17th IEEE International Conference on Control Applications
Part of 2008 IEEE Multi-conference on Systems and Control
San Antonio, Texas, USA, September 3-5, 2008
FrA02.4
Adaptive Observers for Servo Systems with Friction
Amit Dixit and Shashikanth Suryanarayanan
Abstract— We consider the problem of constructing adaptive
observers for servo systems with friction. We show that if
the friction force is modeled using currently popular dynamic
friction models, currently available methods for constructing
adaptive observers can not be applied. We suggest modifications
for one such friction model: the LuGre friction model and point
to several methods of constructing adaptive observers that can
be applied to the modified system. Using one such method, we
construct an adaptive observer to identify parameters of the
model of an experimental setup. The results of the identification
exercise 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 compensation schemes are increasingly being used. Model-based
friction compensation involves using control-oriented models
that predict friction phenomena relevant to motion control applications. 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 modifications 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 Mechanical Engineering, Indian Institute of Technology Bombay, Powai,
Mumbai 400076, India. Email: : [email protected],
[email protected]
978-1-4244-2223-4/08/$25.00 ©2008 IEEE.
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 demonstrate 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 identification 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 include 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:
ẋ(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
960
that the matrix R is of the following form:


0
kT




0

R=

 ..

.
F 
0
adaptive observer:
˙
Γ̂(t) = λ(A0 − K0 C0 )Γ̂(t) + λψ(x̂(t), u(t))
˙
x̂(t)
= A0 x̂(t) + ϕ(x̂(t), u(t)) + ψ(x̂(t), u(t))θ̂(t)
(2)
where k 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) − x̂(t)) and in estimation
of parameters (θ − θ̂(t)) exponentially go to zero:
(3)
= A0 x(t) + ϕ(x(t), u(t)) + ψ(x(t), u(t))θ
y(t) = C0 x(t)
(4)
where x ∈ Rn denotes the state vector, u ∈ Rm 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:


0 1
0


..


.
A0 = 


1
0
0
C0 = 1 0 · · · 0
ϕ(x, u)
ψ(x, u)
where K0 is selected such that A0 − K0 C0 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 systems, 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.
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:
where Ω1 and Ω̄ are sub-matrices of matrix Ω given by ΩT =
[ΩT1 Ω̄T ], 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:
ẋ(t)
(6)
B. Model under consideration
= F V (t) + Ω̄((t))
ϕ(t) = V T (t)k + ΩT1 (ω(t))
˙
θ̂(t) = Γϕ(t)[y(t) − xˆ1 (t)]
˙
x̂(t)
= Rx̂(t) + Ω(ω(t))θ̂(t) + g(t)
"
#
c1 [y(t) − xˆ1 (t)]
+
˙
V (t)θ̂(t)
Λ(λ)−1 [λK0 + Γ̂(t)Γ̂T (t)C0T ][y(t) − C0 x̂(t)]
˙
θ̂(t) = λn Γ̂(t)T C0T [y(t) − C0 x̂(t)]
T
V̇ (t)
+
(ϕ (x , u) ϕ2 (x1 , x2 , u) · · · ϕn (x, u))T
1 1

0
···
0


..
..


.
.
= 
(5)



0
···
0
ψn,1 (x, u) · · · ψn,q (x, u)
J v̇(t) = bu(t) − F (t)
F (t) = σ0 z(t) + σ1 ż(t) + α2 v(t)
|v(t)|z(t)
ż(t) = v(t) −
s(v)
v
s(v) = a0 + a1 e−[ v0 ]
(9)
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 v0
are 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 steadystate (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 construction of adaptive observers for servo systems with friction.
Consider the set of Equations 7-10. The equations can be
rearranged as follows:
=
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
(7)
(8)
v̇(t)
ẇ(t)
= b̄u(t) − w(t) − α¯2 v(t)
σ1
|v(t)|w(t)
−
[σ0 v(t) −
]
Jσ0
s(v)
|v(t)|w(t)
= σ0 v(t) −
s(v)
(11)
(12)
where w(t) = σ0 z(t)/J is the new internal state variable,
s(v) is as defined by equation 10, b̄ = b/J and α¯2 = 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 b̄, σ0 , σ1 , α¯2 , 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 propose the following modifications to the model described by
equations 11-12:
1) Modify the internal state equation 12 as
ẇ(t) = σ0 s̄(v)v(t) − c|v(t)|w(t)
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 damping. 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 coefficient (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 s̄(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
(13)
where s̄(v) = s(v)c is considered to be unit-less while
c has units of (displacement)−1 .
2) Linearly parameterize s̄(v) using a suitable set of basis
functions. That is, express s̄(v) as:
X
s̄(v) = β0 +
βi φi (v)
(14)
i
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:
v̇(t) = b̄u(t) − w(t) − α¯2 v(t)
(15)
X
ẇ(t) = σ0 [β0 +
βi φi (v)]v(t) − c|v(t)|w(t) (16)
a0+a1
i
Equation 16 can be written as:
X
ẇ(t) = σ¯0 v(t) +
σ̄i φi (v)v(t) − c|v(t)|w(t) (17)
Higher sliding velocities
S(V)
a0
i
where σ¯0 = σ0 β0 and σ̄i = σ0 βi .
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
b̄, σ¯0 , σ̄i , α¯2 and c. The number of parameters is equal
to 4 + p, where p is the number of basis functions
used to approximate s̄(v). The unmeasured state w(t)
corresponds to the friction force.
2) At steady-state, i.e. for v̇(t) = 0 and ẇ(t) = 0, we
have
X
1
bu(t) = sgn(v(t)) [σ¯0 +
σ̄i φi (v)] + α¯2 v(t) (18)
c
i
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
1
found to lie between a10 and a0 +a
), with σ̄0 and σ̄i
1
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
V0
V
Fig. 1.
A typical plot of s(v) versus v
functions are listed here:
P
i
• Polynomial: s̄(v) = β0 +
Pi βi |v|
v−v̄i 2
• Gaussian: s̄(v) = β0 +
i βi (exp(−( v̄0i ) ))
where exp is the exponential function and v̄i ,
v̄0i are chosen so as to cover complete range of
velocity for which value of s̄(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 transformation: x1 (t) = v(t) and x2 (t) = v̇(t). Under this
co-ordinate transformation, equations 15 and 17 can be
written as:
962
ẋ1 (t)
ẋ2 (t)
= x2 (t)
= −α¯2 x2 (t) + b̄u̇(t) − σ¯0 x1 (t)
X
−
σ̄i φi (x1 (t))x1 (t)
(19)
i
+ c|x1 (t)|(b̄u(t) − α¯2 x1 (t) − x2 (t))(20)
by considering the extended parameter vector θ =
[α¯2 b̄ σ̄0 σ̄i cb̄ cᾱ2 c]T , the above system can be written as:
ẋ1 (t)
ẋ2 (t)
= x2 (t)
= G(x(t))θ
(21)
(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 realtime differentiation based on Variable Structure
theory [14], signals v(t), v̇(t) and u̇(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 compensation or for friction simulation), construction of
adaptive observer can be simplified by carrying out
the identification procedure in two steps: identification
of parameters ᾱ2 and b̄ 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:
v̇(t) = −α2 v(t) + b̄u(t) − Fc sgnv(t)
(23)
Thus, if the velocity signal is measured, then parameters ᾱ2 and b̄ can be estimated using any standard
linear recursive parameter estimation scheme, for example 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 ᾱ2 and b̄ have been estimated, one method to construct the adaptive observer
is to follow Approach I mentioned in this paper by
treating ᾱ2 and b̄ as known parameters. In this case,
identification of ᾱ2 and b̄ 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 ẋ0 (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, 000
pulses 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.
is significant only for velocities below 500 /sec, following
parameterization was used:
2
s̄(v) = β0 + β1 e−[(v/10) ] + β1 e−[((v−20)/20)
2
]
(24)
E. Identification results
First, parameters ᾱ2 and b̄ 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 Equations 1, 2 and 3, with number of states n = 1, R = 0,
Ω = [−v(t) u(t) − sgn(v(t))], θ = [ᾱ2 b̄ 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 v̂(t) was set to be equal to vc or −vc
depending on sign of v(t).
Figure 4 shows the parameter estimates obtained by implementing the adaptive observer on the experimental data.
Using the estimated values of ᾱ2 and b̄, an adaptive observer
b
α2
8
7
6
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 velocity 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.
5
4
3
2
1
0
−1
−2
0
50
Fig. 4.
100
Time (s)
150
200
Estimated parameters ā2 and b̄
300
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 = (−ᾱ2 x1 (t) + b̄u(t) 0)T , θ = [σ̄0 σ̄1 σ̄2 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 implementing the adaptive observer on the experimental data.
Reference velocity (deg/s)
200
100
0
−100
−200
−300
224
226
Fig. 3.
228
230
Time (s)
232
234
236
All parameters were found to converge at time between
50-60 sec.
Reference velocity signal
F. Validation
D. Choice of basis functions to approximate s̄(v)
A set of two Gaussian functions was used to approximate the s̄(v) curve. Considering that the Stribeck effect
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 estimated values of the parameters were used in the simulation.
964
9
8
σ1
σ0
σ2
7
c
V. SUMMARY AND FUTURE WORK
6
5
4
3
2
1
0
0
20
Fig. 5.
40
Time (s)
60
80
100
Estimated parameters σ̄0 , σ̄1 , σ̄2 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
Measured
Predicted
400
300
200
Velocity (deg/s)
100
0
−100
−200
−300
−400
−500
85
Fig. 6.
90
Time (s)
95
100
Comparison of the predicted and the measured velocity signals
of the error between the two, Mean Square Error (M SE)
defined by
M SE =
2
σerror
× 100%
2
σmeas
(25)
2
was calculated. Here σerror
is the variance of the error
between the predicted and measured velocity signals, and
2
is the variance of the measured velocity signal. The
σmeas
computed values of the M SE for multiple validation trials
were found to lie between 4% and 6%. Note that the simulations 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.
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 dynamic friction make it difficult to apply the adaptive observer 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.
R EFERENCES
[1] B. Armstrong, P. Dupont and Canudas de Wit, A Survey of Models,
Analysis Tools and Compensation Methods for the Control of Machines
With Friction, Automatica, Vol. 30, No.7, pp. 1083-1138;1994.
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