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Nonlinear Model Identification and Observer Design for Thrust
Estimation of Small-scale Turbojet Engines
Affaf Junaid Ahamad Momin1,2, Gabriele Nava1, Giuseppe L’Erario1,3, Hosameldin Awadalla Omer Mohamed1,2,
Fabio Bergonti1,3, Punith Reddy Vanteddu1, Francesco Braghin2, and Daniele Pucci1,3
Abstract— Jet-powered vertical takeoff and landing (VTOL)
drones require precise thrust estimation to ensure adequate
stability margins and robust maneuvering. Small-scale turbojets
have become good candidates for powering heavy aerial
drones. However, due to limited instrumentation available in
these turbojets, estimating the precise thrust using classical
techniques is not straightforward. In this paper, we present
a methodology to accurately estimate the online thrust for the
small-scale turbojets used on the iRonCub - an aerial humanoid
robot. We use a grey-box method to capture the turbojet system
dynamics with a nonlinear state-space model based on the data
acquired from a custom engine test bench. This model is then
used to design an extended Kalman filter that estimates the
turbojet thrust only from the angular speed measurements. We
exploited the parameter estimation algorithm to ensure that
the EKF gives smooth and accurate estimates even at engine
failures. The designed EKF was validated on the test bench
where the mean absolute error in estimated thrust was found
to be within 2% of rated peak thrust.
I. INTRODUCTION
Unmanned Aerial Vehicles (UAVs) feature different types
of propulsion devices ranging from electric-driven propellers
to jet engines. Many modern UAVs employ jet engines
due to their high energy and power densities [
1
], [
2
]. The
iRonCub - a VTOL humanoid robot propelled by four small-
scale turbojets - is a relevant example [
3
], [
4
]. In vehicles
like these, accurate thrust estimation is crucial in order to
guarantee adequate stability margins and perform dynamic
maneuvers. Since direct measurement of in-flight engine
thrust with force sensors is often not feasible due to on-
board complexities, the online thrust is usually calculated
from engine state measurements and engine models. However,
most commercial small-scale turbojets do not feature the
comprehensive instrumentation that is readily available in
full-sized jet engines. Because of limitations in engine state
measurements, estimating the thrust accurately becomes a
challenge. In this paper, we present a methodology to obtain
accurate and robust real-time thrust estimates for small-scale
turbojets using only the live angular speed measurements.
Several physics-based and empirical models of turbojet
engines have been explored in literature [
5
]. While physics-
based models are built exploiting the governing principles of
1
Artificial and Mechanical Intelligence, Istituto Italiano di Tecnologia,
Genova, Italy firstname.surname@iit.it
2
Department of Mechanical Engineering, Politecnico di
Milano, Milan, Italy
francesco.braghin@polimi.it
affafjunaid.momin@mail.polimi.it
3School of Computer Science, Univ. of Manchester, Manchester, U.K.
turbojet [
6
], [
7
], [
8
]; data-driven models are constructed only
from experimental data often disregarding intrinsic system
characteristics [
9
], [
10
], [
11
]. Grey-box or hybrid models that
combine the two approaches have also been explored in [
12
],
[
13
], [
14
]. Basically, all types of engine models use a given
set of measurements like the pressure ratio, fuel flow rate,
etc, to calculate unknown engine states such as the thrust.
Of particular relevance to small-scale turbojets is the work
of L’erario et al [
15
]. They proposed a second-order nonlinear
state-space model to express the thrust dynamics using a
data-driven approach. Their experiment involved applying
input control signals to the turbojet and measuring its thrust
response with a high-bandwidth force sensor in a custom test
bench. However, engine state measurements were not used
in their study. Similar work on a small-scale turbojet engine
was carried out by Jiali et al [16].
Even though turbojet models are abundant in existing
literature, pragmatic approaches to estimate the accurate thrust
considering measurement and model uncertainties are quite
sparse. A data-driven Hammerstein-Wiener model for a UAV
turbojet was proposed in the work of Villarreal-Valderrama
et al [
17
]. They used a linear Kalman Filter to estimate
thrust only from angular speed measurements. However, the
accuracy of the estimator for fast-dynamic maneuvers and its
robustness to engine failures was not investigated.
In this paper, we use an identification technique called
SINDy [
18
] to construct a nonlinear dynamic model for
the turbojet angular speed. We generate key insights into
system characteristics from our experiments on a custom
engine test bench, and incorporate those into the data-driven
model. This model is then used to design a robust EKF that
estimates the accurate online engine thrust only from angular
speed measurements, and exploits the parameter estimation
algorithm to maintain the smoothness and accuracy of thrust
estimates even during events of engine failure.
The paper is organized as follows. Sec. II presents the
notations used, the fundamentals of turbojet physics, and
some technical details of the small-scale turbojet engines
used in this study. Sec. III contains the description of
the experimental setup, the grey-box procedure used for
identifying the nonlinear turbojet models and validation
results. Then, Sec. IV deals with the design of the extended
Kalman Filter to estimate the online turbojet thrust. Sec. V
discusses the performance and validation results of the
designed EKF. Finally, Sec. VI presents the conclusions of
this study and prospects of future work.
arXiv:2205.08330v1 [cs.RO] 17 May 2022
Fig. 1: Turbojet with Centrifugal Compressor. Source [19]
II. BACKGROUND
A. Notation
•x∈Rn- state vector of a system
•u∈Rp- input vector
•y∈Rm- output vector or measurement vector
•Q∈Rn×n- process noise covariance
•R∈Rm×m- measurement noise covariance
•P∈Rn×n- state estimation error covariance
•T- thrust produced by the turbojet [N]
•ω- turbojet angular speed [kRPM]
•u∈[0,100] - input signal to the turbojet
•ma- mass flow rate of air through the turbojet [kg/s]
•Va- volumetric flow rate of air [m3/s]
•ve- velocity of exhaust gas [m/s]
•ρa- air density [kg/m3]
B. Overview of Turbojet Physics
Let us recall the basic principles of the turbojet
engine [
20
], [
21
]. A turbojet is a rotary device that converts
heat energy released by burning fuel into thrust or mechanical
energy. It achieves this by following the Brayton cycle [
22
].
Fig. 1 shows a schematic diagram of a turbojet with
centrifugal compressor.
Assuming a free-stream air inlet velocity of zero and
neglecting the mass flow rate of fuel, the thrust produced by
a subsonic turbojet is given by the following expression [
20
]:
T= ˙mave=ρa˙
Vave.(1)
For a given turbojet compressor operating at a constant
angular speed, the volumetric flow rate of air
˙
Va
depends on
the angular speed [
23
]. The velocity of exhaust
ve
depends
on the angular speed and the fuel flow rate [
20
]. Hence, from
Eq.
(1)
, the thrust produced depends on: the angular speed
of the turbojet, the fuel flow rate, and the air density.
C. Turbojet specifications and architecture
In this study, two different models of small-scale turbojet
engines were used: (a) JetCat P160-RXi-B [
24
], and (b)
JetCat P220-RXi [25].
Both engines have a single-stage centrifugal compressor, a
single-stage axial turbine and an internal fuel pump. Table I
lists the turbojet specifications reported by the manufacturer
at standard atmospheric conditions.
Fig. 2: JetCat Turbojet Engines P160-RXi-B (left) and P220-
RXi (right).
Fig. 3: Internal control scheme of the turbojet
The turbojet is controlled by a digital input control signal
u∈[0,100]
. This input signal is mapped to a desired turbine
angular speed depending on the control mode we choose.
The turbojet has two steady-state control modes, namely:
•
Speed-proportional control where the desired angular
speed is set linearly proportional to u;
•
Thrust-proportional control where the desired angular
speed is set so that the thrust is linearly proportional to
u
.
In both of these modes,
u= 0
corresponds to idle speed
and
u= 100
corresponds to maximum speed. We used the
Thrust-proportional control mode for this study.
A simplified schematic of the internal turbojet control
architecture is shown in Fig. 3. The Engine Control Unit
(ECU) takes
u
as the input signal and maps it to a desired
value of angular speed,
ωdes
. Then, a feedback control loop
alters the fuel flow rate by operating on the pump voltage to
attain the desired angular speed.
III. SYSTEM IDENTIFICATION OF TURBOJET
A. Experimental setup
The experimental test bench setup for this study is the one
used by L’Erario et al. [
15
]. The test bench setup scheme
is presented in Fig. 4. The turbojet is rigidly mounted on a
V-shape that moves on linear bearings. Thrust is measured by
a precision 6-axis Force-Torque (F/T) sensor mounted in the
setup when the V-shape comes into contact with it. The engine
states are measured by internal sensors and streamed into the
data logger. The entire architecture runs at a sampling rate of
100 Hz. Table II contains details on the measured quantities
in the setup. Note that the angular speed measurements are
quantized in steps of 100 RPM.
B. Model structure identification
The objective is to identify a dynamic model with state-
space representation that describes the relationship between
angular speed and input signal of the turbojet. As shown
in Fig. 5, the turbojet was excited with steps, staircases,
TABLE I: Turbojet specifications.
Parameter JetCat P160 JetCat P220
Idle angular speed 33000 RPM 35000 RPM
Max angular speed 123000 RPM 117000 RPM
Thrust at idle speed 7N9N
Thrust at max speed 158 N220 N
Fig. 4: Test bench hardware setup.
sinusoids, chirps (0.05 Hz to 0.5 Hz), and slow ramp signals
in the test bench to collect data for system identification.
From a qualitative analysis of Fig. 5, it is concluded that
a second-order model is an acceptable candidate. Therefore,
the model to be identified is:
¨ω=f(ω, ˙ω, u),(2)
where
ω
,
˙ω
, and
¨ω
are expressed in units of
kRPM
,
kRPM/s
,
and kRPM/s2respectively.
Since the measured angular speed signal is quantized in
steps of 100 RPM, spline interpolation was carried out for
smoothing and calculating its first and second numerical
derivatives. For the thrust measurements, the Savitzky-Golay
filter was used to reduce the noise and calculate the derivative.
We use the SINDy [
18
] tool for model identification. SINDy
is a data-driven sparse identification technique that identifies a
subset of
mlib
functions from a given library of
nlib
functions
using a sequentially thresholded least-squares regression in
order to describe the relationship between input variables
and system dynamics. The library of functions passed to
SINDy can include polynomial combinations of variables or
any other candidate functions. For example, for the
ω−u
dynamic model described in equation (2), it is possible to
use the following function library up to any arbitrary degree
of polynomial:
A={1, ω, ˙ω, u, ω2,˙ω2, u2, ω ˙ω, ω u, ˙ωu, ...}.(3)
The SINDy algorithm then selects the functions from the
given library and computes their coefficients based on the
identification dataset. However, being a data-driven technique,
SINDy has the following issues:
1)
The resulting model is derived only from the
identification dataset. So, the model obtained is heavily
influenced by the training dataset given to SINDy.
2)
If only polynomial terms are chosen for the library, then
no intrinsic information about the system is exploited,
TABLE II: Measured Quantities
Quantity Sensor Units Resolution
Thrust 6-axis F/T N 0.25 N
Rotor Angular Speed Hall effect RPM 100 RPM
Pump Voltage Voltmeter V 0.01 V
Exhaust Gas Temperature Thermocouple oC1oC
0 50 100 150 200 250
0
20
40
60
80
100
20
40
60
80
100
120
Fig. 5: Steps, sinusoid, and chirp Input and Angular Speed
signals
and any relationships or constraints that describe system
characteristics are overlooked.
One way to overcome the above limitations is to incorporate
known system characteristics in the candidate functions. As
discussed in Sec. II, the internal control architecture of the
turbojet maps the input signal
u
to a desired angular speed
ωdes
, which is then achieved by an internal feedback control
loop. Thus, it is worthwhile to investigate the steady-state
characteristics between
ω
and
u
, because this is defined in
the turbojet’s ECU firmware.
From Fig. 6, we see a nonlinear monotonic relationship
between
ω
and
u
at steady state. We would like to model this
relationship with a function
fss(ω, u)=0
. Note that
fss(·)
is
a steady-state expression of the function
f(·)
given in Eq.
(2)
.
We use the function in Eq.
(4)
to fit the data shown in Fig. 6:
fss(ω, u) = ω−a1ub
1−c1= 0.(4)
The constants
a1
and
b1
are identified from regression on
steady-state data as shown in Fig. 6, whereas
c1
is the idle
angular speed of the turbojet in kRPM, i.e., the angular speed
when u= 0. This model fits with a 99.92% R-square value.
So, instead of passing polynomial combinations of
ω
,
˙ω
,
and
u
as candidate terms to SINDy (see Eq.
(3)
), we will
pass the function
fss
, and all polynomial combinations of
terms that necessarily contain
˙ω
. The candidate functions
library will be as follows:
B={fss(ω, u),˙ω , ω ˙ω, u ˙ω, ˙ω2, ω2˙ω, u2˙ω , uω ˙ω, ˙ω3, ...}.(5)
Since the steady-state characteristics are already defined
in the function
fss
, we can use fast dynamic signals (like
sinusoids, and chirps) in the model identification dataset.
The steady-state behaviour will always be preserved in the
dynamic second-order model identified by SINDy as long as
the threshold is carefully tuned. The identification dataset for
SINDy is shown in Fig. 7 for the P220 turbojet. A similar
dataset was used for identifying the P160.
0 20 40 60 80 100
30
40
50
60
70
80
90
100
110
120
Fig. 6: Measured and Modelled angular speed VS input
signal at steady-state.
0 50 100 150 200 250 300 350
0
10
20
30
40
50
60
70
80
90
100
40
50
60
70
80
90
100
110
120
Fig. 7: Identification dataset for the P220 turbojet.
The model structure identified by SINDy is as follows:
¨ω=f(ω, ˙ω, u) = Kss (ω−a1ub1−c1) + Kd˙ω+
+Kwdω˙ω+Kwwdω2˙ω(6)
We also obtain the coefficients (or model parameters)
Kss
,
Kd
,
Kwd
, and
Kwwd
from SINDy. However, these can be
further fine-tuned with parameter estimation EKF as discussed
in the following section.
C. Model parameters estimation with EKF
After obtaining the nonlinear dynamic model in Eq.
(6)
for
angular speed and input signal of the turbojet from SINDy,
we fine-tune the model parameters or coefficients by using the
Extended Kalman Filter algorithm with parameter estimation
on the identification dataset.
The augmented state vector of the dynamic state-space
system is:
x∗
k=ωk˙ωkKss,k Kd,k Kwd,k Kwwd,k >.(7)
The state propagation to the next time step is obtained by
discretizing the model in Eq. (6):
x∗
k+1 =x∗
k+
˙ωk
f(ωk,˙ωk, uk)
[0]4×1
∆t, (8)
yk=ωk.(9)
TABLE III: ω−umodel parameters.
Model Parameter JetCat P160 JetCat P220
a119.36 17.68
b10.3338 0.3332
c133 35
Kss -3.4037 -4.4632
Kw-8.2504 -14.5496
Kwd 0.1365 0.2883
Kwwd -0.0007 -0.00165
TABLE IV: ω−umodel validation.
JetCat P160 JetCat P220 Units
Mean Absolute 1651 1448 [RPM]
Error (1.8%) (1.8%) -
Maximum Error 44167 49730 [RPM]
(49%) (60%) -
The process noise covariance matrix is given by:
Q= Diag 0q1q2q3q4q5.(10)
The noise covariances
q2
,
q3
,
q4
and
q5
are introduced so that
the EKF algorithm can change the values of the parameters
at each time step.
For the initial guesses of model parameters, we use the ones
obtained from SINDy. Then the EKF algorithm is applied
iteratively on the same identification dataset, and the values
of mean absolute error and maximum error between the
modelled and measured turbojet angular speeds are recorded
after each iteration. We keep iterating as long as these error
values decrease with each iteration.
The model parameters for the P160 and P220 are
summarized in Table III. We observe that the model
parameters for the two turbojet engines are in the same
numerical range. This implies that the models are coherent.
The validation data and results of the models are presented
in Fig. 8 and Table IV, respectively. Note that the percentage
errors are reported as fractions of the maximum available
range of turbojet angular speed, i.e., ωmax −ωidle.
D. Angular speed - Thrust Model
We have derived a dynamic model to predict the turbojet
angular speed
ω
from the applied input signal
u
. But since
the ultimate objective is to estimate the turbojet thrust
T
, the
T-ωrelationship has to be investigated.
It was discussed before in Sec. II that the turbojet thrust
depends on its angular speed, fuel flow rate and the air density.
However, for subsonic turbojet with low expansion ratio
nozzles, the angular speed has the most influence on thrust.
Therefore, it is expected that a model relating thrust with
angular speed would give a reasonably accurate prediction of
thrust. This was also concluded by Jiali and Jihong in their
study [
16
]. The
T
-
ω
model is expressed by an empirical
relationship as follows:
T(ω) = a2ωb2+c2,(11)
0 100 200 300 400 500 600
40
50
60
70
80
90
100
110
Fig. 8: Validation results for angular speed model for P220.
30 40 50 60 70 80 90 100 110 120
0
50
100
150
200
250
Fig. 9: Thrust VS Angular speed model for P220.
TABLE V: T−ωmodel coefficients and RMSE.
JetCat P160 JetCat P220
a24.531 ×10−54.928 ×10−5
b23.136 3.205
c24.641 5.477
RMSE 1.20 N 2.05 N
where
T
is in
N
and
ω
is in
kRPM
. The model coefficients
are obtained by performing regression on experimental data
as shown in Fig. 9. The coefficient values and RMSE of the
T−ωmodel are given in Table V.
IV. OBSERVER DESIGN FOR THRUST ESTIMATION
In this section, an Extended Kalman Filter is designed to
estimate the online turbojet thrust by using:
•the angular speed measurements,
•the control input u,
•the ω−udynamic model, and
•the T−ωmodel
A. EKF Design and tuning
In a nutshell, the EKF takes the online angular speed
measurements (which are quantized), smooths them by using
the
ω−u
dynamic process model, and then employs the
T−ωmodel to compute the thrust and thrust derivative.
One obstacle to the implementation of the online EKF is
turbine failure – an occasional event that happens when a
surge of air bubbles enters the fuel line causing the angular
speed and thrust to drop rapidly. This poses an estimation
problem because the
ω−u
dynamic model no longer holds
in this scenario.
Fig. 10: Extended Kalman Filter design schematic.
A simple solution to this problem could be to choose
relatively high covariances for the process noise so that the
EKF gives more weight to the angular speed measurements.
This seems like an attractive choice because the angular speed
measurements are accurate and reliable. However, because
of their quantization, this would lead to noisy estimates
especially for the derivative. Therefore, it seems that we
have to compromise between smoothness and robustness of
the thrust estimates.
B. EKF with parameter estimation
It is possible to simultaneously have both smooth and
robust estimates with the parameter estimation algorithm,
even during engine failures. We have the process model
describing the
ω−u
dynamics in Eq.
(6)
. The parameter
c1
of this dynamic model is the angular speed of the turbojet
in kRPM when the input signal
u= 0
, i.e, the idle angular
speed. Instead of keeping this parameter constant, we can
pass it to the EKF as a state to be estimated online. Thus,
whenever a turbine failure happens and the angular speed
drops suddenly, the EKF observes this and reduces the value
of
c1
. In this way, the estimates of angular speed are smooth
and accurate in any situation. A schematic representation of
the EKF is shown in Fig. 10. The state and measurement
vectors for this EKF are defined as follows:
xk=ωk˙ωkc1,k>,(12)
yk=ωk.(13)
The state transition is given by:
xk+1 =xk+
˙ωk
f(ωk,˙ωk, uk;c1,k)
−Kc1(c1,k −c1)
∆t, (14)
0 100 200 300 400 500 600 700
50
100
150
200
Fig. 11: Validation results for the EKF Thrust estimation for P220.
Fig. 12: EKF performance during engine failure - P160.
where
Kc1
is a non-negative constant,
c1,k
is the online value
of the model parameter
c1
at the
k
-th time step, and
c1
is the
original idle angular speed of the turbojet in kRPM. Notice
that we have imposed first-order dynamics on
c1
in order to
limit its variation by tuning the parameter Kc1.
The process and measurement noise covariance matrices
are tuned with the help of experimental data.
V. OBSERVER VALIDATION
The EKF designed in the previous section was tuned and
deployed on a test bench experiment in order to test its
performance. We have the direct thrust measurements that are
streamed from the F/T sensor in the test bench and filtered by
the Savitzky-Golay filter. The estimated thrust from the EKF
is then compared with the measured and filtered thrust from
the F/T sensor to evaluate the performance and accuracy of
the EKF.
Fig. 11 shows the comparison between the thrust estimated
by the EKF and the one measured by the F/T sensor for the
P220 turbojet. A similar experiment was also run with the
P160 turbojet. The error between EKF estimated thrust and
measured thrust are summarized in Table VI. The percentage
error is calculated as a fraction of the maximum thrust
attainable by the turbojet (see Table I). We observe that the
thrust estimates provided by the EKF are in good agreement
with the F/T measurements.
In Fig. 12, the EKF thrust estimates have been compared
against the measured ground truth thrust during an engine
failure event for the P160 turbojet. The mean absolute error in
estimated thrust was found to be
1.78
N and the max absolute
error was
8.6
N. From this, we conclude that the proposed
EKF generates accurate estimates even during engine failures.
TABLE VI: EKF performance indicators.
Thrust Error JetCat P160 JetCat P220
Mean Absolute 2.52 N 3.96 N
Error (1.7%) (1.9%)
Maximum Error 22.03 N 42.88 N
(13.94%) (19.49%)
VI. RECAP AND CONCLUSIONS
We started by emphasising the importance of accurate and
reliable thrust estimation for jet-powered VTOL drones. The
iRonCub - an aerial humanoid robot that uses four turbojet
engines - is a relevant example that served as the context of
this study. Limitations in engine state measurements of the
turbojets used on the iRonCub posed an obstacle to accurate
thrust estimation using classical techniques. So, we attempted
to solve this problem with a two-step approach.
First, we used a grey-box method to identify the nonlinear
system dynamics of the given small-scale turbojet engines.
From the data and insights collected on our custom turbojet
test bench, we built a second-order nonlinear state-space
model that related the angular speed to the applied input
signal. The mean absolute error of the model obtained was
found to be 1.8% on the validation dataset. We also identified
a static nonlinear model that related the turbojet thrust to its
angular speed.
Second, we designed an EKF that used the nonlinear model
and angular speed measurements to estimate the online thrust.
We exploited the parameter estimation algorithm to ensure
that the EKF generated smooth and accurate estimates even
in cases of turbine failure. It was observed that even for
fast dynamic turbojet control signals, the EKF estimated the
thrust with a high level of accuracy. The mean absolute error
in thrust estimation was found to be less than 2%, even for
scenarios involving engine failure.
Though this study was done in context of the turbojet
engines used on the iRonCub aerial robot, the methodology
presented here can be extended to other kinds of jet engines
and similar nonlinear dynamic systems. Future work would
involve investigating the effects of ambient conditions and free
stream inlet velocity on the given turbojet thrust. Moreover,
high fidelity models that use artificial neural networks
(ANNs) will be explored in order to narrow the gap between
simulations and reality.
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