Appears in Proc. 2012 American Control Conference, pp. 4739-4744, Fairmont Queen Elizabeth, Montreal, Canada, June 27-June 29, 2012.

Friction Compensation Without a Friction Model

Richard J. Vaccaro

Abstract— This paper presents the design of a positioncontrol system for plants that have nonlinear friction. Theapproach uses nested integral tracking loops: an inner velocityloop and an outer position loop. An additional component ofthe proposed approach is the nonstandard use of an observer,or reduced-order observer, in the inner loop to estimate thestate of the velocity integrator. The designs of the inner loop,outer loop, and observer are accomplished using standardlinear pole-placement techniques. The method does not requireknowledge of any friction model, yet results in effective frictioncompensation over a certain bandwidth. The performance ofthe proposed approach is demonstrated using simulation resultsas well as hardware experiments with a belt-driven positioningsystem.

I. INTRODUCTION

The design of position control systems when the plantcontains nonlinear friction is challenging task. Most recentapproaches to this problem use some type of nonlinearmodel for the friction, and either implicitly or explicitlyrely on knowledge of the parameters of the friction model.This knowledge may be obtained from experiments with thephysical plant prior to control, or during real-time controlusing adaptive methods [3–6].

This paper presents a linear, time-invariant control systemapproach that does not require a model for the friction.The method consists of nested, state-space, integral trackingloops: an inner velocity loop and an outer position loop.The velocity loop provides effective friction compensation,allowing the outer position loop to respond like a knownlinear system. A novel feature of the velocity loop is anobserver that is used to estimate the value of the velocityintegrator output. This is not a standard use of an observerand would not be needed to control a linear plant becausethe velocity integrator is part of the control system and itsoutput is available for feedback. However, the presence ofnonlinear friction in the plant causes the velocity integratoroutput to have spikes. The use of an observer, or reduced-order observer, estimates the integrator output largely free ofthese spikes and helps the closed-loop system behave like aknown linear reference system over a certain bandwidth.

The paper is organized as follows. Section II introduces astandard state-space position tracking system architecture forlinear plants. Because this system uses integral control, it willprovide some robustness to plant modeling errors and mildnonlinear friction. However, the standard tracking system haspoor performance in the case of moderate to severe nonlinear

Department of Electrical, Computer, and Biomedical Engineering,University of Rhode Island, Kingston, RI 02881, USA,vaccaro@ele.uri.edu

friction. A figure-of-merit is introduced to quantify the extentto which the tracking system behaves like a known referencemodel. Simulations are provided for a plant with nonlinearfriction showing that the standard tracking system has anunacceptable figure of merit as well as a large peak trackingerror.

In Section III the proposed nested architecture is intro-duced along with a complete specification of the designchoices, e.g. closed-loop poles. Simulation results are pro-vided showing a substantial improvement in the figure-of-merit and a reduction in peak tracking error compared tothe standard tracking system. Section IV contains experi-mental results for a belt-driven positioning system. Althoughthe friction model for this hardware system is unknown,it is severe enough so that the standard tracking systemhas acceptable performance only for an 0.5-Hz bandwidth.The nested tracking system is shown to have acceptableperformance over a bandwidth of 1.5 Hz.

II. STATE FEEDBACK TRACKING SYSTEM

We begin with a standard state-space architecture fortracking system design with a linear plant model that wasintroduced in [1] and is described in textbooks such as [2].Although this type of tracking system is only somewhateffective against nonlinear friction, it is introduced it herefor two reasons. First, it will be used a baseline system.The tracking system with friction compensation developedin Section III will be compared with this baseline system.The second reason is that the proposed friction compensationapproach uses, in part, a nested configuration of two standardtracking systems.

Consider an nth order linear, single-input, single-outputplant with state-space model

x = Ax+ buy = cx.

(1)

In order to design a control system in which y(t) tracks aclass of reference inputs {r(t)} with zero steady-state errorwe must use an additional dynamics system (see Fig. 1):

xa = Aaxa + bae (2)

where the eigenvalues of Aa include the poles of the LaplaceTransform of r(t) [1,2]. For the class of step inputs, only oneeigenvalue is required at s = 0, which gives integral control.That is, for step-input tracking the additional dynamics aregiven by

Aa = 0, ba = 1 (3)

in the standard tracking system architecture shown in Fig. 1.This tracking system will have zero steady-state error toa step input and good tracking performance over somebandwidth.

aΣΣ

L1

x=Ax+buy=cx

r(t) y(t)e(t)

x(t)

Additional Dynamics Plant

ax =A x +b ea a

av=L x 2 a

Fig. 1. A standard state-space tracking system.

In order to compute the 1× n state-feedback gain vectorL1 and the integrator gain L2, a design model consisting ofthe cascade of the plant followed by the additional dynamicsis needed [2]. This design model is given by:

Ad =

[A 0bac Aa

], bd =

[b0

]. (4)

Given a desired set of n + 1 closed-loop poles, call itspoles, a 1 × (n + 1) gain vector Ld is computed (e.g.using the Matlab place command) to obtain

eig(Ad − bdLd) = spoles. (5)

Then L1 consists of the first n elements of Ld and L2 is thelast element of Ld.

A. Plant Model

The plant model for a translational positioning system thatwill be used for simulations later in the paper is given bythe state-space model (A,b, c) shown below:[

x1x2

]=

[0 10 −α

] [x1x2

]+

[0β

]u

x1 = [ 1 0 ]

[x1x2

] (6)

where α and β are positive real numbers, u is the voltageinput to the motor power amplifier, x1 is the cart position(m), and x2 is the cart velocity (m/s). This linear plantmay be simulated using the block diagram shown below,with F (x2) = αx2 representing viscous friction. Simple

β Σ1s

1s

x2F( )

x2 x1u

Fig. 2. Block diagram of plant model. A linear plant with only viscousfriction represented when F (x2) = 1.3x2. A nonlinear function F (x2)such as that shown in Fig. 3 represents nonlinear friction.

representations of nonlinear friction may be obtained bychoosing F (x2) as an appropriate nonlinear function of x2.

Variable Name Pole Locationss1 −4.6200

s2 −4.0530± j2.3400

s3 −5.0093, −3.9668± j3.7845

TABLE IROOTS OF NORMALIZED BESSEL POLYNOMIALS FOR 1ST, 2ND, AND

3RD-ORDER SYSTEMS WITH 1-SECOND SETTLING TIME (FROM [2]).

The particular friction curve that is used for the simulationsin this paper is shown in Fig. 3. It is not important forthis paper that the curve in Fig. 3 is the “right” modelfor friction. Indeed, the “right” model of friction may bea nonlinear dynamical system with an unmeasurable frictionstate [6]. The point of using the curve in Fig. 3 is simply tocreate a significant nonlinear friction effect for the purposeof simulation and to demonstrate the degree to which theproposed approach can ameliorate the effects of friction.

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Velocity (m/s)

For

ce (

N)

Fig. 3. Force vs. velocity curve representing nonlinear friction.

B. Simulation Results for Standard Tracking System

Consider the standard position tracking system shown inFig. 1 with the plant model given in (6) with α = β = 1.3.The desired settling time for this system is TS = 1 second. Inorder to calculate the feedback gains as described in (4) and(5), we choose the closed-loop poles locations to be the rootsof a 3rd-order normalized Bessel polynomial (see Table I):

spoles = s3.

Although this tracking system does not have zero steady-state error to a sinusoidal reference input, the input-outputbehavior of the tracking system is described by the model ofthe closed-loop system. As long as the relationship betweenthe reference input and the plant output can be describedby a known linear system, the tracking system is easy touse either as part of a larger control system, or as is with a

precompensated reference input. From Fig. 1, the closed-loop tracking system from r(t) to y(t) is described bythe following state-space model (the subscript r refers toreference model):

Ar =

[(A− bL1) bL2

−bac Aa

], br =

[0ba

], (7)

cr = [ 1 0 0 ] .

The matrices (A,b, c) come from (6) with Aa = 0, ba = 1for integral control; this model does not contain any non-linear friction. A simulation is performed with the referenceinput chosen to be a sinusoid with amplitude 0.05 m andfrequency 0.5 Hz. The output of the tracking system shownin Fig. 1 is compared with the output of the reference model(7). Of course, these outputs should be identical up to smallerrors due to numerical integration. The results are shown inFig. 4. Note that, after a 1-second settling time, all signals

0 1 2 3 4 5 6 7 8−0.5

0

0.5

Time (sec.)

Plant Input

0 1 2 3 4 5 6 7 8−0.05

0

0.05

Time (sec.)

Reference Model Output (dashed) vs. Actual Position (solid)

0 1 2 3 4 5 6 7 8−5

0

5x 10−5

Time (sec.)

Error Between Reference Model Output and Actual Position

Fig. 4. Simulation results for a standard tracking system controlling thelinear plant (A,b, c) given by (6). The reference input is a sinusoid withamplitude 0.5 m and frequency 0.5 Hz. Upper graph shows the plant input.Middle graph shows the tracking system output y(t) and the reference modeloutput yr(t), which are indistinguishable. Lower graph shows the erroryr(t)− y(t).

are sinusoidal, including the plant input signal, which has anamplitude of about 0.4 volts.

We now change the plant simulation to include the non-linear friction term F (x2) shown in Fig. 3. The controlsystem is not changed in any way. The simulation resultsare shown in Fig. 5. Notice that the plant input is no longersinusoidal, and its peak amplitude is about double the valueit had with only viscous friction.There is now a noticeableerror between the output of the reference model and the plantoutput. The standard position tracking system output is notperfectly predicted by a linear model. To get a quantitativemeasure of the difference between the tracking system outputand the output of the reference model, we introduce a figure-of-merit as follows. Let yr(t) be the output of the reference

0 1 2 3 4 5 6 7 8−1

0

1

Time (sec.)

Plant Input

0 1 2 3 4 5 6 7 8−0.05

0

0.05

Time (sec.)

Reference Model Output (dashed) vs. Actual Position (solid)

0 1 2 3 4 5 6 7 8−0.02

0

0.02

Time (sec.)

Error Between Reference Model Output and Actual Position

Fig. 5. Simulation results for the standard tracking system with the planthaving nonlinear friction.

model (7), y(t) the output of the tracking system, and lete(t) = yr(t)− y(t). Then the figure-of-merit, F , is

F = 100

1−

∫P

e2(t)dt∫P

y2ref (t)dt

(8)

where the integrals are taken over one period, P , of thesteady-state outputs. If F = 100%, it means that the trackingsystem output is perfectly described by the reference model.The value of F for the simulation results shown in Fig. 5 isabout 95%.

III. NESTED TRACKING SYSTEM WITHOBSERVER

Consider using nested tracking systems. The inner loop isa standard tracking system for plant velocity. This closed-loop velocity system may then be used as a new plant in thedesign of a position tracking system. Such an architecture hasbeen referred to as “cascade control,” but the term “nestedtracking system” seems to be a better description becausethe velocity loop is embedded in the position loop. Thedevelopment below uses the specific plant model (6) forease of exposition. This plant model describes the hardwaresystem used in Section IV.

The velocity loop is a standard tracking system, whoseinput is commanded velocity vc. The additional dynamicsfor the velocity loop are given by Aa = 0, ba = 1; that is,integral control. For this plant with velocity as the output,x2 is the only state variable. The gains L1v and L2v arecalculated in the usual way using (4) and (5). The desiredsettling time of the velocity loop, call it Tsv , is chosen to besome fraction of the settling time, Ts, of the position loop,e.g.

Tsv = Ts/4.

a Σx2

Σvc

x2

L1v

e u2= − x + uα β

Velocity Integrator Plant

ax =A x +b ea a

av=L x 2v

Fig. 6. Inner-loop velocity tracking system.

The inner loop is a 2nd-order system, and the closed-looppoles are chosen to be

spoles = s2/Tsv.

The system shown in Fig. 6, augmented with an integratorto have position as the output signal, is the plant that willbe used to design the position tracking system. The velocityintegrator state variable xa is renamed x3 in the followingequation. This complete inner loop system is described by

x1 = x2x2 = − αx2 + β(−L1vx2 + L2vx3)x3 = Aax3 + ba(vc − x2).

(9)

The state-space matrices for this system are denoted(A1,b1, c1) and are given by:

A1 =

0 1 00 −(α+ L1v) βL2v

0 −ba Aa

, b1 =

00ba

, (10)

c1 = [ 1 0 0 ] .

A position tracking system can be designed for the system(A1,b1, c1), as shown in Fig. 7. The complete system is 4th-

aΣΣ

L1

(A , b , c )1 1 1

v (t)cr(t) y(t)e(t)

x(t)

Inner LoopPosition Integrator

ax =A x +b ea a

av=L x 2

Fig. 7. Outer loop position tracking system with embedded inner loopvelocity tracking system. The state vector of the inner loop system containsthe three state variables shown in (9).

order. When selecting the closed-loop poles for this nestedtracking system, the two stable poles of the inner loop arenot changed. The other two closed-loop poles are chosen toachieve a settling time of Ts = 1 second. Thus, the completeset of closed-loop poles is

spoles = [ s2/Tsv s2/Ts ] .

This choice of closed-loop poles, preserving the fast inner-loop poles, is crucial to the proposed approach.

The reference model for this nested tracking system is

Ar =

[(A1 − b1L1) b1L2

−bac1 Aa

], br =

[0ba

], (11)

cr = [ 1 0 0 0 ] .

In a usual state-feedback system, all three state variablesfrom the inner loop system are used for feedback. However,we claim that feeding x3 to the position tracking systemmay not provide effective friction compensation. The reasonis as follows. If, due to nonlinear friction, the actual velocitydiffers from the commanded velocity, the integrator outputwill quickly build up in an attempt to achieve the desiredvelocity. The presence of nonlinear friction will cause theintegrator output to be larger than it would have been if onlyviscous friction were present. This nonlinear effect of theintegrator output means that x3 should not be used for outer-loop position tracking. Instead, an observer may be used toestimate the value of x3. The observer is designed for thevelocity tracking loop shown in Fig. 6, whose input is vcand output is x2. The velocity signal x2 is less effected byfriction that x3. The velocity tracking system removes muchof the effect of friction by creating a plant-input signal toovercome the friction. By using an observer to estimate thevalue of x3, the state variables sent to the outer loop arerelatively free from the effects of friction. This is seen in thesimulation results below and in the experimental results inthe next section.

A standard observer is a system which takes the input andoutput signals of a plant and from them produces an estimateof the state variables of the plant. The observer described inthe previous paragraph is a nonstandard observer in whichthe “plant” is the inner-loop system shown in Fig. 6, whoseinput is vc and output is x2. This system has the followingstate-space model, call it (Ao,bo, co):[

x2x3

]=

[−(α+ L1v) αL2v

−ba Aa

] [x2x3

]+

[0ba

]vc

x2 = [ 1 0 ]

[x2x3

], z =

[x2x3

].

(12)The observer is implemented with the following equations:

z = (Ao − boK)z+ bovc +Kx2x3 = [ 0 1 ] z

(13)

where K is the observer gains vector, which is calculated tomake

eig(Ao − boK) = s2/Tso, Tso = Tsv/3.

The observer (13) is used in the inner loop system shown inFig. 7 and the state vector coming out of the inner loop isx = [x1 x2 x3 ]

T . It is also possible to design a reduced-order observer to estimate x3. This reduced-order observer isa first-order system and the observer pole location is chosento be s1/Tso. The equations for reduced-order observers arefound in [2].

A simulation of the nested tracking system is performedwith the reference input chosen to be a sinusoid withamplitude 0.05 m and frequency 0.5 Hz. The results in Fig. 8may be compared with those in Fig. 5. The peak error isreduced by a factor of five. The figure-of-merit for the nestedtracking system is 99.92%.

0 1 2 3 4 5 6 7 8−1

0

1

Time (sec.)

Plant Input

0 1 2 3 4 5 6 7 8−0.05

0

0.05

Time (sec.)

Reference Model Output (dashed) vs. Actual Position (solid)

0 1 2 3 4 5 6 7 8−2

0

2x 10−3

Time (sec.)

Error Between Reference Model Output and Actual Position

Fig. 8. Simulation results for the nested tracking system with observer inthe inner loop. The reference input is a sinusoid with amplitude 0.05 m andfrequency 0.5 Hz.

The standard and nested tracking systems are compared fora set of sinusoidal reference inputs having amplitude 0.05 mand frequencies ranging from 0.2 Hz to 2.0 Hz. The nestedtracking system is implemented with the full-order observershown in (13) and also with a reduced-order observer. Theresulting figures-of-merit and peak tracking errors are shownin Table II. The data in this Table show that the nestedtracking system without an observer has much better figures-of-merit as well as lower peak tracking errors than thestandard tracking system. Using an observer improves theperformance, and using a reduced-order observer improvesthe performance even more. For the 0.2 Hz reference input,for example, the peak tracking error for the nested systemusing a reduced-order observer is reduced almost by a factorof two over the nested system that uses x3 directly.

If we choose a threshold of 95% for saying that a trackingsystem is well-described by its reference model, then Table IIshows that the standard tracking system F -value exceeds95% only up to 0.5 Hz, while all the nested tracking systemsF -values exceed the threshold up to 1.5 Hz, a 3-fold increasein usable bandwidth. The nested system with the reduced-order observer has the highest figures-of-merit and the lowestpeak tracking errors.

IV. EXPERIMENTAL RESULTS

The standard and nested tracking systems were tested onthe belt-driven cart system shown in Fig. 9. An approximatelinear model for the translational hardware system is givenby (6) with α = 1.3 and β = 1.3. This system, madeby Feedback, Inc., has two rods that may be attached toan axle running through the center of the cart to obtain acart-pendulum system. A control system to regulate the rodsin the upright position may be designed easily using linearcontrol theory. However, the resulting control system doesnot work with this hardware due to the nonlinear friction in

Standard x3 Measured Nested OB Nested ROOFreq.

F F P F P F P

0.2 98.15 99.93 2.56 99.98 1.53 99.98 1.420.5 95.47 99.79 2.41 99.92 1.88 99.93 1.730.8 83.19 99.53 1.99 99.76 1.94 99.80 1.771.0 68.37 99.07 1.82 99.47 1.94 99.56 1.781.5 37.12 95.81 1.70 96.47 1.98 97.06 1.812.0 34.80 88.01 1.75 87.26 1.94 89.37 1.77

TABLE IISIMULATION RESULTS SHOWING THE FIGURE-OF-MERIT F GIVEN IN (8)AND PEAK TRACKING ERROR P (MM) FOR THE STANDARD AND NESTED

TRACKING SYSTEMS. THE REFERENCE INPUT IS A SINUSOID OF

AMPLITUDE 0.05 M AND VARIOUS FREQUENCIES IN HZ. RESULTS FOR

THREE DIFFERENT NESTED TRACKING SYTEMS ARE SHOWN, THE

SYSTEM WITHOUT AN OBSERVER (x3 MEASURED), THE SYSTEM USING

A FULL-ORDER OBSERVER TO ESTIMATE x3 (NESTED OB), AND THE

SYSTEM USING A REDUCED-ORDER OBSERVER TO ESTIMATE x3

(NESTED ROO)

Fig. 9. Photo of a belt-driven cart system.

the cart subsystem. In order to consider only the translationaldynamics, the rods were removed and taped to the top of thecart in order to preserve the mass of the system. (see Fig. 9).

The hardware system was controlled by both the standardand nested tracking systems with a sinusoidal input ofamplitude 0.05 m and frequency 0.5 Hz. Fig. 10 shows theresults for the standard tracking system, which had an F -value of 96.67%. Fig. 11 shows the results for the nestedtracking system with reduced-order observer, which had anF -value of 99.97%.

The standard and nested tracking systems are comparedon the hardware system for a set of sinusoidal referenceinputs having amplitude 0.05 m and frequencies ranging from0.2 Hz to 2.0 Hz. The resulting figures-of-merit are shownin Table III.

The bandwidth provided by the inner loop of the nestedtracking system is sufficient to regulate the completecart/pendulum system with a linear regulator designed for theinner-loop system augmented with the associated pendulumdynamics.

The bandwidth over which the figure-of-merit exceeds95% is primarily determined by the speed (bandwidth) ofthe inner-loop system shown in Fig. 6. Making Tsv smallerincreases the inner-loop bandwidth. However, there are prac-

0 1 2 3 4 5 6 7 8−0.5

0

0.5

Time (sec.)

Plant Input

0 1 2 3 4 5 6 7 8−0.05

0

0.05

Time (sec.)

Reference Model Output (dashed) vs. Actual Position (solid)

0 1 2 3 4 5 6 7 8−0.01

0

0.01

Time (sec.)

Error Between Reference Model Output and Actual Position

Fig. 10. Experimental results for the standard tracking system. Thereference input is a sinusoid with amplitude 0.05 m and frequency 0.5 Hz.

0 1 2 3 4 5 6 7 8−1

0

1

Time (sec.)

Plant Input

0 1 2 3 4 5 6 7 8−0.05

0

0.05

Time (sec.)

Reference Model Output (solid) vs. Actual Position (dashed)

0 1 2 3 4 5 6 7 8−2

0

2x 10

−3

Time (sec.)

Error Between Reference Model Output and Actual Position

Fig. 11. Experimental results for the nested tracking system using areduced-order observer. The reference input is a sinusoid with amplitude0.05 m and frequency 0.5 Hz.

tical limits. For the hardware system, if Tsv is chosen to besmaller than 0.25 seconds, a resonance in the drive belt isexcited. However, choosing Tsv = Ts/4 = 0.25 providesexcellent friction compensation over a bandwidth of 1.5 Hz.

V. CONCLUSIONS

The proposed method of friction compensation consistsof nested, state-space, integral tracking loops. The designof this control system requires only an approximate linearmodel of the plant in the absence of friction. No frictionmodel is required. The nested tracking system uses twodesign parameters: Tsv , the desired settling time of theinner velocity loop, and Ts, the desired settling time ofthe outer position loop. Decreasing Tsv gives better frictioncompensation. However, there are practical considerations,

Standard x3 Measured Nested OB Nested ROOFreq.

F F P F P F P

0.2 99.38 99.98 1.20 99.99 0.90 99.99 0.770.5 96.67 99.86 1.29 99.96 1.15 99.97 1.080.8 89.70 99.39 2.19 99.70 1.55 99.74 1.381.0 85.10 98.77 2.40 99.21 1.65 99.26 1.621.5 71.12 95.64 2.19 96.10 2.05 96.38 1.952.0 57.24 91.31 1.71 90.24 1.73 91.19 1.67

TABLE IIIRESULTS FOR THE HARDWARE SYSTEM SHOWING THE

FIGURE-OF-MERIT F GIVEN IN (8) AND PEAK TRACKING ERROR P (MM)FOR THE STANDARD AND NESTED TRACKING SYSTEMS. THE

REFERENCE INPUT IS A SINUSOID OF AMPLITUDE 0.05 M AND VARIOUS

FREQUENCIES IN HZ. RESULTS FOR THREE DIFFERENT NESTED

TRACKING SYTEMS ARE SHOWN, THE SYSTEM WITHOUT AN OBSERVER

(x3 MEASURED), THE SYSTEM USING A FULL-ORDER OBSERVER TO

ESTIMATE x3 (NESTED OB), AND THE SYSTEM USING A

REDUCED-ORDER OBSERVER TO ESTIMATE x3 (NESTED ROO)

such as the frequency at which unmodeled plant dynamicsbecome significant, that determine how small Tsv can be.

The use of an observer, or reduced-order observer, to esti-mate the state of the velocity integrator in the inner loop wasshown to improve the friction compensation, i.e. increasingthe figure-of-merit and decreasing the peak tracking error.The simulation and experimental results show that the nestedtracking system behaves as a known reference model over acertain bandwidth even though the linear model for the plantis only an approximation and the plant is subject to nonlinearfriction.

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[3] L. Freidovich, A. Robertsson, A. Shiriaev, and R. Johansson, “LuGre-Model-Based Friction Compensation,” IEEE Trans. Control SystemsTechnology, vol. 18, no. 1, pp. 194–200, January, 2010.

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