GLOBAL ASYMPTOTIC VELOCITY OBSERVATION OF

NONLINEAR SYSTEMS

Application to a Frictional Industrial Emulator

R. Guerra

‡

, C. Iurian

⋆

, L. Acho

‡⋆

‡

Centro de Investigaci

´

on y Desarollo de Technolog

´

ıa Digital (CITEDI-IPN), Mexico

⋆

Universitat Polit

`

ecnica de Catalunya, Matem

`

atica Aplicada III, EUETIB, Barcelona, Spain

F. Ikhouane

†

and J. Rodellar

§

†

Universitat Polit

`

ecnica de Catalunya, Matem

`

atica Aplicada III, EUETIB, Barcelona, Spain

§

Universitat Polit

`

ecnica de Catalunya, Matem

`

atica Aplicada III, Barcelona, Spain

Keywords:

Velocity observers, friction, mechanical systems.

Abstract:

Development of velocity observers for mechanical systems with friction deserves a special emphasis, because

as evidenced in numerical and experimental tests when a state-of-the-art observer is armed, friction can in-

duce high-frequency oscillations in the estimated signal. In this short paper, two new velocity-observation

algorithms are designed, based on this previously reported observer, that eliminate the high-frequency oscilla-

tions noted. Numerical and experimental performance comparisons are carried through making use of LuGre

model and a mechanical PID control system that incorporates the estimated velocity into the feedback loop.

1 INTRODUCTION

Velocity-dependent control laws such as PD, PID, and

most robust control laws, among many others, the-

oretically require direct access to velocity. In reality,

there are many applications in which this is not availa-

ble either due to considerable manufacturing savings

in cost, weight, and volume, or because the velocity

measurements are highly contaminated with noise. In

the latter case, for instance when measuring robot

joint velocities, it may not even be desirable to do

so (Arteaga and Kelly, 2004). Consequently, if the

full-state information is missing, it is necessary to

estimate the unmeasurable velocity through the use

of an observer and feed it back into the controller.

Such is the case with the frictional industrial emula-

tor ECP model 220 used in our experiments. It has

been documented when studying mechanical closed-

loop control systems, that friction causes tracking

errors, limit-cycles, and stick-slip motions, among

other difﬁculties and usually unwanted phenomena

(Armstrong-H

´

etlouvry et al., 1994). As evidenced

in (Canudas de Wit and Fixot, 1991), (Canudas de

Wit and Fixot, 1992), (Berghuis and Nijmeijer, 1993),

and (Arteaga and Kelly, 2004) velocity observer de-

sign is a topic that has been and continues to be ex-

tensively studied. State observation of nonlinear ine-

xact plants has been treated by utilizing discontinu-

ous observers (Choi et al., 1999), (Xiong and Saif,

2001), and (Xian et al., 2004). However, little re-

search focused on velocity observation of mechan-

ical systems with friction at low velocities which,

when incorporating existing observers, exhibit high-

frequency oscillations in the velocity-estimation sig-

nal. A state-of-the-art, globally asymptotic, discon-

tinuous velocity-estimation scheme for second-order

mechanical systems has been presented in (Xian et al.,

2004). Though very reliable, this observer is not spe-

cialized for mechanical systems in the presence of

friction. Thus, a high-frequency component of the ve-

locity observation is detected when numerical simu-

lations and experimental testing are performed. The

two newly proposed observers claimed here are an

attempt to alleviate this unwanted oscillatory effect

and try offer an increase in the observation reliability.

The remainder of this document proceeds as fo-

llows: Section 2 postulates two modiﬁed observers

based on the velocity estimator stated in (Xian et al.,

2004). The next section presents numerical simu-

lations evidencing the performance of the observers.

Section 4 begins with a description of the frictional

experimental testbed, the ECP industrial emulator

85

Guerra R., Iurian C., Acho L., Ikhouane F. and Rodellar J. (2007).

GLOBAL ASYMPTOTIC VELOCITY OBSERVATION OF NONLINEAR SYSTEMS - Application to a Frictional Industrial Emulator.

In Proceedings of the Fourth International Conference on Informatics in Control, Automation and Robotics, pages 85-91

DOI: 10.5220/0001621100850091

Copyright

c

SciTePress

(ECP, 1995), and illustrates the results obtained when

implementing of the original observer as well as the

two modiﬁcations proposed in the PID mechatronic

system. Finally, conclusions are drawn in Section 5.

2 VELOCITY OBSERVATION

Consider a class of mechanical systems expressed by

(Xian et al., 2004)

¨x = h(x, ˙x) + G(x, ˙x)u, (1)

where x ∈ R is the system output, u(t) ∈ R is the

control input, and h(x, ˙x) ∈ R as well as G(x, ˙x) ∈ R

are nonlinear functions

1

. The system (1) satisﬁes the

following assumptions (Xian et al., 2004).

Assumption A1. Both h(x, ˙x) and G(x, ˙x) are C

1

functions.

Assumption A2. The control input is a C

1

function

and u(t), ˙u(t) ∈ L

∞

.

Assumption A3. The system state is bounded for all

time; i.e., x(t), ˙x(t) ∈ L

∞

.

The goal of the velocity observer is to estimate the

unmeasurable velocity signal ˙x(t) using only-position

measurement and assuming that h(x, ˙x), G(x, ˙x) and

u(t) are unknown (Xian et al., 2004). Let

˙

ˆx(t) ∈ R

be the estimated velocity and

˙

˜x = ˙x −

˙

ˆx the velocity

estimation error. Then, the objective of the velocity

observer is to ensure that

˙

˜x(t) converges to zero as the

time tends to inﬁnity.

Consider the following velocity observer (Xian

et al., 2004)

˙

ˆx = p+k

0

˜x,

˙p = k

1

sgn( ˜x) + k

2

˜x,

(2)

where k

0

,k

1

, and k

2

are positive constants, and sgn(·)

is the signum function.

Remark 1. System (2) can be expressed as

˙y = f(y,x), y ∈ R

2

, x ∈ R

z = g(y, x), z ∈ R

where y = [ ˆx p]

T

, f(y, x) = [p + k

0

˜x k

1

sgn( ˜x) +

k

2

˜x]

T

, g(y,x) = p + k

0

˜x, and z =

˙

ˆx is the output

velocity estimation.

To state the main result in (Xian et al., 2004), let

N

0

(x, ˙x,t) = h(x, ˙x) +G(x, ˙x)u(t).

1

Without loss of generality, we have assumed a one-

degree-of-freedom (DOF) mechanical system.

Theorem 1 (Xian et al., 2004). The velocity observer

(2) ensures global asymptotic regulation of

˙

˜x(t) (i.e.,

˙

˜x(t) → 0 as t → ∞) provided that k

1

satisﬁes

k

1

> ||N

0

(x, ˙x,t)||

∞

+ ||

˙

N

0

(x, ˙x,t)||

∞

.

For detailed proof, see Theorem 2 in (Xian et al.,

2004). Let us now put forward the following observer

˙

ˆx = p+ k

0

˜x,

˙p = −k

1

sgn( ˜x) + k

2

˜x.

(3)

Theorem 2. The velocity observer (3) ensures global

asymptotic regulation of

˙

˜x(t) (that is,

˙

˜x(t) → 0 as

t → ∞) provided that k

1

satisﬁes exactly the same

conditions as in Theorem 1.

Proof. Identical to that of Theorem 1.

Another alternative proposal to the original observer

is the following

˙

ˆx = p+ k

0

˜x,

˙p = k

1

sgn( ˆx) + k

2

˜x.

(4)

Theorem 3. The velocity observer (4) also ensures

global asymptotic regulation of

˙

˜x(t) (i.e.,

˙

˜x(t) → 0

as t → ∞) on the condition that k

1

satisﬁes the same

restriction as in Theorem 1.

Proof. The same case as for the proof of Theorem 2

above.

These innovative observers are very similar to the one

in Theorem 1, but they present signiﬁcant differences.

The observer in Theorem 2 introduces an inversion of

the sign in the estimation dynamic that produces a ﬁl-

tering effect. The observer in Theorem 3 brings for-

ward an estimation error ˆx inside the signum function,

with the intent of reducing the high-frequency content

the error signal ˜x present in (2).

3 NUMERICAL EXPERIMENTS

Consider a linear motion of unit mass

¨x = u− f, (5)

where f is the friction force. Assuming for a moment

that f = 0 and k

1

= 10, the requirement imposed by

all theorems is satisﬁed. We complete the velocity

observers by setting k

0

= k

2

= 10. We also construct

the following PID controller, which makes the closed-

loop system globally asymptotically stable (if f = 0)

u = −k

p

(x− x

d

) − k

i

(x− x

d

)dt − k

d

˙x, (6)

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

86

with k

v

= 6, k

p

= 3, k

i

= 4, and the position reference

set at x

d

= 1 [m], see details in (Canudas de Wit et al.,

1995). Because friction force can produce limit cy-

cles within the system when the control law has an in-

tegrating action (Canudas de Wit et al., 1995), we in-

corporate a friction force in (5) by invoking the LuGre

model and its standard parameters given in (Canudas

de Wit et al., 1995); i.e., f is obtained as a nonlinear

dynamic. Simulation results are depicted in Figure 1

where the position and velocity of the system are pic-

tured. These plots are a recreation of the experiment

presented in (Canudas de Wit et al., 1995). Obviously,

the PID controller (6) incorporates velocity measure-

ment, which for this simulation was assumed to be

available.

At this point, we repeated the previous simulation

using the true velocity in the control law and imple-

menting the three observers solely to estimate the ve-

locity, obtaining the results shown in Figure 2. Since

in the experimental application the velocity would

not be available for use, the simulation was again re-

peated, this time invoking the observers from Theo-

rems 1, 2, and 3 in the control law (i.e., replacing ˙x by

its corresponding

˙

ˆx in (6)). The results are portrayed

in Figures 3, 4, and 5, respectively. Note that cha-

ttering appears with the observer from Theorem 1 and

not with the others. It is important to stress that cha-

ttering is undesirable in a physical system because the

high-frequency switching can damage the system, as

well as activate unmodelled dynamics (G. Bartolini,

1998) and (Hung, 1993).

Let us return to the previously considered case

where the PID controller (6) is employed using the

exact velocity, which is only observed. If we slightly

modify the observer parameter values to k

1

= 5

and k

0

= k

2

= 1, the velocity observation obtained

with the observer in (2) becomes highly oscillatory

whereas the two observers proposed still provide

good results, as shown in Figure 6. If furthermore

we now feed the corresponding velocity estimation

into the PID controller (6), the ﬁrst observer (stated

in Theorem 1) is much more sensible to variations in

its parameters. This causes instability of the closed-

control-loop (simulation pictures were then omitted

for the ﬁrst case); nevertheless, the other two new ob-

servers yield acceptable good simulation results, see

Figures 7 and 8.

4 APPLICATION TO AN

INDUSTRIAL EMULATOR

To have a more realistic comparison among the ob-

servers stated in Section 2, we proceed to implement

0 10 20 30 40 50 60 70 80 90 100

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Position [m]

Time [s]

0 10 20 30 40 50 60 70 80 90 100

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Velocity [m/s]

Time [s]

Figure 1: PID position control of a second-order system

that incorporates Lugre friction model - see (Canudas de

Wit et al., 1995).

0 10 20 30 40 50 60 70 80 90 100

−0.4

−0.2

0

0.2

0.4

0.6

Velocity [m/s]

Xian et al. Velocity Observer

0 10 20 30 40 50 60 70 80 90 100

−1

−0.5

0

0.5

1

Velocity [m/s]

Proposed Velocity Observer from Theorem 2

0 10 20 30 40 50 60 70 80 90 100

−0.5

0

0.5

1

1.5

Velocity [m/s]

Time [s]

Proposed Velocity Observer from Theorem 3

Estimated velocity

Real velocity

Estimated velocity

Real velocity

Estimated velocity

Real velocity

Figure 2: Comparison of the three estimators (for k

0..

=

k

1

= k

2

= 10): 1) Top with Theorem 1, 2) Middle with

Theorem 2, and 3) Bottom with Theorem 3.

them on an experimentation testbed.

4.1 Experimental Setup

The experiments were performed on an ECP Model

220 industrial emulator which includes a PC-based

control platform and a DC brushless servo system

(ECP, 1995). The mechatronic system includes two

motors, one as servo actuator and the other as dis-

turbance input (not used here), a power ampliﬁer,

and two encoders which provide accurate position

GLOBAL ASYMPTOTIC VELOCITY OBSERVATION OF NONLINEAR SYSTEMS - Application to a Frictional

Industrial Emulator

87

0 10 20 30 40 50 60 70 80 90 100

0

0.5

1

1.5

2

Position [m]

Time [s]

0 10 20 30 40 50 60 70 80 90 100

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Velocity [m/s]

Time [s]

Estimated velocity

Real velocity

Figure 3: PID control incorporating velocity estimation

from Theorem 1.

0 10 20 30 40 50 60 70 80 90 100

0

0.5

1

1.5

2

Position [m]

Time [s]

0 10 20 30 40 50 60 70 80 90 100

−0.5

0

0.5

1

1.5

Velocity [m/s]

Time [s]

Estimated velocity

Real velocity

Figure 4: PID control employing velocity estimation given

in Theorem 2.

measurements; i.e., 4000 lines per revolution with

4× hardware interpolation giving 16000 counts per

revolution to each encoder; 1 count (equivalent to

0.000392 radians or 0.0225 degrees) is the lowest

angular position measurable (ECP, 1995). The sys-

tem was set up to incorporate inertia and friction

brake. The drive and load disks were connected via

a 4 : 1 speed reduction (see Figure 9). In order to

demonstrate that the system is subject to the noto-

rious effects of friction, we calculated according to

the procedure described by (R. Kelly and Campa,

0 10 20 30 40 50 60 70 80 90 100

0

0.5

1

1.5

2

Position [m]

Time [s]

0 10 20 30 40 50 60 70 80 90 100

−0.5

0

0.5

1

Velocity [m/s]

Time [s]

Estimated velocity

Real velocity

Figure 5: PID control utilizing the observer stated in Theo-

rem 3.

0 10 20 30 40 50 60 70 80 90 100

−0.4

−0.2

0

0.2

0.4

0.6

Velocity [m/s]

0 10 20 30 40 50 60 70 80 90 100

−3

−2

−1

0

1

Velocity [m/s]

0 10 20 30 40 50 60 70 80 90 100

−2

0

2

4

Velocity [m/s]

Time [s]

Estimated velocity

Real velocity

Estimated velocity

Real velocity

Estimated velocity

Real velocity

Figure 6: Comparison of the three estimators (for k

0..

=

k

2

= 1 and k

1

= 5): 1) Top with Theorem 1, 2) Middle with

Theorem 2, and 3) Bottom with Theorem 3.

2000) the following friction coefﬁcients for the sys-

tem: F

v

= 0.05772[Nmsec/rad] (viscous friction co-

efﬁcient) and F

c

= 0.43032[Nm] (Coulomb friction).

A Pentium 4, 2.80 GHz CPU, 512 MB RAM,

computer running under Windows XP is programmed

to implement the controller together with the in-

terface medium ECP USR Executive 5.1, a C-like

programming language (ECP, 1995). The system

contains a data-acquisition board for digital to analog

conversion and a counter board to read the position

encoder outputs into the servo system. The minimum

servo-loop closure sampling time T

s

is 0.884 ms.

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

88

0 25 50 75 100 125 150

−1

0

1

2

3

4

Position [m]

Time [s]

0 25 50 75 100 125 150

−4

−3

−2

−1

0

1

2

3

4

5

Velocity [m/s]

Time [s]

Estimated velocity

Real velocity

Figure 7: PID control containing the velocity as given by

Theorem 2.

0 25 50 75 100 125 150

−2

−1

0

1

2

3

Position [m]

Time [s]

0 25 50 75 100 125 150

−4

−3

−2

−1

0

1

2

3

4

5

Velocity [m/s]

Time [s]

Estimated velocity

Real velocity

Figure 8: PID control making use of the observer in Theo-

rem 3.

The output voltage signal generated by the system

is in the range of ±5V and is delivered to the motor

drive via the DAC, the measurement feedback is a

position signal (in counts or radians), measured at the

shaft of each of the two disks by the optical rotary

incremental position encoders, then it is read by the

microcomputer by means of the counter board and

delivered into the PC. A software interface has been

built to easily transfer the raw data collected from the

plant (by means of the ECP USR Executive program)

to the Matlab workspace environment, in order to

Figure 9: Mechanical system with friction.

display the results. The load disk is weighted with

4 masses of 0.50 kg each (at a radius of r = 10.0

cm) while the drive disk remains unweighted (see

Figure 9). It is worth mentioning that the mechanical

system has encoders which give accurate position

measurements, nevertheless no direct velocity sens-

ing is available (ECP, 1995). In this scenario, we

implemented the aforementioned velocity observers,

obtaining the results pictured in Figures 10 – 12.

4.2 Experimental Results

The implemented control law is as follows

u = −k

p

(x− x

d

) − k

i

(x− x

d

)dt − k

d

˙

ˆx,

with k

d

= 0.0011, k

p

= 0.135, and k

i

= 0.4.

The desired reference position was set to

x

d

= 100 [counts] = 0.0392 [rad] = 2.25 [deg].

As it can be noted in Figure 10, when the observer

stated in Theorem 1 is employed, this produces an os-

cillation of relatively high-frequency and amplitude

into the system. The observer (4), after the transient,

eliminates both the amplitude and the frequency of

this oscillation (Figure 11). The estimator from The-

orem 3 further reduces the amplitude and duration of

the transient, even more than observer (4), eliminat-

ing chattering as seen in Figure 12. Moreover, note

that the position limit cycles in Figures 11–12, caused

by friction in PID control of servo drives (Canudas

de Wit et al., 1995), have the same rectangular-like

waveform pattern as in Figure 1, and can be distinctly

identiﬁed.

5 CONCLUSIONS

Two new velocity-observation designs are presented,

and experimentally validated, for use in mechanical

GLOBAL ASYMPTOTIC VELOCITY OBSERVATION OF NONLINEAR SYSTEMS - Application to a Frictional

Industrial Emulator

89

0 2 4 6 8 10 12 14 16 18

0

50

100

150

200

Output Position [count]

Time [s]

0 2 4 6 8 10 12 14 16 18

−200

0

200

400

600

Observed Velocity [count/sec]

Time [s]

Figure 10: Experimental results using the observer from

Theorem 1.

0 2 4 6 8 10 12 14 16 18

0

50

100

150

200

Output Position [count]

Time [s]

0 2 4 6 8 10 12 14 16 18

−200

0

200

400

600

Observed Velocity [count/sec]

Time [s]

Figure 11: Experimental results employing the observer

from Theorem 2.

systems with friction where only-position measure-

ments are available. As it can be appreciated from

numerical and experimental results, the proposed ob-

server schemes are more efﬁcient than their precur-

sor in that chattering is eliminated from the velocity-

observed signal. It is worth emphasizing that the pre-

sented observers (3) and (4) are especially interest-

ing for industrial purposes, for they assure that the

velocity-acquisition hardware can, without difﬁculty,

be replaced by an analogous inexpensive software

performing the same function.

0 2 4 6 8 10 12 14 16 18

0

20

40

60

80

100

120

Output Position [count]

Time [s]

0 2 4 6 8 10 12 14 16 18

0

100

200

300

400

Observed Velocity [count/sec]

Time [s]

Figure 12: Experimental results utilizing the observer from

Theorem 3.

ACKNOWLEDGEMENTS

This work was supported by CICYT through Grant

DPI2005-08668-C03-01. The work of F. Ikhouane

was supported by Spanish Ministry of Science and

Education under “Ram

´

on y Cajal” Program.

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GLOBAL ASYMPTOTIC VELOCITY OBSERVATION OF NONLINEAR SYSTEMS - Application to a Frictional

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91