953

Sensors and Materials, Vol. 31, No. 3 (2019) 953–968
MYU Tokyo
S & M 1827

Online Estimation of Soil Grain Diameter
during Dredging of Hopper Dredger
Using Continuous–Discrete Feedback Particle Filter
Zhen Su,1,2 Zhao-Xu Zhou,2 Meng-Hong Yu,2 Wei Yuan,2 and Jing-Qi Fu1*
1

Department of Automation, College of Mechatronic Engineering and Automation,
Shanghai University, Shanghai 200072, China
2
School of Electronics and Information, Jiangsu University of Science and Technology, Zhenjiang 212003, China(Received November 23, 2018; accepted February 18, 2019)

Keywords: hopper dredge, soil grain diameter, measuring system, mathematical model, estimation,
feedback particle filter, online measurement, sensor

The trailing suction hopper dredger (TSHD) is a ship that excavates sediments from thesea bottom while sailing. Soil properties have a strong effect on the dredging process. The
parameter with the greatest importance is the real-time soil grain diameter dm. This, however,cannot be directly measured by available sensors. In this paper, an alternative method is
proposed to solve this problem. A new estimation method is developed on the basis of an
existing sedimentation model and measuring system data. The loading process with several
grain diameters is simulated to perform a sensitivity analysis of soil type. Simulation results
show that the dredging efficiency is strongly affected by fine soil. This soil-related estimation
problem is solved with a continuous–discrete feedback particle filter (CD-FPF), which is a
recently developed filter for a CD time system. For comparison, a bootstrap particle filter(BPF) is also used to simulate the steplike changes in dm in both the no-overflow and constantvolume loading phases. The results show that the CD-FPF outperforms the BPF in terms ofaccuracy and applicability. Thus, it is recommended to be applied in the estimator of artificial
intelligence (AI) dredging systems.

1.

Introduction

Recent research has shown that smart dredging is an effective approach that can lead tosignificant improvements in production optimization, pollution reduction, and labor saving for
dredging processes. The integrated dredging control system (DCS), developed by the RoyalIHC company, enables the efficient control of the vital dredging process through the use ofadaptive self-learning methods. The artificial intelligence (AI) dredging system,(1) which is thecore structure of the DCS, contains a few estimators and a model predictive controller (MPC) to
provide an optimal control strategy for a given situation.
*

Corresponding author: e-mail: jqfu@staff.shu.edu.cn
https://doi.org/10.18494/SAM.2019.2220
ISSN 0914-4935 © MYU K.K.
https://myukk.org/

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Sensors and Materials, Vol. 31, No. 3 (2019)

The design of the MPC is based on several subsystems such as drag-head excavation, slurrytransport, and hopper sedimentation. In general, the behavior of these systems is strongly
determined by the soil type.(2–4) These systems predict the dredging production by assumingthat the soil characteristics are known. As the soil type changes continuously throughout thedredging cycle, it is often impossible to obtain the parameters associated with soil properties
online by measurements. Instead, a few sophisticated algorithms can be selected to estimate
these immeasurable parameters.(5) The soil properties are estimated by measuring system data.
The measuring system of the hopper dredge consists of pipeline density, pipeline flow, overflow
tube height, draught, and liquid level sensors, and so on (see Figs. 1 and 2).
Most of the dredging-related parameters can be obtained from on-line measurements withthese sensors. Unfortunately, some parameters cannot be obtained by direct measurement.
Moreover, they play a vital role in dredging operations. Therefore, these parameters can only
be estimated using algorithms, models, and data.
One typical application of estimation techniques is the overflow loss estimator,(1,5) which isan AI-based module that is capable of estimating the overflow mixture density ρo and flow rate
Qo. The estimation algorithms that are often used include the unscented Kalman filter (UKF)
and bootstrap particle filter (BPF). It was found that the cascaded observer with the UKF/BPF
for Qo and the BPF for ρo is the best combination and had been implemented in the draught andloading monitor.(6,7) In the case of the drag-head excavation process, more algorithms, suchas the extended Kalman filter (KF), central difference filter, and Gaussian sum filter, were
investigated to solve the problems of cutting estimation and the estimation of cutting and jetting.(8)(a)
Fig. 1.

(a)Fig. 2.

(b)(a) Pipeline density and (b) pipeline flow sensors.

(b)

Hopper mixture level sensor: (a) physical map and (b) installation location of the sensor.

Sensors and Materials, Vol. 31, No. 3 (2019)955

In our previous work, we showed that the pattern search and genetic algorithm can beused to solve the hopper estimation problem. Although these algorithms only provide offline
estimates of soil parameters, the results have shown that both methods are effective and usefulfor different working conditions.
Recently, a real-time soil grain diameter estimator has been developed.(8) Solutions to theestimation problem are obtained by dividing the overall loading process into three separate
phases. The observer of the BPF was designed using the measured values of Qo and ρo for the
no-overflow and constant-volume phases.(9) Moreover, the feedback particle filter (FPF) and
BPF used for estimating immeasurable parameters for the no-overflow phase were compared.(8,10)
The results suggested that the FPF could become a benchmark filter for onboard estimators.
The continuous-FPF is based on mean-field optimal control techniques.(11–13) It applies
a feedback gain structure and avoids the resampling step, which is its main distinction fromthe conventional BPF. Note that the sampling time of the FPF is generally less than 0.05 s toretain the accuracy of outputs.(14) However, the sampling period of the sensors onboard ranges
from 1 to 30 s. Thus, the limitation of the continuous-FPF in terms of sampling rate makes it
unfeasible for the engineering practice of the AI system.
Motivated by the facts that have been presented above, in this paper, we focus on the recently
developed continuous–discrete FPF (CD-FPF),(15) the continuous–discrete version of FPF, toprovide a solution to soil-related estimation problems for the first two loading phases. The
rest of this paper is organized as follows. In Sect. 2, a framework for the online estimation of
the soil grain diameter dm is described. The sensitivity analysis of the effect of the soil grain
diameter dm on loading performance appears in Sect. 3. In Sect. 4, we introduce the CDFPF applied to the CD time nonlinear system. In Sect. 5, we present the results of numericalsimulations and performance comparisons between the CD-FPF and the BPF. We conclude thispaper in Sect. 6.

2.

Estimation Framework

We start by introducing the online estimator based on the sedimentation model to predict theimmeasurable parameters online including the soil grain diameter dm, sand bed height hs, andoverflow density ρo.
The input signals are the incoming flow rate Qi, the density of the incoming mixture, ρi, andthe overflow weir height ho, given by the measuring system (see Fig. 3). The initial soil grain
diameter dm,0, indicating the average soil characteristic of the dredging area, can be calculated
offline. The so-called prior predictions of the total mass mˆ t and height hˆt are calculated withthe state-space equations of the sedimentation model. Once the measurements of mt and ht are
given, dm is calibrated and other immeasurable parameters can be estimated online.

2.1

Sedimentation model

Existing complex sedimentation models provide detailed descriptions of the physicalphenomenon occurring in the hopper.(4,16) These models are based on a large number of partial
differential equations and uncertain environmental parameters, and thus cannot be used in the956

Sensors and Materials, Vol. 31, No. 3 (2019)

Fig. 3.

Process of estimation based on sedimentation model.

AI dredging system. Consequently, we introduce an alternative model, the simplified onedimensional model in the vertical direction (1DV),(17) to describe the sedimentation process.
The sedimentation process can be modeled using the following balance equations:=m t Qi ρi − Qo ρo ,=h (Q − Q ) / S (h ),tiohs = Qs / S (hs ),(1)twhere Qs denotes the settling sand flow rate in the hopper. The horizontal area S depends on
the size of the hopper and is related to the vertical height h [i.e., S = S(h)].
Basically, there are two main methods for loading the hopper, namely, the constant volume
system (CVS) and constant tonnage system (CTS).(18) The CTS has an adjustable overflow weirthat is lowered to keep the tonnage of the mixture constant if the content of the hopper reaches
the maximum tonnage designed. For simplification, we focus on the CVS in this paper, whichmeans that the overflow weir is fixed to make the effective volume of the mixture constant.
The settling sand flow rate Qs, the main factor that affects the sand bed height, is associatedwith the erosion coefficient ke, the terminal settling velocity of particles, vs0, and the hinderedsettling power β as follows.(8,17)
β

2


Qo
Vt − Vs ρ m − ρ w  ρ q − ρ m  ,Qs= [1 −  ]vs 0ht − hs ρ s − ρ m  ρ q − ρ w  ke (ht − hs ) (2)where ρw (1.024 ton/m3) and ρq (2.65 ton/m3) denote the water and quartz densities, respectively.
The sand bed density ρs depends on the in situ soil type. The average density of the mixture
above the sand bed, ρm, can be calculated asρm =mt − ms,
Vt − Vs(3)

Sensors and Materials, Vol. 31, No. 3 (2019)957

where Vt denotes the loading volume. Vs and ms respectively correspond to the volume andmass of the sand bed, which settles at the bottom of the hopper.
Models for the prediction of ρo have been proposed in Ref. 17. Experiments show that thewater-layer model is the most suitable one in terms of practical use. The water-layer model
assumes that a thin water layer is formed on top of the mixture soup layer,(17) formed by upward
water flow caused by the settling of grains. The displacement of a volume of settling grains
invokes the same volume of water to flow in the opposite direction, i.e., the upward water flowrate Qw is equal to the solid flow rate Qsolid:
=
Qw Q=
solid

ρ s − ρmQ.
ρq − ρ w s(4)

The overflow rate Qo is the sum of Qw and the mixture soup flow rate above the sand bed,
Qms; thus, the overflow density ρo is given byρ mQms + ρ wQw
Qw
ρo =
=−
ρm
( ρ m − ρ w ).
Qms + Qw

2.2

Qo

(5)

Soil-type-dependent parameters

Uncertain soil parameters (ke, vs0, and β) that appear in Eq. (2) depend on the properties of
excavated soil. Since it is too difficult to estimate these parameters simultaneously, we need toapproximate them as functions of the soil grain diameter dm. Some functions can be found in
Ref. 17.
424 Rsd d m2
d m ∈ (0,0.1] mm
vs 0 = 8.925( 1 + 95 Rsd d m3 − 1) d m d m ∈ (0.1,1] mm
d m ∈ (1,12.8] mm
87 Rsd d m

β=

(6)

4.7 + 0.41Re0.75
p
1 + 0.175 Re0.75pHere, the relative submerged density Rsd and Reynolds number Rep are given byR=( ρq − ρ w ) / ρ w ,sd
Re p =−2.289 + 41.53d m + 118.6d m2 .

(7)

The erosion coefficient ke is(12)=ke Wsh 8(1 − n) µ gRsd d m / f ,

(8)

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Sensors and Materials, Vol. 31, No. 3 (2019)

where Wsh is the width of the hopper, n denotes the sand bed porosity, and μ is the internal
friction of the sediment. f is the friction factor and g corresponds to the gravitational
acceleration. In general, the accurate calculation of these parameters is not a straightforwardtask. Fortunately, ke can be calculated using the empirical formula as(8,9)=
ke 28.06 d m − 6.35 .
2.3

(9)

Estimation problem

Owing to the time-varying characteristic and high uncertainty of dm, no mathematical model
exists currently to describe the real changes in dm. As a consequence, we model it as a random
walk, which is commonly applied in the stochastic problem(8–10)dm = wdm .

(10)Here, wdm is a Wiener process with the standard deviation σdm.
In this paper, we assume that the sand bed height hs can be measured using the sensors underlaboratory conditions. The values of other parameters, such as Qi, ρi, mt, and ht, are available
through the monitoring system. Considering that the overflow weir height ho is fixed, input andoutput vectors are defined asT=x [ms hs Vs ht mt=
Vt d m ]T ,u [Q=z [mt ht hs ]T .
i ρi ho ] ,The state-space model is given by=
dx1,t Qs ,t ρs ,t dt + dw1,t ,=dx2,t Qs ,t / S ( x2,t )dt + dw2,t ,=dx3,t Qs ,t dt + dw3,t ,dx4,t =
(u1,t − Qo,t ) S ( x4,t )dt + dw4,t ,(11a)

dx5,t =
(u1,t u2,t − Qo,t ρo,t )dt + dw5,t ,dx6,t =
(u1,t − Qo,t )dt + dw6,t ,dx=
0dt + dw7,t7,tz=
x5,t + v1,t ,
1,tz=
x4,t + v2,t ,
2,t

(11b)

z=
x2,t + v3,t ,3,twhere wt and vt correspond to the process noise and measurement noise, respectively. The
settling sand flow rate Qs and overflow density ρo are calculated through the 1DV and waterlayer models, respectively.

Sensors and Materials, Vol. 31, No. 3 (2019)

3.

959

Sensitivity of Soil TypeIn this section, we determine which soil type mainly affects the dredging efficiency.

3.1

Dredging indices

We choose the following four indices to quantify the dredging performance: the sand bed
height hs, the overflow density ρo, the weight of dry solids, TDS, and the sand storage ratio
SSR.
TDSt
=

ρq

t

Q ( ρ − ρ w ) − Qo,τ ( ρo,τ − ρ w )  dτ ,
ρ q − ρ w ∫0  i ,τ i ,τt

∫ Qo,τ ( ρo,τ − ρ w )dτ .
SSRt = 1 − 0t
∫0 Qi,τ ( ρi,τ − ρ w )dτ(12)

The calculations of hs and ρo have been presented in Sect. 2. TDS is a standard index
frequently used in the dredging industry. SSR ranges from 0 to 100%. A low SSR means huge
overflow losses, i.e., poor dredging performance. In contrast, an SSR of 100% indicates that all
the incoming materials remain in the hopper.
3.2

Parameters of loading simulations

The hopper, used for simulating the loading process, is assumed to be a rectangularparallelepiped, approximated with the principal dimensions of the dredger Xin Hai Feng. For
the CVS loading method, Qo is equal to its value in the constant-volume phase (i.e., Qo = Qi
while overflowing). To obtain the true state of simulation, the noise in the process andobservation is considered to be zero. For simplification, Qi and ρi are assumed to be constant.
The initial height of water in the hopper, ht,0, is 4.7 m. The sand bed density ρs is set to1.95 ton/m3. The total simulation time T is set to 90 min. Table 1 shows the vital parameters ofsimulations.
For carrying out the sensitivity analysis, we investigate several soil types with grain sizes
ranging from 0.6 to 1.00 mm (Table 2).

Table 1
Vital parameters of loading simulations.
S (m2) ho (m) ht,0 (m) Qi (m3/s) ρi (ton/m3) ρs (ton/m3) T (min)1080 15.45
4.7
12
1.35
1.95
90

Table 2
Soil grain diameters for simulations.
Soil typedm (mm)Fine
0.06–0.20
Medium
0.20–0.60
0.60–1.00
Coarse960

3.3

Sensors and Materials, Vol. 31, No. 3 (2019)

Sensitivity analysis

Sensitivity analysis is performed to study the effect of soil type on dredging performance.
From Fig. 4, it can be seen that the fine soil with low settling efficiency forms a thin layer
at the bottom of the hopper, resulting in a high initial overflow density. The effect of scour
is clearly visible and ρo increases in the constant-volume phase. The coarse soil with highsettling efficiency leads to a lower mixture density above the sand bed, which means that theinitial overflow density of this soil type is close to that of water. In practice, the loading time
is associated with the soil type: the finer the sand, the longer the loading time. Again, TDS
means the weight of the solids, excluding the pore water and the water on top of the sand bed.
It can be seen that TDS is weakly associated with the soil type in the no-overflow phase, while
it is mainly determined by the overflow density during the constant-volume process. Similarcorrelations with soil type and overflow density are also observed in the case of SSR. Whenρo reaches ρi, the overflow losses become so huge that it is no longer economically feasible tocontinue dredging.
To summarize, the cumulative overflow losses are weakly related to coarse soil. Thus,
the dredging optimization ought to focus on the process of drag-head excavation and slurry

(a)

(b)

(c)

(d)

Fig. 4. Results of dredging evaluations with several dm values: (a) sand bed height, (b) overflow density, (c) TDS,
and (d) sand storage ratio.

Sensors and Materials, Vol. 31, No. 3 (2019)

961

transport. For fine/medium soil, the overflow losses are huge and the optimization of thesedimentation process is indispensable. The particle size distribution of this soil type has aconsiderable effect on the dredging operation. Therefore, it is extremely important to estimate
the soil grain diameter in this case.

4.

Estimation Method

In this section, we introduce the CD-FPF to be applied in engineering practice.
Suppose that the CD time system is=
dxt ft ( xt , ut )dt + d β t ,(13)=zn h( xtn ) + en ,where xt ∈ R d is the state at the time t, which evolves in the form of continuous time, zn ∈ R m
is the discrete-time measurement obtained at the time tn, and ut is the input signal. Moreover,f and h are the state and measurement functions determined using the specified model, and β
and e are the process noise and measurement noise, respectively. Both the process noise andthe measurement noise are assumed to exhibit Gaussian distributions with the zero mean andcovariance matrices Q and R. The objective of filtering is to obtain the posterior distribution ofxt given the observation history ℒ t : = σ{zn: ∀ n, tn ∈ [0,t]}.
Distinguishing from the continuous-FPF, the CD-FPF is a controlled system that transforms
the discrete-time Bayesian measurement update to a continuously evolving process. It works in
two steps: for n = 1, 2, 3, …,
Step 1: Continuous-time evolution of the state equation for t ∈ [tn−1, tn);
Step 2: Discrete-time Bayesian update at the discrete time t = tn.
For t ∈ [tn−1, tn), the evolution of the i-th particle is described as
dxti = ft ( xti , ut )dt +d β ti ,(14)

for i = 1, …, N, where N is the number of particles. The initial state xtin −1 is assumed to be
known and xti− is defined as the right limit of xti : xti− = lim xti.
n

n

t →tn

At the discrete time tn when the n-th measurement zn is given, a particle flow Sni (λ ) is
introduced as
dSni (λ )= K ni (λ ) I ni (λ ),
dλ

(15)

where λ ∈ [0,1] is the pseudo-time variable. The initial condition is set as Sni (0) = xti− , andnthe initial state for the next time interval xtin is set as xtin = Sni (1). K ni (λ ) is the feedback gain

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Sensors and Materials, Vol. 31, No. 3 (2019)

function and the innovation error I ni (λ ) is given by1
I ni (λ ) =zn −  h( Sni (λ )) + hˆ∗ (λ )  ,

2

(16)

N

where hˆ∗ (λ ) := (1 / N )∑ i =1 h( Sni (λ )).

It can be seen that Eq. (16) is similar to the innovation process that appears in the classicalKF. Actually, the CD-FPF provides a generalization of the KF to a general class of continuous–
discrete nonlinear problems. The solution of the feedback gain function K ni (λ ) is the central
problem of the CD-FPF algorithm. In practical use, K ni (λ ) is generally approximated withnumerical algorithms instead of being computed explicitly. In this paper, the constant gain
approximation is chosen for the solution to K ni (λ ),(13) which means that each particle has thesame feedback gain function=
K n (λ )i

K d ×m ]n (λ )[=1 N j ∑ Sn, p (λ )[h( Snj,q (λ )) − hˆq (λ )]  R −1 ,N  j =1
(17)

for p = 1, 2, …, d and q = 1, 2, …, m. Recall that d and m are the dimensions of the state andobservation vector, respectively, and the constant gain function K n (λ ) is a d × m matrix.
To simulate the CD-FPF, we use the Euler discretization scheme for both the real time stepΔt and the pseudo time step Δλ. Note that the gain function should be approximated at eachstep of λ according to Eq. (17). In fact, the algorithm for λ is conceptually the same as that
described for the t of the continuous-FPF.(15) The general CD-FPF is presented in Algorithm 1.

5.

Numerical Simulations

In this section, we provide the estimates of the soil dm obtained with the CD-FPF. For
comparison, we consider the conventional BPF and compare the performance characteristics ofthe two filters. More information about the BPF can be found in Ref. 19 and details are omitted
here to save space.
5.1

Setting

To simulate the dredging operation, the sampling period Δtn of filters ranges from 1 to 10
s. The real-time step Δt and pseudo time step Δλ of the CD-FPF are both set to 0.05 s. For the
CVS loading simulation, true states are generated without the process noise. The BPF applies
the systematic resampling algorithm. For the application of filters onboard, the cost of thecomputation of filters should be considered. Therefore, for the BPF, we use 100 particles, while
we use 20 particles in the experiments associated with the CD-FPF.
In general, the measurement noise is mainly determined by the accuracy of the sensor.
On the other hand, the process noise in system (11a) is crucial for filters to have a chance ofSensors and Materials, Vol. 31, No. 3 (2019)

963

Algorithm 1. Implementation of CD-FPF.
Input: f, h, u, z, Q, R, x0, Δt, ΔλOutput: x
Initialize:
Set t = 0, n = 1
for i = 1, 2, …, N do
Generate a new particle: x0i ∼ x0
end for
Step 1: continuous-time evolution
Set t = tn−1
while t ≤ tn dofor i = 1, 2, …, N do
Generate a sample: ΔV ~ N(0, Q)
xti + ft ( xti , ut )∆t + ∆t ∆V
Calculate: xti + ∆t =end fort = t + Δt
end while
Step 2: discrete-time update
Set λ = 0
for i = 1, 2, …, N do
Set Sni (0) = xti
end for
while λ ≤ 1 doNCalculate: hˆ∗ (λ ) := (1 / N )∑ h( Sni (λ ))i =1for i = 1, 2, …, N do
Calculate: I ni (λ ) =zn − (1 / 2)[h( Sni (λ )) + hˆ∗ (λ )]Approximate: Kn(λ) through Eq. (17)Calculate: Sni (λ + ∆λ )= Sni (λ ) + K n (λ ) I ni (λ )∆λend forλ = λ + Δλend whilefor i = 1, 2, …, N do
Set xti = Sni (1)end forN
Set xt = (1 / N )∑ i =1 xti
n=n+1
return back Step 1

estimating dm. Recall that coarse soil is very weakly sensitive to the dredging performance;
hence, we focus on fine/medium soil where the grain diameter is in the range of dm ∈ [0.1, 0.3]
mm. The process noise of the grain diameter, w7, is selected in several cases to determine thesensitivity of the filters to it. Other parameters of the filters are presented in Table 3.
Owing to the switching nature of the loading process, we split the sedimentation problem
into two phases. In each phase, we consider a scenario where the dredger crosses over an area
where the soil type changes rapidly, with the change in dm set to be steplike. To investigate therobustness of both filters with respect to initial conditions, the initial grain diameters of thefilters are set with an offset of +0.10 mm in each case, which reflects that the off-line calculated
grain diameter deviates from its real value.

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Sensors and Materials, Vol. 31, No. 3 (2019)

Table 3
Parameters of filters.
Number ofFilterparticles, N
BPF
100
CD-FPF20

5.2

Resamplingalgorithm
Systematic
None

Process noise w
w1/ton
10

w2/m
0.005

3

w3/m
5

Observation noise v
w7/mm
0.005–0.020

v1/ton
1

v2/m
0.05

v3/m
0.10

Results and discussion

The performance metrics of the filters we use are the root-mean-square error (RMSE) of dm
over the time tn and the time-averaged RMSE=
RMSEtn1 N i( xˆ7,tn − x7,tn ) 2 ,∑N i =1
1 T 1 N i
RMSE =( xˆ7,tn − x7,tn ) 2 dt .
∑∫T tn =0  N i =1


(18)

i
Here, x7,tn denotes the true simulated state of dm and xˆ7,
tn corresponds to the estimate at thetime tn for the i-th particle.
Through the simulation of steplike changes in the true state of the soil dm, it is possible
to compare the performance characteristics of the filters for both loading processes.
Figures 5 and 6 show the estimates with the process noise w7 = 0.005 and 0.010 mm, where Δtn = 1 s.
Figure 7 depicts the results obtained with the sampling time Δtn = 2 s, where w7 = 0.005 mm.
The RMSEtn values of the two filters with the process noise w7 = 0.005 and 0.010 mm arepresented in Figs. 8 and 9, respectively. For the constant-volume phase, both the BPF and
the CD-FPF perform similarly and produce highly accurate estimates. For the no-overflow
phase, the RMSEtn values of the CD-FPF are significantly smaller than those of the BPF, whichsuggests that the estimates obtained with the CD-FPF are more accurate than those obtainedwith the BPF. At the beginning of the loading, the CD-FPF converges faster to the true statethan the BPF does. With the marked changes in true dm, we can see that the estimates of the
CD-FPF are faster in tracking the true state of the system. Table 4 shows a summary of the
RMSEs of the filters with more values of w7, where all sampling times are set to be 1 s. The
RMSEs of both filters are similar for the constant-volume phase. For the no-overflow phase,the estimates of the BPF are sensitive to w7, while the accuracy of the estimates produced with
the CD-FPF is nearly constant regardless of the choice of w7.
From Figs. 6–10, one can observe that the estimate errors of the BPF are clearly associatedwith the sampling time Δtn for both loading phases. Table 5 gives the summary of RMSEs withsampling periods ranging from 1 to 10 s, where w7 = 0.005 mm. It can be seen that the higher
the frequency of measurements, the higher the accuracy of estimates obtained with the filters.
As the sampling time grows, the estimate error increases. If the sampling time is more than 2 s,
the estimates are relatively noisy and divergent for the BPF, while the RMSEs of the CD-FPF

Sensors and Materials, Vol. 31, No. 3 (2019)(a)Fig. 5.
phase.

(b)

Estimates of filters with w7 = 0.010 mm and Δtn = 1 s: (a) no-overflow phase and (b) constant-volume(a)Fig. 7.
phase.

(b)

Estimates of filters with w7 = 0.005 mm and Δtn = 1 s: (a) no-overflow phase and (b) constant-volume(a)Fig. 6.
phase.

965(b)

Estimates of filters with w7 = 0.005 mm and Δtn = 2 s: (a) no-overflow phase and (b) constant-volume966

Sensors and Materials, Vol. 31, No. 3 (2019)(a)Fig. 8.
phase.

RMSEtn values of filters with w7 = 0.005 mm and Δtn = 1 s: (a) no-overflow phase and (b) constant-volume(a)Fig. 9.
phase.

(b)(b)

RMSEtn values of filters with w7 = 0.010 mm and Δtn = 1 s: (a) no-overflow phase and (b) constant-volume

Table 4
Summary of RMSE values with four choices of w7 (Δtn = 1 s).
No-overflow phase
Algorithm
0.005 mm 0.010 mm 0.015 mm 0.020 mm
BPF
0.15920.08620.05140.0682
CD-FPF
0.04750.02970.02050.0309Constant-volume phase
0.005 mm 0.010 mm 0.015 mm 0.020 mm
0.01710.01490.01510.01690.01150.01140.01010.0111(a)(b)

Fig. 10. RMSEtn values of filters with w7 = 0.005 mm and Δtn = 2 s: (a) no-overflow phase and (b) constant-volume
phase.

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967

Table 5
Summary of RMSEs with four choices of Δtn (w7 = 0.005 mm).
No-overflow phase
Algorithm
1s
2s
5s
10 s
BPF
0.15920.18080.22060.2209CD-FPF
0.04750.06380.09880.11151s
0.01710.0115Constant-volume phase2s5s0.0181
0.09880.01430.014710 s
0.11000.0161are nearly constant in the constant-volume phase. For the no-overflow phase, the estimates of
the CD-FPF are within an acceptable range.

6.

Conclusion

In this paper, we introduce an estimator of the soil grain diameter based on a 1DV
sedimentation model and measuring system data. To determine which excavated soil mainly
affects the dredging operation, we carry out the sensitivity analysis of soil type, which showsthat the dredging performance is sensitive to medium/fine soil.
To overcome the estimation problem, we employ the CD-FPF used in a continuous–discrete
nonlinear system. To compare the performance characteristics of the BPF and CD-FPF, weconsider a steplike change in grain diameter, which models the marked changes in soil type inthe dredging operation. Numerical simulations show that, in general, the estimates producedby the CD-FPF are more accurate than those obtained with the BPF, and that the CD-FPF
converges faster to the true state than the BPF does. On the other hand, the choice of process
noise w7 strongly affects the accuracy of the BPF, especially for the no-overflow phase. At alow signal sampling frequency, the estimates of the BPF diverge from the true state, while thoseof the CD-FPF maintain excellent estimation accuracy.
From simulation results, we conclude that the CP-FPF provides a satisfactory solution to
the soil-related estimation problem and is suggested to be applied in an AI dredging system infuture development.
Note that the loading method we used is the CVS; practically, modern dredgers have beenequipped with a constant-tonnage controller to discharge a low-density mixture before theloading operation ends. Therefore, it is necessary to provide a solution to the estimation
problem of the constant-tonnage loading process. Moreover, in this paper, we consider steplikechanges in soil type during dredging in a real operation; the changes in soil type are smootherand the performance of the CD-FPF should be investigated in this case.
As mentioned previously, the performance of the CD-FPF is strongly determined by theselected feedback gain function. The improvement in estimation accuracy may result in
degraded computational efficiency and the best approach to solving this problem will be studiedin our future research.

Acknowledgments
This research was supported by a grant (No.17511107002) from the Science and Technology
Commission of Shanghai Municipality of China and Natural Science Fund for Colleges andUniversities in Jiangsu Province (No. 18KJB580005).

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Sensors and Materials, Vol. 31, No. 3 (2019)

Author Contributions
Z. Su analyzed the data and wrote most of the manuscript, including quality assurance andcontrol; Z. X. Zhou contributed to writing the manuscript; M. H. Yu and W. Yuan helped torevise the manuscript; J.Q. Fu designed the study.

Conflicts of Interest
The authors declare no conflict of interest.

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