H∞ Loop Shaping for Positioning Control Systemwith Nonlinear Friction
M2011MM067 Yuki SUGIYAMA
Adviser: Isao TAKAMI

1Introduction

2.2

1.1

Background

The motion equation is shown in Eq.(1), where y[m]is the position of the table position, u(t) is the current
input of the motor. Physical parameters are shown in
Table 1.
Kt
u(t)
(1)
M ÿ(t) + Fv ẏ(t) =R

In general, for the precise positioning control, it is necessary to satisfy the speciﬁcation which follows the reference without stationary error and overshoot. However,the friction property of the machine element deteriorates
the positioning performance. The stick-slip motion, thestationary error, overshoots, and the limit cycles are
generated as an inﬂuence of friction. To achieve the friction compensation, many methods are proposed such as
disturbance observer[1] and PD control[2]. Serrarens, etal.[3] suppresses the stick slip motion by the frequency
shaping of H∞ control. H∞ control is used by automatically shaping of the disturbance compensation and the(semi)complementary sensitivity function[4]. H∞ loop
shaping is a control system design method proposed by
McFarlane, et al.[5]. The H∞ loop shaping controller
has a structure with LQ state feedback gain and state
observer, so it is expected to improve the plant responseusing the evaluation function of the LQ control [6]. As
an example of applying the positioning control systemby the H∞ loop shaping, Moon, et al.[7] design the controller that takes the disturbance control and the reference tracking performance into consideration.
1.2

Objectives and Method

In this study, we propose the method of the positioning control for the ball screw system achieved by H∞
loop shaping. The designed controller satisﬁes the friction compensation, the reference tracking performance,
and improvement of plant response by using weightfunctions. The dynamic weight function and the constant matrix are used. The dynamic weight function
is selected to satisfy the friction compensation and thereference tracking performance. The friction compensation is focused on the frequency characteristics of thefriction. The frequency region where a static friction
and the stick-slip motion appears predominantly is identiﬁed. The constant weight is selected to improve theplant response. The selection of the constant weight for
LQ control is easier than selecting appropriate weighting
function for standard H∞ control synthesis framework.
Thus in this study, the constant weight is selected tosimpliﬁed the trial and error for weight function. The
eﬀectiveness of the H∞ loop shaping controller is illustrated by using the simulations with the nonlinear friction and the experiments.

2

Characteristics of Control Plant

2.1

Ball Screw System

In this study, the ball screw system is used as a control
plant. It controls the table position using drive forceof the DC servo motor(rated capacity 110[W], rated
torque 0.221[Nm]) through the ball screw(entire length
100[mm]). The optical encoder(resolution performance0.1[µm]) is used to measure the position.

Modeling of Plant

Table 1 Parameters of ball-screw system
torque constant Kt
0.34[Nm/A]
coeﬃcient of viscosity Fv
5.0 × 103 [Ns/m]mass of table M
0.49[kg]
6.37 × 10−4 [m/rad]ball screw constant R
Let the state vector x(t) = [y(t) ẏ(t)]T , the state space
representation of the ball screw system described by
Eq.(2). In this study, Eq.(2) is used by the control design.
[
]
[
]
0
1
0
ẋ(t) =
x(t) +u(t)Kt
0 − FMv
RM[]
1
0x(t) =
x(t)
(2)
0
1
2.3

Modeling of Nonlinear Friction

Generally, the friction model is consisted of the static
friction, the coulomb friction, and the viscous friction.
Marton et al.[8] compensates the stick slip motion byusing the friction model with the Striebeck eﬀect. Thusin this study, the Tustin model[8], which is the nonlinear
friction model with the Striebeck eﬀect, is used. The
Tustin model is shown in Eq.(3).
|ω|Fn = (Fc + (Fs − Fc )e− ωs ) × sgn(ω) + Fv ω

(3)

Where ω is the table velocity, ωs is the Striebeck velocity, Fs is the maximum static frictional force, Fc isthe coulomb frictional force, Fv is the viscous friction
coeﬃcient, sgn(·) is the signum function. The friction is
identiﬁed by the response experiments which gives thesinusoidal input to the ball screw system. In this study,
the nonlinear simulation is conducted by using the friction model of Eq.(3).
2.4

Characteristics of Nonlinear Friction

Even though the friction itself is nonlinear, the behavior of friction can be considered to have some frequency
or repeatedly movement. In this study we try to extract some frequency that might explain the stick-slip
motion. The stick-slip motion is analyzed by giving thecurrent of 0.4[A] to the plant in open loop. The table
position is shown in Fig.1, and its spectrum is shown in
Fig.2. As can be seen in Fig.2, the stick-slip motion has
some dominant frequency in near 10[rad/sec]. Thus inthis study, the low frequency(0[rad/sec]) for static friction and near 10[rad/sec] for stick-slip are considered for
friction compensation.

1

0.06

3
spectrumposition[m]

−4

4

0.04

the extended system Ps (s) = Q 2 P (s)W (s). Fig.4(b)is the equivalent transformation of Fig.4(a). Final con1
troller C(s) = W (s)C∞ (s)Q 2 is introduced by connecting C∞ (s) and the weight functions once again. In Fig.4,

x 10

0.08

2
1

0.02

zu
0
0

0
5

10
time[s]15

10Fig. 1 Stick slip motion

0

W(s)

u

P(s)

2

10
frequency[rad/sec]

zx

−1

Q2

Ps(s)

δ

Fig. 2 Spectrum

x

+

+ r

-

+

C∞(s)

(a) Closed loop system

3zu

Design of Positioning Control

3.1 H∞ Loop Shaping
The control design via H∞ loop shaping proposed by
McFarlane, et al.[5]. It is well known that the H∞
loop shaping controller C∞ (s) for the design model
Ps (s) = {Ad , Bd , Cd , Dd } is given by Eq.(6) using thepositive solution X and Z for Ricatti equations Eq.(4)and Eq.(5). λmax is the maximum eigenvalue.
(Ad − Bd S −1 DdT Cd )T X + X(Ad − Bd S −1 DdT Cd )−XBd S −1 BdT X + CdT R−1 Cd = 0(4)(Ad − Bd S DdT Cd )Z + Z(Ad − Bd S −1 DdT Cd )T−ZCdT R−1 Cd Z + Bd S −1 BdT = 0
R = I + Dd DdT , S = I + DdT Dd−1[
C∞ (s) =Ad + Bd F + H(Cd + Dd F )
BdT X

H
−DdT

C(s)

W(s)

C ∞(s)

1

Q −2

-

+

Q −2
-1

r

Fig. 4 H∞ loop shaping method
the transfer function matrix from the external input{r, δ} to evaluation output {zu , zx } is given by Eq.(8).
[ ] [
][ ]
zu
P (s) Pzu δ (s) r= zu r(8)zx
Pzx r (s) Pzx δ (s) δ
Pzu r (s) = W −1 (s)(1 + C(s)P (s))−1 C(s)Q− 2
1

Pzx r (s) = Q 2 (I + P (s)C(s))−1 P (s)C(s)Q− 2+ r

Fig. 3 Block diagram of the systemminimizing H∞ norm problem of the transfer function
matrix from δ, r to x, u is given by Eq.(7).
[]I(I + P (s)C(s))−1 [I P (s)]< γ (7)C(s)∞H∞ loop shaping controller which achieves the reference tracking performance and the friction compensation is designed by using the evaluation function Eq.(7).

1Pzx δ (s) = Q 2 (I + P (s)C(s))−1 P (s)W (s)1

Eq.(8) can be explained as follows. Pzu r (s) is not1
1
considered because of Q− 2 ≃ 0 if the value of Q 2 islarge. The control input is not inﬂuenced the disturbance by minimizing norm of Pzu δ (s). Pzx r (s) is theclosed loop transfer matrix. Pzx δ (s) is the disturbance
control characteristic, and the eﬀect of the disturbance1
is minimized by using the constant matrix Q 2 and the
dynamic weight function W (s). Thus the selection of1
the weight function W (s) and Q 2 inﬂuences the control performance widely. W (s) is designed to satisfy thereference tracking performance and the friction compen1
sation and Q 2 is selected to improve the plant response.
3.3

3.2

+

zx

(b) Equivalent closed loop system

(6)

x

-

1

Q −2

1

The designed controller C(s) decreases the eﬀect fromthe friction δ and the reference input r to the control
input u and the state x in the plant P (s). Fig.3 shows
the block diagram of P (s) and C(s). From Fig.3, the

+

+

W(s)

]

1

δ +

x

Pzu δ (s) = (1 + C(s)P (s))−1

H = γ 2 (LT )−1 ZCdT , γ > (1 + λmax (XZ)) 2

P(s)

P(s)

C(s)

δ

(5)

F = −S −1 (DdT Cd + BdT X), L = (1 − γ 2 )I + XZ,

u

u

-1

W(s)

Selection of W (s)From section 2.4, the ball screw system has the stickslip motion in near 10[rad/sec] and the static frictionin the low frequency region. Thus the gain of W (s) is
raised in the low frequency and near 10[rad/sec]. The
1
ﬁnal controller is given by C(s) = W (s)C∞ (s)Q 2 , so thecontroller that has the reference tracking performanceis designed by adding the integrator to W (s). In the
result, W (s) is given by Eq.(9) to raise the gain in the
low frequency and close 10[rad/sec].
W (s) =

Selection of Weight Functions

H∞ loop shaping controller which improves the controlled performance is introduced by multiplying theweight functions to back and forth of the open looptransfer function. In this study, W (s) and the constant1
matrix Q 2 are the weight functions shown in Fig.4(a).
The plant P (s) is extended by the weight functions W (s)1and Q 2 . Then the controller C∞ (s) is synthesized for

3.4

1
10s + 10×
0.01s + 1 s(9)1

Selection of Q 2

The H∞ loop shaping controller is improved the plant
response by using the evaluation function of LQ control. Trial and error for H∞ weight function is notclear compared with traditional LQ control synthesis.
1
Thus in this study, the constant weight Q 2 is selected
to simpliﬁed the trial and error for weight function. Inthis ball screw system, the feedthrough term of the extended system is Dd = 0. Then the H∞ loop shaping controller C∞ (s) is solved by two Riccati equations
1
Eq.(10) and Eq.(11), where Ps (s) = Q 2 P (s)W (s) ={Ad , Bd , Cd , Dd }.
ATd X + XAd − XBd BdT X + CdT Cd = 0(10)Ad Z + ZATd − ZCdT Cd Z + Bd BdT = 0(11)

3.5 Characteristics of Disturbance Rejection
The characteristics of disturbance rejection and controller are analyzed by the gain diagram to verify theeﬀectiveness of the weight functions. Fig.6 shows thegain diagram from the friction δ to the table positiony. From Fig.6, the disturbance rejection is achieved in0

From the solution of X and Z, the controller is given by
Eq.(12), Eq.(13).

Gain[dB]ẋk (t) = (Ad + Bd F + HCd )xk (t) + Hyk (t)uk (t) = F xk (t)

−50

−100

(12)
(13)

−150

ing controller C∞ (s) can be considered as the output
feedback controller consisting of a state feedback andobserver. The block diagram of the closed loop system
is shown in Fig.5. The state feedback gain F in Eq.(12)z u -1

u

W(s)

x

P(s)

1−

Q2

zx

C∞ (s)−200 −5
10

uk

+ +
W(s)

W(s)

+

Bd

Ad

-

xk
Cd

1

−
2
+ Q

-

+

-1

Q 2−

r

observer5

10

the low frequency region and near 10[rad/sec]. This diagram shows the eﬀectiveness of the proposed method.
The obtained controller C∞ (s) is as follows.
[
C∞ (s) =

.
xk 1
+ s

0

10
Frequency[rad/sec]

Fig. 6 Gain diagram from friction to outputHδwith W(s) and Q1/2
without W(s) and Q1/2

Where F = −BdT X, H = γ 2 W1−T ZC T , W1 = I + (XZ −γ 2 I). From Eq.(12) and Eq(13), the H∞ loop shap-

2.7×103 s3 +2.9×107 s2 +2.2×1010 s+5.6×1010
s4 +1.9×104 s3 +1.0×108 s2 +1.4×1010 s+3.5×1010
9.2×103 s3 +9.7×107 s2 +2.9×1010 s+7.3×1010
s4 +1.9×104 s3 +1.0×108 s2 +1.4×1010 s+3.5×1010](18)

The two dimensional controller is obtained by using thebalanced truncation method[9] for C∞ (s).
[ 2.9×102 s+2.0×106 ]state feedback

C∞2 (s) =F

s2 +1.0×104 s+1.2×106
9.5×102 s+2.6×106
s2 +1.0×104 s+1.2×106(19)
1

The ﬁnally obtained controller C(s) = W (s)C∞2 (s)Q 2
is Eq.(20). The gain diagram of C(s) is shown in Fig.7.

Fig. 5 Block diagram of C∞ (s)and Eq.(13) is given F = −BdT X, and X is the solutionof the Ricatti equation Eq.(10).
On the other hand, it is well known that the Ricatti
equation for Eq.(14) is given in Eq.(15), and the state
feedback gain is F = −BdT X.
∫ ∞J=(x(t)T Qx(t) + u(t)T u(t))dt
(14)
0
T
T
(15)
Ad X + XAd − XBd Bd X + Q = 0

C(s) =[]C1 (s)C2 (s)
=

1.0×108 s2 +7.2×1010 s+7.2×1010
5.0×10−2 s4 +5.5×102 s3 +5.9×104 s2 +9.7×105 s
5.5×106 s2 +1.2×109 s+1.2×109
5.0×10−2 s4 +5.5×102 s3 +5.9×104 s2 +9.7×105 s

0

Thus the Ricatti equation for Eq.(16) is equivalent toEq.(10), and F = −BdT X of Eq.(13) is the state feedback gain using LQ control. The state feedback gain Fthat minimizes the the evaluation function
J is obtained.
1
The selection of the constant weight Q 2 for LQ control
is easier than selecting appropriate weighting function
for standard H∞ control synthesis framework. The con1
stant weight Q 2 is given in Eq.(17).
[]14000 0
2
Q =
(17)
0
50



(20)

As can be seen in Fig.7, the controller C(s) is expected
200

1

C1(s)

150

Gain[dB]

When zx (t) is represented zx (t) = Cd xd (t) = Q 2 x(t), xd
is the state of Ps (s). The evaluation function of Eq.(15)
is equivalent to Eq.(16).
∫ ∞J=(x(t)T Qx(t) + u(t)T u(t))dt∫0 ∞=(zx (t)T zx (t) + u(t)T u(t))dt0∫ ∞=(xTd CdT Cd xd (t) + u(t)T u(t))dt
(16)



C2(s)

100

50

0

−50 −510010Frequency[rad/sec]

5

10

Fig. 7 Gain diagram of C(s)to have the reference tracking performance and the friction compensation because the gain of C(s) is risen in
the low frequency region and near 10[rad/sec].

4

Simulations and Experiments

4.1

Step Response

The step response of the ball screw system is veriﬁed
by simulations and experiments. The friction model
shown in section 2.3 is used for the nonlinear simulations. The simulation and the experiment are conductedin the following cases to verify the eﬀectiveness of thefrequency shaping.

• Case 1 : Ŵ (s) = 10s(The same gain as Case 2 in the low frequency region)
5s+10• Case 2 : W (s) = 0.01s+1
× 1s

−6

Ŵ (s) does not take account of the disturbance thatcauses the stick-slip motion near 10[rad/sec]. The simulation result of the table position[m] and the current input[A] are shown in Fig.8 and Fig.9, respectively. From

x 10

0.3

10
5

input[A]position[m]15

ramp reference
Case 1Case 20

0.2
0.1
Case 1Case 2
0

−5
0

5

10
time[s]15

200Fig. 12 Sim of output

5

10
time[s]15

20

Fig. 13 Sim of input

−6

15

x 10

0
−5
0

2

4
6
time[s]8

Case 1Case 2
−6

0

15

−0.2
−0.4
0

10

Fig. 8 Sim of output

5time[s]10

Fig. 9 Sim of input

Fig.8, the step response in Case 1 follows the reference
eventually by the integrator. However the overshoot is
occurred by the friction. On the other hand, the step
response in Case 2 follows the reference without stationary error and overshoot. As can be seen in Fig.9, theinput is applied to reject the static friction at the rising
moment. Thus the design controller achieved reference
tracking performance and friction compensation. Thenthe experiment is run in the same condition of the simulation. The table position[m] is shown in Fig.10, andthe control input[A] is shown in Fig.11.
−6

15

0.3

10

0.2

5
ramp reference
Case 1Case 2

0
−5
0

5

10
time[s]15

Case 1Case 20.1
020

Fig. 14 Exp of output

−0.1
0

5

10
time[s]15

20

Fig. 15 Exp of input

the real friction and friction model. However the experiment result has the same characteristics with thesimulation. From Fig.14, the table position follows thereference without the stick slip motion. As can be seenin Fig.15, the input is given to reject the static friction at the rising moment. From the simulation andthe experiment result, the H∞ loop shaping controller
compensates the stick slip motion by using W (s)

5

x 10

x 10

input[A]step reference
Case 1Case 2is shown in Fig.15. There are some diﬀerence between

0.2

position[m]5

input[A]position[m]0.4
10Conclusion

10
5

step reference
Case 1Case 20−50input[A]position[m]

0.4

2

4
6
time[s]8

0.2
0Case 1Case 2−0.2

10Fig. 10 Exp of output

−0.4
0

5
time[s]10

Fig. 11 Exp of input

As can be seen in Fig.8 and Fig.10, there are some difference between the real friction and our friction model.
However the experiment result has the same characteristics with the simulation. This mean that our friction
model has a certain accuracy. Thus the precise simulation is achieved by getting the friction model used in
section 2.3. From Fig.10, the step response in Case 2
follows the reference without the stationary error andthe overshoot. As can be seen in Fig.11, the input isgiven to reject the static friction at the rising moment.
The eﬀectiveness of the proposed method is veriﬁed.
4.2

Ramp response

The ramp response of the ball screw system is veriﬁed
by simulations and the experiments as well as the previous section. The simulation and the experiment conditions are the same as section 4.1. The simulation resultof the table position[m] and the current input[A] areshown in Fig.12 and Fig.13, respectively. From Fig.12,
the stick-slip motion is observed in Case 1. However the
ramp response in Case 2 follows the reference without
the stick slip motion. As can be seen in Fig.13, the input
is given to reject the static friction at the rising moment.
Based on the simulation result, the experiment is conducted in the same condition of the simulation. The
table position is shown in Fig.14, and the control input

In this study, the H∞ loop shaping controller, which
satisﬁes the reference tracking performance and the friction compensate, is designed for the ball screw system. The static friction and the stick-slip motion
near 10[rad/sec] are compensated by frequency shaping W (s). Even though the framework of the proposed1
method is H∞ control, the other weight Q 2 is easily se1
lected, since the weight Q 2 is equivalent to the weight
for the traditional LQ control framework. The eﬀectiveness of the proposed method is illustrated by thesimulations and the experiments.

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