Adaptive Observers for Servo Systems with Friction
Amit Dixit and Shashikanth Suryanarayanan
Abstract— We consider the problem of constructing adaptiveobservers for servo systems with friction. We show that ifthe friction force is modeled using currently popular dynamicfriction models, currently available methods for constructingadaptive observers can not be applied. We suggest modificationsfor one such friction model: the LuGre friction model and pointto several methods of constructing adaptive observers that canbe applied to the modified system. Using one such method, weconstruct an adaptive observer to identify parameters of themodel of an experimental setup. The results of the identificationexercise are found to be encouraging.
I. INTRODUCTION
Friction is a physical phenomenon that often degrades
performance of precision motion control systems. To counter
the detrimental effects of friction, model-based friction com-
pensation schemes are increasingly being used. Model-based
friction compensation involves using control-oriented models
that predict friction phenomena relevant to motion control ap-
plications. Several such friction models have been proposed
in literature [1], [2], [3], [4].
A typical control-oriented, dynamic friction model consists
of a set of parameterized nonlinear differential equations to
describe friction phenomena. To use such a friction model
for control purposes, the parameters of the model have to be
experimentally identified for the system under consideration.
This identification is typically done using a dedicated set of
experiments. Such an identification method is both costly and
time consuming.
In this paper we consider the problem of construction of
adaptive observers, i.e. observers that estimate both the states
and unknown parameters of a dynamic system, for servo
systems with friction. We consider a system consisting of
a mass acted upon by friction force. The friction force is
assumed to follow the LuGre friction model. We show that
current results related to construction of adaptive observers
can not be applied to this model. We propose certain modifi-
cations to the LuGre model and describe a methodology for
construction of adaptive observer for the modified system.
We demonstrate the application of the methodology by
constructing adaptive observers for estimating the friction
parameters of an experimental setup.
The main contribution of the paper is to show that by
applying some simple modifications to dynamic friction
models it is possible to construct adaptive observers that
provide on-line estimates of the states and parameters of
models of servo systems with friction. Such observers can be
S. Suryanarayanan and A. Dixit are with the Department of Me-chanical Engineering, Indian Institute of Technology Bombay, Powai,Mumbai 400076, India. Email: : [email protected],
attractive alternatives to the traditional method of identifying
the friction parameters based on dedicated experiments, as
well as prove useful in constructing adaptive controllers.
The rest of the paper is organized as follows: In Section II,
we present some preliminaries on adaptive observers and
describe the system under consideration. In Section III, we
present a methodology for construction of adaptive observers
for servo systems with friction. In Section IV, we demon-
strate the methodology by constructing adaptive observers
for identification of parameters of an experimental setup.
Summary of the work and problems for future work are
discussed in Section V.
II. PRELIMINARIES
A. Adaptive Observers
Adaptive observers are used for estimation of states of a
dynamic system whose parameters are either unknown or
time-varying parameters. In addition to estimating states,
adaptive observers estimate the parameters of the system.
Hence, adaptive observers are attractive for system identi-
fication and adaptive control applications. Over the years,
a number of methodologies for construction of adaptive
observers for both linear and nonlinear systems have been
proposed.
A vast majority of the results related to adaptive observers
deal with linear-in-parameters systems. Such methods in-
clude work by Luders and Narendra [5], Kreisselmeier [6]
for linear systems and Bastin and Gevers [7], Marino [8] for
nonlinear systems, where nonlinearities depend only on the
measured input or output. Extension to more general cases
have been proposed under some passivity-like conditions
in Bresancon [9]; using a variable structure approach in
Martinez and Poznyak [10] or using high-gain observer
approach in Besancon et al [11].
Here we briefly summarize two methodologies that are
relevant to the present study. In Bastin and Gevers [7], the
following class of nonlinear systems was considered:
x(t) = Rx(t) + Ω(ω(t))θ(t) + g(t)
y(t) = x1(t) (1)
where x ∈ Rn denotes the state vector, y ∈ R is the
measured output, θ ∈ Rq is a vector of time-varying
unknown parameters, ω ∈ Rs is a vector of known functions
of the input u and the output, Ω is an n × q matrix whose
elements are linear combinations of elements of vector ω and
g ∈ Rn is a vector of known functions of time. It is assumed
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that the matrix R is of the following form:
R =
0 kT
0... F0
(2)
where kT is a 1 × (n − 1) vector of known constants and
F is a stable (n − 1) × (n − 1) matrix. For such class of
systems, it was proved that if the unknown parameters are
constant, i.e. if θ(t) = θ, the following set of equations
implements an exponentially stable adaptive observer, i.e.
errors in estimation of state (x(t) − x(t)) and in estimation
of parameters (θ − θ(t)) exponentially go to zero:
V (t) = FV (t) + Ω((t))
ϕ(t) = V T (t)k + ΩT1 (ω(t))
˙θ(t) = Γϕ(t)[y(t) − x1(t)]˙x(t) = Rx(t) + Ω(ω(t))θ(t) + g(t)
+
[
c1[y(t) − x1(t)]
V (t)˙θ(t)
]
(3)
where Ω1 and Ω are sub-matrices of matrix Ω given by ΩT =[ΩT
1 ΩT ], c1 is an arbitrary positive constant and Γ is an
arbitrary positive definite q × q matrix.
For the class of systems described by the set of equations
in 1, the coefficients of unknown parameters of the system
(i.e. matrix Ω) are assumed to be known. This condition was
relaxed in Bresancon et al [11], which considered systems
of the following form:
x(t) = A0x(t) + ϕ(x(t), u(t)) + ψ(x(t), u(t))θ
y(t) = C0x(t) (4)
where x ∈ Rn denotes the state vector, u ∈ R
m denotes the
input vector, y ∈ R is the measured output, and θ ∈ Rq is
a vector of constant unknown parameters. It is assumed that
A0, C0, ψ(x, u) and ϕ(x, u) are of the following form:
A0 =
0 1 0. . .
10 0
C0 =[
1 0 · · · 0]
ϕ(x, u) = (ϕ1(x1, u) ϕ2(x1, x2, u) · · · ϕn(x, u))T
ψ(x, u) =
0 · · · 0...
...
0 · · · 0ψn,1(x, u) · · · ψn,q(x, u)
(5)
For the above class of systems, under certain persistence
of excitation condition on signals, it was proved that the
following set of equations implement an exponentially stable
adaptive observer:
˙Γ(t) = λ(A0 −K0C0)Γ(t) + λψ(x(t), u(t))˙x(t) = A0x(t) + ϕ(x(t), u(t)) + ψ(x(t), u(t))θ(t)
+ Λ(λ)−1[λK0 + Γ(t)ΓT (t)CT0 ][y(t) − C0x(t)]
˙θ(t) = λnΓ(t)TCT
0 [y(t) − C0x(t)] (6)
where K0 is selected such that A0 − K0C0 is stable, λ,
a positive constant, is a design variable and Λ(λ)−1 =diag(1, λ, λ2...λn−1).
Apart from the above results for linear-in-parameters sys-
tems, some results for nonlinear-in-parameters systems have
also been recently proposed [See, for example, [12], [13]].
However, these results are generally applicable to a restricted
class of systems.
B. Model under consideration
We consider model of a single inertia system under action
of a controlled force (which corresponds to the input force)
and the friction force. The friction is assumed to be modeled
by the LuGre friction model [2]. The model is assumed to
be described by the following set of equations:
Jv(t) = bu(t) − F (t) (7)
F (t) = σ0z(t) + σ1z(t) + α2v(t) (8)
z(t) = v(t) −|v(t)|z(t)
s(v)(9)
s(v) = a0 + a1e−[ v
v0]2
(10)
Here v(t) is the velocity, J is the inertia, bu(t) is the input
force, F (t) is the friction force, σ0, σ1, α2, a0, a1 and v0are parameters of the LuGre friction model. The friction
force dynamics are captured using the internal state variable
z(t), which is an unmeasured signal. Equation 10 models
the Stribeck curve, i.e. the relation between the steady-
state (constant) velocity and the steady state friction force.
The above model typically describes two distinct regimes of
motion: one at close to zero velocities (pre-sliding motion)
and the other at higher sliding velocities (sliding motion). A
number of motion control systems can be described using
the above set of equations.
III. ADAPTIVE OBSERVERS FOR SERVO
SYSTEMS WITH FRICTION
In this section, we propose a methodology for construc-
tion of adaptive observers for servo systems with friction.
Consider the set of Equations 7-10. The equations can be
rearranged as follows:
v(t) = bu(t) − w(t) − α2v(t)
−σ1
Jσ0[σ0v(t) −
|v(t)|w(t)
s(v)] (11)
w(t) = σ0v(t) −|v(t)|w(t)
s(v)(12)
where w(t) = σ0z(t)/J is the new internal state variable,
s(v) is as defined by equation 10, b = b/J and α2 = α2/J . It
can be seen that the system described by above two equations
961
is nonlinear in states v(t) and w(t) as well as nonlinear in
parameters b, σ0, σ1, α2, a0, a1 and v0. Consequently, none
of the methodologies for construction of adaptive observers
mentioned in Section II-A can be applied to the above
system.
In order to bring the system into a form to which the
adaptive observer design methods can be applied, we pro-
pose the following modifications to the model described by
equations 11-12:
1) Modify the internal state equation 12 as
w(t) = σ0s(v)v(t) − c|v(t)|w(t) (13)
where s(v) = s(v)c is considered to be unit-less while
c has units of (displacement)−1.
2) Linearly parameterize s(v) using a suitable set of basis
functions. That is, express s(v) as:
s(v) = β0 +∑
i
βiφi(v) (14)
where φi(v) is a suitable set of basis functions.
3) Set σ1 = 0, i.e. assume zero bristle damping.
Using the above modifications we get:
v(t) = bu(t) − w(t) − α2v(t) (15)
w(t) = σ0[β0 +∑
i
βiφi(v)]v(t) − c|v(t)|w(t) (16)
Equation 16 can be written as:
w(t) = σ0v(t) +∑
i
σiφi(v)v(t) − c|v(t)|w(t) (17)
where σ0 = σ0β0 and σi = σ0βi.
We make the following comments about the modified
system described by equations 15 and 17:
1) The model is linear in the new set of parameters
b, σ0, σi, α2 and c. The number of parameters is equal
to 4 + p, where p is the number of basis functions
used to approximate s(v). The unmeasured state w(t)corresponds to the friction force.
2) At steady-state, i.e. for v(t) = 0 and w(t) = 0, we
have
bu(t) = sgn(v(t))1
c[σ0 +
∑
i
σiφi(v)] + α2v(t) (18)
Thus, for constant velocity motion, it can be seen
that the input force predicted by the model equals the
friction force corresponding to the Stribeck curve.
3) The authors have confirmed, through simulation, that
for appropriate choice of parameter “c” (which we have
found to lie between 1a0
and 1a0+a1
), with σ0 and σi
correspondingly chosen to model the Stribeck curve,
the pre-sliding motion (i.e. the motion in the sticking
regime) predicted by the model matches well with the
corresponding motion predicted by equations 11-12.
4) Equation 17, that describes the dynamics of the internal
state variable, retains the properties of dissipativity and
boundedness of the map v(t) 7→ w(t) exhibited by
the LuGre model (described by equation 12). This can
be proven in a straightforward manner using methods
similar to the ones used in [2].
5) We have chosen to set σ1 = 0, i.e. ignore bristle damp-
ing. The parameter σ1 is used in the LuGre model to
introduce damping in the pre-sliding motion. However,
it has negligible effects on the sliding motion and even
on the pre-sliding motion if the linear damping coeffi-
cient (α2) is high enough. If the model is to be used for
velocity tracking applications, then most of the motion
is expected to take place in sliding regime. Further, the
proportional control action often provides large linear
damping. Consequently, the effects of σ1 on the motion
can be considered to be negligible. As it turns out,
ignoring the bristle damping significantly simplifies the
design of the adaptive observer. However, if required,
it may be possible to take into account the effect of σ1
by modifying the above model. This will be a part of
the future work.
6) Several reasonable choices exist for basis functions
used for modeling s(v) = s(v)/c (where s(v) is given
by Equation 10). A typical plot of s(v) versus v is
shown in Figure 1 to give an idea of the general shape
of the s(v) curve. Some of the possible choices of basis
a0+a1
a0S(V)
Higher sliding velocities
V0
V
Fig. 1. A typical plot of s(v) versus v
functions are listed here:
• Polynomial: s(v) = β0 +∑
i βi|v|i
• Gaussian: s(v) = β0 +∑
i βi(exp(−(v−vi
v0i
)2))where exp is the exponential function and vi,
v0i are chosen so as to cover complete range of
velocity for which value of s(v) is significant.
We now describe two approaches to construction of adaptive
observers for system described by equations 15 and 17 with
velocity v(t) as the measured output signal.
1) Approach I: Let us assume that all parameters of the
system are unknown. Consider the co-ordinate trans-
formation: x1(t) = v(t) and x2(t) = v(t). Under this
co-ordinate transformation, equations 15 and 17 can be
written as:
x1(t) = x2(t) (19)
x2(t) = −α2x2(t) + bu(t) − σ0x1(t)
−∑
i
σiφi(x1(t))x1(t)
+ c|x1(t)|(bu(t) − α2x1(t) − x2(t))(20)
962
by considering the extended parameter vector θ =[α2 b σ0 σi cb cα2 c]
T , the above system can be writ-
ten as:
x1(t) = x2(t) (21)
x2(t) = G(x(t))θ (22)
where G(x(t)) can be obtained from Equation 20.
For the above overparameterized system, the following
options for constructing adaptive observer are possible:
• Using recently developed techniques for real-
time differentiation based on Variable Structure
theory [14], signals v(t), v(t) and u(t) can be
computed on-line in real-time. In such a case,
the complete state x(t) can be assumed to be
measurable, and adaptive observer presented in
Zhang [15] can be applied.
• Using a piecewise polynomial approximation for
the input signal u(t) and treating velocity signal
as the only measurable output, an observer can be
constructed using methods presented in Besancon
et al [11].
2) Approach II: If the main objective of constructing the
Adaptive Observer is to carry out system identification
(e.g. developing model for feedfoward friction com-
pensation or for friction simulation), construction of
adaptive observer can be simplified by carrying out
the identification procedure in two steps: identification
of parameters α2 and b at high sliding velocities and
identification of rest of the parameters at low sliding
velocities.
At high sliding velocities, i.e. velocities sufficiently
higher than the Stribeck velocity v0 (refer Figure 1),
friction force is approximately constant and equals the
Coulomb friction force Fc, i.e. F (t) ≈ Fc. Thus,
motion at high sliding velocities can be adequately
described by the following equation:
v(t) = −α2v(t) + bu(t) − Fcsgnv(t) (23)
Thus, if the velocity signal is measured, then param-
eters α2 and b can be estimated using any standard
linear recursive parameter estimation scheme, for ex-
ample the recursive least squares parameter estimation
filter [16] or the adaptive observer presented in Bastin
and Gevers [7]. The parameter estimates should be
updated only for v(t) > vc, where vc is chosen to be
sufficiently higher than v0. A heuristic for estimating
the Stribeck velocity v0 is presented in Section IV-B.
Once the values of parameters α2 and b have been esti-
mated, one method to construct the adaptive observer
is to follow Approach I mentioned in this paper by
treating α2 and b as known parameters. In this case,
identification of α2 and b has the effect of avoiding
the overparameterization used in Approach I.
It is also possible to identify the rest of the parameters
without using the co-ordinate transformation used in
Approach I by using the results presented in Bresancon
et al [11] (details of this method are briefly discussed
in Section II-A ).
Note that both of the methods discussed above can be
easily extended to the case where only the position is
measured. The only modification that will be required is to
add another state variable x0(t) with x0(t) = v(t) and by
considering this equation along with equations 15 and 17 for
the design of the adaptive observer.
IV. EXPERIMENTAL RESULTS
In this section we describe the application of an adaptive
observer constructed using Approach II discussed in the last
section for identification of parameters of an experimental
setup.
A. Description of the experimental setup
A labeled photograph of the experimental setup is shown
in Figure 2. It consists of a D.C. motor driving an inertia
Fig. 2. Experimental setup
load in the form of a metal disc, in presence of friction.
The friction force is generated by pressing a spring-loaded
metal button on the edge of the disc, forming a line contact.
The interface between the metal button and the disc was
lubricated with grease. By varying the compression in the
loading spring, the normal load at the contact interface
between the disc and the button can be varied. It is well
known that the friction parameters are a function of the
normal load. Thus, by varying the normal load, different
friction conditions can be created. The diameter of the metal
disc was 75mm while its moment of inertia was of the order
of 0.001 kgm2. The torque constant of the d.c. motor was
40mN/A. The angle of rotation of the motor was measured
by an incremental encoder with a resolution of 10, 000pulses per revolution, which, after quadrature decoding, is
equivalent to a resolution of 0.009o. The normal load at the
contact can be measured by a load cell. The encoder was
interfaced with a dSPACE signal processing unit for data
acquisition. The dSPACE unit was also used to drive the
motor by varying the voltage applied across terminals of the
motor through an H-bridge amplifier. The dynamics of this
system can be described by equations 15 and 17 with voltage
signal across the motor terminals as the input signal (u(t)).
963
B. A heuristic for estimation of vc
As mentioned in the last section, construction of an
adaptive observer using Approach II requires selection of
a cut-off velocity vc such that the motion of the system at
velocities above this velocity can be adequately described
by Equation 23. One method of determining such a value
of vc is described here. As discussed earlier, for a constant
input signal (u(t)), velocity settles down to a constant value.
While this phenomenon is observed at high sliding velocities
(velocities sufficiently greater than v0), such uniform motion
is not observed at low velocities (velocities less than v0). So,
an estimate of v0 can be obtained by observing the response
of the system to constant step inputs. For higher values of the
step inputs, system is expected to settle down to a constant
velocity while for lower values of input, motion of the system
will be non-uniform. The value of vc can then be chosen to
be the lowest value of velocity that corresponds to a uniform
motion. The choice of step input (i.e., high rate of application
of input force) ensures that effect of stiction is minimized.
This is because the break-away force (the force necessary to
initiate sliding motion) decreases with the rate of application
of input force [1].
The above method was used to choose the value of vc for
the experimental setup. From the experimental observations,
vc was chosen to be equal to 500/sec.
C. Choice of input signal
For successful identification, it is important that all
regimes of motion are adequately excited. To ensure that
this is achieved, a Proportional-Integral (PI) controller was
implemented to track a velocity reference signal. The veloc-
ity reference signal, computed by passing a random signal
through a low-pass filter with a cut-off frequency of 5 rad/s.
A sample of the reference signal is shown in Figure 3. The
measured velocity signal and the corresponding input signal
were recored using the dSPACE unit with a sampling period
of 1ms.
224 226 228 230 232 234 236
−300
−200
−100
0
100
200
300
Time (s)
Re
fere
nce
ve
locity (
de
g/s
)
Fig. 3. Reference velocity signal
D. Choice of basis functions to approximate s(v)
A set of two Gaussian functions was used to approxi-
mate the s(v) curve. Considering that the Stribeck effect
is significant only for velocities below 500/sec, following
parameterization was used:
s(v) = β0 + β1e−[(v/10)2] + β1e
−[((v−20)/20)2] (24)
E. Identification results
First, parameters α2 and b were identified as described
in Approach II in Section III. An adaptive observer was
constructed for the system described by equation 23 using
method of Bastin and Gevers [7], as described in Equa-
tions 1, 2 and 3, with number of states n = 1, R = 0,
Ω = [−v(t) u(t) − sgn(v(t))], θ = [α2 b Fc]T . Design
parameters c1 and Γ were chosen as c1 = 100 and Γ =diag(0.001, 0.001, 0.0001). Initial values of all parameters
and states was set to be zero. The parameters were updated
only for |v(t)| > vc. At every transition from |v(t)| < vc to
|v(t)| > vc, estimate v(t) was set to be equal to vc or −vc
depending on sign of v(t).Figure 4 shows the parameter estimates obtained by im-
plementing the adaptive observer on the experimental data.
Using the estimated values of α2 and b, an adaptive observer
0 50 100 150 200−2
−1
0
1
2
3
4
5
6
7
8
Time (s)
bα
2
Fig. 4. Estimated parameters a2 and b
was constructed for estimating the rest of the parameters
using the method of Bresancon et al [11], discussed in
Section II-A of this paper. The observer was constructed for
system described by equations 15 and 17 using equations 5
and 6 with n = 2, x1(t) = v(t), x2(t) = −w(t),ϕ = (−α2x1(t) + bu(t) 0)T , θ = [σ0 σ1 σ2 c]
T . Design
parameters were chosen as K0 = [100 1000]T and λ = 50.
Initial values of all parameters and states was set to be zero.
Figure 5 shows the parameter estimates obtained by im-
plementing the adaptive observer on the experimental data.
All parameters were found to converge at time between
50-60 sec.
F. Validation
To gauge the effectiveness of the estimated models, the
model of the system (equations 15 and 17) was simulated
with same input signal as that of the collected data. The esti-
mated values of the parameters were used in the simulation.
964
0 20 40 60 80 1000
1
2
3
4
5
6
7
8
9
Time (s)
σ1
σ0
σ2
c
Fig. 5. Estimated parameters σ0, σ1, σ2 and c
The velocity signal predicted by the model was compared
with the measured velocity signal. The data set used for
this validation exercise was different from the one used for
estimation of the parameters. A sample plot that shows the
comparison between the velocity predicted by the model and
the measured velocity is presented in Figure 6. As a measure
85 90 95 100−500
−400
−300
−200
−100
0
100
200
300
400
Time (s)
Ve
locity (
de
g/s
)
MeasuredPredicted
Fig. 6. Comparison of the predicted and the measured velocity signals
of the error between the two, Mean Square Error (MSE)
defined by
MSE =σ2
error
σ2meas
× 100% (25)
was calculated. Here σ2error is the variance of the error
between the predicted and measured velocity signals, and
σ2meas is the variance of the measured velocity signal. The
computed values of the MSE for multiple validation trials
were found to lie between 4% and 6%. Note that the simula-
tions were carried out in open-loop, i.e. the predicted values
of the simulation were independent of the past measured
values. The values of MSE indicate that the identified models
are of acceptable quality.
V. SUMMARY AND FUTURE WORK
In this paper a methodology for construction of adaptive
observers for servo systems with friction was discussed.
It was shown that the current methods of modeling dy-
namic friction make it difficult to apply the adaptive ob-
server design techniques. Modifications were proposed for
a popular friction model, the LuGre friction model and
several methodologies that can be used for constructing
adaptive observers for the resulting modified system were
pointed out. The methodology was used to construct an
adaptive observers to identify parameters of the model of
an experimental setup. The encouraging performance of the
identified models indicates that the adaptive observers based
friction identification method can be a good alternative to the
traditional friction identification methods based on dedicated
experiments.
Since a number of adaptive observer techniques can be
used for the model of the servo system, it will be interesting
to perform a comparative study of the different methods.
Possibilities of constructing adaptive observers based on
other dynamic friction models such as the GMS friction
model [3] can also be investigated.
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