforcement might be achieved through numerical
simulations and trial-and-error tuning of the controller parameters. For the path
following control with roll constraints considered in this dissertation, both the cross-
tracking error and heading error are controlled by the rudder angle as an under-
actuated problem and roll constraints need to be enforced simultaneously. The Model
Predictive Control (MPC) [48, 58], which has the capability of handling input and
state constraints explicitly, has been proposed to achieve satisfactory performance.
The details of path following with roll constraints will be presented in Chapter 4 and
Chapter 5 and also have been reported in [42, 43].
1.2 Contributions
The contributions of this dissertation on modeling and control of path follow-
ing with roll constraints for marine surface vessels in wave elds are summarized as
follows:
A numerical test-bed combining the ship dynamics and wave eects (both rst-
order excitation and second-order drift loads) on vessels has been established
to test the performance of the ship motion control systems in a wave eld.
This numerical test-bed is established in MATLAB, which is the most popular
development environment for control community. Most importantly, this nu-
merical test-bed is generic and can be widely used in many other ship motion
10
control applications, such as course keeping, roll stabilization and dynamical
positioning.
A novel robust feedback dominance back-stepping (FDBS) controller for path
following of marine surface vessels has been developed. The novelty of the ap-
proach presented in this dissertation lies in the following aspects: (a) The back-
stepping nonlinear controller design is based on feedback dominance, instead of
feedback linearization and nonlinearity cancelation; (b) Additional design pa-
rameters are employed in the Lyapunov function that lead to simplication of
the controller in the design procedure and normalization of dierent variables
in the Lyapunov function to improve the controller performance; (c) Relying on
feedback dominance and the introduction of the additional parameters in the
Lyapunov function, the resulting controller is almost linear, with very benign
nonlinearities allowing for analysis and evaluation; (d) The performance of the
nonlinear controller, in terms of path following, is analyzed for robustness in the
presence of model uncertainties. Simulation results are presented to verify and
illustrate the analytic development and the eectiveness of the resulting con-
trol against rudder saturation and rate limits, delays in the control execution,
as well as measurement noise. Furthermore, the control design is validated by
experimental results conducted in a model basin using a model boat.
The novel robust path following controller was evaluated by the proposed nu-
merical test-bed for its performance in wave elds. Several issues, such as
steady state errors and rudder oscillations, have been identied, thereby moti-
vating controller modication and gain re-tuning. Mitigating strategies, such
as gain re-tuning and gain scheduling, for improving the controller performance
are proposed and numerically evaluated. The simulation results show that the
performance of the modied controller can be substantially improved in wave
11
elds.
A standard Model Predictive Control (MPC) approach for path following with
roll constraints of marine surface vessels in calm water using the rudder as
the control input has been proposed. The focus is on satisfying all the input
(rudder) and state (roll) constraints while achieving satisfactory path following
performance. The path following performance of the proposed MPC controller
and its sensitivity to the major controller parameters, such as the sampling time,
predictive horizon and weighting matrices in the cost-function, are analyzed by
numerical simulations. This study is the rst reported MPC application in path
following for marine surface vessels, to the best knowledge of the author.
To investigate the benets as well as the associated cost, in terms of both
path following and computational complexity, of using multiple actuator for
path following control, the propeller is used as the second control actuator,
in addition to the rudder angle, for solving the path following problem with
roll constraints. MPC, where the design is based on multiple linear models,
is used to handle the multi-variable control problem and roll constraints. The
simulation results verify the eectiveness of the resulting two-actuator controller
and show the advantage of the proposed controller over the one-input controller
with a reduced roll responses.
The eectiveness of the MPC path following controller in wave elds is also
studied by simulation on the numerical test-bed. The feasibility issue due to roll
constraint violations is identied and the mitigating strategies, such as gain re-
tuning and constraint tightening and softening, are then proposed to guarantee
the satisfaction of roll constraints. The satisfactory performance of the modied
MPC controller is shown by the simulations on the numerical test-bed.
Motivated by the constraint violation and feasibility issues of a MPC controller
12
for systems in the presence of disturbances, a novel disturbance compensating
MPC (DC-MPC) algorithm has been proposed to guarantee the state constraint
satisfaction in the presence of environmental disturbances. The eectiveness of
the proposed algorithm has rst been analyzed theoretically. The performance
of DC-MPC algorithm in terms of constraint enforcement and error conver-
gence is validated by numerical simulations, demonstrated on a ship heading
control application. The DC-MPC algorithm has the potential to be applied to
other motion control problems with environmental disturbances, such as ight,
automobile and robotics control.
1.3 Dissertation Overview
The dissertation is organized as follows:
Chapter 2 rst introduces the vessel dynamical models adopted in the path fol-
lowing controller design and evaluation, respectively. The numerical test-bed
facilitating ship motion controller evaluation in wave elds is then presented,
followed by the description of the experimental test-bed for the controller test.
Chapter 3 derives the path following control law based on back-stepping method
using feedback dominance. Then the unmodeled dynamics are considered for
the robustness analysis of the resulting control system. The simulation results
are also presented and the experimental validation is summarized. Furthermore,
the feedback dominance back-stepping (FDBS) controller is re-tuned to achieve
satisfactory system performance in wave elds.
Chapter 4 applies the standard MPC algorithm to address the path following prob-
lem for marine surface vessels with input (rudder and propeller) and state (roll)
constraints. The one-input (rudder) and two-input (rudder and propeller) MPC
13
controllers are both developed to achieve constrained path following and com-
pared by simulations. The controller parameter tuning is also studied by simu-
lations in this Chapter. Furthermore, the standard MPC controller is evaluated
in wave elds and the state constraint violation is identied.
Chapter 5 aims at the state constraint satisfaction of the path following for marine
surface vessels in wave elds. By the methods of gain re-tuning and constraint
softening and tightening, the path following with roll constraints is achieved in
wave elds. For both cases, the roll constraints is enforced successfully at the
expense of slight slower path following convergence speed.
Chapter 6 presents a novel disturbance compensating MPC (DC-MPC) scheme.
The capability of the DC-MPC algorithm is rst analyzed theoretically. Then
the proposed DC-MPC algorithm is applied to the ship heading control of ma-
rine surface vessels. The simulation results compared with standard MPC and
time varying MPC schemes show the constraint satisfaction capability and good
performance of the DC-MPC controller. The features and limitations of DC-
MPC are also analyzed, which identify several suitable applications while at the
same time rule out the path following control with roll constraints for marine
surface vessels as a viable application.
Chapter 7 presents conclusions and future plans.
14
CHAPTER 2
Modeling and Controller Evaluation Test-bed
Development
This dissertation focuses on the path following controller design and analysis for
marine surface vessels in wave elds. For successful model based control design and
analysis, the proper modeling of the dynamical system is critical and necessary. For
our purposes, namely path following control with roll constraints, the appropriate
modeling of ship dynamics and environmental loads are a pre-requisite.
In this chapter, ship dynamical models for control design and evaluation are rst
introduced, together with a wave model to calculate both the rst- and second-order
wave loads on ships. Based on that, the numerical test-bed, combining the ship
dynamics and the ship-wave interactions, is developed to facilitate the ship motion
controller evaluation in wave elds. Finally, the experimental test-bed for the con-
troller performance validation and verication is presented.
2.1 Modeling of Marine Surface Vessels
The maneuvering and ship motion control community was largely inuenced by
Fossens signicant work on modeling of ship dynamics. He established the framework
15
of nonlinear dynamic equations of motion in six DoF in terms of Newtonian and
Lagrangian formalism. Models developed in his books [23, 24] are widely adopted by
subsequent researchers in this community. In this dissertation, his framework of ship
dynamics modeling is also employed.
As mentioned in Chapter 1, two types of ship dynamical models are used for ship
motion control [52]: high-delity models and control-design models. This section
introduces the corresponding high-delity and control-design models adopted in this
dissertation.
2.1.1 High-delity Model: A4-DoF Nonlinear Container Ship
For maneuvering of surface vessels, normally 3-DoF are discussed, namely for
surge, sway and yaw. When linearized, the surge is decoupled and 2-DoF are left. In
this dissertation, in order to address the path following problem with roll constraints,
a 4-DoF model is needed, including 3-DoF discussed in traditional maneuvering and
the additional DoF focusing in seakeeping characteristics, namely the roll.
Fossen [23] summarized a nonlinear mathematical model for a single-screw high-
speed container ship (so-called S175) in surge, sway, roll and yaw based on the results
of [67]. The geometric parameters for the container ship modeled are given in Ta-
ble 2.1.
The nonlinear equations of motion (surge u, sway v, roll p and yaw r) are given
by:
(m
+m
x
) u
(m
+m
y
)v
= X
, (2.1)
(m
+m
y
) v
+ (m
+m
x
)u
+m
y
r
y
l
y
p
= Y
, (2.2)
(I
x
+J
x
) p
y
l
y
v
x
l
x
u
+W
GM
= K
, (2.3)
(I
z
+J
z
) r
+m
y
v
= N
G
. (2.4)
16
Table 2.1: Main parameters of the container ship
Length (L) 175.00(m)
Breadth (B) 25.40 (m)
Draft fore (d
F
) 8.00 (m)
aft (d
A
) 9.00 (m)
mean (d) 8.50 (m)
Displacement volume 21222 (m
3
)
Height from keel to transverse metacenter (KM) 10.39 (m)
Height from keel to center of buoyancy 4.62 (m)
Block coecient (C
b
) 0.559
Rudder area (A
R
) 33.04 (m
2
)
Aspect ratio () 1.822
Propeller diameter (D) 6.53 (m)
Here, m
x
and m
y
denote the added mass in the x and y
directions respectively. I
x
and I
z
denote the moment of inertia and J
x
and J
z
denote
the added moment of inertia about the x and z axes, respectively. Furthermore,
y
denotes the x-coordinate of the center of m
y
; l
x
and l
y
the z-coordinates of the centers
of m
x
and m
y
respectively. W
G
is the location of the center of gravity in the x-axis. All the primes mean the
corresponding dimensionless terms, please see Appendix E.1.3 in [23] for the details.
The hydrodynamic forces X
, Y
and moments K
, N
= X
uu
u
2
+ (1 tt)T
(J) +X
vr
v
+X
vv
v
2
+X
rr
r
2
+X
2
+c
RX
F
N
sin
, (2.5)
Y
= Y
v
v
+Y
r
r
+Y
p
p
+Y
+Y
vvv
v
3
+Y
rrr
r
3
+Y
vvr
v
2
r
+Y
vrr
v
2
+Y
vv
v
+Y
v
v
2
+Y
rr
r
+Y
r
r
2
+ (1 + a
H
)F
N
cos
, (2.6)
17
K
= K
v
v
+K
r
r
+K
p
p
+K
+K
vvv
v
3
+K
rrr
r
3
+K
vvr
v
2
r
+K
vrr
v
2
+K
vv
v
+K
v
v
2
+K
rr
r
+K
r
r
2
(1 +a
H
)z
R
F
N
cos
, (2.7)
N
= N
v
v
+N
r
r
+N
p
p
+N
+N
vvv
v
3
+N
rrr
r
3
+N
vvr
v
2
r
+N
vrr
v
2
+N
vv
v
+N
v
v
2
+N
rr
r
+N
r
r
2
+ (x
R
+a
H
x
H
)F
N
cos
, (2.8)
where, tt is the thrust deduction factor. c
RX
, a
H
and a
H
x
H
are interactive forces and
moment coecients between hull and rudder. x
R
and z
R
are the location of rudder
center of eort in x and z direction, respectively. All the coecients in X
, Y
, K
and N
are the corresponding hydrodynamic derivatives, and their values for S175
are given in the Appendix E.1.3 of [23].
The rudder force F
N
, according to [23] can be resolved as:
F
N
=
6.13
+ 2.25
A
R
L
2
(u
R
2
+v
R
2
)sin
R
, (2.9)
R
=
+tan
1
(v
R
2
/u
R
2
), (2.10)
u
R
= u
_
1 + 8kK
T
/(J
2
), (2.11)
v
R
= v
+c
Rr
r
+c
Rrrr
r
3
+c
Rrrv
r
2
v, (2.12)
where, is the rudder aspect ratio and A
R
is rudder area. L is the ship length. u
R
and v
R
are inow velocities of the rudder in x and y directions, respectively.
R
is the
relative angle between rudder and its inow.
P
is the advance
speed. and k are adjustment constants. K
T
is the propeller thrust coecient and J
is the open water advanced coecient. , c
Rr
, c
Rrrr
and c
Rrrv
are the corresponding
18
hydrodynamic derivatives. Furthermore:
K
T
= 0.527 0.455J, (2.13)
J = u
P
U/(nD), (2.14)
u
P
= cosv
[(1 w
p
) +((v
+x
p
r
)
2
+c
pv
v
+c
pr
r
)], (2.15)
where, U =
u
2
+v
2
. n is the propeller shaft speed and D is the propeller diameter.
w
p
is the wake fraction number and is an adjustment constant. c
pv
and c
pr
are the
corresponding hydrodynamic derivatives. The propeller force T
= 2D
4
K
T
n
[n
[, (2.16)
where is the water density. Also, the dynamics of the rudder and propeller are
incorporated by:
= (
c
)/T
, (2.17)
n = (n
c
n)/T
n
. (2.18)
T
and T
n
are time constants. And the saturation of the rudder and propeller speed
is given by [[
max
and 0 n n
max
.
The motion equations can be transformed into the control-oriented dynamics equa-
19
tions as follows:
_
_
u
v
r
x
y
n
_
_
=
_
_
X
11
U
2
/L
(m
33
m
44
Y
+m
32
m
44
K
+m
42
m
33
N
)
detM
U
2
/L
m
42
m
33
Y
+m
32
m
42
K
+m
22
m
33
N
22
2
N
detM
U
2
/L
2
(u
cos
sin
cos
)U
(u
sin
cos
cos
)U
(r
cos
)U/L
m
32
m
44
Y
+m
22
m
44
K
42
2
K
+m
32
m
42
N
detM
U
2
/L
2
p
U/L
n
_
_
(2.19)
where, m
11
= m
+m
x
, m
22
= m
+m
y
, m
32
= m
y
l
y
, m
42
= m
y
, m
33
= I
x
+J
x
,
m
44
= I
z
+J
z
and detM
= m
22
m
33
m
44
m
32
2
m
44
m
42
2
m
33
.
When environmental forces are neglected, the model (2.19) can be simulated for
dierent rudder angles and the same forward speed. The results are given in Fig-
ure 2.1.
It is worth noting that while larger rudder angles normally lead to smaller turning
diameter (as shown in Figure 2.1(a)), larger surge velocity reduction (Figure 2.1(b)),
larger sway, yaw and roll motion (Figure 2.1(c)(d)(e)), some of these trends are re-
versed for very large rudder angles (say 35 deg), which are shown in Figure 2.1(c)(e).
This is due to the decrease in centrifugal force with a large rudder angle input, which
causes great forward velocity deduction.
This 4-DoF nonlinear container ship model is one of the most comprehensive ship
models available in open literature. It captures the fundamental characteristics of
the ship dynamics and covers a wide range of operating conditions. However, due
to the complexity of this original nonlinear model and the non-ane input terms,
20
-500 0 500 1000
-200
0
200
400
600
800
1000
1200
1400
1600
Trajectory of Ship with Inertial Conditions: u =7 m/s, v=w =p =r =0
x (m)
y
(
m
)
=10 (deg)
=20 (deg)
=30 (deg)
0 100 200 300 400 500 600
3.5
4
4.5
5
5.5
6
6.5
7
S
u
r
g
e
V
e
l
o
c
i
t
y
(
m
/
s
)
time (sec)
0 100 200 300 400 500 600
-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
time (sec)
S
w
a
y
V
e
l
o
c
i
t
y
(
m
/
s
)
Figure 2.1: Open-loop simulation results of the original nonlinear container model.
21
the model is not amenable to model-based control design methodology and tools. In
our work, this 10th-order nonlinear model is used for simulation and performance
evaluation. For control design, however, a reduced-order model is used to make the
problem tractable.
2.1.2 Control-design Model Development
For typical path following, which seeks to control the ship in the horizontal plane,
only three degrees, namely the surge u, sway v and yaw r, are critical and the heave
w, roll p and pitch q are normally neglected. The most widely used ship model in the
path following research was developed in [23], which is in the following form:
u =
m
22
m
11
vr
d
11
m
11
u +
1
m
11
u
1
, (2.20)
v =
m
11
m
22
ur
d
22
m
22
v, (2.21)
r =
m
11
m
22
m
33
uv
d
33
m
33
r +
1
m
33
u
2
, (2.22)
where the parameters m
ii
> 0 are given by the ship inertia and added mass eects.
The parameters d
ii
> 0 are given by the hydrodynamic damping. The available
controls are the surge force u
1
and the yaw moment u
2
.
Many dierent nonlinear design methodologies have been applied to the above
model (2.20)-(2.22) [7, 8, 14, 16, 25, 31, 37, 56]. However, due to the nonlinearities
involved, the resulting control laws often have complex expressions, making the con-
troller dicult for gain tuning and its performance sensitive to model parameters. In
our work, the following reduced-order linear model is developed based on (2.20)-(2.22)
to facilitate the model-based approach:
v = a
11
v +a
12
r, (2.23)
22
r = a
21
v +a
22
r +b
2
, (2.24)
where a
11
, a
12
, a
21
, a
22
and b
2
are constant parameters. The validity of the model
(2.23-2.24) is demonstrated through performance evaluation of the resulting control
system on the high-delity simulation.
Remark 2.1 The 2-DoF linear model (2.23)-(2.24) is developed from the non-
linear 3-DoF model (2.20)-(2.22) based on the assumption that surge velocity u
is constant and the yaw moment u
2
is proportional to the rudder angle . An
independent control system can be used to maintain the ship surge speed. Even
without such a control system, the surge velocity deduction is not signicant if
the engine is operating at its rated power. The constant surge velocity assump-
tion is adopted by many researchers [56, 65]. Notice that the rudder angle is the
control input in the linear model (2.23)-(2.24), while the yaw moment is used
as the input in (2.20)-(2.22). However, the former is the real actuator variable,
but the latter is not.
For the path following problem without roll constraints, the control design, sum-
marized in Chapter 3, is based on the linear system (2.23)-(2.24). However, if we
consider path following with roll constraints, the additional DoF, namely roll, should
be also included. Based on the high-delity nonlinear container ship model, which is
described in Section 2.1.1, a 3-DoF (sway, yaw and roll) linear system is developed
and adopted in the control design in Chapter 4. The model has the following form:
v = a
11
v +a
12
r +a
13
p +a
14
+b
1
, (2.25)
r = a
21
v +a
22
r +a
23
p +a
24
+b
2
, (2.26)
= r, (2.27)
p = a
31
v +a
32
r +a
33
p +a
34
+b
3
, (2.28)
23
= p, (2.29)
where a
11
, a
12
, a
21
, a
22
, a
31
, a
32
, a
33
, a
34
, b
1
, b
2
and b
3
are constant parameters. Note
agian that the surge speed is also assumed to be constant and the surge dynamics are
neglected.
2.2 Modeling of Wave Disturbances
Marine surface vessels maneuvered in a seaway will be bearing loads from environ-
mental disturbances, such as waves, wind and current. In this dissertation, we only
consider the wave disturbances, which are the dominant environmental disturbances
in the course-keeping or path following problem [53].
The wave-induced loads can be represented by the sum of the rst-order and
second-order eects, where the rst-order terms correspond to the wave excitation
load while the second-order terms represent the wave drift load [21]. The summation
of these two terms serves as the total wave loads acting on the vessels.
The coordinate system used in the wave force calculation is shown in Figure 2.2.
, the ship heading angle with respect to the wave heading angle, is dened as:
=
wave
ship
. (2.30)
where
wave
and
ship
are the wave heading angle and ship heading angle in the inertial
frame, respectively.
2.2.1 First-Order Wave Excitation Force and Moment
The calculation of the rst-order wave excitation forces is very involved. An
irregular sea [38] surface can be expressed as a sum of single frequency waves with
dierent frequencies (
i
), wavenumbers (k
i
), and uniformly distributed random phase
24
X
0
wuc
0
shp
Figure 2.2: Wave angle denition.
angles (
i
):
(x
, t) =
N
i=1
A
i
cos(
i
t k
i
x
+
i
), (2.31)
where A
i
is the corresponding wave amplitude and x
, t) =
N
i=1
A
i
[H
j
(
i
, , u)[ cos(
i
t k
i
x
+
i
+
j
(
i
, , u)), (2.32)
for directional index j = 1, 2, 3, 4, 5, 6, which stand for surge, sway, heave, roll, pitch
and yaw, respectively. H
j
(
i
, , u) is the response amplitude operator (RAO) in the j
direction, with a magnitude [H
j
(
i
, , u)[ and a phase angle
j
(
i
, , u). By denition,
the RAO (such as H
j
(
i
, , u)) is the response of the ship system, such as the ship
motion variable, to wave forces, per wave height due to a wave of frequency , a wave
heading and ship speed u [38]. For our purpose, only the index 1, 2, 4 and 6 are
used for surge, sway forces and roll, yaw moments, respectively. For further details,
please see [5].
In the calculation of rst-order wave loads, a quasi-steady approach is adopted
25
where the transient eects are neglected in order to greatly simplify the computation,
because the calculation of transients involves computational convolution integrals.
The similar approach is employed in [54].
2.2.2 Second-Order Wave Drift Force and Moment
For the wave-induced drift force in the surge direction (
f
1
), we approximate the
second-order drift forces by an empirical equation [18] which is a sixth-order polyno-
mial function of forward speed u (m/s) and relative ship heading angle (rad):
f
1
= C
0
+C
10
u +C
01
+C
20
u
2
+C
11
u +C
02
2
+C
21
u
2
+C
12
u
2
+C
03
3
+C
22
u
2
2
+C
13
u
3
+C
04
4
+C
23
u
2
3
+C
14
u
4
+C
24
u
2
4
, (2.33)
where C
0
, C
10
, C
01
, C
20
, C
11
, C
02
, C
21
, C
12
, C
03
, C
22
, C
13
, C
04
, C
23
, C
14
and C
24
are
the empirical coecients tted using data generated from a detailed ship numerical
simulation program (see [18] for the details). For the container ship S175 in sea state
5 (corresponding to 3.25 m of signicant wave height), the coecients in (2.33) are
summarized in Table 2.2.
Table 2.2: Empirical coecients for calculation of second order drift wave force in
surge.
C
0
C
10
C
01
C
20
C
11
84.988 32.040 -487.122 -2.436 40.734
C
02
C
21
C
12
C
03
C
22
1076.446 -0.348 -135.610 -577.089 5.582
C
13
C
04
C
23
C
14
C
24
77.390 91.061 -3.558 -12.646 0.606
For the drift sway force and the drift yaw moment, the following empirical equa-
tions were developed in [12]:
f
2
=
1
2
gL
2
sin C
Y
, (2.34)
26
f
6
=
1
2
gL
2
2
sin C
N
, (2.35)
where is the water density, g is the acceleration due to gravity, L is the ship length,
is the mean wave amplitude, C
Y
and C
N
are the corresponding empirical coecients,
whose expressions are given as follows:
C
Y
= 0.46 + 6.83
L
11.65(
L
)
2
+ 8.44(
L
)
3
, (2.36)
C
N
= 0.11 + 0.68
L
0.79(
L
)
2
+ 0.21(
L
)
3
, (2.37)
where is the mean wave length. However, (2.36) and (2.37) are regressed from the
data of the specic ship with zero speed in [20]. For dierent ships with nonzero
speed, these two coecients need to be corrected. We introduce speed dependent
correction coecients C
f2
(u) and C
f6
(u) to adjust the second order drift loads by the
following equations:
f
2
= C
f2
(u)
f
2
, (2.38)
f
6
= C
f6
(u)
f
6
. (2.39)
It was pointed out in [21] that the ratio between the magnitudes of the mean wave
drift forces and linear rst-order wave forces is about /100. For dierent speeds, the
coecients C
f2
(u) and C
f6
(u) adopted in this dissertation are calculated using the
magnitude of the rst-order excitation loads, which is calculated based on the detailed
information of the ship hull form using the linear seakeeping theory. Specically, given
the vessel speed and mean wave amplitude , we rst calculate the mean amplitudes
of the rst-order wave loads in sway and yaw, namely F
f2
and F
f6
, for equal to 45,
90 and 135 deg. Then the correction coecients C
f2
(u) and C
f6
(u) are obtained by
averaging the corresponding three values of F
f2
/(100[
2
[) and F
f6
/(100[
6
[). For
the container ship S175 with a speed of 10 m/s in sea state 5, the values of C
f2
(u)
27
and C
f6
(u) are 0.2535 and 0.5211, respectively.
2.3 Numerical Test-bed for Controller Evaluation
in wave elds
A numerical test-bed for controller evaluation in wave elds, based on the high-
delity ship model S175 presented in Section 2.1.1 and the wave load calculation
described in Section 2.2, is developed in this section.
As mentioned before, the forces/moments X
, Y
, K
and N
w
= X +f
1
+
f
1
, (2.40)
Y
w
= Y +f
2
+
f
2
, (2.41)
K
w
= K +f
4
, (2.42)
N
w
= N +f
6
+
f
6
, (2.43)
where X
w
, Y
w
, K
w
and N
w
are the corresponding hydrodynamic forces and mo-
ments in the incident wave eld and will replace X
, Y
, K
and N
in the equations
(2.1)-(2.4) when waves exist. As mentioned in Section 2.2, f
1
, f
2
, f
4
and f
6
are
the corresponding dimensionless rst-order wave loads and
f
1
,
f
2
,
f
4
and
f
6
are the
corresponding dimensionless second-order wave loads. Note that the second-order
moment in roll is neglected. Using (2.40)-(2.43) in (2.1)-(2.4), the wave induced loads
are incorporated into the ship dynamics, with the assumption that the damping and
added mass of the ship are unchanged in wave elds. The similar approach is em-
ployed to incorporate the wave loads into calm water maneuvering model in the work
28
of [23, 24, 53].
Figure 2.3 shows the block diagram of the overall model. The wave load program
calculates the wave induced forces and moments based on the wave eld information
(sea state, dominant wave direction) and the ship states (position, heading and speed).
The ship maneuvering model is driven by the wave forces and moments, together with
the control input (rudder angle calculated based on a control law using the current
ship state measurement or estimation).
RACE Lab
1
Block Diagram of the Test-Bed
Ship State
Ship Maneuvering
Model
Wave Load
Calculation Program
Controller
Rudder angle
Wave Force
Wave Info
Wave field
Figure 2.3: Block diagram of the simulation model.
As an example, Figure 2.4 shows the wave-induced rst-order excitation forces
and moment for four dierent heading angles, namely following sea ( = 0 deg), stern
quartering sea ( = 45 deg), beam sea ( = 90 deg) and head sea ( = 180 deg).
In simulations, the JONSWAP spectrum [21] was adopted with 3.25 m signicant
wave height (sea state 5), 9.53 sec peak period and default Gamma peak factor 3.3.
The ship used in the simulation is the container ship S175, which is widely used in
research and is described in Section 2.1.1. The ship velocity is maintained at 10 m/sec.
From Figure 2.4, we can see that the wave load calculation program captures the key
characteristics of the wave excitation loads on vessels in a short crested wave eld.
For example, the head sea has the highest encounter frequency while the following
has the lowest frequency; the beam sea has the largest sway force and roll moment
29
among these four cases while these loads are relatively small in the following sea and
head sea cases; the head sea has the largest surge force and the following sea and
head sea have very small sway force.
0 10 20 30 40 50 60 70 80 90 100
-3
-2
-1
0
1
2
3
x 10
6
s
u
r
g
e
f
o
r
c
e
f
1
(
N
)
0 10 20 30 40 50 60 70 80 90 100
-2
-1
0
1
2
x 10
7
s
w
a
y
f
o
r
c
e
f
2
(
N
)
0 10 20 30 40 50 60 70 80 90 100
-1
0
1
x 10
7
r
o
l
l
m
o
m
e
n
t
f
4
(
N
*
m
)
0 10 20 30 40 50 60 70 80 90 100
-5
0
5
x 10
8
y
a
w
m
o
m
e
n
t
f
6
(
N
*
m
)
time (sec)
=0 deg
=45 deg
=90 deg
=180 deg
Figure 2.4: First-order wave excitation forces and moments with dierent wave head-
ing angles.
The proposed numerical test-bed is used to evaluate the path following controller
in this dissertation. This numerical test-bed is established in MATLAB, which is the
30
most popular software in the control community. The program calculating the rst-
order wave induced loads is coded in FORTRAN and called from the main program
in MATLAB for computational eciency. It should be pointed out that this model
is generic and can be used in many other applications, such as course keeping, roll
stabilization and dynamical positioning, and its utility is independent of the control
design methodology used.
2.4 Experimental Test-bed Introduction
To support the control development, a fully instrumented model ship was designed
so that the control algorithm developed in Section 3 could be experimentally evalu-
ated. The model ship is actuated with two opposite rotating main propellers with
two rudders aft. Propellers and rudders are actuated by two DC servo motors tted
with encoders and tachometers, respectively. The main parameters of the model ship
are given in Table 2.3.
Table 2.3: Principal particulars of the model ship.
Item Symbol Value
Length L 1.60 m
Beam B 0.38 m
Draft H 0.17 m
Mass m 38 kg
Inertia I
z
2.7 kgm
2
When the model ship is tested in the towing tank (320ft.22ft.10.5ft.) located
in the Marine Hydrodynamics Laboratories (MHL) of the University of Michigan,
GPS signals are not available (even if available, the accuracy of GPS signals is not
high enough for the model test). Thus, four infra-red cameras are used, in lieu of the
GPS system, to provide the feedback signal to the control system. A picture of the
instrumented model is shown in Figure 2.5.
The real-time feedback control is accomplished using a PC-based PC104 hardware
31
Note that |e
[1]
| |e
| and
r
:=
o
e(t) , we obtain
W (
o
+k
e
) U
sin(
~
+)
+
e
2
(t) ||e
(t)||
2
+U
sin(
~
+)
+
||e(t)|| ||e
(t)||
(
o
+ k
e
) U e
2
(t) ||e
(t)||
2
+U ||e(t)|| ||e
(t)||.
(19)
We assume that
+ varies over (+
o
,
o
) where
o
is
a small positive constant, and dene = min(
sin(
~
+)
+
) =
sin(
o
)
o
, Eq. (19) can be further formulated in a quadratic
form as follows:
||e||
||e
||
T
A
COM
||e||
||e
||
(20)
where
A
COM
=
(
o
+ k
e
) U
U
2
U
2
1
. (21)
Using the standard Lyapunov stability argument, one can
show that if the following condition is satised:
o
+k
e
>
U
4
, (22)
we have
W negative denite and therefore the overall system
is exponentially stable. It can be seen that the sway velocity v
adversely affects the system stability since a large v implies
a large
U
r
approach to
zero. In addition, k
e
, which is a design parameter in the
inner loop control law, can be properly selected to meet
certain performance criteria of the outer loop system. This
observation can be made by plugging
r
= e
[1]
k
e
e+
into Eq. (7) as follows:
V
o
(t) = (
o
+ k
e
) U
sin(
~
+)
+
e
2
(t) + U
sin(
~
+)
+
e(t)e
[1]
.
(23)
IV. EXPERIMENTAL PLATFORM AND ITS
MATHEMATICAL MODEL DEVELOPMENT
To support the control development, the fully instrumented
model ship is designed so that the control algorithm de-
veloped in Section II can be experimentally evaluated. The
model ship, which is a 1:50 scaled model of an offshore
supply vessel, has a length of 1.6 m, a mass of 38 kg, and
its breadth is 0.3 m. It is actuated with two contra-rotating
main propellers and two rudders aft. Propellers and rudders
are actuated by two DC servo motors tted with encoders
and tachometers, respectively.
When the model ship is tested in a towing tank where
GPS signals are not available, four infra-red cameras are
used, in lieu of the GPS system, to provide the position
feedback signal to the control system. Meanwhile, an off-
the-shelf gyro is installed on-board the model ship to get the
information of the ship orientation in real time. A picture of
the instrumented model is shown in Fig. 2.
Fig. 2. A system overview of the fully instrumented model ship.
Fig. 3. Wireless link between devices.
The real-time feedback control is accomplished using a
PC-based PC104 hardware which runs the QNX real-time
operating system. PC104 communicates, through a wireless
LAN, to a host PC, on which the control algorithm is pro-
grammed and tuned, data acquisition function is performed,
and ship position signals are collected from the camera
system and transmitted to PC104. This model ship will
be used to validate the control algorithm proposed in the
previous section. In the sequel, we rst describe the modeling
process and present a mathematical model for the platform.
This model will allow the algorithm to be evaluated rst in
the simulation environment before it is nally tested on the
real hardware.
A. Development of Mathematical Model
Note that for the generic model, the terms
X
h
, Y
h
, N
h
,
x
,
y
and
z
in Eq. (1) are not specied
in Section II. In our modeling effort, we determine these
Figure 2.5: A system overview of the fully instrumented model ship.
which runs the QNX real-time operating system. PC104 communicates, through a
wireless LAN, to a host PC, on which the control algorithm is programmed and tuned,
data acquisition function is performed, and ship position signals are collected from
the camera system and transmitted to PC104. The key control and communication
devices are shown in Figure 2.6, together with the connections among the devices.
For the experimental validation conducted later in Chapter 3, the onboard com-
puter controls and coordinates the motion of two propeller motors and two rudder
motors according to the control algorithm. At the higher level, the desired path to be
tracked is specied and communicated wirelessly to the on-board controller. In the
MHL towing tank where the experiments were conducted, the speed of the towing
carriage (on which the camera position tracking system is mounted) and the ship
position data captured by the camera motion tracking system are transmitted in real
time to PC104 through two pairs of wireless RF modems. The actuators have their
low level inner loop control for the propeller speed and the rudder angle. PI con-
trollers with the gains K
prop
= (0.05, 1.0) and K
rudder
= (3, 5) are used. The control
32
Note that |e
[1]
| |e
| and
r
:=
o
e(t) , we obtain
W (
o
+k
e
) U
sin(
~
+)
+
e
2
(t) ||e
(t)||
2
+U
sin(
~
+)
+
||e(t)|| ||e
(t)||
(
o
+ k
e
) U e
2
(t) ||e
(t)||
2
+U ||e(t)|| ||e
(t)||.
(19)
We assume that
+ varies over (+
o
,
o
) where
o
is
a small positive constant, and dene = min(
sin(
~
+)
+
) =
sin(
o
)
o
, Eq. (19) can be further formulated in a quadratic
form as follows:
||e||
||e
||
T
A
COM
||e||
||e
||
(20)
where
A
COM
=
(
o
+ k
e
) U
U
2
U
2
1
. (21)
Using the standard Lyapunov stability argument, one can
show that if the following condition is satised:
o
+k
e
>
U
4
, (22)
we have
W negative denite and therefore the overall system
is exponentially stable. It can be seen that the sway velocity v
adversely affects the system stability since a large v implies
a large
U
r
approach to
zero. In addition, k
e
, which is a design parameter in the
inner loop control law, can be properly selected to meet
certain performance criteria of the outer loop system. This
observation can be made by plugging
r
= e
[1]
k
e
e+
into Eq. (7) as follows:
V
o
(t) = (
o
+ k
e
) U
sin(
~
+)
+
e
2
(t) + U
sin(
~
+)
+
e(t)e
[1]
.
(23)
IV. EXPERIMENTAL PLATFORM AND ITS
MATHEMATICAL MODEL DEVELOPMENT
To support the control development, the fully instrumented
model ship is designed so that the control algorithm de-
veloped in Section II can be experimentally evaluated. The
model ship, which is a 1:50 scaled model of an offshore
supply vessel, has a length of 1.6 m, a mass of 38 kg, and
its breadth is 0.3 m. It is actuated with two contra-rotating
main propellers and two rudders aft. Propellers and rudders
are actuated by two DC servo motors tted with encoders
and tachometers, respectively.
When the model ship is tested in a towing tank where
GPS signals are not available, four infra-red cameras are
used, in lieu of the GPS system, to provide the position
feedback signal to the control system. Meanwhile, an off-
the-shelf gyro is installed on-board the model ship to get the
information of the ship orientation in real time. A picture of
the instrumented model is shown in Fig. 2.
Fig. 2. A system overview of the fully instrumented model ship.
Fig. 3. Wireless link between devices.
The real-time feedback control is accomplished using a
PC-based PC104 hardware which runs the QNX real-time
operating system. PC104 communicates, through a wireless
LAN, to a host PC, on which the control algorithm is pro-
grammed and tuned, data acquisition function is performed,
and ship position signals are collected from the camera
system and transmitted to PC104. This model ship will
be used to validate the control algorithm proposed in the
previous section. In the sequel, we rst describe the modeling
process and present a mathematical model for the platform.
This model will allow the algorithm to be evaluated rst in
the simulation environment before it is nally tested on the
real hardware.
A. Development of Mathematical Model
Note that for the generic model, the terms
X
h
, Y
h
, N
h
,
x
,
y
and
z
in Eq. (1) are not specied
in Section II. In our modeling effort, we determine these
Figure 2.6: Wireless link between devices.
system runs at a sample rate of 1 mSec. In the experiments, the same controller
parameters used in the simulation are downloaded to the onboard processor.
The initial conditions of the model ship are not specied. Instead, the boat is
rst running in the manual mode where the operator is trying to position the ship
to the launching area. Then the system is switched to the autonomous mode where
the rudders are controlled according to the specic algorithm tested. In the test, the
control algorithm was tested with a constant propeller speed, and the rudder was
constrained to turn only 30 deg to prevent the potential mechanical damage to the
model ship.
Using this experimental test-bed, a successful evaluation of a back-stepping path
following controller (please see details in Chapter 3) and a path following controller
using dynamic surface control technique [51] have been conducted. Other path fol-
lowing controllers, such as MPC and disturbance compensating MPC, will be tested
in the future.
33
CHAPTER 3
Path Following without Roll Constraints for
Marine Surface Vessels
In this chapter, the simplied 2-DoF linear model is rst presented along with
the Serret-Frenet formulation to facilitate the path following control design without
roll constraints. Then the path following control law based on the back-stepping
method using feedback dominance is derived, and the control law is shown to have
a simple expression. The robustness of the resulting control system is also analyzed,
where unmodeled dynamics are considered, followed by the simulation results and
experimental validation. The controller evaluation and modication in wave elds
are nally summarized in Section 3.6.
3.1 Control-design Model for Path Following with-
out Roll Constraints
As mentioned in Section 1.1.3, two dierent approaches have been adopted to
address the path following problem for marine surface vessels: one treats it as a
tracking control problem [14, 15, 19, 31, 37, 57], and another simplies the tracking
control problem into the regulation control problem by adopting proper path following
34
error dynamics [79,35,56,60,65]. In this chapter, the latter approach with the Serret-
Frenet error dynamics [46, 61] is employed to design the path following controller.
The error dynamics [65] based on the Serret-Frenet equations are given by equa-
tions (1.1) and (1.2) (please see details in Section 1.1.3). For the most common
straight line or way-point path for marine surface vessels in open sea, the heading
error dynamics (1.1) can often be simplied as equation (1.3) since the path curvature
in these cases is zero.
In this study, the path following error dynamics adopted for control design are
(1.2) and (1.3). Notice the control objective for path following of marine surface
vessels is to drive e and
to zero.
As discussed in Section 2.1.2, many path following controllers in the literature
[7, 8, 14, 16, 25, 31, 37, 56] based on the most popular design model (2.20)-(2.22) have
complex expressions, making the controller dicult for gain tuning to achieve good
performance and its performance sensitive to model parameters. To address these
issues, in our work, the reduced-order 2 DoF linear model (2.23) and (2.24) are used
to facilitate the control design.
Before proceeding to the controller design, an additional assumption can be made
to further simplify the model and to avoid the under-actuation problem. In ship
maneuvering, the sway velocity is relatively small compared with other motion vari-
ables [65]. Therefore we assume that the sway velocity was small enough to be
neglected, which means v = 0. With this assumption, the nal model for control
design, that captures the dominant ship maneuvering dynamics and path following
error dynamics, with one control variable , has been simplied into:
e = usin
, (3.1)
= r, (3.2)
35
r = a
22
r +b
2
. (3.3)
This overall model (3.1)-(3.3) will be used in this chapter to design the path
following controller without roll constraints for marine surface vessels.
3.2 Feedback Dominance Back-Stepping Controller
Design
With u being treated as a constant, which is a common assumption for many de-
sign [56, 65], the dynamic system (3.1)-(3.3) assumes a triangular structure where the
control action inuences only r while r aects
and in turn
inuences e. This
triangular structure naturally renders the back-stepping control design [34]. However,
given the nonlinearity of the dynamics and the explosive nature of the back-stepping
design approach, the controller resulting from the standard back-stepping design ap-
proach involves many nonlinear terms [65]. As such, the controller may be susceptible
to unmodeled dynamics and implementation errors.
In this work, in an attempt to enhance system robustness and implementation
ease, we propose a design approach that will result in a relatively simple control
law. Instead of feedback linearization and nonlinearity cancelation, our back-stepping
design is based on feedback dominance. The design procedure is delineated as follows:
Step 1:
Dene the rst Lyapunov function as:
V
1
:=
1
2
e
2
> 0. (3.4)
Dierentiating V
1
with respect to time yields:
V
1
= e e = ue sin
. (3.5)
36
Normally, according to the back-stepping design procedure, a virtual control of
= arcsin(c
1
e) would be chosen to stabilize (3.1). However, this approach will
result in a very complex controller when
is dierentiated in the subsequent design
steps. In our work, the stabilizing virtual control
1
:= c
1
e, c
1
> 0 is selected, then
for
(, ),
V
1
becomes:
V
1
= c
1
ue
2
sin
+ue
sin
1
). (3.6)
For
=
1
, this becomes:
V
1
= c
1
ue
2
sin
0. (3.7)
In deriving the inequality for V
1
in (3.7), the property sin(x)/x > 0, x (, )
has been used.
Step 2:
Let z
2
=
1
and dierentiating with respect to time, giving:
z
2
=
1
= r +c
1
u
sin
(z
2
+
1
). (3.8)
Augment the rst Lyapunov function into the second Lyapunov function as V
2
:=
V
1
+
p
1
2
z
2
2
> 0 where p
1
is a positive constant, whose role will become apparent in the
subsequent analysis. Dierentiating V
2
with respect to time yields:
V
2
= e e +p
1
z
2
z
2
= c
1
ue
2
sin
+p
1
z
2
[r +c
1
uz
2
sin
+ (
1
p
1
c
2
1
)eu
sin
]. (3.9)
37
If p
1
= 1/c
2
1
, equation (3.9) becomes:
V
2
= c
1
ue
2
sin
+
1
c
2
1
z
2
[r +c
1
uz
2
sin
]. (3.10)
Remark 3.1. Note that the introduction of p
1
in V
2
allows us to eliminate the
cross-product term in
V
2
that involves nonlinearity. Without this exibility in
V
2
, one has to rely on the virtual control to cancel the nonlinear term to achieve
V
2
0. It should be noted that since both c
1
and p
1
are design parameters and
no model parameters are involved in meeting the condition p
1
= 1/c
2
1
, therefore
the equation (3.10) is not subject to modeling errors.
To design the next virtual control for the (e, z
2
) dynamics, feedback dominance
is used instead of feedback linearization (which will involve the exact cancelation of
nonlinearities) to form the stabilizing virtual control. By selecting
2
:= c
2
z
2
and
making c
2
satisfy c
2
> c
1
u,
V
2
becomes:
V
2
= c
1
ue
2
sin
c
2
c
2
1
z
2
2
[1
c
1
u
c
2
sin
] +
1
c
2
1
z
2
(r
2
). (3.11)
If r =
2
, this becomes:
V
2
= c
1
ue
2
sin
c
2
c
2
1
z
2
2
[1
c
1
u
c
2
sin
] 0, (3.12)
where the inequality in (3.12) is a direct result of c
2
> c
1
u and 0 < sin(
)/
< 1.
Note that in (3.12), the linear virtual control c
2
z
2
is used to dominate, instead of
canceling, the nonlinear term, thus the name feedback dominance.
Step 3:
Dene z
3
= r
2
and dierentiating with respect to time, giving:
z
3
= r
2
= a
22
r +b
2
2
. (3.13)
38
Further augmenting the second Lyapunov function as V
3
:= V
2
+
p
2
2
z
2
3
> 0, where
p
2
is a positive constant,
V
3
becomes:
V
3
= c
1
ue
2
sin
c
2
c
2
1
z
2
2
[1
c
1
u
c
2
sin
] +p
2
z
3
[
1
c
2
1
p
2
z
2
+a
22
r +b
2
2
]. (3.14)
If is selected to be
=
1
b
2
(c
3
z
3
1
c
2
1
p
2
z
2
a
22
r +
2
), (3.15)
then (3.14) turns into:
V
3
= c
1
ue
2
sin
c
2
c
2
1
z
2
2
[1
c
1
u
c
2
sin
] p
2
c
3
z
2
3
0, (3.16)
and the second inequality holds if
(, ) and c
2
> c
1
u. Furthermore,
V
3
= 0
only if (e, z
2
, z
3
) = (0, 0, 0), which means the control law (3.15) renders the system
(3.1), (3.2), (3.3) asymptotically stable.
Substituting the expressions for z
2
, z
3
and
2
in terms of the original states, (3.15)
becomes:
= c
1
e c
2
c
3
r c
4
usin
, (3.17)
where
c
1
=
1
b
2
(c
1
c
2
c
3
+
1
c
1
p
2
), (3.18)
c
2
=
1
b
2
(c
2
c
3
+
1
c
2
1
p
2
), (3.19)
c
3
=
a
22
+c
2
+c
3
b
2
, (3.20)
c
4
=
c
1
c
2
b
2
. (3.21)
As shown in the equation (3.17), the nal control law has a very simple structure.
The rst three terms in the control law are linear and the only nonlinearity comes
39
from the last term.
Remark 3.2. Besides their functions as mentioned in Remark 2, namely, to
eliminate the nonlinear term in the design procedure and therefore simplify the
resulting control, the parameters p
1
and p
2
also serve to normalize the eects of
dierent variables in the Lyapunov function. For example, three variables in V
3
are e, z
2
=
1
and z
3
= r
2
, which have the order of magnitude of 1000,
and 0.01, respectively in the case of the four DoF container ship. Thus, p
1
and p
2
have the order of 10
5
and 10
11
to make all the three variables have the
comparable inuences on the V
3
.
In contrary, the nonlinear control designed using the standard back-stepping and
feedback linearization, instead of feedback dominance, would have the form (3.22):
=
1
b
2
(k
3
z
3
+ cos(arcsin( z
2
+
1
)) z
2
+a
22
r
2
). (3.22)
where
1
= k
1
e, (3.23)
z
2
= sin
1
, (3.24)
2
=
ue k
2
z
2
+
1
cos(arcsin( z
2
+
1
))
, (3.25)
z
3
= r
2
, (3.26)
and k
1
, k
2
and k
3
are positive control parameters. When z
2
, z
3
,
1
and
2
are replaced
by their corresponding expressions in terms of the original states, the control law
40
(3.22) becomes:
=
1
b
2
[(a
22
+k
1
u +k
2
+k
3
)r + (k
1
u +k
2
)r tan
2
+(k
3
u +k
1
k
2
k
3
)e sec
+ (u +k
1
k
2
)er sin
sec
+(u
2
+k
1
k
2
u +k
2
k
3
k
1
k
3
u) tan
+sin
cos
+k
1
e cos
]. (3.27)
The resulting control law (3.27) has a lengthy expression comprising of many
nonlinear terms, most of which are due to the non-ane function of the input that
the feedback design is trying to cancel. The complexity will not only make the
controller dicult to tune, but also make it susceptible to implementation errors and
model uncertainties.
Remark 3.3. Note that the controller (3.17) has the proportional, integral
and derivative terms when e and r are expressed in terms of
, the tuning
of the controller gains are relatively easy in the sense that the eects of each
parameter on the system dynamics and control saturation can be interpreted in
physical variables and many of the PID tuning algorithms can be used.
3.3 Robustness Analysis of the Resulting FDBS
Controller
In deriving the path following controller, the linear vessel model was used and
the sway velocity was neglected. To incorporate the nonlinearities and non-zero sway
velocities, the following model is used in robustness analysis:
e = usin
+v cos
, (3.28)
41
= r, (3.29)
r = a
21
v +a
22
r +b
2
+ , (3.30)
where captures the unmodeled dynamics.
For the uncertainties , it is assumed that:
Assumption 3.1. There exist positive constants
0
,
v
and
r
such that:
[[
0
+
v
[v[ +
r
[r[. (3.31)
For the sway dynamics, which is not considered in the controller design, the fol-
lowing assumption is made:
Assumption 3.2. There exist positive constants
0
and
r
such that:
[v[
0
+
r
[r[. (3.32)
Remark 3.4. Comparing (2.24) with the nonlinear yaw dynamical equation
(2.22) and assuming a
22
=
d
33
m
33
, yields:
= (
m
11
m
22
m
33
u a
21
)v +
1
m
33
u
2
b
2
. (3.33)
Given the constant surge speed assumption, the term
m
11
m
22
m
33
ua
21
is zero since
a
21
=
m
11
m
22
m
33
u. However, in the actual maneuvering, the varying surge speed
will lead to nonzero values for
m
11
m
22
m
33
u a
21
. Therefore,
v
in equation (3.31)
is introduced to capture this surge speed eect and other parameter uncertain-
ties. On the other hand, the assumption that the yaw moment induced by the
rudder action is proportional to the rudder angle is an approximation for the
ship physical responses observed in simulations of the high delity model and
42
experiments. Recognizing that the rudder induced yaw moment also depends on
the ship state such as u, v and r, we introduce the
0
term to capture the higher-
order but bounded nonlinear terms in control input (the dierence between b
2
and
1
m
33
u
2
) and other model dynamics. Furthermore, the eects of parameter
uncertainties in the r term are captured by
r
.
Remark 3.5. For surface vessel maneuvering in calm water, the nonzero rud-
der angle, which is the only control input considered here, will result in corre-
sponding nonzero sway velocity v and yaw rate r. Normally, the magnitudes of
v and r are both related to the magnitude of rudder angle. Extensive simulation
using a high-order nonlinear model shows that there exists a phase lag between
the response v and r. Therefore, we use
0
in Assumption 3.2 to capture this
lag and
r
for the proportional relation between v and r.
To study the stability of the closed loop system with the proposed controller imple-
mented to the system (3.28)-(3.30), the Lyapunov function V
3
used in the controller
derivation in the previous section is adopted. Dierentiating V
3
with respect to time,
yields,
V
3
= c
1
ue
2
sin
c
2
c
2
1
z
2
2
[1
c
1
u
c
2
sin
] p
2
c
3
z
2
3
+ve cos
+
1
c
1
vz
2
cos
+p
2
c
1
c
2
vz
3
cos
+p
2
a
21
vz
3
+p
2
z
3
. (3.34)
Dening d
1
:= c
1
u
sin
, d
2
:=
c
2
c
2
1
(1
c
1
u
c
2
sin
) and d
3
:= p
2
c
3
, it follows from (3.34)
that:
V
3
d
1
e
2
d
2
z
2
2
d
3
z
2
3
+[v[([ cos
[[e[ +
[ cos
[
c
1
[z
2
[ +p
2
c
1
c
2
[ cos
[[z
3
[
+p
2
[a
21
[[z
3
[) +p
2
[z
3
[[[. (3.35)
43
If Assumption 3.1 and 3.2 are satised and notice that [r[ [z
3
[ + c
2
[z
2
[, (3.35)
leads to:
V
3
d
1
e
2
d
2
z
2
2
d
3
z
2
3
+l
1
[e[ +l
2
[z
2
[ +l
3
[z
3
[
+n
1
z
2
2
+n
2
z
2
3
+q
1
[e[[z
2
[ +q
2
[e[[z
3
[ +q
3
[z
2
[[z
3
[, (3.36)
where
l
1
=
0
[ cos
[, (3.37)
l
2
=
0
[ cos
[
c
1
, (3.38)
l
3
= p
2
(c
1
c
2
0
[ cos
[ +
0
[a
21
[ +
0
+
0
v
), (3.39)
n
1
=
c
2
r
[ cos
[
c
1
, (3.40)
n
2
= p
2
(c
1
c
2
r
[ cos
[ +
r
[a
21
[ +
r
+
r
v
), (3.41)
q
1
= c
2
r
[ cos
[, (3.42)
q
2
=
r
[ cos
[, (3.43)
q
3
= p
2
c
2
(c
1
c
2
r
[ cos
[ +[a
21
[
r
+
r
v
+c
1
r
) +
c
3
r
c
1
. (3.44)
Using the arithmetic mean geometric mean inequality in (3.36), we have:
V
3
d
1
e
2
d
2
z
2
2
d
3
z
2
3
+l
1
[e[ +l
2
[z
2
[ +l
3
[z
3
[. (3.45)
where
d
1
= d
1
q
1
2
q
2
2
, (3.46)
d
2
= d
2
n
1
q
1
2
q
3
2
, (3.47)
d
3
= d
3
n
2
q
2
2
q
3
2
. (3.48)
44
From (3.45), it follows that the system will converge to a region around the origin
that is characterized by (e, z
2
, z
3
) [ [e[
l
1
d
1
, [z
2
[
l
2
d
2
, [z
3
[
l
3
d
3
. By a proper
selection of the controller gains c
1
, c
2
, c
3
and p
2
, we can make the region very small
if there are no signicant unmodeled dynamics, which means
0
,
v
and
r
are small,
and the sway velocity is relatively small, which means
0
and
r
are small.
Remark 3.6. In order to eliminate the steady state error in cross-track error
e when environmental disturbances exist (such as the lateral current), we could
design a controller with an integral term e by augmenting the system dynamics
(3.1)-(3.3) with e
I
= e, where e
I
is the integral of cross-track error e. However,
the feedback dominance technique for simplifying the backstepping controller is
not applicable in this case. Thus, the resulting controller derived by following
the standard backstepping technique is very complex, which defeats our purpose
to develop a simple controller for easy gain-tuning and analysis. In fact, an
integral term c
4
e
I
( c
4
is positive gain) could be directly added into the control
law (3.17) to achieve good performance with environmental disturbances. How-
ever, rigorous analysis would be dicult to establish stability and converging
performance in this case.
3.4 Simulation Results in Calm Water
To verify and illustrate the theoretical results, the proposed control law is im-
plemented and simulated with the 4-DoF nonlinear container model (S175 described
in Section 2.1.1) along with the corresponding reduced-order model. The actuator
saturation and its rate limits ([[ 20 degree and [
(
d
e
g
)
Rudder Angle
0 sec
1 sec
5 sec
Figure 3.3: Simulation results with time delay.
controller are [0.018
2
, 0, 0; 0, 3.5
2
, 0; 0, 0, 0] and 1, respectively. The gains adopted in
FDBS and linearized FDBS are c
1
= 0.0028, c
2
= 0.1, c
3
= 0.1 and p
2
= 10
11
. The
simulation results for three dierent controllers are shown in Figure 3.5. It can be
seen from Figure 3.5 that the FDBS controller achieves the fastest path following
convergence speed with the least rudder action, while the LQR controller has slower
path following and more rudder action than the FDBS controller. We also notice that
49
-500 0 500 1000 1500 2000 2500 3000 3500 4000
-1000
-500
0
500
1000
1500
2000
2500
x (m)
y
(
m
)
Trajectory
p =0
p =0.5
p =1
0 50 100 150 200 250 300
-5
0
5
10
15
20
time (sec)
(
d
e
g
)
Rudder Angle
p =0
p =0.5
p =1
Figure 3.4: Simulation results with measurement noises.
the linearized FDBS controller has a slight overshoot in both cross-track and heading
errors.
To further verify the controller performance and prepare for the experimental
validation, the numerical simulations of a model boat (the details of this boat has
been given in Section 2.4) with the controller proposed in this paper were conducted.
The numerical model of the model boat has 2 DoF of sway and yaw by assuming the
50
0 5 10 15 20 25 30
-2
-1
0
1
C
r
o
s
s
-
t
r
a
c
k
E
r
r
o
r
(
m
)
[1(m) -10(deg)]
[-1(m) 30(deg)]
[-2(m) 0(deg)]
0 5 10 15 20 25 30
-20
0
20
40
H
e
a
d
i
n
g
E
r
r
o
r
(
d
e
g
)
0 5 10 15 20 25 30
-50
0
50
R
u
d
d
e
r
A
n
g
l
e
(
d
e
g
)
time (sec)
Figure 3.6: Simulation results of the model boat for dierent initial conditions (num-
ber in the brackets are the initial cross-track error and heading error).
3.5 Experimental Results in Calm Water
The experimental validation was conducted in the Marine Hydrodynamics Labo-
ratories (MHL) towing tank with a model boat, which has been introduced in Sec-
tion 2.4.
In all experiments, the model ship could converge to the desired path, regardless
of where it actually started in the towing tank. Figure 3.7 shows one example of the
experiment results, giving the time evolution of the states y,
and the rudder angle
. The rst and second plots given in Figure 3.7 show that the ship tracks the path
well given that the cross-track and heading errors are intended to approach zero. The
control algorithm was also validated to be eective under other dierent propeller
speeds.
Figure 3.7 also shows the comparison between the simulation and experimental
validation results of the model boat. Figure 3.7 shows the good match between the
simulation and experimental results. In Figure 3.7, the simulation result is shown to
have less rudder eort than the experimental results and this error could be attributed
52
) +vcos(
), (3.49)
= r, (3.50)
v = a
11
v +a
12
r +b
1
+
m
66
(
f
2
+f
2
) m
62
(
f
6
+f
6
)
m
22
m
66
m
2
62
, (3.51)
r = a
21
v +a
22
r +b
2
+
m
22
(
f
6
+f
6
) m
62
(
f
2
+f
2
)
m
22
m
66
m
2
62
, (3.52)
where (3.49) and (3.50) are the original path following error dynamics. Moreover,
m
22
= m +m
y
, m
66
= I
z
+J
z
and m
62
= m
y
y
.
The equilibrium point of the overall system (3.49)-(3.52) in the average sense is
the solution of the equations:
usin(
) +vcos(
) = 0, (3.53)
a
11
v +b
1
+
m
66
f
2
m
62
f
6
m
22
m
66
m
2
62
= 0, (3.54)
a
21
v +b
2
+
m
22
f
6
m
62
f
2
m
22
m
66
m
2
62
= 0. (3.55)
Notice that r = 0 and the zero-mean oscillating 1st-order wave loads are neglected.
According to the control law (3.17), is a function of e and
. Therefore, the above
three equations (3.53)-(3.55) have three unknowns, namely v, e and
. And the
solution of these three equations depends on the controller gains c
1
, c
2
, c
3
and p
2
. If
gain set 1 is selected, the steady state error given by the (3.53)-(3.55) is:
e
0
= 16.67 (m), (3.56)
0
= 4.6 (deg), (3.57)
56
v
0
= 0.64 (m/s), (3.58)
which match the simulation result given in Figure 3.8. To reduce or eliminate the
steady state error, the gains c
1
,c
2
,c
3
and p
2
should be properly selected. The proper
gains that eliminate the steady state cross-track error should satisfy the following two
equations:
a
11
v
0
b
1
b
2
[(c
2
c
3
+
1
c
2
1
p
2
)
0
+c
1
c
2
usin
0
] +
m
66
f
2
m
62
f
6
m
22
m
66
m
2
62
= 0, (3.59)
a
21
v
0
(c
2
c
3
+
1
c
2
1
p
2
)
0
c
1
c
2
usin
0
+
m
22
f
6
m
62
f
2
m
22
m
66
m
2
62
= 0. (3.60)
Notice that e
0
is set to zero to derive (3.59)-(3.60) from (3.53)-(3.55). To nd the
proper parameters so that (3.59)-(3.60) are satised, let
0
= 4.6 (deg) and v
0
=
0.64 (m/s), which is the same as the solution of gain set 1. For a given wave
eld with a specic sea state, the steady state errors in the heading error and sway
velocity should be the same to counteract the same wave drift loads, regardless of
the controller gains. For this particular case, the control gain c
1
, c
2
, c
3
and p
2
should
satisfy the following condition:
(c
2
c
3
+
1
c
2
1
p
2
) + 9.9893c
1
c
2
0.0039 = 0. (3.61)
The reason for the rudder oscillations is the state oscillations induced by the rst-
order wave excitation load, especially the yaw rate r. The correlation between the
rudder angle and yaw rate can be clearly seen in Figure 3.9. One intuitive solution
to reduce the rudder oscillations is to use the low gain corresponding to the yaw
rate term to make the controller insensitive to the oscillating state. More specically,
proper c
2
and c
3
should be selected to make the coecient of the r term (a
22
+c
2
+c
3
)
in control law to be zero or small. For this particular case, the following condition
57
should be satised:
c
2
+c
3
0.1068 0. (3.62)
WaveDirection
DesiredPath
Vessel Starting Point
2190
2180
2170
2160
2150
2140
2130
2120
2110
620
2100
625 630 635 640 645 650
Figure 3.10: Simulation results of gain scheduling controller to reduce the steady
state error and rudder oscillations.
The proposed path following gain scheduling controller has the great advantage of
easy re-tuning to address the environmental disturbance because of its simple form.
Furthermore, since no adaptation mechanism and signal lters are adopted in the gain
scheduling controller, the complexity of the control system can be largely reduced.
59
3.7 Summary
In this chapter, a control system for marine surface vessel path following was
proposed. The back-stepping design, based upon feedback dominance, leads to a
simple control law that achieves asymptotic path following. Robustness analysis
against unmodeled dynamics was also performed. Simulation results revealed that
the controller enhanced the path following capability while demonstrating robust
performance against model uncertainties, communication delays and measurement
noise. The eectiveness of the designed controller was also validated by the successful
experimental results. Because of the simple form of the controller, it is expected that
very limited on-board computational power will be required to implement the control.
The FDBS controller for marine surface vessels was also evaluated in wave elds.
Since a steady state cross-track error and the rudder oscillations were observed in the
evaluation, controller tuning was performed to modify the system response to reduce
the steady state cross-track error and rudder oscillations. Simulation validated that
the re-calibrated controller with gain scheduling achieved satisfactory performance in
terms of both path following convergence speed and steady state behavior.
60
CHAPTER 4
Path Following with Roll Constraints for Marine
Surface Vessels
This chapter presents a Model Predictive Control (MPC) design of the path fol-
lowing for an integrated model of the surface vessel dynamics and path following kine-
matics. The focus is on satisfying all the inputs and state constraints while achieving
satisfactory path following performance. The one-input (rudder) and two-input (rud-
der and propeller) MPC controllers are both developed to achieve constrained path
following and their performance are compared by simulations. The path following per-
formance of the proposed MPC controller and its sensitivity to the major controller
parameters, such as the sampling time, predictive horizon and weighting matrices in
the cost-function, are analyzed by numerical simulations.
4.1 Introduction
One challenge for path following of marine surface vessels stems from the fact that
the system is often underactuated. Conventional ships are usually equipped with one
or two main propellers for forward speed control, and rudders for course keeping
of the ship. For ship maneuvering, such as path following and trajectory tracking,
61
where we seek control for all three degrees of freedom (surge, sway and yaw), the two
controls can not inuence all three variables independently, thereby leading to under-
actuated control. Another challenge in the path following of marine surface vessels
is the inherent physical limitations in the control inputs, namely the rudder satura-
tion and rudder rate limit. More recently, given that the roll motion produces the
highest acceleration and is considered as the principal villain for the seasickness and
cargo damage [24], enforcing roll constraints while maneuvering in a seaway becomes
an important design consideration in surface vessel control. While typical nonlinear
control methodologies do not take these input and output constraints explicitly into
account in the design process, the constraint enforcement is often achieved through
numerical simulations and trial-and-error tuning of the controller parameters. Few
other control methodologies, such as the MPC [48, 58] and reference governor [28],
have a clear advantage in addressing input and state constraints explicitly. [74] con-
siders rudder saturation in its MPC controller for tracking control of marine surface
vessels and [55] achieves the roll reduction for the heading control problem using an
MPC approach. For the path following control problem considered in this disserta-
tion where both the cross-track error and heading error are controlled by the rudder
angle as an under-actuated problem and rudder limitation and roll constraints need
to be enforced simultaneously, MPC applications have not been found in the open
literature, to the best knowledge of the author.
MPC, also known as the receding horizon control (RHC), is a control technique
which embeds optimization within feedback to deal with systems subject to con-
straints on inputs and states [48, 58]. Over the last few decades, MPC has proven
to be successful for a wide range of applications including chemical, food process-
ing, automotive and aerospace systems [58]. Using an explicit model and the current
state as the initial state to predict the future response of a plant, it determines the
control action by solving a nite horizon open-loop optimal control problem on-line
62
at each sampling interval. Furthermore, because of its natural ability to treat multi-
variable systems, MPC can handle underacuated problem gracefully by combining all
the objectives into a single objective function.
4.2 MPC for Path Following of Marine Surface
Vessels using Rudder
4.2.1 Control-design Model for MPC using Rudder
The Serret-Frenet error dynamics (1.2) and (1.3) described in Section 1.1.3 are
employed to design the MPC path following controller. The control objective is to
drive e and
to zero.
For the path following problem without roll constraints, the control design, sum-
marized in Chapter 3, is based on the linear system (2.23)-(2.24). However, the
additional DoF roll should be also included for path following with roll constraints.
Therefore, the corresponding 3-DoF (sway, yaw and roll) linear system (2.25)-(2.29),
described in Section 2.1.2, will be adopted in the one-input MPC control design.
Notice that the surge speed is assumed to be constant and the surge dynamics are
neglected.
The performance of the control system designed using the reduced-order model
((1.2)-(1.3) and (2.25)-(2.29)) will be presented to justify the utility of the reduced-
order model when the same controller is applied to the full-order model S175 (see
Section 2.1.1).
4.2.2 MPC Formulation for Path Following using Rudder
This section presents the formulation of the MPC for the path following problem of
marine surface vessels using rudder. For notational convenience, the ship dynamics
63
(2.25)-(2.29) together with linearized path following error dynamics (1.1)-(1.2) are
written into the matrix form:
x =
A x +
B, (4.1)
where
x =
_
_
e
v
r
p
_
, (4.2)
A =
_
_
0 u 0 0 0 0
0 0 0 1 0 0
0 0 a
11
a
12
a
13
a
14
0 0 a
21
a
22
a
23
a
24
0 0 a
31
a
32
a
33
a
34
0 0 0 0 1 0
_
_
,
B =
_
_
0
0
b
1
b
2
b
3
0
_
_
. (4.3)
Given a specic sampling time T
s
, the plant (4.1) is easily transformed into its
discrete-time version:
x
k+1
= A x
k
+B
k
. (4.4)
Using the discrete-time plant (4.4), the future state of the plant can be predicted
by:
X
k
= F x
k
+HU
k
, (4.5)
64
where
X
k
=
_
_
x
k+1
x
k+2
.
.
.
x
k+Np
_
_
, U
k
=
_
k+1
.
.
.
k+Np1
_
_
, (4.6)
F =
_
_
A
A
2
.
.
.
A
Np
_
_
, H =
_
_
B 0 0 0
AB B 0 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
A
Np1
B A
Np2
B A
Np3
B B
_
_
, (4.7)
and N
p
is the predictive horizon.
Then the MPC online optimization problem can be formulated as follows: at each
time k, nd the optimal control sequence
k
,
k+1
, ,
k+Np1
to minimize the
following cost function (4.8):
J(U
k
, x
k
) =
Np
j=1
( x
T
k+j
Q x
k+j
+
T
k+j1
R
k+j1
), (4.8)
subject to
max
k+j
max
, j = 0, 1, , N
p
1, (4.9)
max
k+j
k+j1
max
, j = 0, 1, , N
p
1, (4.10)
x
max
x
k+j
x
max
, j = 1, 2, , N
p
, (4.11)
where (4.9), (4.10) and (4.11) stand for rudder saturation, rudder rate limit and state
limit respectively. Q and R are the corresponding weighting matrices and N
p
is the
predictive horizon. The control law is given by
k
=
k
.
Since the cost function (4.8) is quadratic in x and and all the constraints are
linear, we can use quadratic programming (QP) to solve the optimization problem.
In this study, the optimization and simulation are performed in MATLAB.
65
4.2.3 Simulation Results and Controller Parameter Tuning
The proposed control law is implemented and simulated on the full-order nonlinear
model. The actuator saturation and its rate limits ([[ 35 deg and [
[ 5 deg/sec)
are incorporated in simulations, while dierent roll constraints are imposed to evaluate
the eectiveness of the MPC and the trade-os between tightening the roll constraint
and achieving path following. For all simulations of the MPC controller using the
rudder, the propeller speed is maintained as constant (99.50 RPM). Since only the
relative penalty on x and will inuence the performance, Q and R are chosen to
have the form of Q = 0.0001, c
1
, 0, 0, 0, 0, R = c
2
, namely, the cost function is
J =
Np
j=1
(0.0001e
k+j
2
+c
1
2
k+j
+c
2
2
k+j1
), with c
1
, c
2
being positive constants. The
numerical values of these dierent gains used for simulations are listed in Table 4.1.
Table 4.1: Controller gains for simulations of MPC path following controller.
G1 G2 G3 G4 G5
c
1
8 1.6 40 8 8
c
2
1 1 1 0.1 10
Selection of the Sampling Time
The general guideline for selecting the sampling rates for discrete-time dynamical
system is about 4-10 samples per rise time [3], which is about 18 second for the
roll dynamics (which is the fastest among yaw, sway and roll) of the container ship.
Therefore, a rational choice of the sampling is between 1 to 4 seconds. For the MPC
application, small sampling times provide more timely feedback but require more
frequent optimization, and a good trade-o between the path following performance
and real-time implementation consideration can be achieved through the sensitivity
analysis. The roll, sway and yaw responses of the closed-loop system with the MPC
corresponding to dierent sampling times are summarized in Figure 4.1. For each
simulation, the predictive time window is set to 120 seconds (considering that the
66
time constant for the maneuvering dynamics is around 20 seconds), which leads to
dierent predictive steps N
p
for dierent sampling intervals. The gain set G1 is
employed in this simulation. From Figure 4.1, the responses with 1 second and 2
second sampling period can be seen to be almost indierentiable, while the responses
with 3 or 4 second sampling interval start to deviate. Figure 4.1 shows that T
s
= 2
sec is a good choice for the implementation of MPC controller for the container ship
under consideration. Simulations performed for many other gain sets yield the same
conclusion.
0 5 10 15 20 25 30 35 40
-20
-15
-10
-5
0
5
10
15
20
time (sec)
r
o
l
l
a
n
g
l
e
(
d
e
g
)
G1, 20 Deg Roll Constraints
N
p
=120, Ts =1 sec
N
p
=60, Ts =2 sec
N
p
=40, Ts =3 sec
N
p
=30, Ts =4 sec
0 5 10 15 20 25 30 35 40
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
time (sec)
s
w
a
y
v
e
lo
c
it
y
(
m
/
s
)
0 5 10 15 20 25 30 35 40
75
80
85
90
time (sec)
y
a
w
a
n
g
le
(
d
e
g
)
Figure 4.1: Simulation results of the ship response with dierent sampling time.
67
Prediction Horizon
The length of the predictive horizon N
p
is a basic tuning parameter for MPC
controllers. Generally speaking, controller performance improves as N
p
increases, at
the expense of additional computation eort [58]. The eects of predictive horizon
N
p
on the path following performance are studied by simulations with results given
in Figure 4.2. The gain set G1 is employed in this simulation. It is clear from
Figure 4.2 that longer predictive horizon leads to faster path following and avoids
over-steering, but the benets of extending the prediction horizon diminishes beyond
N
p
= 40. Given the heavy computational cost associated with long prediction horizon
(in our simulations, the computational time for each optimization with N
p
= 160 and
N
p
= 80 are about 16 and 4 times of the one for N
p
= 40, respectively), it can be
concluded that a value of 40-60 achieves a good trade-o for the predict horizon N
p
,
given 2 seconds as the sampling period. The same conclusion can be drawn from
simulations performed for many other gain sets.
Putting it all in the context of computational eort required for MPC implemen-
tation for a marine surface vessel path following control, the optimization problem
with 2 second sampling interval and 60 step predictive horizon can be solved in about
0.6 second in simulations on a desktop computer with P4 2.4 CPU and 2G RAM.
Experience with the real-time optimization implementation shows that this number
can be substantially reduced, using real-time computing technology, to a small frac-
tion of sampling time. This moderate computational demand makes the MPC path
following control promising for real-time implementation.
Eects of Weighting Matrices Q and R
The weighting matrices Q and R are used as the main tuning parameters to shape
the closed-loop response for desired performance [48]. Investigating the performance
sensitivity to the weighting matrix leads to useful insights that will be discussed in
68
x =
A x +
B
u, (4.12)
73
=
_
_
0 0
0 0
b
11
b
12
b
21
b
22
b
31
b
32
0 0
_
_
, u =
_
n
_
_
. (4.13)
For control design, three linear models, obtained by linearizing the original nonlin-
ear model S175 (see Section 2.1.1 (2.19)) at dierent equilibrium points, are used to
facilitate the model-based design. To incorporate the propeller eects on the system
dynamics, the nonlinear model is linearized around the equilibrium points correspond-
ing to
0
= 0 (for the operating range corresponding to the rudder angle from -10
to 10 deg),
0
= 15 (for the operating range corresponding to the rudder angle large
than 10 deg) and
0
= 15 deg (for the operating range corresponding to the rudder
74
angle less than -10 deg). Those three models will be denoted as M
1
, M
2
and M
3
,
respectively, in the sequel. The dierent values of matrix
A and
B
of three linearized
models are given in Table 4.2.
Table 4.2: System parameters for dierent linear models.
M
1
M
2
M
3
a
11
-0.02276 -0.04062 -0.04062
a
12
-2.7910 -0.1899 -0.1899
a
13
-0.09211 -0.06664 -0.06664
a
14
-0.1169 -0.09348 -0.09348
a
21
-0.0009168 0.0001167 0.0001167
a
22
-0.1068 -0.1468 -0.1468
a
23
0.009949 0.007198 0.007198
a
24
0 -0.0008284 -0.0008284
a
31
0.002032 0.002594 0.002594
a
32
-0.3058 -0.3051 -0.3051
a
33
-0.01982 -0.01434 -0.01434
a
34
-0.04486 -0.04471 -0.04471
b
11
-0.05699 -0.04575 -0.04575
b
12
0 -0.0002503 0.0002503
b
21
0.002838 0.002279 0.002279
b
22
0 0.00001247 -0.00001247
b
31
0.004081 0.003277 0.003277
b
32
0 0.00001793 -0.00001793
The reason of adopting multiple linear models instead of using a single linear
model employed in Section 4.2 is that the single linear model linearized around the
equilibrium point
0
= 0, designated as M
1
, can not capture the dynamic relation
between the system response and propeller speed. If a single linear model M
1
is
adopted, the values of b
12
, b
22
and b
32
, shown in Table 4.2 are all zeros. For the linear
model M
1
, the propeller speed has no eective inuence on state variables. However,
for M
2
and M
3
(linearized at ,= 0), the nonlinearities of the dynamics render some
control authority to the propeller speed on vessel responses for all the considered
degrees of freedom sway v, yaw r and roll p. In order to incorporate this nonlinearity
without incurring substantial computational penalty, the multiple linear models are
75
adopted for the MPC design.
4.3.2 MPC Formulation for Path Following Control using
Rudder and Propeller
This section presents the formulation of the MPC for the path following problem
of marine surface vessels. For notational convenience, we rewrite the multiple linear
models into the matrix form:
x =
A(M) x +
B(M) u. (4.14)
Notice that the matrix
A and
B are now dependent on which linear model is employed.
Specically, one of the three linear models will be selected in the MPC optimization,
based on the yaw rate r, as follows:
M =
_
_
M
1
, if r
c
< r < r
c
;
M
2
, if r r
c
;
M
3
, otherwise,
(4.15)
where r
c
is the steady state value of the yaw velocity when the rudder angle is -10
deg. The reason of adopting yaw velocity as the criteria of model-switching is that
the vessel forward speed eects on the roll response are largely inuenced by the yaw
rate, which can be clearly seen from Figure 4.6 that shows the simulation results of
S175 for dierent yaw rates and the same propeller speed. Yaw rates 0.0058, 0.0078
and 0.0089 rad/sec are the steady state values when the rudder angle is set to 5, 10
and 15 deg, respectively.
Given a sampling time T
s
and yaw rate (which determine M), the plant (4.14)
can be discretized as:
x
k+1
= A(M) x
k
+B(M) u
k
. (4.16)
76
j=1
( x
T
k+j
Q x
k+j
+ u
T
k+j1
R
u
k+j1
), (4.17)
subject to state equation (4.16) and
u
min
u
k+j1
u
max
, j = 1, 2, , N
p
, (4.18)
u
max
u
k+j1
u
max
, j = 1, 2, , N
p
, (4.19)
x
max
x
k+j
x
max
, j = 1, 2, , N
p
, (4.20)
where (4.18), (4.19) and (4.20) stand for the control input saturation, input rate
limit and state limit, respectively (the inequalities (4.18), (4.19) and (4.20) have to
be satised element-by-element). Q and R
[ 5
deg/sec), are incorporated in the simulations. No rate limit is imposed on the change
of the propeller speed.
Sampling Time and Prediction Horizon Choices
The philosophy of selecting the sampling time and prediction horizon for the
MPC using rudder and propeller is basically the same as the case using rudder only.
Considering the rise time of the fastest dynamics, the sampling time T
s
= 2 sec is
still a rational choice. For the selection of the prediction horizon, simulations were
performed using dierent prediction horizons, which are shown in Figure 4.7. The
gain set G1 and c
3
= 0 are employed in these simulations with the sampling time
of 2 seconds. Figure 4.7 shows that longer predictive horizon leads to faster path
following but the benets of extending the prediction horizon diminishes beyond
N
p
= 60. Considering the associated computational cost, N
p
= 60 was selected for
this particular case, given 2 seconds as the sampling period. The same conclusion
can be drawn from simulations performed for many other gain sets.
78
(
d
e
g
)
0 50 100 150 200 250 300
0
20
40
60
80
p
r
o
p
e
l
l
e
r
s
p
e
e
d
{
R
P
M
)
time (sec)
N
p
=20
N
p
=40
N
p
=60
N
p
=80
Figure 4.7: Simulation results of the ship response with dierent prediction horizon.
Eects of Weighting Matrix R
The guidelines given in Section 4.2 for tuning matrix Q of the MPC path following
controller using just the rudder are also useful for the tuning of controller using
rudder and propeller. Additional simulations are conducted to study the eect of
matrix R
= c
2
, c
3
on the system response. The impact of c
2
has been analyzed in
Section 4.2, which is similar in two-input case as shown in simulations. In this section,
focus is on investigating the performance sensitivity to the gain c
3
. With gain set
G1, three values of c
3
, namely 0, 0.00002 and 0.2, are adopted in simulations, which
are summarized in Figure 4.8. Please notice the dierent magnitudes for values of
rudder angle ( /5) and propeller speed ( 70). Thus, with c
3
= 0.2, the penalty
on propeller speed is signicantly large compared with the rudder angle penalty with
c
2
= 1.
79
700 800 900 1000 1100 1200 1300 1400 1500
0
200
400
600
800
1000
1200
1400
1600
1800
2000
y
(
m
)
x (m)
Trajectory
c
3
=0
c
3
=0.00002
c
3
=0.2
0 50 100 150 200 250 300
-100
0
100
200
300
400
500
c
r
o
s
s
-
t
r
a
c
k
e
r
r
o
r
(
m
)
time (sec)
(
d
e
g
)
0 50 100 150 200 250 300
0
20
40
60
80
P
r
o
p
e
l
l
e
r
s
p
e
e
d
n
(
R
P
M
)
time (sec)
Figure 4.8: Simulation results of the ship response with dierent penalties on the
propeller speed.
The simulation results employing dierent values of c
3
are illustrated in Figure 4.8.
Figure 4.8 shows that the value of c
3
has signicant eect on the propeller response.
A large value of c
3
will prevent the change of propeller speed. Specially, if the c
3
is extremely large (c
3
= 0.2), the propeller speed almost can not be changed and it
results in the MPC controller using rudder, which is discussed in Section 4.2. It is
also shown in Figure 4.8 that smaller c
3
results in faster path following convergence
speed and smaller maximum roll angle. Because the propeller has more freedom to
slow down the ship speed, which helps to make a faster turn and reduce roll action,
when c
3
is small.
80
Comparisons of One-input (Rudder) and Two-input (Rudder and Pro-
peller) MPC
The MPC designed using multiple reduced-order linear models with rudder and
propeller as inputs is implemented and simulated with the full-order original nonlinear
model S175 and compared with simulations of the one-input case.
First the performance of these two controllers are compared when no roll constraint
is imposed. In this simulation, the gain set G1 is employed and c
3
is set to be zero
to maximize the capability of the propeller speed in changing the vessels forward
speed to aect the roll response. The simulation results are summarized in Figure 4.9.
Figure 4.9 shows that the introduction of additional propeller control helps to enhance
the roll reduction capability when making abrupt turns. Moreover, this improvement
is achieved without compromising the path following convergence speed. When the
vessel makes large turns, the two-input controller predicts that the large roll motion
will happen and thus the propeller speed is slowed down in order to reduce the vessel
forward speed, not signicantly. As the result of forward speed reduction, the vessel
has the capability to make the easier turn while keeping the roll motion small. In this
case, the propeller speed never exceed the initial value because the linear model M
1
is used when the vessel approaches the path, which can not reect the propeller eect
in system responses. If a better prediction model is adopted, which can involve the
dynamic relation between propeller speed and ship states, the propeller speed might
exceed the initial value to make the path convergence speed even faster.
To quantitatively evaluate the controller performance, four performance indices
are introduced, namely maximum roll angle
max
, Root Mean Squares (RMS) roll
angle
RMS
and path convergence time t
con
, with the denitions given by:
RMS
=
1
T
final
_
T
final
0
2
dt, (4.21)
81
(
d
e
g
)
0 50 100 150 200 250 300
0
20
40
60
80
p
r
o
p
e
l
l
e
r
s
p
e
e
d
(
R
P
M
)
time (sec)
Figure 4.9: Comparisons of one-input and two-input MPC performance without roll
constraints.
and path convergence time t
con
is the time the vessel nally approaches the path
(cross-track error less than 10 m). T
final
is the total simulation time, which is 300 sec
in this case.
The summary of these three performance indices without roll constraints is given
82
in Table 4.3, which shows that the roll action of two-input case is largely reduced,
both in maximum and RMS roll angle. The path following performance depends on
which performance index is considered. The two-input case has smaller convergence
time compared with one-input case.
Table 4.3: Comparisons of performance indices for one-input and two-input MPC
without roll constraints.
max
RMS
t
con
[deg] [deg] [sec]
One-input 7.78 1.44 188
Two-input 6.60 1.30 172
Change Percent -15.17 -9.72 -8.51
Furthermore, these two controllers implemented in the original nonlinear system
were compared with roll constraints. In simulations, the maximum allowed roll angle
is set to 5 deg. The corresponding results are summarized in Figure 4.10. As shown in
Figure 4.10, these two controllers both achieve path following while satisfying the roll
constraints. The path following convergence speeds for the two cases are very close.
Although they have the same maximum roll angle, which is due to the constraint
enforcement capability of MPC, the two-input MPC controller has less overall roll
motion because it slows down the vessel forward speed when making the turns.
The performance index comparisons for these two controllers with 5 deg roll con-
straints are summarized in Table 4.4. Table 4.4 shows that the introduction of ad-
ditional propeller control helps to reduce the roll response. However, the two-input
case approaches the path with a slightly slower speed.
Finally, the simulations are performed with tighter rudder saturation to further
compare the performance of one-input and two-input MPC controllers. These results
are shown in Figure 4.11. In the simulations, the maximum rudder angle allowed is
20 deg. Figure 4.11 shows that the two-input MPC controller can eectively reduce
the roll actions compared with the one-input case. Meanwhile, the path following
83
700 800 900 1000 1100 1200 1300 1400 1500
0
200
400
600
800
1000
1200
1400
1600
1800
2000
y
(
m
)
x (m)
Trajectory
one-input
two-input
(
d
e
g
)
0 50 100 150 200 250 300
30
40
50
60
70
p
r
o
p
e
l
l
e
r
s
p
e
e
d
{
R
P
M
)
time (sec)
Figure 4.10: Comparisons of one-input and two-input MPC performance with roll
constraints.
convergence speed is very close. Furthermore, due to the tightened rudder constraints,
the overshoot is observed in both cases.
Table 4.5 presents the performance index comparisons for these two controller with
20 deg rudder saturation. Table 4.5 shows that the advantage of the introduction
84
Table 4.4: Comparisons of performance indices for one-input and two-input MPC
with 5 deg roll constraints.
max
RMS
t
con
[deg] [deg] [sec]
One-input 4.99 1.29 198
Two-input 4.99 1.23 202
Change Percent 0 -4.65 2.02
of the propeller control is more pronounced when there is a tighter rudder limit.
Particularly, the RMS of the roll angle is reduced by 21.60 percent, which is achieved
with almost the same path following performance.
Table 4.5: Comparisons of performance indices for one-input and two-input MPC
with tighter rudder saturations (20 deg).
max
RMS
t
con
[deg] [deg] [sec]
One-input 6.35 1.62 266
Two-input 5.34 1.27 260
Change Percent -15.59 -21.60 -2.26
To sum up all the comparisons, the additional propeller control helps to reduce
the roll response, and this improvement is achieved without compromising the path
following convergence speed. Using a desktop computer with P4 2.4 CPU and 2G
RAM, the optimization problem of two-input MPC with 2 second sampling interval
and 60 step predictive horizon can be solved in about 0.9 second in simulations,
compared to around 0.6 second for one-input case. Real-time implementation should
not be a problem given this moderate computational demand.
4.3.4 Summary
The two-input (rudder and propeller) MPC design of the path following controller
with roll constraints for an integrated model of the surface vessel dynamics and 2-
DoF path following kinematics was presented. Two inputs, namely the rudder angle
85
700 800 900 1000 1100 1200 1300 1400 1500
0
200
400
600
800
1000
1200
1400
1600
1800
2000
y
(
m
)
x (m)
Trajectory
(
d
e
g
)
0 50 100 150 200 250 300
0
50
100
p
r
o
p
e
l
l
e
r
s
p
e
e
d
(
R
P
M
)
time (sec)
Figure 4.11: Comparisons of one-input and two-input MPC performance with tighter
rudder saturation.
and propeller speed, were employed and coordinated to control the vessel. Multiple
3-DoF simplied linear vessel models were adopted in the controller design and a
corresponding 4-DoF nonlinear container model was used in simulations in order
to investigate the interactions between the path following maneuvering control and
86
roll dynamics. The path following performance and roll response were analyzed by
numerical simulations and compared with the one-input (rudder) MPC performance.
The simulations show that the two-input controller has the advantage over the single-
input controller with improved roll response. Moreover, the improvement in the roll
response was achieved without compromising the path following performance in terms
of convergence speed.
87
CHAPTER 5
Path Following with Roll Constraints for Marine
Surface Vessels in Wave Fields
This chapter rst evaluates the standard MPC path following controller using
rudder as the input, developed in Chapter 4, in wave elds by the numerical test-
bed introduced in Chapter 2. Since roll constraint violation and feasibility issues
were found in the evaluation, the mitigating strategies such as gain re-tuning and
constraint tightening and softening are then proposed to guarantee the feasibility of
MPC scheme and satisfaction of roll constraints. The satisfactory performance of the
modied MPC controller is shown by the simulations on the numerical test-bed.
5.1 MPC Controller Evaluation in wave elds
The MPC path following controller using rudder, developed based on the reduced-
order linear ship model in calm water, is implemented and simulated in the numerical
test-bed (original nonlinear model of S175) incorporating the wave eects (described
in Section 2.3) to evaluate the performance. The ship maneuvering model is driven by
the wave forces and moments, together with the control input. In the evaluation, the
propeller speed is maintained as constant (99.50 RPM). The signicant wave height
88
in the simulations is 3.25 m, which corresponds to sea state 5, and kept the same in
all simulations of this chapter.
500 600 700 800 900 1000 1100 1200 1300 1400 1500
0
500
1000
1500
2000
y
(
m
)
x (m)
Trajectory (Np =50, Ts =2 sec, G2)
Without Roll Constraints
20 Deg Roll Constraints
DesiredPath
WaveDirection
Vessel Starting Point
0 50 100 150 200 250
-40
-20
0
20
40
r
u
d
d
e
r
a
n
g
l
e
(
d
e
g
)
0 50 100 150 200 250
-30
-20
-10
0
10
20
30
R
o
l
l
A
n
g
l
e
(
d
e
g
)
time (sec)
Figure 5.1: Simulation results of the ship response with one-input MPC path following
controller in wave elds.
The evaluation results of the one-input MPC path following controller in wave
elds without roll constraints and with 20 degree roll constraints are summarized in
Figure 5.1. In this simulation, the controller gain G2 is employed (see Chapter 4
89
Table 4.1 for the information on G2). Figure 5.1 shows that the proposed MPC
controller achieves path following while satisfying the roll constraints. Without roll
constraints, the extreme roll angle reaches 26 degrees, while the maximum roll angle
with the constraints is 19 degrees.
It is worthy to point out that the proposed MPC controller is addressing the
roll motion induced by maneuvering, not that due to wave impacts, since the model
embedded here can not predict the future wave loads on the vessel. As a result, the
robustness of the standard MPC controller without incorporating the wave eects in
the design is vulnerable. If we further tighten the roll constraints, the constraints
might be violated and thus the feasibility issue emerges. The simulation result with
15 deg roll constraints is shown by Figure 5.2, where the roll constraint violations
can be clearly seen. Furthermore, this feasibility issue can not be solved by simply
introducing additional propeller control in the design. Therefore, research to address
the feasibility of the path following MPC controller in wave elds has been motivated
and the progress is summarized in the remainder of this chapter.
Remark 5.1. In the simulation of MPC with 15 deg roll constraints, which
shown in Figure 5.2, the optimization problem of MPC has no solution to satisfy
all the constraints in many time steps. In such circumstances, we temporarily
remove the roll constraints to avoid the breakdown of the MPC controller so that
the simulation can continue.
5.2 Roll Constraint Satisfaction in wave elds
One of the primary merits of MPC is the input and state constraint enforcement.
The input constraints often come from physical limitations of the actuators and it
is always benecial to enforce them in the control design. Even if these constraints
are not enforced by the control design, they will be satised in the physical imple-
90
500 600 700 800 900 1000 1100 1200 1300 1400 1500
0
500
1000
1500
2000
y
(
m
)
x (m)
Trajectory (Np =50, Ts =2 sec, G2)
20 Deg Hard Roll Constraint
15 Deg Hard Roll Constraint
DesiredPath
WaveDirection
Vessel Starting Point
0 50 100 150 200 250
-40
-20
0
20
40
r
u
d
d
e
r
a
n
g
l
e
(
d
e
g
)
0 50 100 150 200 250
-20
-10
0
10
20
R
o
l
l
A
n
g
l
e
(
d
e
g
)
time (sec)
Figure 5.2: Simulation results of the ship response with one-input MPC path following
controller with 15 deg hard roll constraints in wave elds.
mentation because of the physical constraints of the actuator. However, the state
constraints are normally associated with the safety and device protection, and they
can not be enforced directly in physical implementation (unless additional hardware
is included). If these state constraints are enforced as hard constraints in the op-
timization, the feasibility problem may arise, especially if the system is subject to
91
large disturbances. As an example, the hard roll constraint enforcement results in
infeasibility of the MPC scheme in wave elds, as shown in Figure 5.2.
Although many robust MPC algorithms have been proposed in the open litera-
ture ( [44, 48, 58] and references therein), the industrial applications of such schemes
are still very limited due to their conservatism and computational complexity [58].
The feasibility guarantee in industrial applications is often achieved by re-tuning pa-
rameters [75] and using soft constraint approaches [63, 77]. The latter approach is
by far the most popular one to avoid feasibility issues in real applications. In all
commercial algorithms, the hard state constraints are softened by introducing slack
variables and augmenting the cost function [58]. In this study, these two approaches
have been explored, namely gain re-tuning and constraint softening, to deal with the
roll constraints in wave elds.
5.2.1 Gain Re-tuning for Roll Constraint Satisfaction
In Chapter 4, the guideline for tuning the controller gains, namely c
1
and c
2
, has
been presented to achieve good path following performance in calm water. In that
case, the guidelines were: 1) Set c
2
= 1, and vary c
1
to achieve desired path following
performance; 2) Fix c
1
as selected in 1), vary c
2
to tune for dierent rudder and roll
responses.
Similar guidelines can be used here for gain re-tuning to reduce roll motion in
wave elds, where the feasibility can not always be guaranteed. The selection of c
2
,
the penalty on rudder angle, to achieve the desired roll response is the focus here. On
one hand, the roll constraints are removed from the optimization problem, thus the
feasibility of resulting MPC scheme can be always guaranteed. On the other hand,
the roll constraints are satised by the proper selecting of c
2
. The input constraints
are still enforced to improve the controller performance because they do not induce
the feasibility issue. For this strategy, the roll constraint satisfaction is achieved
92
by trial-and-error. However, such an MPC scheme is still expected to have better
performance than traditional control methodologies such as PID and LQR, because
it considers the input limitations, which are normally neglected in the design process
and imposed afterwards in the traditional control design.
For gain re-tuning strategy, c
1
= 1.6 is adopted, since it achieves good performance
in calm water. By proper selection of the rudder gain c
2
, the roll response of the vessel
is shaped and kept within the desired limits.
The simulation of the re-tuned one-input MPC path following controller is sum-
marized in Figure 5.3. The goal here is to achieve 15 and 10 deg roll constraint
satisfaction, respectively. Figure 5.3 shows that although no roll constraints are en-
forced in the MPC scheme, the roll is reduced by the proper choice of gain c
2
as the
penalty for rudder action. With the value of 5 and 20, respectively, the roll con-
straints of 10 and 15 deg can be satised, respectively. Figure 5.3 also shows that the
roll reduction and roll constraint satisfaction are achieved at the expense of slightly
lower path following convergence speed.
5.2.2 Constraint Softening and Tightening for Roll Constraint
Satisfaction
Soft-constraint MPC has wide and successful applications because it is easy to
implement and there is no feasibility issue. In soft-constraint MPC, violations of
the state constraints are allowed, while an additional term is introduced in the cost
function, which penalizes the constraint violations.
In reality, many constraints can be violated for a short period of time. It is the
long period of sustained violation that causes detrimental problems to the system.
The roll constraint in the marine surface vessel path following problem is of this type.
Slight violation of roll constraints for short periods, caused by big waves, normally
will not deteriorate ship performance greatly, nor endanger the ship safety. But large
93
500 600 700 800 900 1000 1100 1200 1300 1400 1500
0
500
1000
1500
2000
y
(
m
)
x (m)
Trajectory (Np =50, Ts =2 sec, c
1
=1.6)
c
2
=5
c
2
=20
DesiredPath
WaveDirection
Vessel Starting Point
0 50 100 150 200 250
-40
-20
0
20
40
r
u
d
d
e
r
a
n
g
l
e
(
d
e
g
)
0 50 100 150 200 250
-20
-10
0
10
20
R
o
l
l
A
n
g
l
e
(
d
e
g
)
time (sec)
Figure 5.3: Simulation results of the ship response with re-tuned one-input MPC path
following controller in wave elds.
roll motion lasting over time should be avoided [53].
The soft-constraint MPC approach transfers the original cost function (4.8) into
the following form [77]:
J(U
k
, x
k
) =
Np
j=1
( x
T
k+j
Q x
k+j
+
T
k+j1
R
k+j1
+
T
P), (5.1)
94
where P is the constant weighting matrix to penalize the constraint violations. Cor-
respondingly, the original state constraints (4.11) become:
x
max
x
k+j
x
max
+, j = 1, 2, , N
p
. (5.2)
The value of the weighting matrix P determines how soft (or hard) the state
constraints are. Setting P to zero removes the state constraints while increasing P
results in increasingly hard constraints. As discussed in [63], tuning the matrix P
can be counterintuitive because of the mismatch between the predicted states based
on the nominal system and the actual system states. Therefore the matrix P normally
does not serve as the performance parameter to shape the system response [77]. In this
study, P = 10 gives satisfactory performance. If P is too small, say P 1, the state
violation will be too large because it eectively removes the state constraints, while
P is too large, say P 100, it introduces constraints that are too hard resulting
in poor performance, in terms of extremely slow path following convergence. One
advantage of the soft-constraint method is the computational eciency, because just
a single quadratic program needs to be solved. Another advantage is that it normally
will not induce instability [63].
By the introduction of soft state constraints, the feasibility issue of the MPC
optimization is eliminated [63]. However, the roll constraints might be violated.
To achieve roll constraint satisfaction, the constraint tightening technique can be
employed. Normally, the constraint tightening technique requires the knowledge of the
disturbance bound [11, 45]. As an example, in a given sea state, the wave amplitude
can be estimated, thus the bound on the roll angle induced by wave disturbances
can be estimated. With the knowledge of the disturbance bound, the amount of the
state constraints should be tightened can be determined. For example, to make the
maximum roll angle less than 10 degree when the maximum roll angle induced by
95
waves is known to be 5 degrees, the roll constraints can be tightened to 5 degrees.
Initially, the tightened soft constraint MPC scheme tries to make the roll angle within
5 degrees. With the wave disturbance entering the system, the roll angle will probably
violate the 5 degree roll constraints, which is allowed by the soft constraints. However,
the actual roll angle will be still within 10 degrees, the pre-set goal, if proper gains
are adopted.
In order to successfully enforce roll constraints in wave elds, the estimation of
the maximum wave induced roll angle is needed. The wave induced roll response may
be the largest in the beam seas, which is the study focus in this case. The roll angle
history in wave elds without rudder control ( = 0) is shown in Figure 5.4, where
the ship forward speed is maintained at 10 m/s and the wave heading angle is 90
deg. From Figure 5.4, we can see the maximum roll angle is about 5 deg, which is
used to tighten the roll constraints.
500 600 700 800 900 1000 1100 1200 1300 1400 1500
0
500
1000
1500
2000
y
(
m
)
x (m)
Trajectory (Np =50, Ts =2 sec, G2)
10 Deg Soft Roll Constraints
5 Deg Soft Roll Constraints
DesiredPath
WaveDirection
Vessel Starting Point
0 50 100 150 200 250
-40
-20
0
20
40
r
u
d
d
e
r
a
n
g
l
e
(
d
e
g
)
0 50 100 150 200 250
-15
-10
-5
0
5
10
15
R
o
l
l
A
n
g
l
e
(
d
e
g
)
time (sec)
Figure 5.5: Simulation results of the ship response with one-input constraint tightened
and softened MPC path following controller in wave elds.
97
5.3 Summary
By the methods of gain re-tuning and constraint softening and tightening, the path
following with roll constraints was achieved in wave elds. For both cases, the roll
limits were satised at the expense of slight slower path following convergence speed.
While the gain re-tuning technique more or less relies on trial-and-error, the constraint
softening and tightening strategy supplies a more systematic method to achieve the
constraint satisfaction. However, more information about the system, especially the
wave disturbance magnitude, is needed to perform successful constraint tightening.
With the development of sophisticated robust MPC algorithm, this problem can be
attacked from other angles.
98
CHAPTER 6
Disturbance Compensating MPC Scheme:
Development and Applications
This chapter presents a disturbance compensating model predictive control (DC-
MPC) algorithm to satisfy state constraints for linear systems with environmental
disturbances, which has been motivated by state constraint violation and feasibility
issues identied in the evaluation of standard MPC scheme for path following applica-
tion in wave elds. The proposed algorithm focuses on modifying the standard MPC
to satisfy state constraints while achieving good system performance with low addi-
tional computational eort. The capability of the novel DC-MPC algorithm is rst
analyzed theoretically. Then the proposed DC-MPC algorithm is applied to solve the
ship heading control problem and its performance is compared with a time varying
MPC controller. The simulation results show the constraint satisfaction capability
and good performance of the DC-MPC controller. Furthermore, the limitations of
proposed DC-MPC scheme are discussed and future research is suggested.
99
6.1 Motivation
One of the primary reasons for the success of MPC in industrial applications is
its capability in enforcing of various types constraints on the process [63]. However,
it may happen, because of model mismatches or disturbances, that the constrained
optimization problem considered in MPC becomes infeasible at particular time steps.
Namely, no solution can be found that satises all constraints. As an example, wave
disturbances may cause infeasibility of the standard MPC path following controller,
which is shown in Chapter 4.
To address the feasibility issues in MPC applications, numerous studies on robust
MPC have been pursued and the eorts have led to extensive publications in the
literature ( [44, 48, 58] and references therein). The typical Robust MPC approaches
[11,36,45,62,76] often consider bounded disturbances, assuming that they are conned
to a compact set and allowed to take values within the set. However, this assumption
may be conservative in the case where the disturbance dynamics are known or can
be estimated. Aiming to reduce the controller conservativeness by explicit mitigation
of disturbances, [27] estimates the disturbance based on current states and previous
states and controls, and compensates for the disturbance to avoid constraint violation
and conservativeness.
Inspired by the work of [27], this study proposes a novel disturbance compen-
sating model predictive control (DC-MPC) algorithm to guarantee state constraint
satisfaction and successive feasibility for linear systems with environmental distur-
bances while achieving good system performance with low additional computational
eort compared with standard MPC, with an assumption that the disturbance is
measurable.
100
6.2 Problem Statement
Consider a discrete-time linear time-invariant system with disturbances:
x(k + 1) = Ax(k) +Bu(k) +w(k), w W, (6.1)
where x R
no
is the system state, u R
n
i
is the control and w R
no
is an unknown
disturbance taking values in the set W.
The standard MPC considers optimization problem T(x(k)) as follows:
min
u(|k)
Np
j=1
[x(k +j[k)
T
Qx(k +j[k) +u(k +j 1[k)
T
Ru(k +j 1[k)], (6.2)
subject to
x(k +j +1[k) = Ax(k +j[k) +Bu(k +j[k); x(k[k) = x(k), j = 0, 1, , N
p
1, (6.3)
Cx(k +j + 1[k) D, j = 0, 1, , N
p
1, (6.4)
Su(k +j[k) T, j = 0, 1, , N
p
1, (6.5)
where (6.3) is the nominal system dynamic equation used to predict the future states,
(6.4) and (6.5) are general state and input constraints, respectively. Q and R are the
corresponding weighting matrices and N
p
is the predictive horizon. x(k + j[k) and
u(k +j[k) are the state and control, respectively, j steps ahead of the current time k
as a reference, x(k) is the (measured) state at time k.
If the optimization problem T(x(k)) is feasible, then the optimal solution is given
by u
(k[k), u
(k +1[k), , u
(k +N
p
1[k). Accordingly, the predicted optimal
states are x
(k + 1[k), x
(k + 2[k), , x
(k + N
p
[k). For the standard MPC
approach, the control action for the system (6.1) is chosen to be the rst vector in
101
the optimal sequence, i.e.,
u(k) = u
(k[k). (6.6)
With disturbances (w ,= 0), even if the optimization problem T(x(k)) is feasible,
the feasibility of T(x(k+1)) can not be guaranteed if the control action for the system
(6.1) is given by (6.6). More specically, Cx(k + 1) D can not be guaranteed.
One goal of this study is to ensure that Cx(k+1) D is always satised if Cx(k)
D is satised. The other goal of this study is to make the system response with
disturbances as close as possible to the system response without disturbances, since
the performance without disturbances is always satisfactory if the MPC controller is
properly implemented. Mathematically, we want to make x(k + 1) x
(k + 1[k).
6.3 Disturbance Compensating Model Predictive
Control
If we can accurately estimate the future disturbances, even just in the current time
step k, it is possible to improve the system performance by utilizing the additional
disturbance information.
The disturbances at time step k 1, i.e., w(k 1), can be estimated by the
following equation if the state and control are measurable [27]:
w(k 1) = x(k) Ax(k 1) Bu(k 1). (6.7)
When the sampling time T
s
is small and/or the disturbance changes slowly with
time, we can make the following assumption:
Assumption 6.1. The disturbance at time step k, i.e. w(k), can be estimated
by:
w(k) = w(k 1) +, (6.8)
102
where V and V W.
Remark 6.1. If the sampling rate is very fast compared with the disturbance
changing rate, the disturbance variate will be very small and the bound on V
will be much tighter than that for W. One important consideration in selecting
the sampling rate of the MPC is the available computation recourses. Assump-
tion 6.1 is valid for applications where computational resource is not an issue
and fast sampling can be implemented.
With Assumption 6.1, the following disturbance compensating MPC scheme is
proposed:
Step 1: At time step k, calculate the disturbance w(k 1) of the previous time
step k 1 using the equation of (6.7), and measured values of x(k), x(k 1)
and u(k 1).
Step 2: Calculate the disturbance compensation control u by solving the
following low-dimension optimization problem T
( w(k 1)):
min
uR
n
i
[[CBu +C w(k 1)[[, (6.9)
subject to
CBu C w(k 1) E, (6.10)
Su T, (6.11)
where E = max(C) with V . Suppose the corresponding optimal solution
for T
( w(k 1)) is u
(x(k), u
):
min
u(|k)
Np
j=1
[x(k +j[k)
T
Qx(k +j[k) +u(k +j 1[k)
T
Ru(k +j 1[k)], (6.12)
subject to
x(k +j +1[k) = Ax(k +j[k) +Bu(k +j[k); x(k[k) = x(k), j = 0, 1, , N
p
1,
(6.13)
Cx(k +j + 1[k) D, j = 0, 1, , N
p
1, (6.14)
Su(k[k) T Su
, (6.15)
Su(k +j[k) T, j = 1, , N
p
1. (6.16)
Suppose the solution of T
(x(k), u
) is u
(k[k), u
(k +1[k), , u
(k +N
p
(k+1[k), x
(k+2[k), , x
(k+
N
p
[k).
Step 4: Implement the following control to the system (6.1):
u(k) = u
(k[k) + u
. (6.17)
Proposition 6.1. If the optimization problems T
(w(k1)) and T
(x(k), u
)
are both feasible, the state constraint satisfaction, i.e., Cx(k + 1) D, can
always be guaranteed if the control law (6.17) is applied to the linear system
(6.1).
Proof. If the optimization problems T
(x(k), u
) are feasi-
ble, we have the corresponding optimal solutions and the following constraints are
satised:
CBu
Cw(k 1) E, (6.18)
104
Cx
(k + 1[k) D, (6.19)
Su
(k[k) T Su
. (6.20)
From (6.20), it follows that S(u
(k[k) + u
(k[k) + u
) +w(k)
= x
(k + 1[k) +Bu
+ w(k 1) +. (6.21)
Notice that inequality (6.18) and (6.19) are already satised. By adding each side of
them together, we have
CBu
+Cx
(k + 1[k) +CBu
+C w(k 1) +E D. (6.23)
Since E = max(C), C E, then
Cx(k + 1) = Cx
(k + 1[k) +CBu
+C w(k 1) +C
Cx
(k + 1[k) +CBu
+C w(k 1) +E
D. (6.24)
Therefore, the state constraints Cx(k + 1) D are satised.
Remark 6.2. The computational eort needed for the DC-MPC scheme is
105
similar to the standard MPC scheme. In addition to the quadratic programming
(QP) problem (which has the same structure as for the standard MPC) solved in
Step 3, the DC-MPC scheme also solves an m-dimensional optimization problem
in Step 2, where m is the dimension of the control input. Compared with the
QP problem in Step 3, which has a dimension of N
p
n
i
, such a low-dimension
optimization problem does not involve much additional computational cost. As
a result, the proposed DC-MPC scheme is much more computational ecient
than the robust MPC algorithm discussed in [44, 48, 58].
Remark 6.3. The minimization of the cost function in T
( w(k 1)) is
meaningful in terms of tracking the system response achieved without distur-
bances, which is always desired in real applications. Specically, if [[CBu +
C w(k1)[[ = 0 is satised, x(k+1) = x
(x(k), u
(x(k), u
(x(k), u
) might be infeasible.
Another approach, which might be more straightforward than the DC-MPC scheme,
would be to utilize the disturbance information directly in the optimization problem
in Step 3, instead of solving an additional optimization problem. Specically, the
following optimization problem T
j=1
[x(k +j[k)
T
Qx(k +j[k) +u(k +j 1[k)
T
Ru(k +j 1[k)], (6.25)
subject to
x(k + 1[k) = Ax(k[k) +Bu(k[k) +w(k), x(k[k) = x(k), (6.26)
x(k +j + 1[k) = Ax(k +j[k) +Bu(k +j[k), j = 1, , N 1, (6.27)
Cx(k +j + 1[k) D, j = 0, 1, , N 1, (6.28)
Su(k +j[k) T, j = 0, 1, , N 1. (6.29)
The rst element of the optimal sequence of T
_
0.1068 0
1 0
_
_
, B
c
=
_
_
0.0026
0
_
_
. (6.32)
Normally, the rudder saturation has to be enforced due to the physical limit. Fur-
thermore, to avoid abrupt turns, which may induce unexpected ship motion, a yaw
rate limit is considered in the control design. Therefore, the corresponding matrices
108
C, D, S and T are given by:
C =
_
_
1 0
1 0
_
_
, D =
_
_
0.006
0.006
_
_
, S =
_
_
1
1
_
_
, T =
_
_
35/180
35/180
_
_
. (6.33)
The proposed DC-MPC scheme is rst implemented in the linear system with
sinusoidal and constant disturbances to illustrate the constraint satisfaction capability
compared with the standard and TV-MPC scheme. Then the DC-MPC scheme is
applied to the original nonlinear system in wave elds for the performance validation.
6.4.2 Simulation Results: Linear System with Constant and
Sinusoidal Disturbances
Two kinds of disturbances are considered in this case. One is sinusoidal and the
other is constant, which mimic the rst-order and second-order wave disturbances.
In this particular study, the rudder constraints are [[ 35 deg and the yaw rate
constraints are [r[ 0.006 rad/sec (0.34 deg/sec).
The standard MPC scheme is rst studied by the simulations, which are sum-
marized in Figure 6.1. Figure 6.1 shows that although the standard MPC scheme
achieves good performance in calm water in terms of constraint satisfaction and de-
sired heading tracking, the performance of the standard MPC with disturbances is not
satisfactory. First, the yaw constraint violations are found with both constant and si-
nusoidal disturbances. Second, a steady state error exists in the constant disturbance
case, while heading angle oscillations are observed with the sinusoidal disturbance.
The TV-MPC and DC-MPC are also implemented with dierent prediction hori-
zons to study their performance with constant and sinusoidal disturbances. The sim-
ulations of the TV-MPC are summarized in Figure 6.2 and Figure 6.3, while those
of the DC-MPC shown in Figure 6.4 and Figure 6.5, for constant and sinusoidal dis-
109
( w(k
113
(x(k), u
Figure 6.6: Comparisons of the standard MPC, TV-MPC and DC-MPC ship heading
controller.
mance indices for the DC-MPC and TV-MPC under disturbances, both constant and
sinusoidal, is summarized in Table 6.1. It is shown from Table 6.1 that the DC-MPC
scheme has better performance in terms of less steady state and cumulative errors
with constant and sinusoidal disturbances, respectively.
The dierent approaches adopted by the TV-MPC and DC-MPC lead to the per-
formance dierences. The TV-MPC scheme minimizes the cost function based on
115
Table 6.1: Performance index comparisons of DC-MPC and TV-MPC.
N
p
DC-MPC TV-MPC
1 0 19.07
Steady State Error [deg] 2 0 5.16
(Constant Disturbance) 5 0 1.51
20 0 1.38
1 1.6502 2.1637
_
t
final
0
[[dt [degsec10
3
] 2 1.5504 1.9154
(Sinusoidal Disturbance) 5 1.4997 1.6035
20 1.4987 1.5916
the predictions of the nominal system (considering only the disturbance in one time
step), thus the mismatch of the nominal system and real system results in the steady
state error (constant disturbance) or state oscillations (sinusoidal disturbance). In
contrast, the DC-MPC scheme is trying to track the desired no-disturbance perfor-
mance (minimize the distance between the actual states and the predicted stated
without disturbance), which results in steady state error elimination and state oscil-
lations reduction. The DC-MPC algorithm has the potential to be applied to other
motion control problems with environmental disturbances, such as ight, automobile
and robotics controls, since in these cases the system response without disturbances
is always designed to be desirable.
6.4.3 Simulation Results: Nonlinear System with Wave Dis-
turbances
To further validate its performance, the DC-MPC scheme is evaluated in the
numerical test-bed developed in Chapter 2. The simulation results, compared with
the standard MPC without yaw constraints, are summarized in Figure 6.7. In the
simulations, sea state 5 is used and the initial wave heading is 0 deg and the nal
wave heading is 30 deg. Sampling time T
s
= 1 sec and the prediction horizon
116
N
p
= 40. Figure 6.7 shows that the DC-MPC scheme successfully enforces the yaw
rate constraints. The initial course changing speed for the DC-MPC is slower than
the standard MPC without yaw constraints. However, the nal convergence speeds
for both cases are similar.
10
-4
10
-3
10
-2
10
-1
10
0
10
1
-100
-50
0
50
100
M
a
g
n
i
t
u
d
e
(
d
B
)
10
-4
10
-3
10
-2
10
-1
10
0
10
1
-200
-100
0
100
200
frequency (rad/sec)
P
h
a
s
e
(
d
e
g
)
yaw angle
roll angle
Roll Constraints
Figure 6.9: Bode plot from rudder angle to yaw angle and roll angle.
For the path following problem with roll constraints for marine surface vessels, the
roll angle is a constrained state. However, compared with the signicant eect of the
rudder on yaw motion, the roll motion is not as easily inuenced by the rudder input.
A Bode plot from rudder angle to yaw angle and roll angle, as shown in Figure 6.9,
shows that the low authority of rudder on roll motion in the low frequency range
(the vertical line in Figure 6.9 indicates the highest frequency the rudder action can
achieve for S175, namely 5 deg/sec or 0.087 rad/sec). Therefore, the application of
the DC-MPC scheme in path following control for the roll constraint satisfaction in
wave elds will not be successful. In fact, the disturbance eects on roll motion in
wave elds can not be compensated by the limited authority rudder actions.
To address this issue, namely the lack of control authority in constrained states,
one possible approach is to predict the disturbance for longer future time. With
119
enough knowledge of future disturbance, the disturbance can be incorporated into
prediction in predicting of the future states to avoid feasibility issue. However, the
accurate prediction of future disturbance is normally dicult and deserves future
research.
6.6 Summary
The DC-MPC scheme was motivated and developed. An simple disturbance es-
timation method was rst introduced and discussed. The theoretical analysis shows
that DC-MPC can satisfy state constraints and achieve good performance if certain
assumptions hold. The DC-MPC scheme was applied to the ship heading control
with a linear system model and compared with the standard and time-varying MPC.
The simulations show that the DC-MPC can mitigate the drawbacks of the standard
MPC by satisfying the state constraints, eliminating the state error and reducing
the state oscillations. The simulation results also showed the better performance of
the DC-MPC over the TV-MPC scheme. Furthermore, the performance of the DC-
MPC scheme was validated by simulations of original nonlinear system with wave
disturbances. Finally, the limitation of DC-MPC scheme was discussed.
120
CHAPTER 7
Conclusions and Future Work
This dissertation has addressed the path following of marine surface vessels in
wave elds. The control design model, together with the numerical and experimental
test-bed for controller evaluation, has been introduced in Chapter 2. The design,
robustness analysis and evaluation (both numerical and experimental) of the novel
feedback dominance back-stepping for path following without roll constraints have
been presented in Chapter 3. The numerical evaluation and modication of the
FDBS controller in wave elds have been also included in Chapter 3. The model
predictive control approach for path following with roll constraints has been explored
in Chapter 4. Both one-input (rudder) and two-input (rudder and propeller) MPC
schemes have been developed and analyzed by simulations. Chapter 5 has evaluated
the standard MPC path following controller in wave elds, followed by introduction
of mitigating strategies to satisfy roll constraints and guarantee feasibility. The theo-
retical development and application of disturbance compensating MPC scheme have
been nally reported in Chapter 6.
7.1 Conclusions
The main work and results are summarized as follows:
121
Developed a numerical test-bed for ship motion controller evaluation in wave
elds - The numerical test-bed introduced in Chapter 2 combines the ship dy-
namics and both rst- and second-order wave eects on vessels. This numerical
test-bed, established in MATLAB, is generic and can be widely used in many
other ship motion control applications, such as course keeping, roll stabilization
and dynamical positioning.
Designed a novel robust feedback dominance back-stepping path following con-
troller for marine surface vessels - The resulting controller, proposed in Chap-
ter 3, is almost linear, with very benign nonlinearities facilitating analysis and
evaluation. The performance of the nonlinear controller, in terms of path fol-
lowing, has been analyzed for robustness in the presence of model uncertainties.
The simulation results have veried and illustrated the analytic development
and the eectiveness of the resulting control against rudder saturation and rate
limits, delays in the control execution, as well as measurement noise. Further-
more, the control design has been validated by experimental results conducted
in a towing tank using a model ship.
Evaluated and modied the novel robust path following controller in wave elds
- Several issues, such as steady state errors and rudder oscillations, have been
identied in the evaluation by the numerical test-bed, thereby motivating con-
troller modication and gain re-tuning. Mitigating strategies, i.e., gain re-
tuning and gain scheduling, for improving the controller performance has been
proposed and numerically evaluated in Chapter 3. The simulation results
showed that the performance of the modied controller can be substantially
improved in wave elds in terms of steady state error elimination and rudder
oscillation reduction.
Proposed a model predictive control approach for path following with roll con-
122
straints of marine surface vessels - The focus of Chapter 4 was on satisfying
all the input (rudder) and state (roll) constraints while achieving satisfactory
path following performance. The path following performance of the proposed
MPC, both one-input (rudder) and two-input (rudder and propeller), and its
sensitivity to the major controller parameters, such as the sampling time, pre-
dictive horizon and weighting matrices in the cost-function, have been analyzed
by numerical simulations. This study is the rst reported MPC application in
path following for marine surface vessels.
Evaluated and improved MPC path following controller in wave elds - Roll con-
straint violation and feasibility issues have been found in the numerical eval-
uation summarized in Chapter 5, thus motivating the research eort to seek
mitigating strategies for state constraint satisfaction and feasibility guarantee.
By the methods of gain re-tuning and constraint softening and tightening, the
path following with roll constraints has been achieved in wave elds. For both
cases, the feasibility of the MPC scheme was guaranteed and the roll constraints
were satised at the expense of slightly slower path following convergence speed.
Developed a disturbance compensating MPC scheme for state constraint satis-
faction with disturbances - Motivated to overcome the constraint violation and
feasibility issues of the MPC controller for system with disturbances, a novel
DC-MPC algorithm has been proposed in Chapter 6 to guarantee the state
constraint satisfaction in the presence of environmental disturbances. The ef-
fectiveness of the proposed algorithm was rst analyzed theoretically. The state
constraint satisfactory of the DC-MPC scheme was validated by numerical sim-
ulations, i.e., the applications in ship heading control. The DC-MPC scheme
has the potential to be applied to other motion control problems with environ-
mental disturbances, such as ight, automobile and robotics controls.
123
7.2 Future Work
Although substantial progress has been made on the ship motion control, enor-
mous research opportunities as well as challenges still exist in each frontier of this
rapidly evolving eld. The work presented in this dissertation can be usefully ex-
tended in a number of dierent aspects.
Integrating path following with wave measurement and prediction and optimal
path planning
This dissertation discussed path following for marine surface vessels when a pre-
determined path is given. However, the problems of path following and path
planning might be coupled in wave elds. Therefore, path following for marine
surface vessels should be intergraded with wave measurement and prediction
and optimal path planning to improve the vessel performance in wave elds, in
terms of minimizing the time for the vessel to reach a target without violating
vessel motion constraints.
Experimental Validation of the MPC Path Following Controller
Although the MPC path following has satisfactory performance in the numerical
simulations, further validation on established experimental platform is needed
to study real-time implementation issues. Only with the successful experimental
validation, could the feasibility of real commercial or military applications of
MPC path following controller be claimed.
Seakeeping Criteria Development
Only using the maximum roll angle as the performance constraint might not
always be thorough enough, because roll velocity and acceleration could also
induce unpleasant riding experience or cargo damage. However, such guidelines
on acceptable level of roll velocity and acceleration are still lacking, although
124
there are some seakeeping criteria based on statistical measurement, which nor-
mally can not be employed in control design.
Combining Path Following and Roll Stabilization
As discussed in Chapter 6, the rudder action has limited authority in roll motion,
which might result in poor performance in wave elds, especially when wave
elds is very rough. Therefore exploring the use of other control actuators
such as active stabilizing ns and uid tank to reduce the roll motion will be a
rational next step. The coordination of the path following and roll stabilization
aiming at improving the overall ship performance also represents an interesting
direction that advanced control technology can make substantial impact.
Robust MPC Development and Application in Ship Motion Control
In Chapter 5, the path following with roll constraints has been achieved in wave
elds by the methods of gain re-tuning and constraint softening and tightening.
However, the gain re-tuning technique more or less relies on trial-and-error and
the constraint softening and tightening strategy needs a good estimation of the
wave disturbance. Therefore, robust MPC algorithms, which can attack this
problem in a more systematic and sophisticated way, are much needed.
Extension from 2D to 3D Path following
Finally, it is possible to extend the 2D path following for marine surface vessels
to the 3D path following for underwater vehicles, which has attracted a lot
of research interest as exploring the ocean resource becomes more and more
appealing.
125
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