Load speed regulation in compliant mechanical transmission systems using feedback and feedforward control actions

Load speed regulation in compliant mechanical transmission systems
using feedback and feedforward control actions
P.R. Raul a
, R.V. Dwivedula b
, P.R. Pagilla c,n
a
Mechanical & Aerospace Engineering, Oklahoma State University, Stillwater, OK 74078, United States
b
Department of Mechanical Engineering, Sree Vidyanikethan Engineering College, Tirupati, Andhra Pradesh 517102, India
c
Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843, United States
a r t i c l e i n f o
Article history:
Received 19 February 2015
Received in revised form
19 November 2015
Accepted 13 March 2016
Available online 25 April 2016
This paper was recommended for publica-
tion by Didier Theilliol
Keywords:
Adaptive feedforward
Load speed regulation
Belt–pulley mechanical transmission
a b s t r a c t
The problem of controlling the load speed of a mechanical transmission system consisting of a belt-
pulley and gear-pair is considered. The system is modeled as two inertia (motor and load) connected by a
compliant transmission. If the transmission is assumed to be rigid, then using either the motor or load
speed feedback provides the same result. However, with transmission compliance, due to belts or long
shafts, the stability characteristics and performance of the closed-loop system are quite different when
either motor or load speed feedback is employed. We investigate motor and load speed feedback
schemes by utilizing the singular perturbation method. We propose and discuss a control scheme that
utilizes both motor and load speed feedback, and design an adaptive feedforward action to reject load
torque disturbances. The control algorithms are implemented on an experimental platform that is
typically used in roll-to-roll manufacturing and results are shown and discussed.
& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
Mechanical transmissions are widely used in various industries
where the mechanical power is typically transmitted from motor
shafts to load shafts by utilizing transmission systems. Examples
include manufacturing, power generation, and transportation
systems. Power transmission with speed reduction and variable
torque requirement is made possible with mechanical transmis-
sion systems. Belt–pulley and gear transmission systems are
commonly used. In many applications, a mechanical transmission
system containing a combination of belt–pulley and a gear-pair is
very convenient over a purely gear transmission system. When the
center distance between the driving (motor) shaft and the driven
(load) shaft is too large for use of a single gear-pair, using a belt to
transmit motion/power may be the only practical alternative.
Further, such an arrangement is advantageous because coupling
the drive motor directly to the process end mandates very accurate
collinearity of the axes and takes considerable amount of time;
also, there is no guarantee that collinearity is maintained over
extended period of time due to load disturbances. Belt driven
transmission systems offer considerable ﬂexibility as small
inaccuracies in alignment can be absorbed into compliance of the
belt. However, compliance of the belt introduces additional
dynamics into the system. The belt driven power transmission
system is common in roll to roll manufacturing. The presence of
compliance from transmissions and the stiffness of web material
[1] will pose different levels of severity in properly transporting
the web.
Control of load speed is essential in many applications. When
rigid transmissions are employed, there is no dynamic relation
between the motor shaft and the load shaft, and typically the
motor shaft speed is controlled to control the speed of the load
shaft. However, due to the transmission dynamics, resulting from
the compliance of belt as well as long shafts in the transmission,
regulating load shaft speed is not the same as regulating motor
shaft speed. In the presence of such a transmission, practicing
engineers are often confronted with the question of whether to
use (i) motor speed feedback to control load speed as is done in
conventional practice, or (ii) use load speed feedback, or (iii) use a
combination of motor and load speed feedback.
There is a large body of literature on the characteristics of belt
drives and design of mechanisms using belt drives. Much of this
work focused on the mechanism of motion/power transfer, loca-
tion and extent of slip-arc, nature of frictional contact, efﬁciency
limit of the belt-drive system, and methodology of design/selec-
tion of belt-drive components [2–11]. In [12], modeling and con-
trol of a belt-drive positioning table is discussed, and in [13], direct
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/isatrans
ISA Transactions
http://dx.doi.org/10.1016/j.isatra.2016.03.005
0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
n
Corresponding author. Tel.:þ1 979 458 4829; fax: þ1 979 845 3081.
E-mail addresses: pramod.raul@okstate.edu (P.R. Raul),
ramamurthy@vidyanikethan.edu (R.V. Dwivedula),
ppagilla@tamu.edu (P.R. Pagilla).
ISA Transactions 63 (2016) 355–364

drive control of X–Y table is presented. However, no specific model
is reported for including the effect of compliance of the belt;
system identification techniques were used to obtain the system
dynamics, to be later used in tuning of the feedback gains. Simi-
larly in [14], a composite fuzzy controller, consisting of a feedback
fuzzy controller and a feed-forward acceleration compensator, is
proposed to control a belt drive precision positioning table; the
effects of belt compliance were not included in this paper. In [15],
a robust motion control algorithm for belt-driven servomechanism
is reported. In this paper, the belt-stretch dynamics is assumed to
contribute a pair of purely imaginary poles to the transfer function
of the system; the fact that the belt serves as an interconnection
from load-side to the motor-side is ignored in this paper. Modeling
of belt–pulley and gear-pair transmission system with gear back-
lash is given in [16]. Analysis and control of speed drive systems
with torsional loads is reported in [17–20].
The following are the contributions of this work: based on the
model of the two inertias (motor and load) connected by a belt–
pulley and gear-pair transmission system, we have investigated
the effect of using either motor or load feedback to control the
load speed by utilizing the singular perturbation method. In each
case, we consider a PI controller that is typical in the industry for
the feedback controller. The small parameter in the singular per-
turbation method is proportional to the reciprocal of the square
root of the belt compliance. The singular perturbation analysis
revealed that the controller using pure load feedback results in an
unstable system. Therefore, use of pure load feedback must be
avoided. To directly control the load speed, we also propose a
control scheme that uses both the motor speed and load speed
feedback and show that such scheme results in a stable closed-
loop system. Since feedback action is not sufficient in rejecting
periodic disturbances that commonly act on the load, we also
consider an adaptive feedforward compensation action that is
based on adaptive estimation of the coefficients of the periodic
disturbance as suggested in [21]. This adaptive feedforward action
is quite suitable for this application because it preserves closed-
loop stability achieved with the feedback controller. Experiments
were conducted to evaluate the performance of the various control
schemes on an industrial grade transmission system that is com-
mon in roll-to-roll manufacturing.
The remainder of the paper is organized as follows. The model
of the system is described in Section 2. Sections 3 and 4 describe
the motor speed feedback only and load speed feedback only
cases, respectively. A control scheme that utilizes both motor and
load speed feedback is discussed in Section 5. An add-on adaptive
feedforward compensation to reject load speed disturbances is
discussed in Section 6. Section 7 provides a description of the
experimental platform and a comparison of the results with the
various control schemes. Conclusions are given in Section 8.
2. Model of the system
A schematic of the belt–pulley and gear transmission system
connecting the motor with the load is shown in Fig. 1. In the
schematic, Ji denote the inertias, bi denote the viscous friction
coefficients, Ri denote the radii of the pulleys and gears, θi denote
the angular displacements of the inertias, τm denotes the motor
torque, τL denotes the torque disturbance on the load, and Kb
denotes the stiffness of the belt.
To derive the governing equations for this system we consider
the action of the belt in transmitting power. For a given direction
of rotation of the pulley, the belt has a tight side and a slack side as
shown in Fig. 1. It is assumed that the transmission of power is
taking place on the tight side and the transport of the belt is taking
place on the slack side. Under this assumption, the net change in
tension on the slack side will be much smaller than that in the
tight side and thus may be ignored. The tight side of the belt can
then be modeled as a spring with spring constant of Kb. For given
angular displacements θm and θL, the net elongation of the tight
side of the belt can be written as ðR1θm ÀGRR2θLÞ. Because of this
elongation, the driving pulley experiences a torque of ðR1θm ÀGR
R2θLÞKbR1 and the driven pulley experiences a torque of
ðR1θm ÀGRR2θLÞGRR2Kb. Under the assumption that the inertias of
the pulleys and gears are much smaller than the motor and the
load, the governing equations of motion for the motor-side inertia
and the load-side inertia are given by
Jm
€θm þbm
_θm þR1Kb R1θm ÀGRR2θL
À Á
¼ τm; ð1aÞ
JL
€θL þbL
_θL ÀGRR2Kb R1θm ÀGRR2θL
À Á
¼ τL: ð1bÞ
A block diagram representation of the system given by (1) is
provided in Fig. 2; note that this block diagram represents the
open-loop system and the two “loops” appearing in the block
diagram that represent the interconnections in (1). The open-loop
transfer functions from the motor torque signal τm to the motor
speed ωm and load speed ωL are given by
Gτmωm
ðsÞ9
ωm sð Þ
τm sð Þ
¼
JLs2
þbLsþG2
RR2
2Kb
D sð Þ
; ð2aÞ
Gτmωm ðsÞ9
ωL sð Þ
τm sð Þ
¼
GRR1R2Kb
D sð Þ
; ð2bÞ
where
DðsÞ ¼ JmJLs3
þ bLJm þJLbm
À Á
s2
þ KbJeq þbmbL

sþKbbeq; ð3aÞ
Jeq ¼ G2
RR2
2Jm þR2
1JL; ð3bÞ
beq ¼ G2
RR2
2bm þR2
1bL: ð3cÞ
θ2
Sprocket 1
Jm
τm
Belt
Kb
2R2
Sprocket 2
Gear 1
Gear2
bm
JL
bL
τL
θL
2R1
θm
2Rg1
2Rg2
Slackside
Tightside
Fig. 1. Schematic of a belt–pulley and gear-pair transmission system.
P.R. Raul et al. / ISA Transactions 63 (2016) 355–364356

The goal is to control load speed. In the following we will
discuss the closed-loop control systems that consider three sce-
narios: (i) pure motor speed feedback or (ii) pure load speed
feedback or (iii) a combination of motor and load speed feedback.
3. Motor speed feedback control scheme
It is common to control load speed by using measurement of
motor speed ωm as feedback. This control scheme is shown in
Fig. 3. The control structure is designed to regulate motor speed
ωm to the reference ωrm, and thereby indirectly regulate load
speed ωL.
We consider the often used Proportional–Integral (PI) control
action which is widely used in industrial environments. The
feedback law is given by
τm ¼ Kpm ωrm Àωmð ÞþKim
Z
ωrm Àωmð Þ dτ: ð4Þ
With this control law, the closed-loop transfer function from ωrm
to ωL is obtained as
ωLðsÞ
ωrmðsÞ
¼
ðGRR1R2Kb=JmJLÞðsKpm þKimÞ
ψmðsÞ
; ð5Þ
where
ψmðsÞ ¼ s4
þc3s3
þc2s2
þc1sþc0;
c3 ¼
ðbmJL þJmbL þKpmJLÞ
JmJL
;
c2 ¼
ðKbJeq þbmbL þKpmbL þKimJLÞ
JmJL
; ð6Þ
c1 ¼
ðKbbeq þG2
RR2
2KbKpm þKimbLÞ
JmJL
;
c0 ¼
G2
RR2
2KbKim
JmJL
:
Note the the coefﬁcients c0 to c3 depend on the controller gains.
We consider the singular perturbation method for analyzing such
a system with the small parameter proportional to the reciprocal
of the square root of the belt stiffness Kb. For conduction singular
perturbation analysis, we need to express the equations in the
form
_x ¼ A11xþA12z; x t0ð Þ ¼ x0
ð7aÞ
ϵ_z ¼ A21xþA22z; z t0ð Þ ¼ z0
ð7bÞ
where x and z are the states of the slow and the fast subsystems,
respectively, and ε is the small parameter; for our system we will
consider ε2
¼ 1=Kb. The elements of matrices Aij may depend on ε.
However, to use the singular perturbation method, the matrix A22
needs to be nonsingular [22] at ε ¼ 0. A natural choice of the state
variables for the singular perturbation analysis is θm, _θm, θL and _θL.
However, with this choice of the state variables, the matrix A22
becomes singular at ε ¼ 0. To obtain a state-space representation
in the form that would enable the use of the singular perturbation
method, we consider the following transformation of variables:
θc 9
Jmθm þJLGRðR2=R1ÞθL
Jm þJL
; ð8aÞ
θs 9θm ÀGRðR2=R1ÞθL: ð8bÞ
The variable θc is a weighted average of angular displacements
(θm and θL) referred to the motor side and the variable θs is dif-
ference between the angular displacements (θm and θL) referred to
the motor side; transformations similar to these have been used in
prior studies of two inertia systems, see for example [23]. The idea
of the weighted average of the displacements arises naturally in
the case of a translational system wherein θc represents the
position of the centroid of the masses. Now, choosing the state
variables as x ¼ ½θc; _θcŠ
and z ¼ ½θs=ε2
; _θs=εŠ
, the state space
representation of the system is obtained in the form given by (7)
where
A11 ¼
0 1
f 1 f 3
#
; A12 ¼
0 0
ϵ2
f 21 þf 22 ϵf 4
#
;
+
-
+
-
Jms + bm JLs + bLm
1 +
-
1
BRKbR2
1
s
BR
ωm
ωL
τm
τL
Fig. 2. Block diagram of the belt–pulley and gear transmission system; BR denotes the overall speed ratio, BR ¼ ðR2=R1ÞGR.
+
-
+-
1 +
-
1
s
Controller
+
-
BR
Motor Speed Feedback
BR
Jms + bm
KbR2
1
JLs + bLm ωL
τL
ωm
τm
ωrm
Fig. 3. Motor speed feedback control scheme.
P.R. Raul et al. / ISA Transactions 63 (2016) 355–364 357

A21 ¼
0 0
g1 g3
#
; A22 ¼
0 1
ϵ2
g21 þg22 ϵg4
#
; ð9Þ
f 1 ¼ ÀKim=J0; f 21 ¼ ÀKimJL=J2
0;
f 22 ¼ G2
RR2
2 ÀR2
1

=J0; f 3 ¼ À Kpm þbm þbL
À Á
=J0;
f 4 ¼ bLJm ÀbmJL ÀKpmJL
À Á
=J2
0;
g1 ¼ ÀKim=Jm; g21 ¼ ÀKimJ2
L = JmJLJ0
À Á
;
g22 ¼ À R2
1JL þG2
RR2
2Jm

= JmJL
À Á
;
g3 ¼ bLJm ÀbmJL ÀKpmJL
À Á
= JmJL
À Á
;
g4 ¼ À KpmJ2
L þbmJ2
L þbLJ2
m

= JmJLJ0
À Á
;
where J0 ¼ Jm þJL, and 1=ε2
¼ Kb. Notice that det A22 ϵð Þ ϵ ¼ 0j Þ ¼ð
Àg22 a0, thus satisfying the requirement of non-singularity of the
matrix A22 at ε ¼ 0. The characteristic equation for the system
given by (9) can be factored as [22]
ψmðs; εÞ %
1
ε2
ψmsðs; εÞψmf ðp; εÞ ¼ 0 ð10Þ
with
ψmsðs; εÞ9det½sI2 ÀðA11 ÀA12LðεÞÞŠ ð11aÞ
ψmf ðp; εÞ9det½pI2 ÀðA22 þεLðεÞA12ÞŠ ð11bÞ
where ψmsðs; εÞ is the characteristic polynomial for the slow sub-
system and ψmf ðp; εÞ is the characteristic polynomial of the fast
subsystem exhibited in the high-frequency scale p ¼ εs. The matrix
LðεÞ is obtained using the iterative scheme given in [22].
Using the matrices given by Eq. (9), the slow and the fast
characteristic polynomials are obtained as
ψmsðs; εÞ % s2
þα1sþα0; ð12aÞ
ψmf ðp; εÞ % p2
þα0
1pþα0
0 ð12bÞ
where
α1 ¼
G2
RR2
2bm þR2
1bL þG2
RR2
2Kpm
G2
RR2
2Jm þR2
1JL
;
α0 ¼
G2
RR2
2Kim
G2
RR2
2Jm þR2
1JL
; ð13Þ
α0
1 ¼
G2
RR2
2KpmJL
JmðG2
RR2
2Jm þR2
1JLÞ
ε;
α0
0 ¼
G2
RR2
2JL þR2
1Jm
JmJL
:
Eq. (12) indicates that both the fast and the slow subsystems are
stable for all Kpm; Kim 40. The result is true even without the
approximation introduced by LðεÞ as shown in Theorem 1 in the
Appendix.
4. Load speed feedback control scheme
One can employ the load speed feedback scheme shown in
Fig. 4, where the measured variable is ωL. This seems to have the
advantage of directly controlling load speed and attenuating the
effect of the disturbance τL. The feedback law is given by
τm ¼ KpL ωrL ÀωLð ÞþKiL
Z
ωrL ÀωLð Þ dτ; ð14Þ
and the closed-loop transfer function from ωrL to ωL is obtained as
ωLðsÞ
ωrLðsÞ
¼
ðGRR1R2Kb=JmJLÞðsKpL þKiLÞ
ψLðsÞ
ð15Þ
where
ψLðsÞ ¼ s4
þd3s3
þd2s2
þd1sþd0;
d3 ¼
ðbmJL þJmbLÞ
JmJL
;
d2 ¼
ðKbJeq þbmbLÞ
JmJL
; ð16Þ
d1 ¼
ðKbbeq þGRR1R2KbKpLÞ
JmJL
;
d0 ¼
GRR1R2KbKiL
JmJL
:
eqno
rightlefthskip 12pt
Singular perturbation analysis pertaining to this control
scheme results in the following slow and fast characteristic poly-
nomials:
ψlsðs; εÞ % s2
þβ1sþβ0 ð17aÞ
ψlf ðp; εÞ % p2
Àβ0
1pþβ0
0 ð17bÞ
where
β1 ¼
G2
RR2
2bm þR2
1bL þGRR2R1KpL
G2
RR2
2Jm þR2
1JL
;
β0 ¼
GRR2R1KiL
G2
RR2
2Jm þR2
1JL
;
β0
1 ¼
G2
RR2
2bm þR2
1bL þG2
RR2
2KpL
G2
RR2
2Jm þR2
1JL
ε; ð18Þ
β0
0 ¼
G2
RR2
2JL þR2
1Jm
JmJL
:
Note that the slow subsystems are stable for all KpL, KiL 40.
However, the fast subsystem is unstable for all KpL 40 and KiL 40.
Also, note that the characteristic polynomials given by Eqs. (12b)
and (17b) are identical when ε ¼ 0. Thus, analyzing the limiting
+
-
+-
1 +
-
1
s
τ
L
Controller
+
-
BR
Load Speed Feedback
BR
ωrL τm
Jms + bm
ωm
KbR2
1
JLs + bL
ωL
Fig. 4. Load speed feedback control scheme.

case of an infinitely stiff belt, that is, ε ¼ 0 will not reveal the
instability exhibited by (17b).
Remark 1. Eq. (1b) may be used to give an interpretation of the
foregoing analysis. Differentiating (1b), we obtain
JL €ωL þbL _ωL þR2
2KbωL ¼ R1R2Kbωm; ð19Þ
which indicates that ωL can attain steady-state only when ωm
attains steady-state first. Even after ωm attains steady-state, ωL
continues to exhibit damped oscillations. Thus, by measuring only
ωL and using the control law given by the (14), it will be difficult to
distinguish whether the oscillations in ωL are due to fluctuations
in motor speed or, the oscillations are indeed damped oscillations.
In such a situation, the controller attempts to react to the damped
oscillations also, and in this process, changes ωm, which in turn
affects ωL because of the dynamics given by the (19). Thus, the
control law given by (14) does not present a desirable situation.
5. Simultaneous motor and load speed feedback control
scheme
In this scheme, the load speed control corrects directly the
torque input to the system as shown in Fig. 5. The closed-loop
transfer function from ωrL to ωL is given by
ωLðsÞ
ωrLðsÞ
¼
αmLt
ψmLtðsÞ
ð20Þ
where
αmLtðsÞ ¼ a1sþa0;
a1 ¼
ðGRR1R2KbKpL þGrR2Kpm=R1Þ
JmJL
; ð21Þ
a0 ¼
ðGRR1R2KbKiL þGrR2Kim=R1Þ
JmJL
:
ψmLtðsÞ ¼ s4
þf 3s3
þf 2s2
þf 1sþf 0;
f 3 ¼
ðbmJL þJmbL þJLKpmÞ
JmJL
;
f 2 ¼
ðKbJeq þbmbL þKpmbL þJLKimÞ
JmJL
; ð22Þ
f 1 ¼
ðKbbeq þG2
RKbKpm þKimbL þGRR1R2KbKpLÞ
JmJL
;
f 0 ¼
KimG2
RR2
2Kb þGRR1R2KbKiL
JmJL
:
Note that the coefficients f0 to f3 depend on the gains of the
control law. Singular perturbation analysis for this case results in
the slow and fast characteristic polynomials as
ψmlsðs; εÞ % s2
þγ1sþγ0 ð23aÞ
ψmlf ðp; εÞ % p2
þγ0
1pþγ0
0 ð23bÞ
where
γ1 ¼
G2
RR2
2bm þR2
1bL þGRR2R1KpL þG2
RR2
2Kpm
G2
RR2
2Jm þR2
1JL
;
γ0 ¼
G2
RR2
2Kim þGRR2R1KiL
G2
RR2
2Jm þR2
1JL
;
γ0
1 ¼
G2
RR2
2bm þR2
1bL þG2
RR2
2KpL þG2
RR2
2KpmðJL=JmÞ
G2
RR2
2Jm þR2
1JL
ε; ð24Þ
γ0
0 ¼
G2
RR2
2JL þR2
1Jm
JmJL
:
Therefore, the slow and fast subsystems are stable for all
positive controller gains. Note that the outputs of both load speed
and motor speed controller combine to form a torque input to the
motor; this is typically referred to as the torque mode in practice
when multiple loops such as this are employed. Another strategy
is to use the output of the load speed controller as the motor speed
reference correction; a block diagram of such a scheme is provided
in Fig. 11 in Appendix B. This strategy results in an unstable system
which is shown in Appendix B.
6. Adaptive Feedforward (AFF) compensation to reject load
disturbances
The use of feedforward compensation to reject known dis-
turbances by direct cancelation or unknown disturbances by their
estimation has been known to be effective in attenuating dis-
turbances. We consider the rejection of periodic disturbances on
the load by using an adaptive feedforward action based on load
speed error. The control scheme that utilizes the feedforward
action is shown in Fig. 6. We use an adaptive feedforward algo-
rithm given in [21] that is particularly applicable in this situation
as the feedforward action preserves the stability of the overall
system with the feedback controller with simultaneous motor and
load speed feedback. The approach is briefly discussed as applic-
able to this problem; the details are given in [21]. The idea is to
estimate the amplitude and phase of the disturbance for a known
frequency of the disturbance. The disturbance can be expressed in
the form
d ¼ θÃ
1 cos ðωtÞþθÃ
2 sin ðωtÞ
≔ϕðtÞwÃ
0 ð25Þ
where ω is a known frequency, θÃ
1 and θÃ
2 are unknown
+-
+-
1 +-
1
s
+-
ωrm
PI
Controller
+
-
PI
Controller
+
+
BR
Load Speed Feedback
BR
Jms + bm
ωm
KbR2
1
JLs + bL ωL
τL
ωrL
τm
Fig. 5. Simultaneous motor and load speed feedback scheme: torque mode.

parameters. The adaptation laws for the unknown parameters θÃ
1
and θÃ
2 are given by the following simple pseudo-gradient algo-
rithm:
_θ1 ¼ γeðtÞ cos ðωtÞ; ð26aÞ
_θ2 ¼ γeðtÞ sin ðωtÞ; ð26bÞ
where θ1 and θ1 are the parameter estimates, eðtÞ ¼ ωrL ÀωL is the
load speed error, and γ is the adaptation gain. Using the estimated
parameters, the feedforward control action is given by
uf ¼ Àθ1 cos ðωtÞÀθ2 sin ðωtÞ: ð27Þ
The estimation of the disturbance and its cancelation when the
load speed error contains a sinusoidal component with frequency
ω may be intuitively explained as follows. If the load speed error is
eðtÞ ¼ eðtÞþeθ1
sin ðωtÞ, in the adaptive algorithm the product eðtÞ
sin ðωtÞ will generate a positive eθ1
sin
2
ðωtÞ term. This will result
in a parameter drift which results in the attenuation of dis-
turbance until it reaches its nominal value. At this point the load
speed error is free of the sinusoidal component as the disturbance
is compensated by feedforward control uf. With the compensation,
the product term eðtÞ sin ðωtÞ in the parameter adaptive law
becomes zero and the parameters converge. Since the regressor
vector ϕðtÞ is persistently exciting, the parameter vectors converge
to zero. In fact, this adaptive feedforward action with estimation of
disturbance parameters using the pseudo-gradient algorithm has
been shown to be equivalent to the use of the internal model of
the disturbance in [21].
7. Experiments
A picture of experimental setup is shown in Fig. 7. It consists of an
AC motor shaft connected to the load shaft (roll) via a belt–pulley and
gear-pair transmission. A 15 HP (11.19 KW) AC motor with a rated
speed of 1750 RPM is employed. The belt ratio (BR ¼ ðR2=R1ÞGR) for
the transmission is 3.825. An encoder on the motor shaft is employed
to measure the motor shaft speed and a laser sensor is used to
measure the load shaft speed. The real-time hardware, including the
drives, controller, and communication network, was provided by
Rockwell Automation (Allan-Bradley). All the real-time hardware
components of the machine are connected through a ControlNet
communication network. The network is updated every 5 ms (Net-
work Update Time) and data is communicated to the network every
10 ms (Request Package Interval). A brake is attached on the other
side of the load shaft to inject periodic torque disturbances; a mag-
netic clutch brake (Magpower GBC 90) that can apply 26 lb-ft torque
is used.
7.1. Design and implementation guidelines for the proposed load
speed regulation scheme
The following guidelines should provide assistance to practi-
cing engineers in the design and implementation of the proposed
load speed regulation scheme:
1. Select the PI gains for the motor speed PI controller by tuning
them for the desired regulation performance without the load.
These PI gains will provide a baseline for the motor speed PI
controller.
2. Select the PI gains for the load speed feedback PI controller by
placing stable closed-loop poles for the fast and slow dynamics
given in Section 5. It may be necessary to re-tune the PI gains of
the motor speed PI controller to obtain the desired closed-
loop poles.
3. Adaptive feedforward compensation is generated based on the
disturbance parameter estimates given in Section 6. Although
some tuning of the adaptation gain γ may provide better
Load Speed
Reference Two Inertia
System
Feedforward
Compensation
PI
Controller
Disturbance
+
−
PI
Controller
+
+
−
+
Load Speed Feedback
Motor Speed
Reference
Load Speed
Output
+
+
Fig. 6. Control scheme with feedback and feedforward compensation.
Gear Box
Laser Sensor
Load
Brake
AC Motor
Timing Belt
Gear Box
Laser Sensor
Load
Brake
AC Motor
Fig. 7. Picture of the experimental platform. Top view: load side. Bottom view:
motor side.

transient response, in theory any positive gain and reasonable
initial conditions for the estimates would work. For simplicity,
we have chosen the values of γ ¼ 1 and the initial conditions for
both estimates to be zero; this approach has worked in a
number of experiments we have conducted with different
disturbances.
4. Ensure that the the bandwidth of the drive system is sufficient
enough to attenuate the targeted disturbance frequency.
5. The adaptive feedforward compensation requires the knowl-
edge of the disturbance frequency. This can be obtained in
multiple ways. One can obtain the frequency spectrum of the
load speed to determine all the disturbance frequencies in the
frequency region of interest. It is possible to attenuate more
than one frequency by simply cascading multiple adaptive
feedforward compensation blocks with each targeting a parti-
cular frequency. In the case of a roll-to-roll system, the dis-
turbance frequency may be simply obtained from the line speed
and the radius of the roller.
The PI controller gains for the motor speed loop were chosen to
be Kpm ¼ 15 and Kim ¼ 3:09 and for the load speed loop to be KpL
¼ 0:07 and KiL ¼ 0:001. A number of experiments were conducted
at different reference speeds to evaluate the performance of pro-
posed control scheme. In each experiment, the brake provides an
external periodic disturbance torque of the form AþB sin ðωdtÞ
ðA ¼ 2; B ¼ 1:5Þ. The following disturbance frequencies were
injected to evaluate the control schemes: ωd ¼ 0:05; 0:15; 0:25 Hz.
These disturbances are typical of the disturbances that are
observed in roll-to-roll manufacturing machines where such
transmission systems are typically employed. The adaptation gain
γ ¼ 1 is chosen and the initial values of the estimates are set
to zero.
Fig. 8 shows the evolution of the load speed (reflected to the
motor side) in the presence of disturbance with frequency 0.25 Hz
when the reference speed is 719 RPM without the use of the
adaptive feedforward action. Fig. 9 shows the Fast Fourier Trans-
form (FFT) of the load speed for the two cases. It is evident that the
control scheme with the AFF action (shown in Fig. 6) can provide
significantly improved load speed regulation. Fig. 10 controls tor-
que input corresponding to the two cases, without and with
adaptive feedforward compensation. It is evident that the torque
input is larger when the adaptive feedforward is employed. Table 1
shows the standard deviation of the load speed signal from its
reference for the various schemes. It is clear that the employing
load speed feedback in addition to motor speed feedback can
improve performance. Further, use of the adaptive feedforward
0 20 40 60
712
714
716
718
720
722
724
726
Time (Sec)
Speed(RPM)
Load Speed Response without AFF
Reference Speed
Load Speed
0 20 40 60
712
714
716
718
720
722
724
726
Time (Sec)
Speed(RPM)
Load Speed Response with AFF
Reference Speed
Load Speed
Fig. 8. Load speed response with 0.25 Hz torque disturbance. Top: without AFF.
Bottom: with AFF.
0.2 0.4 0.6 0.8 1
0
0.5
1
1.5 X: 0.2522
Y: 1.603
Frequency (Hz)
SpeedVariation(RPM)
FFT of Load Speed Response without AFF
0.2 0.4 0.6 0.8 1
0
0.5
1
1.5
X: 0.24
Y: 0.2559
Frequency (Hz)
SpeedVariation(RPM)
FFT of Load Speed Response with AFF
Fig. 9. FFT of load speed response with and without AFF.

action based on load speed feedback can significantly improve the
regulation performance.
8. Conclusions
We have investigated the problem of regulating load speed in a
mechanical transmission with a compliant belt. Several speed control
strategies that rely on either motor speed feedback or load speed
feedback or both are investigated. A singular perturbation approach
with the inverse of the belt compliance as the small parameter is
employed to analyze different control strategies. It is shown that the
system is unstable when pure load speed feedback is employed. A
control system that considers both motor speed feedback and load
speed feedback in the torque mode is stable and can provide improved
closed-loop performance. Since the feedback control action is not
sufficient to reject periodic load disturbances, an adaptive feedforward
algorithm is designed to estimate the disturbance and generate a
compensation term to attenuate periodic disturbances of known fre-
quency and unknown amplitude. Experiments were conducted on an
industrial grade transmission system to evaluate the control schemes
and compare their performance. Although we have used only belt
compliance as the compliant element in the transmission system,
torsional compliance due to long shafts can also be included and the
analysis conclusions will remain the same.
Acknowledgments
This work was supported by the Web Handling Research Center at
Oklahoma State University and the US National Science Foundation.
Appendix A
A.1. Stability of the System with only Motor Speed Feedback
The following theorem provides the stability of the system with
only motor feedback that is discussed in Section 3.
Theorem 1. The closed-loop system defined by (4) and (1) is stable
and ωm approaches ωrm for all Kpm; Kim 40.
Proof. Substituting (4) into (1) and neglecting backlash effect, we
obtain dynamics of the closed-loop system as
Kpm ωrm Àωmð ÞþKim
Z
ωrm Àωmð Þ
dτ ¼ Jm
€θm þbm
_θm

þR1Kb R1θm ÀGRR2θL
À Á
; ð28aÞ
GRR2Kb R1θm ÀGRR2θL
À Á
¼ JL
€θL þbL
_θL

: ð28bÞ
Differentiate (28) to obtain
ÀKpm _ωm þKim ωrm Àωmð Þ ¼ Jm €ωm þbm _ωm
À Á
þR1Kb R1ωm ÀGRR2ωLð Þ; ð29aÞ
GRR2Kb R1ωm ÀGRR2ωLð Þ ¼ JL €ωL þbL _ωL
À Á
: ð29bÞ
Defining errors, em ¼ ωm Àωrm and eL ¼ ωL ÀðR1=GRR2Þωrm, (29)
may be written as
ÀKpm _em ÀKimem ¼ Jm €em þbm _em þR1Kb R1em ÀGRR2eLð Þ; ð30aÞ
GRR2Kb R1em ÀGRR2eLð Þ ¼ JL
€eL þbL _eL ð30bÞ
Choose
V tð Þ ¼
1
2
Jm
_e2
m þJL
_e2
L þKbðR1em ÀGRR2eLÞ2
þKime2
m

: ð31Þ
Then, the time derivative of V along the trajectories defined by
(30) is obtained to be
dVðtÞ
dt
¼ Àðbm þKpmÞ_e2
m ÀbL _e2
L : ð32Þ
Thus, V(t) is a Lyapunov function and em; eL; _em; _eL AL1 which
implies, from (30), that €em; €eL AL1. From (31) and (32), we con-
clude that because V(t) is bounded from below and is non-
increasing with time, it has a limit, i.e., limt-1VðtÞ ¼ V1. Now
0 20 40 60
−1.5
−1
−0.5
0
0.5
1
1.5
Time (sec)
TorqueInput(lbf−ft)
Control Input without AFF
0 20 40 60
−1.5
−1
−0.5
0
0.5
1
1.5
Time (sec)
TorqueInput(lbf−ft)
Control Input with AFF
Fig. 10. Control input with 0.25 Hz torque disturbance. Top: without AFF. Bottom:
with AFF.
Table 1
Comparison of different control schemes.
Disturbance fre-
quency (Hz)
Standard deviation
Only motor
feedback
Motor þ load
feedback
Motor þ load
feedback þ AFF
0.25 2.09 1.35 0.34
0.15 4.71 3.53 0.87
0.05 3.89 2.47 0.68

from (32), we have
lim
t-1
Z t
0
ðbm þKpmÞ_e2
m þbL _e2
L ¼ V0 ÀV1 o1 ð33Þ
Therefore, _em; _eL AL2 and by Barbalat's Lemma, we have _em-0
and _eL-0. Thus, ωm and ωL tend to become constants as t-1
and from (29), we see that ωm-ωrm and ωL-ðR1=GRR2Þωrm. □
A.2. Speed correction based simultaneous motor and load speed
feedback control scheme
The control scheme that utilizes both motor and load speed
feedback discussed in Section 5 considers the output of the load
speed and motor speed controllers as torque correction. There is
also another control scheme that is employed in practice where
the outer load speed loop provides a speed reference correction to
the inner motor speed which is shown in Fig. 11. In the following
we show that such a control scheme results in an unstable system,
and thus must be avoided. For this analysis, we employ a simple
proportional control action for the load speed controller and a PI
controller for the motor speed loop. The closed-loop transfer
function from ωrL to ωL for this strategy is obtained as
ωLðsÞ
ωrLðsÞ
¼
ðGRR1R2Kb=JmJLÞαmLs
ψmLsðsÞ
ð34Þ
where
αmLsðsÞ ¼ KpmKpLsþKimKpL ð35Þ
ψmLsðsÞ ¼ s4
þe3s3
þe2s2
þe1sþe0;
e3 ¼
bmJL þJmbL þJLKpm
À Á
JmJL
;
e2 ¼
KbJeq þbmbL þbLKpm þJLKim

JmJL
;
e1 ¼
Kbbeq þG2
RR2
2KbKpm þbLKim

JmJL
þ
GRR1R2KbKpmKpL
À Á
JmJL
ð36Þ
e0 ¼
G2
RR2
2KbKim þGRR1R2KbKimKpL

JmJL
:
Singular perturbation analysis results in the following slow and
fast characteristic polynomials:
ψlsðs; εÞ % s2
þδ1sþδ0 ð37aÞ
ψlf ðp; εÞ % p2
Àδ0
1pþδ0
0 ð37bÞ
where
δ1 ¼
G2
RR2
2bm þR2
1bL þG2
RR2
2Kpm þðG2
RR2
2=R1ÞKpmKpL
G2
RR2
2Jm þR2
1JL
δ0 ¼
G2
RR2
2Kim þðG2
RR2
2=R1ÞKimKpL
G2
RR2
2Jm þR2
1JL
δ0
1 ¼
G2
RR2
2bm þR2
1bL þG2
RR2
2KpLKpm þG2
RR2
2KpmðJL=JmÞ
G2
RR2
2Jm þR2
1JL
ε; ð38Þ
δ0
0 ¼
G2
RR2
2JL þR2
1Jm
JmJL
:
Note that the slow subsystem is stable for all Kpm, Kim, and KpL.
However, the fast subsystem is unstable for all KpL 40. The instability
of the system is also evident from simple root locus analysis of the
closed-loop characteristic with varying KpL, which is shown in Fig. 12.
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Fig. 11. Simultaneous motor and load speed feedback scheme: speed mode.
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