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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: : adixit@me.iitb.ac.in,

shashisn@iitb.ac.in

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

17th IEEE International Conference on Control ApplicationsPart of 2008 IEEE Multi-conference on Systems and ControlSan Antonio, Texas, USA, September 3-5, 2008

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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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