PreprintPDF Available
Preprints and early-stage research may not have been peer reviewed yet.

Abstract and Figures

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.
Content may be subject to copyright.
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.
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 [
], [
]. The
iRonCub - a VTOL humanoid robot propelled by four small-
scale turbojets - is a relevant example [
], [
]. 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 [
]. While physics-
based models are built exploiting the governing principles of
Artificial and Mechanical Intelligence, Istituto Italiano di Tecnologia,
Genova, Italy
Department of Mechanical Engineering, Politecnico di
Milano, Milan, Italy
3School of Computer Science, Univ. of Manchester, Manchester, U.K.
turbojet [
], [
], [
]; data-driven models are constructed only
from experimental data often disregarding intrinsic system
characteristics [
], [
], [
]. Grey-box or hybrid models that
combine the two approaches have also been explored in [
], [
]. 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 [
]. 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 [
]. 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
] 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]
A. Notation
xRn- state vector of a system
uRp- input vector
yRm- output vector or measurement vector
QRn×n- process noise covariance
RRm×m- measurement noise covariance
PRn×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 [
], [
]. 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 [
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 [
T= ˙mave=ρa˙
For a given turbojet compressor operating at a constant
angular speed, the volumetric flow rate of air
depends on
the angular speed [
]. The velocity of exhaust
on the angular speed and the fuel flow rate [
]. Hence, from
, 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 [
], 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
. 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
In both of these modes,
u= 0
corresponds to idle speed
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
as the input signal and maps it to a desired
value of angular speed,
. Then, a feedback control loop
alters the fuel flow rate by operating on the pump voltage to
attain the desired angular speed.
A. Experimental setup
The experimental test bench setup for this study is the one
used by L’Erario et al. [
]. 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)
, and
are expressed in units of
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 [
] tool for model identification. SINDy
is a data-driven sparse identification technique that identifies a
subset of
functions from a given library of
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
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:
The resulting model is derived only from the
identification dataset. So, the model obtained is heavily
influenced by the training dataset given to SINDy.
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
Fig. 5: Steps, sinusoid, and chirp Input and Angular Speed
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
to a desired angular speed
, which is then achieved by an internal feedback control
loop. Thus, it is worthwhile to investigate the steady-state
characteristics between
, because this is defined in
the turbojet’s ECU firmware.
From Fig. 6, we see a nonlinear monotonic relationship
at steady state. We would like to model this
relationship with a function
fss(ω, u)=0
. Note that
a steady-state expression of the function
given in Eq.
We use the function in Eq.
to fit the data shown in Fig. 6:
fss(ω, u) = ωa1ub
1c1= 0.(4)
The constants
are identified from regression on
steady-state data as shown in Fig. 6, whereas
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
as candidate terms to SINDy (see Eq.
), we will
pass the function
, and all polynomial combinations of
terms that necessarily contain
. The candidate functions
library will be as follows:
B={fss(ω, u),˙ω , ω ˙ω, u ˙ω, ˙ω2, ω2˙ω, u2˙ω , ˙ω, ˙ω3, ...}.(5)
Since the steady-state characteristics are already defined
in the function
, 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
Fig. 6: Measured and Modelled angular speed VS input
signal at steady-state.
0 50 100 150 200 250 300 350
Fig. 7: Identification dataset for the P220 turbojet.
The model structure identified by SINDy is as follows:
¨ω=f(ω, ˙ω, u) = Kss (ωa1ub1c1) + Kd˙ω+
We also obtain the coefficients (or model parameters)
, and
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.
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:
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):
k+1 =x
f(ωk,˙ωk, uk)
t, (8)
TABLE III: ωumodel 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: ωumodel 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
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
from the applied input signal
. But since
the ultimate objective is to estimate the turbojet thrust
, 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 [
]. The
model is expressed by an empirical
relationship as follows:
T(ω) = a2ωb2+c2,(11)
0 100 200 300 400 500 600
Fig. 8: Validation results for angular speed model for P220.
30 40 50 60 70 80 90 100 110 120
Fig. 9: Thrust VS Angular speed model for P220.
TABLE V: Tωmodel coefficients and RMSE.
JetCat P160 JetCat P220
a24.531 ×1054.928 ×105
b23.136 3.205
c24.641 5.477
RMSE 1.20 N 2.05 N
is in
is in
. The model coefficients
are obtained by performing regression on experimental data
as shown in Fig. 9. The coefficient values and RMSE of the
Tωmodel are given in Table V.
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 ωudynamic model, and
the Tωmodel
A. EKF Design and tuning
In a nutshell, the EKF takes the online angular speed
measurements (which are quantized), smooths them by using
dynamic process model, and then employs the
Tωmodel 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
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
dynamics in Eq.
. The parameter
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
. 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:
The state transition is given by:
xk+1 =xk+
f(ωk,˙ωk, uk;c1,k)
Kc1(c1,k c1)
t, (14)
0 100 200 300 400 500 600 700
Fig. 11: Validation results for the EKF Thrust estimation for P220.
Fig. 12: EKF performance during engine failure - P160.
is a non-negative constant,
is the online value
of the model parameter
at the
-th time step, and
is the
original idle angular speed of the turbojet in kRPM. Notice
that we have imposed first-order dynamics on
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.
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
N and the max absolute
error was
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%)
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.
D. Pucci, S. Traversaro, and F. Nori, “Momentum control of an
underactuated flying humanoid robot,” IEEE Robotics and Automation
Letters, vol. 3, no. 1, pp. 195–202, 2017.
“Fusion flight - jetquad,” 2022. [Online]. Available: https://fusionflight.
G. Nava, L. Fiorio, S. Traversaro, and D. Pucci, “Position and attitude
control of an underactuated flying humanoid robot,” in 2018 IEEE-
RAS 18th International Conference on Humanoid Robots (Humanoids).
IEEE, 2018, pp. 1–9.
H. A. O. Mohamed, G. Nava, G. L’Erario, S. Traversaro, F. Bergonti,
L. Fiorio, P. R. Vanteddu, F. Braghin, and D. Pucci, “Momentum-based
extended kalman filter for thrust estimation on flying multibody robots,”
IEEE Robotics and Automation Letters, vol. 7, no. 1, pp. 526–533,
H. Asgari, X. Chen, and R. Sainudiin, “Modelling and simulation of
gas turbines,” Int. J. of Modelling, vol. 20, pp. 253 270, 01 2013.
D. Klein and C. Abeykoon, “Modelling of a turbojet gas turbine
engine,” in 2015 Internet Technologies and Applications (ITA), 2015,
pp. 200–206.
J. Göing, A. Kellersmann, C. Bode, and J. Friedrichs, System Dynamics
of a Single-Shaft Turbojet Engine Using Pseudo Bond Graph, 01 2020,
pp. 427–436.
I.-C. Andrei, A. Toader, G. Stroe, and F. Frunzulica, “Performance
analysis and dynamic modeling of a single-spool turbojet engine,” AIP
Conference Proceedings, vol. 1798, no. 1, p. 020005, 2017. [Online].
N. Chiras, C. Evans, and D. Rees, “Nonlinear gas turbine
modeling using feedforward neural networks,” American Society of
Mechanical Engineers, International Gas Turbine Institute, Turbo Expo
(Publication) IGTI, vol. 2, 01 2002.
H. Asgari, X. Chen, M. B. Menhaj, and R. Sainudiin, “Artificial Neural
Network–Based System Identification for a Single-Shaft Gas Turbine,
Journal of Engineering for Gas Turbines and Power, vol. 135, no. 9, 07
2013, 092601. [Online]. Available:
A. Ruano, P. Fleming, C. Teixeira, K. Rodriguez, and C. Fonseca,
“Nonlinear identification of aircraft gas-turbine dynamics,”
Neurocomputing, vol. 55, pp. 551–579, 10 2003.
R. M. Catan˘
a, G. Dediu, C. M. T˘
abîc, and H. M.
“Performance calculations of gas turbine engine components based on
particular instrumentation methods,” Applied Sciences, vol. 11, no. 10,
2021. [Online]. Available:
O. Lyantsev, A. Kazantsev, and A. Abdulnagimov, “Identification
method for nonlinear dynamic models of gas turbine engines
on acceleration mode,” Procedia Engineering, vol. 176, pp.
409–415, 2017, proceedings of the 3rd International Conference
on Dynamics and Vibroacoustics of Machines (DVM2016) June
29–July 01, 2016 Samara, Russia. [Online]. Available: https:
E. Mohammadi and M. Montazeri-Gh, “A New Approach to the
Gray-Box Identification of Wiener Models With the Application of
Gas Turbine Engine Modeling, Journal of Engineering for Gas
Turbines and Power, vol. 137, no. 7, 07 2015, 071202. [Online].
G. L’Erario, L. Fiorio, G. Nava, F. Bergonti, H. A. O. Mohamed,
E. Benenati, S. Traversaro, and D. Pucci, “Modeling, identification and
control of model jet engines for jet powered robotics, IEEE Robotics
and Automation Letters, vol. 5, no. 2, pp. 2070–2077, 2020.
Y. Jiali and Z. Jihong, “Dynamic modelling of a small scale turbojet
engine,” in 2015 European Control Conference (ECC), 2015, pp. 2750–
F. Villarreal-Valderrama, C. Santana-Delgado, P. Robledo, and
L. Amezquita-Brooks, “Turbojet direct-thrust control scheme for full
envelope fuel consumption minimization, Aircraft Engineering and
Aerospace Technology, vol. ahead-of-print, 12 2020.
S. L. Brunton, J. L. Proctor, and J. N. Kutz, “Discovering
governing equations from data by sparse identification of nonlinear
dynamical systems,” Proceedings of the National Academy of
Sciences, vol. 113, no. 15, pp. 3932–3937, 2016. [Online]. Available:
“Centrifugal turbojet engine,” 2009. [Online]. Available: https:
R. D. Flack, Fundamentals of Jet Propulsion with Applications, ser.
Cambridge Aerospace Series. Cambridge University Press, 2005.
N. Cumpsty and A. Heyes, Jet Propulsion: A Simple Guide to the
Aerodynamics and Thermodynamic Design and Performance of Jet
Engines, 3rd ed. Cambridge University Press, 2015.
C. Wu, Thermodynamics and Heat Powered Cycles: A Cognitive
Engineering Approach. Nova Science Publishers, 2007. [Online].
Available: MghCpQC
R. Brown, “Fan laws, the user and limits in predicting centrifugal
compressor off design performance. 1991.
“Jetcat p160-rxi-b,” 2021. [Online]. Available:
[25] “Jetcat p220-rxi,” 2019. [Online]. Available:
ResearchGate has not been able to resolve any citations for this publication.
Full-text available
This paper presents an analytical method to determine various main parameters or performances of engine components when those parameters cannot be directly measured and it is necessary to determine them. Additionally, some variants of instrumentation methods are presented, for example: engine inlet, compressor, turbine or jet nozzle instrumentation. The purpose of the instrumentation methods is to directly measure the possible parameters, which are then used as inputs in a model to determine other parameters or performance metrics. This model is based on gasodynamic process equations, and it is used to compute the air and gas parameters, such as enthalpy and entropy, which are described in polynomial form, thus leading to a more realistic calculation. At the end, this paper presents a practical example of instrumentation applied on a Klimov TV2-117A turboshaft, with a series of experimental results, following the engine testing on the test bench.
Full-text available
The paper contributes towards the modeling, identification, and control of model jet engines. We propose a nonlinear, second order model in order to capture the model jet engines governing dynamics. The model structure is identified by applying sparse identification of nonlinear dynamics, and then the parameters of the model are found via gray-box identification procedures. Once the model has been identified, we approached the control of the model jet engine by designing two control laws. The first one is based on the classical Feedback Linearization technique while the second one on the Sliding Mode control. The overall methodology has been verified by modeling, identifying and controlling two model jet engines, i.e. P100-RX and P220-RXi developed by JetCat, which provide a maximum thrust of 100 N and 220 N, respectively.
Full-text available
The form and identification method of nonlinear dynamic gas turbine models on acceleration mode on the base of experimental transition processes is proposed. The nonlinear dynamic model of a gas turbine engine is represented as a system of nonlinear differential equations as in the normal form of Cauchy. The parameters of the system of differential equations are presented as a function of the acceleration parameter α, and the identification method is based on the determination of parameter α using numerical optimization method, allowing to simulate an acceleration mode with high accuracy. The experimental data are approximated by cubic splines for reduction of identification errors. The developed fast-calculating real-time nonlinear dynamic model is approved to implement in the software for hardware-in-the-loop test-beds and in the on-board software for automatic control systems of gas turbines.
Full-text available
The ability to discover physical laws and governing equations from data is one of humankind's greatest intellectual achievements. A quantitative understanding of dynamic constraints and balances in nature has facilitated rapid development of knowledge and enabled advanced technological achievements, including aircraft, combustion engines, satellites, and electrical power. In this work, we combine sparsity-promoting techniques and machine learning with nonlinear dynamical systems to discover governing physical equations from measurement data. The only assumption about the structure of the model is that there are only a few important terms that govern the dynamics, so that the equations are sparse in the space of possible functions; this assumption holds for many physical systems. In particular, we use sparse regression to determine the fewest terms in the dynamic governing equations required to accurately represent the data. The resulting models are parsimonious, balancing model complexity with descriptive ability while avoiding overfitting. We demonstrate the algorithm on a wide range of problems, from simple canonical systems, including linear and nonlinear oscillators and the chaotic Lorenz system, to the fluid vortex shedding behind an obstacle. The fluid example illustrates the ability of this method to discover the underlying dynamics of a system that took experts in the community nearly 30 years to resolve. We also show that this method generalizes to parameterized, time-varying, or externally forced systems.
Purpose-Reducing fuel consumption of unmanned aerial vehicles (UAVs) during transient operation is a cornerstone to achieve environment-friendly operations. The purpose of this paper is to develop a control scheme that improves the fuel economy of a turbojet in its full operating envelope. Design/methodology/approach-A novel direct-thrust linear quadratic integral (LQI) approach, comprised by an optimal observer/controller satisfying specified performance parameters, is presented. The thrust estimator, based in a Wiener model, is validated with the experimental data of a micro-turbojet. Model uncertainty is characterized by analyzing variations between the identified model and measured data. The resulting uncertainty range is used to verify closed-loop stability with the circle criterion. The proposed controller provides stable responses with the specified performance in the whole operating range, even with after considering plant nonlinearities. Finally, the direct-thrust LQI is compared with a standard thrust controller to assess fuel economy and performance. Findings-The direct-thrust LQI approach reduced the fuel consumption by 2.1090% in the most realistic scenario. The controllers were also evaluated using the environmental effect parameter (EEP) and transient-thrust-specific fuel consumption (T-TSFC). These novel metrics are proposed to evaluate the environmental impact during transient-thrust operations. The direct-thrust LQI approach has a more efficient fuel consumption according to these metrics. The results also show that isolating the thrust dynamics within the feedback loop has an important impact in fuel economy. Controllers were also evaluated using the EEP and T-TSFC. These novel metrics are proposed to evaluate the environmental impact during transient-thrust operations. The direct-thrust LQI approach has a more efficient fuel consumption according to these metrics. The results also show that isolating the thrust dynamics within the feedback loop has an important impact in fuel economy. Originality/value-This study shows the design of an effective direct-thrust control approach that minimizes fuel consumption, ensures stable responses for the full operation range, allows isolating the thrust dynamics when designing the controller and is compatible with classical robustness and performance metrics. Finally, the study shows that a simple controller can reduce the fuel consumption of the turbojet during transient operation in scenarios that approximate realistic operating conditions.
The system performance of a single-shaft turbojet engine is modelled with the pseudo bond graph approach in this paper. This theory is implemented in the in-house software tool ASTOR (AircraftEngine Simulation of Transient Operation Research) to simulate the overall dynamic of the turbojet model engine P200SX. In ASTOR, the transient performance is calculated with dynamic and individual control volumes to determine the three conservation equations.
This paper takes the first step towards the development of a control framework for underactuated flying humanoid robots. We assume that the robot is powered by four thrust forces placed at the robot end effectors, namely the robot hands and feet. Then, the control objective is defined as the asymptotic stabilization of the robot centroidal momentum. This objective allows us to track a desired trajectory for the robot center of mass and keep small errors between a reference orientation and the robot base frame. Stability and convergence of the robot momentum are shown to be in the sense of Lyapunov. Simulations carried out on a model of the humanoid robot iCub verify the soundness of the proposed approach.