Analysis of the Gains Tuning Problem in a Backstepping Controller
Applied to an Electrohydraulic Drive
HONORINE ANGUE MINTSA
Department of Mechanical Engineering,
Polytechnic School of Masuku, University of Sciences and Technologies of Masuku, Franceville,
REPUBLIC OF GABON
GEREMINO ELLA ENY
Department of Physics,
Faculty of Sciences, University of Sciences and Technologies of Masuku, Franceville,
REPUBLIC OF GABON
NZAMBA SENOUVEAU
Department of Electrical Engineering,
Polytechnic School of Masuku, University of Sciences and Technologies of Masuku, Franceville,
REPUBLIC OF GABON
ROLLAND MICHEL ASSOUMOU NZUE
Department of Mechanical Engineering,
Polytechnic School of Masuku, University of Sciences and Technologies of Masuku, Franceville,
REPUBLIC OF GABON
Abstract: This paper highlights the problem of tuning the gains of a non-adaptive backstepping controller in an
electrohydraulic servo system. While the other non-adaptive controllers in the literature have precise gains
tuning methods, the non-self-tuning backstepping controller has no rigorous gain tuning method. The proposed
study aims to analyze the contribution of each backstepping controller gain in the closed-loop performance. Our
final goal is to establish a rigorous gains-tuning method for the non-adaptive backstepping controller. The study
starts with the development of three-stage gains backstepping controller using a non-conventional time
derivative Lyapunov function. This particular Lyapunov function makes it possible to analyze the response of
the system when all the controller gains are cancelled. Then, we analyze the effect of each gain by cancelling
out the values of the others. The first simulation results show that the convergence of the tracking error to zero
is not maintained when all gains are set to 0 despite the presence of a negative definite of the Lyapunov
function time derivative. In this case, the equilibrium point is not the expected one as time goes to infinity. The
second set of results indicates that adjusting the gain related to the feedback of the actual output only ensures
the asymptotic convergence of the tracking error to zero as time goes to infinity. However, developing a
heuristic tuning of the three controller gains like Ziegler Nichols tuning remains a challenge.
Key-Words: - tuning gains, backstepping controller, electrohydraulic servo system, heuristic method, chattering
effect
Received: November 19, 2022. Revised: April 24, 2023. Accepted: May 18, 2023. Published: June 16, 2023.
1 Introduction
Electro-Hydraulic Servo Systems (EHSS) are used
to handle large mechanical loads with a fast,
accurate, and robust response. In these systems,
pressurized hydraulic oil is used to transmit power.
Some industrial EHSS applications include
aerospace actuation, [1], automobile actuation, [2],
and machine tools, [3]. Most of these
electrohydraulic actuators are implanted using PID
control laws because of their well-known and
flexible methodology. PID controller consists of
tuning three gains. The literature identifies rigorous
approaches for non-self-tuning controllers like
Ziegler Nichols, [4]. Good results but only in a
limited operating point range are achieved using the
PID control strategy, [4]. The EHSS dynamics have
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.17
Honorine Angue Mintsa, Geremino Ella Eny,
Nzamba Senouveau, Rolland Michel Assoumou Nzue
E-ISSN: 2224-2856
166
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strong nonlinearities, [5], making the linear control
theory inadequate to guarantee satisfactory
performances over a large panel of operating points.
PID control may be combined with some methods
like fuzzy logic, [6], sliding mode, [7], fractional
order strategy, [8], and optimization tools, [9], to
improve the performance. However, when linear
control is used on nonlinear systems, it is difficult to
ensure both improved performances and expanded
working conditions, [10].
Of all the control laws encountered in the
literature, the backstepping control has advantages,
especially when faced with system nonlinearities.
Unlike feedback linearization control, this approach
allows choosing which nonlinearity to cancel, [11],
thus improving the robustness of the closed-loop
system. Moreover, the recursive construction of the
Lyapunov function allows the flexibility of the
architecture of the control law, [12]. Backstepping
control consists of dismantling the system into first-
order subsystems where a state variable is
considered as a control signal, [13]. These virtual
controls or desired system variable states, [14], are
chosen to ensure the negative definition of the
Lyapunov function time derivative. Experimental
and numerical results show that the backstepping
controller is more efficient than the PID controller,
[15].
There are two drawbacks to using the
backstepping approach. The first one is the
explosion of complexity due to repeated calculations
when the plant model has a high order, [15]. The
second drawback, and the one discussed in this
paper, is the lack of a rigorous gains-tuning
approach. At each recursive step in the design of the
backstepping controller, a gain to be adjusted
appears. In this paper, we focus on non-self-
adjusting backstepping controller gain strategies. In
[16], the authors show that there is a trade-off
between the chattering effect and the convergence
of the tracking error while adjusting the gains of the
backstepping controller. An optimal gain is difficult
to find in the absence of a rigorous tuning method.
The literature identifies rigorous methods for tuning
the parameters of non-self-tuning controllers like
Ziegler Nichols, [17], for PID controllers and pole
placement for feedback linearizing controllers, [18].
However, to our knowledge, authors in the literature
adjust the gains of the non-self-tuning backstepping
controller via trial and error. Few authors try to
analyze the effect of the gain in the closed loop
performance. Authors, [19], show that the
backstepping controller gains affect the robustness
against parametric uncertainties. For each gain, they
found a minimum value, an optimum value and a
maximum value to guarantee convergence of the
error. However, the contribution of each gain is not
highlighted. In [20], authors show that the three
gains of the backstepping controller affect the
performance of the electrohydraulic brake system by
varying one gain and setting the others to zero. They
found that two of the three gains affect overshoot
and steady state. However, their backstepping
control law contains two input variables that operate
alternately. The three gains appear in these input
variables. The complex conditions of these inputs
weaken the actual influence of the three gains.
The main contributions of this article are listed
below:
- an unconventional Lyapunov function that
makes possible the analysis of the performance
while the controller gains are set to 0;
- an actual analysis of each gain contribution in
the closed loop performance using a simple non-
self-tuning backstepping control law with one input
variable;
- a discussion of the perspective of tuning
methods.
2 System Modelling
Fig. 1 shows the electrohydraulic servo system
considered in this study. It is the same system
presented in our previous work, [21]. It consists of a
hydraulic motor that drives a rotating load. The
hydraulic unit includes the pump, tank, pressure
relief valve, and accumulator. It provides hydraulic
oil flow at constant pressure. The electrohydraulic
servo-valve is the interface between the operative
part and the control part. The electric signal u(t) acts
on the servo-valve spool by varying the oil flow into
the hydraulic motor. Because a mechanical load is
attached to the motor, a pressure difference PL(t) is
noted across the hydraulic motor lines. The
objective of the control law is that the actual angular
velocity tracks the desired angular velocity. The
measure of the actual angular velocity is sent to the
control law via a sensor. The state-space equation
(1) is developed from three equations. The first
equation describes the motion equation of the load.
The second equation is the continuity equation
through the motor lines. The third equation shows
the dynamics of the servo valve.
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Fig. 1: Electro-hydraulic servo system
33
1
1 2 1
2 3 3 2 1 2
-
4
1
-
- - -
d
mm
ds m sm
m
dB
tJJ
c
x
K
x t u t x t
y t x t
x x t x t
c
t x t P sigm x t x t d x t c x t
V
(1)
Where
1
xt
is the angular velocity
t
2
xt
is the motor pressure difference due to the load
3
xt
is the servo-valve opening area due to the
input signal
ut
is the control current input
J
is the hydraulic motor's total inertia
m
d
is the volumetric displacement of the motor
is the fluid bulk modulus
m
V
is the total oil volume of the hydraulic motor
d
c
is the servo-valve discharge coefficient
is the fluid mass density
sm
c
is the leakage coefficient of the hydraulic motor
s
P
is the supply pressure at the inlet of the servo
valve
K
is the servo-valve amplifier gain
is the servo-valve time constant
To satisfy the Lipschitz condition in this paper,
we choose to approximate the sign function to the
continuous function (2) proposed in the work of
[22].
2
xt
sign x t sigm x t
xt
(2)
3 Backstepping Controller Design
In this section, the angular velocity backstepping
controller is derived. It is the same controller
presented in [21]. Here, we focus on the controller
gains locations and tuning. The desired state
variables are denoted by xid(t).
Now, consider the first subsystem
1 2 1
-
mm
dB
tJJ
x x t x t
of the state space model (1).
The first candidate Lyapunov function for this
subsystem is
2
11
1
2
V t e t
(3)
Where
1 1 1d
e t x t x t
. The time derivative of this
Lyapunov function gives
1 1 2 2 1 1 1
m m m m
d d d
d d b b
V t e t e t x t e t x t x
J J J J
(4)
If we choose x2d(t) as the first virtual control such
that
2 1 1 1 1
m
d d d
m
b
J
x t x t x k e t
dJ
(5)
Where the first gain controller
10k
, we obtain
2
1 1 1 2 1
mm
bd
V t k e t e t e t
JJ
(6)
Now, we choose the second candidate Lyapunov
function for the second subsystem of (1)
2 3 3 2 1 2
4- - -
dds m sm
m
c
xc
t x t P sigm x t x t d x t c x t
V
as
22
2 1 2
11
22
V t e t e t
(7)
Its time derivative gives
2
2 1 1
1 3 3 2
2
3 3 2 1 2 2
4
4 4 4
m
md
s
m
d m sm
d s d
mm
m
b
V t k e t
J
dc
e t e t P sigm x t x t
Jv
et c d c
x t P sigm x t x t x t x t x t
vv
v
(8)
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If we choose
3d
xt
as the second virtual control
such that
11
3
32 2 2 2 2
4
4
4
mm
m
m
dsm
ds dd
m
dd
e t x t
Jv
v
xt c
c P sigm x t x t x t x t k e t
v
(9)
Where the second controller
20k
, we obtain
22
2 1 1 2 2
2 3 3 2
4
4
m sm
m
ds
m
bc
V t k e t k e t
Jv
ce t e t P sign x t x t
v
(10)
Finally, consider the third subsystem
33
1K
x t u t x t
. Choose the final candidate
Lyapunov function for this subsystem as
2 2 3
3 1 2 2
111
222
V t e t e t e t
(11)
The time derivative of this Lyapunov function gives
22
3 1 1 2 2
2 3 2
3
3 3 3
4
4
11
m sm
m
ds
m
dd
bc
V t k e t k e t
Jv
cK
e t P sign x t x t u t
v
et
e t x t x t
(12)
If we choose the control signal
ut
such that
3 3 2 3 2 3 3
4
1d
d d s
m
c
u t x t x t e t P sigm x t x t k e t
Kv
(13)
Where the third controller gain
30k
, we obtain
2 2 2
3 1 1 2 2 3 3
41
m sm
m
bc
V t k e t k e t k e t
Jv
(14)
Fig. 2 shows the implementation of the
backstepping controller in Matlab /Simulink with
the highlight of the three steps.
Fig. 2: Closed-loop controlled system block diagram
in Matlab /Simulink environment
3.1 Tuning Issue Analysis
One can note that the gains of our backstepping
controller are chosen such that the time derivative of
the final Lyapunov function gives (14). In most
works, [13], [19], [20], these gains are located such
that the other terms
m
b
J
,
4sm
m
c
v
and
1
do not
appear in the time derivative of the Lyapunov
function. Because (14) leads to inequality (15), the
Lasalle principle states that the tracking errors
12
,e t e t
and
3
et
go to zero as time goes to infinity.
2 2 2
3 1 1 2 2 3 3 0V t k e t k e t k e t
(15)
Unlike the feedback linearization controller or
PID controller, the backstepping controller does not
show a weighted sum of the angular velocity error
and its time derivatives. The different tracking
errors
i
et
are not related in a direct sense. The
angular velocity tracking error occurs in the first
virtual control. The pressure difference tracking
error occurs in the second virtual control and so on.
The gains of the different tracking errors,
1
k
,
2
k
and
3
k
must be adjusted in a given order shown by the
work of [16], where certain dynamics of the state
variables are neglected. In [20], the authors show
that varying the controller gains one by one may
affect the robustness of the closed-loop response
under parametric uncertainties. Their backstepping
controller has two variable inputs. Our control law
has one variable input. Additionally, no disturbance
is present and the focus is on the effect of each gain
in the closed-loop performance. We vary the value
of each gain while the others are set to zero. Our
objective is to see if a heuristic method like the one
of Ziegler Nichols can be developed.
4 Results
In this section, the results of the numerical
simulation are presented. The performances of the
proposed backstepping controller are obtained in the
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Matlab/ Simulink environment using the closed-
loop block diagram of Fig. 2. The sampling time is
0.01 seconds and the simulation is performed in 20
seconds. Table 1 lists the numerical value used for
the simulation.
Table 1. Numerical values used for the simulation
Symbol
Description
Value and
units
EHSS
Sigmoid function constant
102
Servo valve time constant
0.01 s
K
Servo valve amplifier gain
8.10-7m2/mA
Vm
Total oil volume of the
hydraulic motor
3 10-4m3
Fluid bulk modulus
8 108 Pa
cd
Flow discharge coefficient
0.61
Ps
Supply pressure
9 106
csm
Leakage coefficient
9 10-13 m5/
(N.s)
dm
Volumetric displacement of the
motor
3 10-6m3/rad
Fluid mass density
900 Kg/m3
J
Total inertia of the motor and
the load
0.05 N.m.s2
B
Viscous damping coefficient
0.2 N.m.s
Fig. 3 shows the reference signal describing the
desired trajectory of the angular velocity used for
the simulation. In the first ten seconds, the desired
angular velocity is a step of amplitude 1 rad/s. In the
last ten seconds, the desired angular velocity has a
sinusoidal waveform with an amplitude of 1 rad/s
and a frequency of 2 rad/s.
Fig. 3: Reference signal used for the simulation
The first sets of simulations present the
performances of the backstepping controller when
the gains
1
k
,
2
k
and
3
k
equal 0. Fig 4 shows that
angular velocity tracking error does not converge to
zero as time goes to infinity. To appreciate the
infinity response behaviour, the time of simulation
is extended to 80 seconds. The tracking error does
not go to infinity as time goes to infinity. It is seen
that its value is limited to approximately 90 rad/s.
According to (14), the time derivative of the
deducing Lyapunov function is negative define.
Hence, the equilibrium state
0
i
et
is
asymptotically stable. In Fig 5, at the start of the
simulation, the zoomed view shows that the tracking
error is negligible before 4 seconds. The tracking
error converges to 0 before 4s. However, without a
disturbance in the closed loop system at 3,5 seconds,
the system response starts to diverge from the
desired output. We can conclude that the
equilibrium state is not asymptotically stable and
diverges to a limit cycle or other equilibrium state.
Fig. 4: System response when
1 2 3 0k k k
Fig. 5: Zoomed view of system response when
1 2 3 0k k k
4.1 k2 and k3 Gains Tuning Results
The next set of figures shows the backstepping
controller performances where the first gain
1
k
is 0.
The gains
2
k
and
3
k
are varied to see the effect of
these gains on the backstepping controller
performance. Fig. 6 and Fig. 7 show that the
response obtained with
10k
gives the same result
obtained in the previous section. The angular
velocity tracking error converges to 0 at the start of
the simulation. As time goes to infinity, the tracking
error diverges from zero but remains bounded
around
90
rad/s. The variation of
2
k
and
3
k
gains
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does not affect the backstepping controller
performances.
Fig. 6: System response when
13
0kk
a)
20k
b)
2100k
and c)
21000k
Fig. 7: System response when
12
0kk
a)
30k
b)
3100k
and c)
31000k
4.2 k1 Gain Tuning Results
In this section, the response of the proposed
backstepping controller is analyzed by varying the
1
k
gain while the other gains are set to 0. Fig 8
shows that the
1
k
gain strongly affects the behaviour
of the closed-loop response. The convergence of the
angular velocity varies while changing the tuning of
the
1
k
gain. The closed-loop response displays the
same behaviour encountered in the previous section
when
10k
. The response behaviour drastically
changes when
10k
. In Fig.8 a and Fig. 8 b, we note
that the response remains limited at 90 rad/s.
Meanwhile, in Fig 8 c, when
11000k
, high-
frequency sustained oscillations with amplitude less
than 40 rad/s are visible in the response. In Fig 9,
the
1
k
gain is varied with a larger sweep to observe
further changes. However, beyond the value of
1000, the response shows the same profile.
Fig. 10 shows that the tracking error converges to 0
when the value of the
1
k
gain is close to 27. In Fig
10. b, the tracking error is negligible when
127.2k
.
In Fig 10. a, the tracking error shows large
overshoots at 10 s when the reference signal
changes its form. The zoomed view of the response,
when
127k
according to Fig. 11, shows that the
tracking error is small on either side of the change in
the profile of the reference. In Fig 10. c, the
chattering effect or high frequency sustained
oscillations with small amplitude occurs in the
backstepping controller response when
127.5k
.
According to [16], there is a trade-off between the
robustness and the chattering effect in the closed-
loop response while adjusting the k1 gain. The
robustness, in the simulations where k1 is about 27,
is described by the change in the reference signal.
As one can see, large overshoots occur in Fig.10 a
while no overshoot is visible in Fig. 10 c.
Fig. 8: System response when
32
0kk
a)
10k
b)
1100k
and c)
11000k
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Fig. 9: System response when
32
0kk
a)
11000k
b)
12000k
and c)
13000k
Fig. 10: System response when
32
0kk
a)
127k
b)
127.2k
and c)
127.5k
Fig. 11: Zoomed view of system response when
32
0kk
and
127k
5 Discussion and Gains Tuning
Method Perspective
In this section, we discuss two issues encountered in
the results section.
The first problem concerns the change in the
response profile by varying the k1 gain while no
change is noted with the other gains. In (5), the k1
gain is the coefficient of the feedback term
appearing in the second virtual control. When k1 is
0, the first virtual control does not give the desired
second state variable. Since the second virtual
control depends on the first one via the backstepping
effect, the desired third state variable is also biased.
The k1 value provides feedback on the actual
angular velocity in the backstepping controller. This
feedback link guarantees the convergence of the
tracking error to zero. However, without this link,
the tracking error does not maintain the convergence
as time goes to infinity.
The second problem concerns the limit values
indicated in certain simulations. This problem may
be related to the first one because the biased virtual
control leads to biased tracking errors. Indeed, our
tracking error is calculated using the reference
signal. However, without the feedback link, the
equilibrium point in the Lyapunov function is not
the desired one. The limit value noted in some
simulations may be the biased equilibrium point.
6 Conclusion
In this paper, we address the problem of adjusting
the gains of the non-self-tuning backstepping
controller. The objective is to analyze the effect of
each controller gain in the closed-loop response to
propose a standardized approach to tune these gains.
The results show that only one of the three gains
affects the convergence of the tracking error to zero
as time tends to infinity. This gain provides the
feedback link of the actual angular velocity in the
backstepping controller. The calculated virtual
controls are not those expected when this gain is not
well adjusted. Future works will investigate the
analytic value of this critical gain. The effect of the
other gains once the critical gain is adjusted, will be
studied to propose a rigorous gain tuning method.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
-Honorine Angue Mintsa and Rolland Michel
Assoumou Nzué carried out the design and the
Matlab Simulink implementation of the
backstepping controller.
-Gérémino Ella Eny was responsible for the
literature review and the analysis of the results
shown in Section 4.
-Nzamba Senouveau executed the numerical results
by varying the different gains in the controller.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflict of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.17
Honorine Angue Mintsa, Geremino Ella Eny,
Nzamba Senouveau, Rolland Michel Assoumou Nzue
E-ISSN: 2224-2856
173
Volume 18, 2023
