2011 4th International Conference on Mechatronics (ICOM), 17-19 May 2011, Kuala Lumpur, Malaysia

978-1-61284-437-4/11/$26.00 ©2011 IEEE

Bounded Constrained Optimization of Performance Weighting Function for Precise Robust Positioning

Control System Safanah M.Raafat

Mechatronics Engineering Dept. International Islamic University Malaysia IIUM

Kuala lumpur, Malaysia E-mail safanamr@gmail.com

Rini Akmeliawati Mechatronics Engineering Dept.

International Islamic University Malaysia IIUM Kuala lumpur, Malaysia

E-mail rakmelia@iium.edu.my

Abstract— In this paper, we present a robust control design and analysis for a single axis servo positioning system. A new method that is based on inequalities and bounded constrained optimization technique is developed for tuning the performance weighting function. The effect of parameter changes in the system model are treated as a set of parametric uncertainties. It is shown that the proposed method can simplify the design procedure of H� robust controller design. The developed controllers are implemented experimentally; high robustness, precision, and high bandwidth are achieved.

Keywords- performance weighting function, servo positioning system; structured (parametric) uncertainty

I. INTRODUCTION High precision motion control has become an essential

requirement in today’s advanced manufacturing systems such as machine tools, micro-manipulators, surface mounting robots, etc. High precision motion control is first challenged by the presence of friction. In addition, other uncertainties which may also be regarded as parasitic effects are often present in real-world systems. These effects include: Parametric uncertainty, such as parameter changes due to, for example, different operating conditions and load changes and actuator/sensor nonlinearities, such as hysteresis, dead-zone and saturation.

Many interesting research work were developed to have precise robust positioning systems [1-4]. Improved robust stability against different uncertainties and improved tracking performance are still challenging requirements. Nevertheless, suitable formulations of the robust control design problem can lead to the achievement of these requirements. Therefore, some research works are developed on optimal H� feedback control methods for positioning systems, as in the following research works; standard nominal H� control combined model reference and H� controllers [5], robust control with parametric uncertainties, using DK iteration for evaluating the control optimization problem [6], H� optimization and feed forward control [7], Glover-McFarlane loop-shaping scheme [8]. However, there is still the difficulty of selecting the necessary weighting functions for the robust controller design. This usually requires a rather complicated procedure and often

needs many iterations as well as fine-tuning. In addition, it is hard to find a general formula for the weighting functions that can work in every case [9].

In this paper, the performance weighting function is automatically tuned using constrained optimization procedure. The optimized weighting function guarantees the development of robust controlled system with best possible performance. The structured (parametric) uncertainty is considered in the design.

This paper is organized as follows: In Section II, the modeling of the servo positioning system is described; this is followed by the H� robust control design in Section III. The demonstration of experimental uncertainty formulation, identification, and validation, and control design application are presented in Section IV. Discussion and robust stability and performance analysis via μ analysis is summarized in V. We conclude with final observations in Section VI.

II. PRESISE SERVO POSITIONING SYSTEM

The single axis feed drive system under investigation has two large inertias; a motor inertia and a table inertia, and they are connected by a ball screw. The primary sources of elasticity in the system are the ball screw, flex coupling, and bearing supports. A simplified model of the single axis positioning system is shown in Fig.1.

The equation of motion can be derived analytically to form the following equations:

dlm TTTBJ −−=+ θθ ���1

(1)

p

ll l

TFBxM ==+ 2�� (2)

where � is the angular position, x is the measured table position, �m is the motor torque, �l is the load torque, �d is the torque disturbances, Fl is the equivalent force acting on the positioning table, J is the rotational inertia that combines the motor shaft, the coupling, and ball screw mass inertias, B1 the viscous damping contributed by the ball nut and rotational bearings , lumped together, lp is the screw pitch that serves as

the transformation factor from rotational to linear motion, B2 is the mechanical damping from the linear bearings.

The simplified transfer function of the positioning table will be expressed as:

)()()()(

α+==

ssK

sUsXsPn (3)

where K and � are linearized functions of the system’s parameters. The dynamic friction effect has been neglected in this model. In this work, these parameters are identified experimentally by applying Off-line identification on measured input-output data. Prediction Error Method (PEM) has been applied on experimental input-output data. The selected model structure is a type one second order process model as follows

)1()()(

sTsK

sEsx

p+= (4)

where, x is the output displacement as measured by an incremental encoder, K is a constant and Tp is the time constant in second.

Figure1. Simplified model of positioning stage.

III. H� ROBUST CONTROL SYNTHESIS

A. Parametric Uncertainties Description: To formulate the parametric uncertainty of the system into

the H� framework, equation (3) will be described first by the following second order differential equation:

ubxaxa 012 =+ ��� (5) where x and u are respectively the system state variable (position) and the input variable (control), a2=f(J,M), a1=f(B1,B2) and b0 are constant parameters that can be identified.

The uncertainties in parameters that are derived from linearized approximation of nonlinearities (e.g. friction, inertia, cogging and hysteresis) variations in material properties, perturbations, and disturbances will be taken into account; equation (5) will be reformulated as a monic vector differential equation with parametric perturbations as follows:

u)bb(x)aa(x)aa( 001122 ΔΔΔ +=+++ ��� (6) where �a2, �a1 and �b0 are the uncertainties in a2, a1 and b0 respectively and posses the following forms

2222 aapaa δΔ = ,

1111 aapaa δΔ = and

000 bbo pbb δΔ = (7)

where 2a , 1a and 0b are the nominal values of a2, a1, b0. 2ap ,

1ap and 0bp and2aδ ,

1aδ and 0bδ represent the possible fixed

and unknown perturbations on these three parameters. The three parameters can be represented as an upper linear

fractional transformation (LFT) [10] as

),M(F)p(aa aau

aa22

221

11

22

δδ

=+

= (8)

with

����

�

�

����

�

�

−

−=

2

21

1

2

2

2

ap

ap

Ma

a

a (9)

Similarly, the parameters a1 and b0 can be represented as an upper LFT in

1aδ and 0bδ as:

),M(Fa aau 111 δ= (10) with

��

���

�=

1

1

11

0apa

Ma

a (11)

and ),M(b bb 000 δ= (12)

with

���

�

���

�

−=

0

0

0

0

0bpb

Mb

b (13)

The inputs and outputs of2aδ ,

1aδ and 0bδ are denoted

as 012 baa y,y,y and

012 baa u,u,u respectively, as shown in Fig. 2. The equations governing the system behavior can be

obtained as:

��������

�

�

��������

�

�

����������

�

�

����������

�

�

−−−

−−−

=

��������

�

�

��������

�

�

uuuuxx

ba

aap

paa

aap

paa

yyyyxx

b

a

aaa

aa

b

a

a

0

1

21

2

1

2

0

1

2

2

1

0

1

222

1

222

1

2

1

0000010000000000

100

100

000010

��

(14)

���

�

�

���

�

�

���

�

�

���

�

�

=���

�

�

���

�

�

0

1

2

0

1

2

0

1

2

000000

b

a

a

b

a

a

b

a

a

yyy

uuu

δδ

δ (15)

Let P denotes the input/output dynamics of the system, which takes into account the uncertainty of parameters. P has

four inputs ( uuuu baa ,,,012

), four outputs ( yyyy baa ,,,012

) and two states ( 21, xx ). The state space representation of P is:

���

�

�

���

�

�=

22212

12111

21

DDCDDCBBA

P (16)

where��

�

�

��

�

�

−=2

1010

aaA ,

��

�

�

��

�

�

−−= 0

000

2

1 12 a

ppB a

a ��

�

�

��

�

�=

2

210

aB,

�����

�

�

�����

�

� −

=00

0

0

1

2

1

1 aaa

C ,

������

�

�

������

�

�−−

=

0

2

11

00000

01

2

b

ap

p

D

aa

,

�����

�

�

�����

�

�

=00

1

2

12

aD , [ ]012 =C , [ ]00021 =D , 022 =D

The uncertain behavior of the original system is described by an upper LFT representation [9]

uPFy u ),( Δ= (17) with structured diagonal uncertainty matrix ),,(

012 baadiag δδδ=Δ .

The bode plot of the family of perturbed systems for -1 ≤

2aδ , 1a

δ ,0bδ ≤1 is shown in Fig. 3. It shows that the

bandwidth of the system varies in accordance to parameter variations. In some cases, the system becomes slower. When the bandwidth decreases, the phase margin decreases, thus, the system oscillates and becomes unstable in some cases.

x�� x�u x2aM0bM

1aM

+

−�

2aδ0bδ

1aδ

0bu0by

1ay1au

2ay2au

Figure 2. Block diagram of the servo motion system with uncertain

parameters.

B. The Optimized Performance Weighting Function The performance weighting Function We is used to shape

the sensitivity function and also limit its infinity norm. It will be represented in this paper by the following form [9]

αωω

b

bse s

MsW+

+= / (18)

where Ms is the maximum value of the sensitivity function in all frequencies. � is a small number to approximate the integral part of the filter with a pole near the origin, and �b the system bandwidth is selected to have a good robust performance by satisfying the condition of that the nominal plant should be less than 1 on all frequencies.

100 101 102 10310-10

10-5

100

Log

Mag

nitu

de

Frequency (radians/sec)

100 101 102 103-200

-150

-100

-50

Phas

e (d

egre

es)

Frequency (radians/sec)

uncertain system

uncertain system

Figure 3. Open loop frequency response characteristics with parameters

uncertainty.

For the current application of robust control, optimization algorithm is utilized to ensure proper selection of the weighting function parameters in terms of improved maximum singular value of the robust controller.

The weight formula is selected first as given in (18). Then the parameters of the weighting function eW , is optimized using a Constrained Optimization (Inequalities and Bounded) technique [11]. The selection of sM and bω is formulated in the following constrained optimization problem:

Minimize f0 (x) = 1/|We (x)| (19) Subject to f1(x) � Ta, f2(x) � �max,

where x is a vector of decision variables (Ms and �b ), and f0 is the objective function, f1 is the allowable tolerance between the sensitivity function and the reciprocal of the norm of the performance weighting function, Ta is the maximum allowable tolerance between the sensitivity function and the reciprocal of the norm of the performance weighting function and f2 is the required singular value of the closed-loop controlled system, �max is the maximum allowable singular value of the closed-loop controlled system. Ta and �max are selected based on performance requirements. As a result, the sensitivity function of the closed-loop controlled system is ensured to be shaped in its best condition, as shown in Fig. 4. The constraints are selected as the required maximum singular value of the closed- loop controlled system and the allowable tolerance between the sensitivity function and the reciprocal of the norm of the performance weighting function. For the given values of

Ta and �max, the optimized values were obtained within 48 iterations with an objective function of 0.0225.

C. The Control Weighting Function The weighting function for the control Wu was chosen to

reduce the high frequency content of the control signal. In order to ensure that the servo signals within saturation limits, this weighting function is better to be selected as [9]

bc

ubcu s

MsW

ωαω

++

=1

/

(20)

where �bc , the bandwidth of the controller, Mu , the maximum value of the control signal are selected according to the control signal requirements, and �1 is selected as a small number.

B o d e D ia g ra m

F re q u e n c y (ra d /s e c )1 0

-21 0

01 0

2-1 0 0

-8 0

-6 0

-4 0

-2 0

0

2 0

Mag

nitu

de (

dB)

1 / | W e (jw )|| S | n o c o n tro l| S | H in f

Figure 4. Plot of the sensitivity with and without H� control against 1/|We(j�)|

D. The Controller Design The H� controller was designed so that the H�-norm from

input w= d to output ��

���

�=

2

1

zz

z is minimized, where d

represents the disturbing effect. The entire connection configuration of (Fig.5) is developed, where the parametric uncertainty representation is considered to reflect the performance objectives into optimal control setting. The H� controller will minimize the infinite norm of

),( ∞KPF Nl overall stabilizing controllers K�, where ),( ∞KPF Nl is the transfer function matrix of the nominal

closed loop system from the disturbance to the error signals.

NP

Figure 5. The closed-loop controlled system with structured uncertainties.

The lower linear fractional transformation of the generalized

plant PN and the controller K� can be described by [10]:

( ) ��

���

�=∞ KSW

SWK,PF

u

eNl (21)

The objective of H� is to find the controller K� that internally stabilizes the system such that the transfer function of the system from input w to output z,

∞zwT is minimized.

∞zwT can be expressed as:

∞∞ �

�

���

�=

KSWSW

Tu

ezw (22)

This cost function will be minimized by the robust controller using - iterations, such that γ<

∞zwT . To find the optimal value of , the constrained optimization

algorithm as described in Section (III.B) is used to tune the parameters of the performance weighting function.

This particular weighting function has a significant effect on the design of the robust controller, since the other control weighting function Wu parameters can be easily fixed according to actuator requirements.

The proposed constrained optimization algorithm for obtaining the optimal values of We parameters is described by the flow chart shown in Fig. 6.

Identify the system nominal model GN

Define the structure of We, Wu according to equations

(18) and (20)

Find K(s) by H�

optimization algorithm

Start

End

Initialize the parameters of We

Define the constraints

Construct the overall controlled system

Set the weighting functions then find the controller

Find best We through constrained optimization procedure

The best We and K(s) values are applied to system

Figure 6. Flow-chart for tuning the weighting functions using constrained

optimization

IV. EXPERIMENTAL WORK AND RESULTS A. System Description

Hardware setup of the overall motion control scheme for the motor-table direct drive system is shown in Fig. 7. The basic

hardware consists of a host PC, DC servo motor, and the motor-table mechanism. The currently used machine has an operating range of 225mm. It is capable of 1�m resolution for measurements. In the system, position feedback signal is the only sensing available, which is obtained via an incremental encoder. The desired control signal is generated by the designed H� controller. The control signal is sent to the servo power amplifier to regulate the actuator’s position.

Figure 7. Experimental Setup of the Single - axis Positioning System.

In this work, the parameters of system’s nominal model are

identified experimentally using offline identification of the nominal model parameters. Prediction Error Method (PEM) is applied on experimental input/output data. The nominal model of the positioning system is identified using PEM method as:

ss

G9401.29

0928.13920

+= (23)

The input-output data is collected from the positioning system using MATLAB’s xPC Target two PC-type desktop computers in a host-target configuration, and a NI BNC-2110 data acquisition card (DAQ) with 16 bit A/D and 16 bit D/A channels. B. Simulation and Experimental Results

The parametric representation of uncertainties is applied first. As a result, equation (6) will be expressed as

uxx baa )4.00928.139()5.09401.29()3.01(012

δδδ +=+++ ��� (24)

where 302

.p a = , 501

.p a = 401

.p b = and it is assumed that -1≤

2aδ ,1aδ ,

0bδ ≤1. Note that this represent up to 30% uncertainty in the parameter a2, 50% uncertainty in the parameter a1 and 40% uncertainty in the parameter b0.

The performance weighting function is optimized for best performance as

)1485.0()0707.00501.0(

++=

ssWe

(25) And, the control weighting function is selected according to

(20) as

)20001.0()0.20(

++=s

sWu (26)

The obtained H� robust controller for this case is

)5300.20)(1668.0()1640.0(3717.159

+++=∞ ss

sK (27)

The obtained value is 0.7945. The corresponding sensitivity transfer function of the closed loop controlled system is shown in Fig. 8. The design is stable and achieves an acceptable tracking performance since the sensitivity function lies below the inverse of We. Then the controller of (27) is implemented on the host PC–target using MATLAB/ Simulink xPC target tools.

10-3 10-2 10-1 100 101 102 103

100

101

Frequency (rad/sec.)M

agni

tude

(dB

)

1/|We(jw)||S(jw)|

Figure. 8 The sensitivity function of the closed loop robustly controlled

system with 1/|We(j�)| Triangular signals are tracked first using the implemented

robust controller, as shown in Figs. 9-11. The performance of the controlled system is robust; the controlled system tracked the reference signal with very slight error even at zero velocity. Similarly, another signal is tested for tracking as shown in Figs.12-14. Although faster rate of reference change is implemented the tracking error is small and the controlled system almost perfectly tracks the reference signal.

V. DISCUSSION Table I presents the mean and standard deviations for each

of the maximum error and the measured control input signal for 10 experiments. In addition, the Root Mean Square Value (RMS) of the resulted tracking error and measured control input signal are presented.

0 2 4 6 8 10-200

0

200

400

600

800

1000

Time (sec.)

Positio

n (M

m)

ReferenceDisplacement

Figure 9. Experimental results of applying triangular signal,

using parametric uncertainty representation; the transient response of the closed loop controlled system

0 2 4 6 8 10-30

-20

-10

0

10

20

30

Time (sec.)

The

Trac

king

Erro

r (M

m)

Figure10. Experimental results of applying triangular signal,

using parametric uncertainty representation; the tracking error

0 2 4 6 8 10-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Time (sec.)

The

Con

trol S

igna

l (V

)

Figure 11. Experimental results of applying triangular signal, using parametric uncertainty representation; the control signal

0 2 4 6 8 10-1000

-500

0

500

1000

Time (sec.)

Positio

n (M

m)

DisplacementReference

Figure 12. Experimental results of applying special signal, using parametric

uncertainty representation; the transient response of the closed loop controlled system

0 2 4 6 8 10-40

-30

-20

-10

0

10

20

30

40

50

Time (sec.)

Trac

king

Error (M

m)

Figure13. Experimental results of applying special signal, using parametric

uncertainty representation; the tracking error

0 2 4 6 8 10-1

-0.5

0

0.5

1

Time (sec.)

Con

trol S

igna

l (V)

Figure 14. Experimental results of applying special signal, using parametric

uncertainty representation; the control signal

Similarly, another set of experiments is conducted on the system under an added load of 1.777 Kg. Table II represents the obtained results; although the tracking error is increased, the control signal is slightly affected by the presence of load indicating the robustness of the designed controller. However, to avoid the problem of increased tracking error it is better to design the controller within gain schedule structure, where another controller for the system under load effect should be designed.

Table I

Performance measures of the robustly controlled system using parametric uncertainty representations

Input signal

Tracking error (μm) Control Signal (V) Mean

Of Max.

Stand. Dev. R.M.S.

Mean Of

Max.

Stand. Dev.

R.M.S.

Triangu-lar

28.121 0.505 18.311 0.642 0.015 0.458

Special signal

42.020 2.254 20.203 0.932 0.057 0.497

Table II

Performance measures of the robustly controlled system using parametric uncertainty representations, under load variations

Input signal

Tracking error (μm) Control Signal (V) Mean

Of Max.

Stand. Dev. R.M.S.

Mean Of

Max.

Stand. Dev.

R.M.S.

Triangu-lar

73.558 0.776 47.812 0.663 0.009 0.477

Special signal

127.20 5.460 54.733 1.116 0.044 0.532

The resulted μ analysis of robust stability for the robustly

controlled system with parametric uncertainty indicates that the upper bound for robust stability is 1.111 and the lower bound is 1.1066. The system can tolerate up to 111% of the modeled uncertainty. The system is robustly stable to modeled uncertainties. Similarly, the μ analysis of the robust performance gives an upper and lower bounds of 0.9545 and 0.9499. These results mean that the system is stable with all parameters uncertainty, that is, the robust stability has been satisfied and the disturbance attenuation specifications have been met.

VI. CONCLUSIONS

In this paper, an H� robust controller is designed to assure

robust stability and robust performance of the uncertain single axis servo actuator system even in the presence of load variation. Structured Parametric uncertainty representation is utilized to synthesize an H�

robust controller. A new method is

based on constrained optimization is used to tune the performance weighting function for robust controller design requirements. The developed controller is implemented experimentally; the controller can achieve robust stability with acceptable resolution. Further, it can be concluded that the H�

optimal control is a powerful technique to design a robust control for the positioning servo actuator system with structured uncertainty and disturbances. The requirement for only measuring the position sensor to develop the identification and control in a straightforward procedure indicates also a good cost performance. Further improvements

on better resolution and smaller control signal are working on using a specially designed integral robust controller scheme.

REFERENCES [1] H.-S.,Park,P.H. Chang, and D.Y.Lee, ” Concurrent design of continuous

zero phase error tracking controller and sinusoidal trajectory for improved tracking control”, ASME J. Dyn. Syst., Meas. Control, vol. 123, pp.127-129, 2001.

[2] Y. Honga and B.Yao,” A globally stable saturated desired compensation adaptive robust control for linear motor systems with comparative experiments”, Automatica, 43, pp. 1840 – 1848, 2007.

[3] L. Xu, and B.Yao. “Output Feedback Adaptive Robust Precision Motion Control of Linear Motors”, Automatica, 37, pp.1029-1039, 2001.

[4] W.F. Xie, “Sliding-mode-observer-based adaptive control for servo actuator with friction”, IEEE Trans. Ind. Electron., vol.54, no.3, pp.1517-1527, June 2007.

[5] K.B. Choi, S.H. Kim, and B.W. Choi, “Moving-magnet type precision miniature platform for fine positioning and compliant motion”, Mechatronics, Technical note, Elsevier Science Ltd, 921-937, 2001.

[6] Shin H.J., Pahk, H.J., “Development of the Precision Planer Motor Stage Using a New Magnet Array and Robust Control with Parametric Uncertainty Model”, Int. J. Adv. Manuf. Technolo, 33, pp.643-651,2007.

[7] Z.Z.Liu, F.L. Luo, and M.A. Rahman,” Robust and precision Motion Control System of Linear- Motor Direct Drive for High-Speed X-Y Table Positioning Mechanism”, IEEE Trans. On Industrial Electronics, Vol.52, No.5, 1357-1363, 2005.

[8] A. Sebastian, and S. M. Salapaka, “Design methodologies for robust nano-positioning”, IEEE Trans. Control systems Technol., vol.13, no.6, pp.868-876, Nov. 2005.

[9] K .Zhou and J.C. Doyle,” Essentials of Robust Control”, Prentice-Hall, Inc, 1998.

[10] D.-W. Gu, P.Hr. Petkov and M.M. Konstantinov,” Robust Control Design with MATLAB”, Springer-Verlage London Limited, 2005.

[11] MATLAB (R2008b), Version 4.1 Optimization Toolbox Software, MathWorks™.