Position Control of Belt-drive System
Using Robust LQ Controller
M2014SC009 Toma KATO
supervisor：Isao TAKAMI

Abstract
In this study, a controller which has high applicability to the actual situation of industry is proposed bydiscussing about robust control of belt-drive system.
First, the control target is represented by mathematicalmodel. Second, the variation and uncertainty of the system are represented by polytope. Third, the controller
which guarantees robust stability is designed by using
LMI solution to the optimal regulator problem. Fourth,
the robustness of the controller is veriﬁed by conducting simulation. Finally, experiment is conducted. In
the experiment, the robustness and superiority of proposed method is established by comparing with a popular method.

1Introduction

Power train means the mechanical device which conveys rotary power by using belt, chain, gear and soon. It is applied in many sectors of industry, for example, the timing belt of car engine, the belt conveyer
of factory and so on. Therefore, to regulate the position or velocity conveyed by power train accurately is
the exceedingly important problem for industrial application of the control method. However, in most casesof industrial application, the control target has someundesirable elements for designing the controller. For
example, inﬂuence of disturbances, wear of parts, uncertain parameter and so on. Various methods for the
motion control were proposed by researchers. For example, propotional-integral-derivative (PID) control ischosen in most industrial applications. However, it has
vulnerability to the variation of system. Then, various
methods for the robust stability were proposed. However, most of them are based on the complicated theory.
Then, this study propose the method which has robustness, and designing that is comparatively easy.

2Modeling

2.1Control Target

Table 1 Physical Constants and Variables
rdrpdrplrlθdθpθl
JdJpJlτcdcl

Radius of Drive Disc gear
Radius of Idler Pulleys gear
(connected to Drive Disc)
Radius of Idler Pulleys gear
(connected to Load Disc)
Radius of Load Disc gear
Angle of Drive Disc
Angle of Idler Pulleys
Angle of Load Disc
Inertia of Drive Disc
Inertia of Idler Pulleys
Inertia of Load Disc
Torque acting on Drive Disc
Friction coeﬃcientof Drive Disc
Friction coeﬃcientof Load Disc

[m]
[m]
[m]
[m]
[rad]
[rad]
[rad]
[kg·m2 ][kg·m2 ][kg·m2 ][Nm][Nm/(rad/s)][Nm/(rad/s)]

It consists of three discs and two belts mainly. The
discs are called Drive Disc, Idler Pulleys and Load Disc
respectively. A motor connected to the Drive Disc gives
input torque to the system. Then, the Idler Pulleys
and the belts transmit the power to the Load Disc. By
putting some weights on the Drive Disc and the LoadDisc, their inertias Jd and Jl are changed. In otherwords, we can verify the eﬀectiveness of a controller withvarious cases. The subject of this study is to control theLoad Disc angle θl (t) by the input torque τ (t) acting onthe Drive Disc.
2.2

Dynamical Equation

We treat three angles θd , θp and θl here. Let us derive
three dynamical equations corresponding to each of theangles. First, let us focus on the Drive Disc.

The structure of the experimental device used in thisstudy is shown in Fig. 1. Further, Table 1 shows physical constants and variables.

Figure 2 Enlarged Figure of the Drive DiscHere, F1 (t) and F2 (t) are tension of the belt between
the Drive Disc and the Idler Pulleys, and cd is the friction coeﬃcient of the Drive Disc. From Fig. 2, the dynamical equation of the Drive Disc is derived as Eq.(1).
Jd θ̈d (t) = τ (t) + {F1 (t) − F2 (t)}rd − cd θ̇d (t)Figure 1 Structure of the Experimental Device

Second, let us focus on the Idler Pulleys.

(1)

Figure 3 Enlarged Figure of the Idler Pulleys

Here, F3 (t), F4 (t) are tension of the belt between theIdler Pulleys and the Load Disc. From Fig. 3, thedynamical equation of the Idler Pulleys is derived asEq.(2).
Jp θ̈p (t) = {F2 (t) − F1 (t)}rpd + {F4 (t) − F3 (t)}rpl (2)Finally, let us focus on the Load Disc.

Figure 5 Belt between Idler Pulleys and Load DiscThe following equations are obtained by substituting
Eq.(6), eq:F4 into Eq.(5), (3) respectively.
Jd θ̈d (t) = τ (t) − cd θ̇d (t)rd [+−2kpl {rpl θp (t) − rl θl (t)}rpl
rpd− Jp θ̈p (t)Jl θ̈l (t) = 2kpl {rpl θp (t) − rl θl (t)}rl − cl θ̇l (t)](8)(9)

Now, let us deﬁne a constant k as follows for simplicity.
△k = 2kpl rl2(10)

The following equations are obtained by applying k toEq.(8) and Eq.(9).
Figure 4 Enlarged Figure of the Load Disc
Here, cl is the friction coeﬃcient of the Load Disc.
From Fig. 4, the dynamical equation of the Load Discis derived as Eq.(3).
Jl θ̈l (t) = {F3 (t) − F4 (t)}rl − cl θ̇l (t)2.3Jd θ̈d (t) = τ (t) − cd θ̇d (t){}2rplrpl
rd− 2 kθp (t) +kθl (t) − Jp θ̈p (t)+rpdrlrl(11)Jl θ̈l (t) =(3)2.4

Spring Constant of the Belt

We obtain the following equation by transforming
Eq.(2).
]
1 [{F4 (t) − F3 (t)}rpl − Jp θ̈p (t)rpd(4)
By substituting this equation into Eq.(1), we obtainthe following equation.
{F1 (t) − F2 (t)} =Jd θ̈d (t) = τ (t) − cd θ̇d (t)]rd [+{F4 (t) − F3 (t)}rpl − Jp θ̈p (t) (5)rpd
By regarding the belt between the Idler Pulleys andLoad Disc as the spring whose spring constant is kpl asshown in Fig. 5, we obtain Eq.(6), (7). Here, F0 is theinitial tension of the belt.
F3 (t) = F0 + kpl {rpl θp (t) − rl θl (t)}(6)

F4 (t) = F0 − kpl {rpl θp (t) − rl θl (t)}

(7)

rpl
kθp (t) − kθl (t) − cl θ̇l (t)
rl

(12)

Gear Ratio

Now let gr′ be the gear ratio of the Drive Disc and theIdler Pulleys as shown in Eq.(13). Furthermore, let gr
be the gear ratio of the Drive Disc and Load Disc as
shown in Eq.(14).
△ rpd
gr ′ =(13)rd()
rpd rlrl△=·(14)
gr = gr ′ ·rplrd rpl
Then, the following equations hold as a fundamental
property of the gear mechanism for power transmission.
θd (t) = gr ′ θp (t), θ̇d (t) = gr ′ θ̇p (t), θ̈d (t) = gr ′ θ̈p (t) (15)θd (t) = gr θl (t) , θ̇d (t) = gr θ̇l (t) , θ̈d (t) = gr θ̈l (t) (16)The following equations are obtained by applying gr′
and gr to Eq.(11) and Eq.(12).
τ (t) =Jd θ̈d (t) + gr ′−2Jp θ̈d (t) + cd θ̇d (t)+ gr −2 kθd (t) − gr −1 kθl (t)0 = Jl θ̈l (t) + cl θ̇l (t) + kθl (t) − gr −1 kθd (t)

(17)
(18)

Now let us deﬁne a constant Jd∗ as follows.
△Jd∗ = Jd + gr ′−2

Jp(19)

The following equation is obtained by applying Jd∗ toEq.(17).
τ (t) = Jd∗ θ̈d (t) + cd θ̇d (t) + gr −2 kθd (t) − gr −1 kθl (t) (20)
2.5State Space Representation

The following equations are obtained by representing
Eq.(18) and Eq.(20) by matrices.
[][][]θ̈d (t)θ̇d (t)θd (t)
T =D+E+F(21)θl (t)
θ̈l (t)θ̇l (t)[T =[E=cd0

τ (t)
0]
0,cl]

[,D=
[F =Jd∗0

kgr −2−kgr −10Jl]−kgr −1k]

The state space representation of the control target isobtained from Eq.(21) as follows.[1]{ẋ(t) = Ax(t) + Bu(t)(22)y(t) = Cx(t)0100−2−1
 − kgr∗− Jcd∗ kgJr∗
0 
Jddd
A=
(23)
0001 −1kgr
0− Jkl − Jcll
Jl0
 1∗ J
(24)
B= 0d  , C = [ 0 0 1 0 ]0θd (t) θ̇d (t) x(t) = (25) θl (t)  , u(t) = τ (t)θ̇l (t)3Control Design

In this section, we design a controller which guarantees
robust stability to variation of the inertias Jd , Jl anduncertainty of the friction coeﬃcients cd , cl .
3.1

Polytopic Representation

The ranges of the variation parameters Jd , Jl are asshown in Eq.(26), (27). On the other hand, let us deﬁne
the range of the uncertain parameters cd , cl as Eq.(28),(29).

In Eq.(22), all of terms including the varying parameters Jd∗ , Jl and uncertain parameters cd , cl are multiaﬃne for them. Therefore, if the stability at the vertexes
shown in Eq.(30) is guaranteed, the robust stability toJd∗ , Jl , cd and cl is guaranteed among the ranges sownin Eq.(26) to (29).
3.2

LQ Optimal Control

Let us deﬁne u(t) as the state feedback controller. LetQ be a weight matrix to state variable, and R a weight
to input. Then, let us minimize the cost function Jdeﬁned as following equation.
∫ ∞{x(t)T Qx(t) + u(t)T Ru(t)}dt
J=(30)
0Here, by using a feedback gain given as K = Y X −1 ,let us deﬁne u(t) = Kx(t). Then, the system is stabilized by u(t) if there exist X and Y satisfying followingequations.[2]
1
He[Ai X + Bi Y ] X T (Q 2 )T Y T R
1
Q2 X−IO  ≺ 0 (31)RYO−R
(i = 1, 2, 3, · · · , 16)[ZI

IX]≻0 ,trace[Z] < γ△(32)△Here, X, Y are deﬁned as X = P −1 , Y = KX. When
we deﬁne the weights for state variable and input asEq.(33), the feedback gain K is obtained as Eq.(34).

1 00000  0 1Q=,R = 1800 (33)
0 0 900 0 0 001Kopt = [ −1.8324 −0.1696

46.4070 0.4093 ] (34)Simulation

In this section, the simulation is conducted with theproposed method. Here, the reference of the Load Discangle θl (t) is π2 [rad]. By changing the condition of simulation model variously, the robustness of the controller isveriﬁed with various cases.[3] Table 2 shows conditionsof the simulation.
Table 2 Simulation Conditions
Jd∗

Jl

cdcl

Nominal case

min

min

0.0043

0.053

Case 1max

min0.00430.053] (26)

Case 2minmax

0.00430.053

Jl ∈ [Jl,min , Jl,max ] = [8.3 × 10−3 , 2.8 × 10−2 ] (27)Case 3max

max

0.0043

0.053

Case 4minminminmin

Case 5minminmaxmax∗Jd∗ ∈ [Jd,min,∗Jd,max] = [4.2 × 10−4 ,−3

5.5 × 10

cd ∈ [cd,min , cd,max ] = [2.8 × 10−3 , 5.7 × 10−3 ] (28)−2cl ∈ [cl,min , cl,max ] = [3.8 × 10−2, 6.8 × 10] (29)Then, Based on these ranges, let us deﬁne the vertexesof system matrices A and B as follows.
Ai (i = 1, 2, 3, · · · , 16) , Bi (i = 1, 2, 3, · · · , 16)Here, the ”nominal case” assumes the state that Drive
Disc and Load Disc have no weights, and the value ofcd , cl is the middle point of their assumed range. The
simulation results are shown by Fig. 6, 7.

Figure 6 Simulation Result of Case 1 to 3

Figure 9 Experiment Result of Case 2

Figure 7 Simulation Result of Case 4, 5

Figure 10 Experiment Result of Case 3This section shows the typical results, however thatis not all. The simulation is conducted in all endpointsof the variation range which is deﬁned as Eq.(26) toEq.(29). Then, the controller makes Load Disc angle
to follow the reference in all cases correctly. Thus therobustness of the controller is veriﬁed with simulation.

5Experiment

By changing the set up of experimental device variously, the robustness of the controller is established withvarious cases. Furthermore, by comparing the propose
method and nominal LQ, the superiority of the proposed
method is proved. Table 3 shows conditions of the experiment.
Table 3 Experiment Conditions
Drive Disc weight

Load Disc weight

Case 100Case 240

Case 304

Case 422The experiment results are shown by Fig. 8 to 11.

Figure 11 Experiment Result of Case 4
In some cases, the nominal LQ controller makes overshoot or oscillation. By contrast, the proposed methoddose not make them in all cases. Thus the robustness
and the superiority of the controller is established withthe experiment.

6Conclusion

In this study, robust LQ controller which guarantees
robust stability to variation and uncertainty is designed
by using polytopic representation and LMI solution tothe optimal regulator problem. Further, the robustness
and superiority of the proposed method is veriﬁed bysimulation and experiment.

References[1] Manual for Model 220 Industrial Emulator/ServoTrainer, Educational Control Products, 5725 Ostin Avenue, Woodland Hills, CA 91367, 1995.
[2] H.kataoka, T. Mizuno, H.Yamazaki, G.Chen, I. Takami:
Robust Stabilization of Antilock Braking System withLQ Control. JSME, 2014, pp.1B24 1-11, 2014.

Figure 8 Experiment Result of Case 1[3] Zhiqiang Gao, Shaohua Hu, and Fangjun Jiang: A NovelMotion Control Design Approach Based On Active Disturbance Rejection. Cleveland State University, Cleveland OH.