H∞ control of Industrial Emulator with backlash
M2014SC003 Masatsugu HIBINO
Supervisor：Isao TAKAMIAbstract
The objective of this study is improving the performance of positioning control by using H∞ control theory. In general,a backlash which has nonlinear characteristic exists between a gear and gears or gear andbelts.Therefore, considering backlash is necessary to realize accurate performance of position control. In thisstudy,Industrial Emulator is used. The inertia of it can
ﬂuctuate by putting some weights on Drive Disc andLoad Disc. A controller which guarantees robust stability to variation of the inertia. By using H∞ control, thedisturbance is guaranteed. The eﬀectiveness of proposed
method is veriﬁed by simulation results.

1Introduction

A machine which convey rotary power with belt andgear are used to timing belt of a car engine, machine
tool, belt conveyor, and so on. In the spot of a manufacturing, it is indispensable. A lot of studies of themachine which convey rotary power with belt and gear
are considered[2] [3] [4] [5]. There is a gap between gearand gears or gear and belts. The gap expresses backlash. However, the backlash has a big problem. Whenrotation of the gear turned over, the gear cannot introduce the rotary power into the size of the backlash.The
backlash cause a position gap and limit cycle. Therefore,
considering backlash is necessary to realize accurate performance of position control [2] [3] [4]. In the real spot,a parameter variation by abrasion of a part and disturbance is considered. It is necessary to guarantee therobust stability[5]. The object of this study is to improve a performance of positioning control for Industrial
Emulator. The modeling including backlash is derived.
And the backlash is modeled by using Dead-zone Model[2] [3]. Furthermore, the controller in consideration ofbacklash and parameter variation caused by variation ofinertia moment is designed. Here, for the backlash, H∞
controller is designed. And, for the parameter variation,robust controller is designed. Finally, the eﬀectivenessof proposed method is veriﬁed by simulation results.

2modeling

2.1

Physical Parameter

Table 1 Physical Parameters
Drive Disc angle
θd[rad]Load Disc angle
θl[rad]Idler Pulleys angle
θp[rad]Radius of Drive Discrd[m]Radius of Load discrl[m]Radius of Bottom Pulley
rpd[m]
Radius of Top Pulley
rpl[m]
Input torque to Drive Discτ[Nm]
Moment of Drive DiscJd
[kgm2 ]
Moment of Load DiscJl
[kgm2 ]
Moment of Idler Pulleys
Jp[kgm2 ]
Viscosity friction coeﬃcient cd [Nm/(rad/s)]of Drive Disc
Viscosity friction coeﬃcient cl[Nm/(rad/s)]of Load Disc

Fig. 1 Industrial Emulator

The structure of control target is shown in
Fig.1[1] .Physical parameters used in this modeling isshown in Table1.
2.2Motion Equation

Fig.2 shows a magniﬁed ﬁgure of Drive Disc.
Where,F1 (t), F2 (t) is the tensions of the belt between
Drive Disc and Idler Pulleys. cd is viscosity friction
coeﬃcient of Drive Disc. Motion equation of Drive Discis shown as follows.
Jd θ̈d (t) = τ (t) − cd θ̇d (t) + {F1 (t) − F2 (t)}rd
Fig.3 shows a magniﬁed ﬁgure of Load Disc.

Fig. 2 Drive Disc

(1)
Where,F3 (t), F4 (t) is the tension in the belt between
Idler Pulleys and Load Disc. cl is viscosity friction coeﬃcient of Load Disc. Motion equation of Load Disc isFrom Eq.(6), (9) and (10), motion equation are rewritten as follows.
rdJd θ̈d (t) +Jp θ̈p (t) = τ (t) − cd θ̇d (t)(11)
rpd2rplrdrpl−K{ 2 θp (t) −θl (t)}
(12)
rpd
rl
rl
Jl θ̈l (t) = −cl θ̇l (t) + K{
Fig. 3 Load discshown as follows.
Jl θ̈l (t) = −cl θ̇l (t) + {F3 (t) − F4 (t)}rl(2)

Fig.4 shows a magniﬁed ﬁgure of Idler Pulleys.

rplθp (t) − θl (t)}rl(13)

Here, the gear ratio between Drive Disc to Load Discand between Drive Disc and Idler Pulleys are deﬁned asfollows.
rl rplgr =(14)rpd rdrpd(15)gr′ =rdFrom Eq.(14), (15), the following equation is expressed angle of the Drive Disc and Load Discθd = gr θl(16)

The following equation is expressed the Drive Disc andIdler Pulleys
θd = gr′ θp(17)
From Eq.(14) (15), the following equation are obtained.
(Jd + gr′ Jp )θ̈d (t) = τ (t) − cd θ̇d (t) − gr−1 K{gr−1 θd (t) − θl(18)(t)}Jl θ̈l (t) = −cl θ̇l (t) + K{gr−1 θd (t) − θl (t)}Fig. 4 Idler Pulleys
Motion equation of Idler Pulleys is shown as follows.
Jp θ̈p (t) = {F2 (t) − F1 (t)}rpd + {F4 (t) − F3 (t)}rpl (3)Here, the belt has elasticity in a property. Therefore,
the belt regard as spring. The spring constant of thebelt is deﬁned as kL . Then, let us regard the belts as aspring F3 and F4 are obtained as follows
F3 (t) = F0 + kL (rpl θp (t) − rl θl (t))(4)

F4 (t) = F0 − kL (rpl θp (t) − rl θl (t))(5)

Where,the following expression is deﬁned to simplify
Eq.(20).
Jd∗ ∼(20)= Jd + gr′−2 Jp
From Eq.(20), Eq.(18) is rewritten as follows.
Jd∗ θ̈d (t) = τ − cd θ̇d (t) − K(gr−2 θd (t) − gr−1 θl (t)) (21)Based on Eq.(19), (21), let us regard the belt as transmission. Then, the transmission torque is deﬁned asfollows.
T (t) := K{gr−1 θd (t) − θl (t)}(22)
Eq.(19) (21) is rewritten as follows by using (22).

Where, motion equation of Load Disc and Idler Pulleys are rewritten as follows.
Jl θ̈l (t) = −cl θ̇l (t) + 2kL {rpl θp (t) − rl θl (t)}rl
1F2 (t) − F1 (t) =[Jp θ̈p (t) + 2kL {rpl θp (t)rpd−rl θl (t)}rpl ](19)

Jd∗ θ̈d (t) = τ − cd θ̇d (t) − gr−1 T (t)(23)

Jl θ̈l (t) = −cl θ̇l (t) + T (t)

(24)

(6)
2.3

Dead-zone Model

When rotation of the gear turns over, backlash delays
transmission of torque. Therefore, in this study, I consider that T (t) in Eq.(23), (24) is inﬂuenced by backlash.
(7)
(8)
(9)

For the simpliﬁcation of the equation, let K be asfollows.
K := 2kL rl2(10)

Backlash nonlinearity is captured here using the Deadzone model. Dead-zone model can be expressed as follows.
 −1gr θd − θl − α (gr−1 θd − θl > α)Dα (t) = 0(|gr−1 θd − θl | < α) (25)
 −1gr θd − θl + α (gr−1 θd − θl < −α)2.5Servo System

I designed a controller which makes to follow the control target. In order to remove a steady-state error, aservo system is used.Let e(t) be the error between theoutput y(t) and the reference r(t). The state∫ variable ofthe servo system satisfy xe (t)T = [x(t) e(t)])T . The
servo system can be expressed as follows.
ẋe (t) = Ae xe (t) + B1e w(t) + B2e u(t)y(t) = C1 exe (t)Where,matrices Ae , B1e , B2e and C1 are given as follows.
]][[[]A 0B1 0B2
Ae =, B1e =, B2 =−C 00 10Fig. 5 Backlash
Where, α is backlash angle.T (t) can be expressed asfollows.
T (t) = KDα (θd (t), θl (t))(26)
And Eq.(25) is rewritten as follows.
Dα (t) = gr −1 θd (t) − θl (t)+dα (t)(27)

C1e = [ 0 03

(gr−1 θd − θl > α)(|gr−1 θd − θl | < α)(gr−1 θd − θl < −α)Jl ∈ [Jl,min , Jl,max ] = [8.3 ∗ 10−3 , 2.8 ∗ 10−2 ] (34)Then, the combination of ﬂuctuation is shown as follows.
∗∗{Jd,min, Jl,min }, {Jd,max, Jl,min },

(29)

(30)

State Space Representationx(t) = [θd , θl , θ̇d , θ̇l ]T u(t) = τ, w(t) = dα (t)Kgr−1Jl
B1 = 


0−Kgr−1
Jd∗

0KJl





00− JKl

H∞ Control

H∞ controller is designed. For Eq.(31), the generalplant is deﬁned as follows.
{ẋe (t) = Ae x(t) + B1e w(t) + B2e u(t)(36)z(t) = C2 xe (t) + D2 u(t)[C2 =00 1 − Jcll

Wx
000We0][, D2 =00Wu]

Where, z(t) is cost function. Wx ≻ 0, We ≻ 0 andWu ≻ 0 are weight matrices for the state variable, integral and input. For the generalized plant, let us consider
to minimize H∞ norm from disturbance w(t) to the cost
output z(t). Where, H∞ norm is deﬁned as follows.

 1  B2 =  Jd∗  , C = [ 0 0 0 
0If each endpoint is stabilized, it may be said that it isstabilized at the endpoint interval.
3.2

Then, the state space representation of industrial emulator is given as follows.
{ẋ(t) = Ax(t) + B1 w(t) + B2 u(t)(31)y(t) = Cx(t)0By using polytopic representation,the endpoints of variation range of matrices Aei , B1ei , B2ei are shown as
follows.
Aei (i = 1, 2, 3, 4), B1ei (i = 1, 2, 3, 4), B2ei (i = 1, 2, 3, 4)

State variable, input and disturbance is deﬁned as follows.

Kgr−1Jd∗(35)∗∗{Jd,min, Jl,max }, {Jd,max, Jl,max }Jl θ̈l (t) = −cl θ̇l (t) + Kgr−1 θd (t)−Kθl (t) + Kdα (t)1− Jcd∗d00

Polytopic Representation∗∗Jd∗ ∈ [Jd,min, Jd,max] = [1.1 ∗ 10−5 , 6.9 ∗ 10−5 ] (33)+Kgr−1 θl (t) − Kgr−1 dα (t)0 − Kg∗r−2
Jd
A=0
0 ]

Control Design(28)Jd∗ θ̈d (t) = τ (t) − cd θ̇d (t) − Kgr−2 θd (t)0The range of the variation parameter Jd∗ , Jl are asshown in Eq.(33), (34).

By Eq.(25)−(28),motion equation of Drive Disc andLoad Disc is given as follows.

2.4

1In this section, I design a controller which guarantees
robust stability to variation of the inertias Jd∗ , Jl .
3.1

Where, dα (t) is deﬁned as follows.
−αdα (t) = −gr−1 θd + θl

α(32)

10 ]

∥ Gzω (s) ∥= sup∥ z(t) ∥∞∥ ω(t) ∥∞(37)If there exist ∥ Gzω (s) ∥≺ γ∞ , the system is stabilized.

3.3

LMI Condition

The problem is formulated as follows.
Lemma 1 :If there exist X and Y satisf ying thef ollowing LM Is, the system is stabilized by u(t) =Kxe (t) = Y X −1 xe (t).
minimize :γsubjectto X ≻ 0

XATei + Ae iX + Y T B2ei + B2ei Y
C2 X + D2 YT
B1ei

XC2T + Y T D2T−γ∞ I0


B1ei
≺00
−γ∞ I

Fig. 7 Jdmax , Jlmin(i = 1, 2, 3, 4)(38)

4Simulation

In this section, the simulation results are shown. Furthermore, the proposed method is compared with therobust LQ controller. In this study, the backlash angle
is deﬁned as 0.6[deg]:=0.0105[rad] by system identiﬁcation. A reference of the Load Disc angle is assigned π2[rad]. Simulation results of the Load Disc angle considering variation of the moment of Drive Disc and Load
Disc are shown in Fig.6-9. The red line shows the Load
Disc angle controlled by the H∞ controller. The blue
solid line shows the Load Disc angle controlled by therobust LQ. The black dot line shows reference.

Fig. 6 Jdmin , Jlmin
As can be seen in Fig.6-9, the Load Disc angle stably
follows the reference with the proposed method. However, the Load Disc angle does not follow the referencewith the robust LQ. Even if there is the parameter variation,the Load Disc angle stably follows by H∞ controller.

5Conclusion

In this study, the modeling including backlash is derived by using Dead-zone model. H∞ controller is designed for Industrial Emulator with backlash and parameter variation. The eﬀectiveness of the proposed
method is veriﬁed by simulation.

6References[1] Educational Control Products, 1995. Manual for
Model 220 Industrial Emulator/Servo Trainer, California 91367, USA

Fig. 8 Jdmin , Jlmin

Fig. 9 Jdmax , Jlmax[2] L.Acho,F.Ikhouane, and G.Pujol ”Robust Control
Design for Mechanisms with Backlash”JCET Vol.3
Iss.4 October 2013 PP. 175-180[3] M. Nordin, P.O. Gutman, Controlling mechanicalsystems with backlash: A survey , Automatica, Vol.
38, 2002, pp. 1633-1649.
[4] M. Nordin, J. Galic and P.O. Gutman, New models
for backlash andgear play , International Journalof Adaptive Control and Signal Processing, Vol. 11,
No. 1, 1997, pp. 49-63. [doi: 10.1002/(SICI)10991115(199702)11:1][5] Orlov, Y. , Aguilar, L.T. , Alvarez, J, ”Robust tracking control of servo systems with backlash: Nonsmooth H∞ control vs. linear H∞ control”AmericanControl Conference (ACC), July 2015 pp.2051-2056