Abstract— This paper proposes an optimized performance of

an intelligent H∞ robust controller of a single axis positioning

system. The objective is to achieve wider bandwidth, better

resolution, and robustness to modeling uncertainties. The main

contribution is the combination of intelligent uncertainty

weighting function and optimized weighting function in an H∞

robust controller design. The main distinguishing features of

this approach are: the accurate, fast identification of the

uncertainty bounds using an adaptive neuro fuzzy inference

system and the automatic tuning of the performance weighting

function in accordance to performance requirements. v-gap

measure is utilized to validate the intelligent identified

uncertainty bounds for wider stability region. Then the

methodology is demonstrated through both simulation and

experiments on the practical system. Experimental results also

demonstrate the robustness against load variations.

I. INTRODUCTION

ecently, research has focused on robust control

techniques to address the control challenges associated

with high precision positioning systems like achieving

higher resolution and faster response. Different H∞ design

methods of robust controllers were developed, like [1], [2];

the required weighting functions were selected according to

the designer knowledge and experience in a lengthy trial and

error procedure. Although these controllers achieve large

improvements in performance and robust to operating

conditions, still there are some important issues that need

more attention. The uncertainties need to be studied,

identified and analyzed systematically to improve the

performance of the system without developing overly

conservative robust controller.

The main concept of applying robust control in practical

problems is to consider a physical system as an uncertain

model which may be represented as a family of

mathematical models. Using robust control techniques, all

models in this family will be stabilized in an appropriate

manner. This family is described by a nominal model and a

bounded uncertainty. Thus it is customary to identify not

S.M.Raafat is with the International Islamic University Malaysia, Kuala-

Lumpur, 50728, Malaysia (phone: +017 259 6473; e-mail: safanamr@

gmail.com).

R. Akmeliawati is with the International Islamic University Malaysia, Kuala-Lumpur, 50728, Malaysia. (e-mail: rakmelia@liiu. edu.my).

Late Prof.Wahyudi was with the International Islamic University

Malaysia, Kuala-Lumpur, 50728, Malaysia.

only a nominal model, but also an uncertainty bound

associated to this nominal model. Identification methods

producing a nominal model and its associated uncertainty

are known as "Robust Identification”. Various techniques of

system identification exist that provide a nominal model and

an approximate uncertainty bound. Specific approaches

include stochastic embedding; set membership and model

error modeling [3]. Additionally a large number of

uncertainty descriptions are available from robust control

theory as e.g. a H∞ norm bounded additive or multiplicative

uncertainty on the plant model [4]; uncertainties bounded in

the gap or v-gap metric [5], and parametric uncertainty sets

[6].

“Soft computing” method was also developed to estimate

uncertainty bounds for dynamic systems [7], [8]. The main

purpose is to identify more accurate and less conservative

uncertainty bounds for robust active magnetic bearing

controller. On the other hand, adaptive fuzzy and neuro

fuzzy model for approximation of unknown function prove

to be efficient and stable [9, 10], satisfying engineering

requirements for design feasibility and simplicity of

computational effort, maintenance, and real-time

effectiveness [11]. In addition, a combination of neural

networks (NNs) and fuzzy logic systems can make good use

of both sensory numerical data and expert linguistic

information, as clearly presented in [12]. In a previous work,

we proposed an adaptive neuro fuzzy inference system as a

tool for estimating the uncertainty bounds for robust control

of servo systems [13]; ANFIS is used as its architecture can

identify near-optimal membership functions of fuzzy logic

for achieving desired input-output mapping. An uncertainty

set will be estimated for the identified nominal model and

uncertainty weighting function. The entire frequency domain

behavior of the physical plant is completely contained in that

of the uncertainty set. The main gained advantage over that

in [8] is considerably less computation time, reduced

computational difficulties and larger stability margin of

robust controllers as measured by the v-gap metric.

Therefore, the intelligent feedback ANFIS approach, as

developed in [14] will be applied in this work to identify a

non- conservative and accurate uncertainty weighting

function for robust control design.

In the H∞ robust controller design, it is also essential to

shape the sensitivity function and limit its infinity norm. The

performance weighting function must be adjusted to meet

these requirements in a lengthy trial and error procedure. In

this paper, a constrained optimization method will be

utilized to optimize the performance weighting function for

Enhanced Servo Performance of a single Axis Positioning System in

an Intelligent Robust Framework

Safanah M. Raafat, Student Member, Rini Akmeliawati, Senior Member, and Wahyudi Martono,

Member, IEEE

R

2010 IEEE International Symposium on Intelligent ControlPart of 2010 IEEE Multi-Conference on Systems and ControlYokohama, Japan, September 8-10, 2010

978-1-4244-5361-0/10/$26.00 ©2010 IEEE 2450

a better sensitivity function and to further improve the

performance of the resulted controlled system.

Then, to guarantee precise reference tracking in practical

application, a conventional integral controller is added to the

closed loop controlled system in a simple scheme.

Experimental results prove the validity of the applied

approach for robust stability and precise positioning

controlled system.

This paper is organized as follows. Section II provides the

system modeling and identification of the single axis

positioning system. Uncertainty bound identification and

validations for robust stability are discussed in Section III.

Section IV presents the robust controller design, the

optimized performance weighting function and the integral

robust scheme. Section V demonstrates the implementation

of the intelligent robust control. Section VI gives some

concluding comments.

II. SYSTEM MODELING AND IDENTIFICATION

A. System Modeling

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.

J lp M

B1 B2

xRotational motion Linear motion

θω

τm Flx.

Fig.1 Simplified model of positioning stage.

The equation of motion can be derived analytically to

form the following equations:

𝐽𝜃 + 𝐵1𝜃 = 𝜏𝑚 − 𝜏𝑙 − 𝜏𝑑 (1)

𝑀𝑥 + 𝐵2𝑥 = 𝐹𝑙 =𝜏𝑙

𝑙𝑝 (2)

where θ is the angular position, x is the measured table

position, τm is the motor torque, τl is the load torque, τd is the

torque disturbances, Fl is the equivalent force acting on the

positioning table, M is the overall mass of the stage, 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:

)()(

)()(

pn

Tss

K

sU

sXsP

(3)

where K and Tp are linearized functions of the system’s

parameters. The dynamic friction effect has been neglected

in this model.

B. System Identification

Since there is no available information about the system’s

model parameters, it is necessary to obtain the required

nominal model by identification. 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 as given in

(1). 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.

III. UNCERTAINTY OF THE SYSTEM

High precision motion control is first challenged by the

presence of friction. The problems caused by friction

primarily result in unacceptable tracking/positioning errors

which cannot simply be eliminated by introducing an

integral action in the controller. Unmodeled dynamic friction

and other unmodeled nonlinearities like hysteresis are

modeled in this paper as unstructured uncertainties because

they are difficult to express in highly structured

parameterized forms.

A. Modeling and Identification of Uncertainty Bounds

Additive description is used in this paper for its explicit

formulation of the uncertainty sets. "Model Error Modeling"

(MEM) methodology will be applied to prepare the required

data for intelligent identification of the uncertainty bounds

[3]. The key idea is to estimate an unbiased model of the

error between the nominal model and the true system, using

validation data. Additive description of uncertainty is used in

this paper for its explicit formulation of the uncertainty sets.

The related family of generated plants is:

)()()()(: ssWsPsP aNa

)]}()([{max)( max sPsPsW Nk

ka (4)

where PN(s) is the nominal model, σmax (.) is the maximum

singular value of (.), Wa is the weighting function that

describes the frequency dependent characteristics of the

uncertainty and defines a neighborhood about the nominal

model Pn(s) inside which the actual infinite order plant

resides [4]. and Δ(s) is the perturbation such that σmax (Δ) ≤1.

The drawback of this technique is that it leads to

conservative uncertainty sets because it is based on worst

case assumptions. Conservative uncertainty sets can result in

a sluggish performance of the designed controlled system.

However, this problem was treated in this paper by using

intelligent identification of uncertainty bounds.

B. Intelligent model uncertainty identification using

ANFIS

To construct least possible conservative uncertainty

weighting functions, ANFIS is used in this paper to estimate

model error.

2451

ANFIS (Adaptive Neuro-Fuzzy Inference System) uses a

combination of adaptive method and fuzzy method [15]. The

adaptive method works with numerical data, and the fuzzy

method uses linguistic data. Rather than choosing the

parameters associated with a given membership function

arbitrarily, these parameters could be chosen so as to tailor

the membership functions to the input/output data in order to

account for these types of variations in the data values. This

learning method works similarly to that of neural networks.

Nevertheless, the implementation of ANFIS model is less

complicated than that of sophisticated identification and

optimization procedures.

The main purpose of intelligent uncertainty identification

is to estimate the upper magnitude bound of the model error

frequency response function Pe (jω), which can be defined

as [8]

)(

)()()()(

jU

jEjPjPjP nre (5)

where Pr (jω) is the measured frequency response function

of the actual system, and Pn (jω) is the frequency response

function of the nominal linear model of the system.

Fig.2 shows the applied approach to estimate the model

error frequency response |Pe (jω)| using the experimental

input-output data and ANFIS of four rules. The trained data

contains elements of the model error frequency response

function Pe (jω), and the previous-iteration identified

uncertainty bound. The error err between these two

functions will be minimized iteratively

),_()( ),( fismatdatatrainedANFISjestafis ji (6)

,])()([)( ),(),( jijie jestafisjPierr (7)

.,..,1;,..,1 mjni

where estafis(jω) is the ANFIS trained estimated function,

trained_data is the vector of training data including err(i-1),

fismat is the Sugeno-type FIS for ANFIS training , ρ is a

pre-specified very small numerical value, e.g.< 1e-3

, n is

number of iterations and m is the number of evaluated

Pe(jω) from a data set.

The shape of the identified uncertainty region depends on

the experimental conditions under which the model and its

uncertainty region have been identified. Therefore it is

useful to determine experimental conditions for which it is a

priori guaranteed that the obtained uncertainty region is

small enough for the design of a robust controller achieving

a given level of performance.

C. Uncertainty Model Validation and Robust Stability

Measure

The intelligent identified uncertainty weighting function is

validated utilizing v-gap to ensure the stability of the

designed H∞ controlled system. It is the maximum distance

between the frequency response loci of two systems G1 and

G2. The v-gap metric provides a measure of the size of the

controllers that guarantees to stabilize the closed loop

controlled system.

The size of the set of controllers that guarantees to

stabilize both G1 and G2 is related to the v-gap δv (G1, G2), as

),(| 21,1GGbC vCG , where

))(/1),((min 1,1

jj

CG eCeGkb (8)

is a generalized stability margin of the stable loop [G1, C],

where

2

2

2

1

2121

|)(|1|)(|1

|)()(|))(),((

jj

jjjj

eGeG

eGeGeGeGk

(9)

This relation shows that the smaller the v-gap between the

nominal plant G1 and the perturbed plant G2, the larger is the

set of controllers stabilizing G1 that also stabilizing G2.

A1

A2

B1

B2

Π

Π

N

N

∑ f

x

y

x

x y

y1layer 2layer 3layer 4layer 5layer

1

2

11 f

22 f2

1

|Pe(jω)|

|E(jω)|

|U(jω)|

|Ge(jω)|

ANFIS

E(t)

U(t)

Frequency (jω)Time to

Frequency

Conversion

Time to

Frequency

Conversion

/ +

ANFIS

Fig. 2 Intelligent model error identification, using ANFIS.

IV. DESIGN OF THE CONTROL SYSTEM

H∞ control is a robust control technique that seeks to

calculate a controller such that the effects of model

uncertainty, steady state error, disturbance, and noise effects

are minimized according to performance specifications. H∞

control allows for frequency dependent bounds to be placed

on each of these signals during controller synthesis to

specify admissible levels of these undesirable effects [4].

A. H∞ robust control

In order to reflect the performance objectives into optimal

control setting, the configuration of Fig.3 is considered. The

main idea of this setup is to shape the closed loop transfer

functions S and T with weighting functions We, Wu and Wa to

achieve robust stability, disturbance rejection, and noise

attenuation, and to make the closed loop response close to

target reference response r. The closed-loop matrix transfer

function from the exogenous variables ω=[r n]T to the

regulated variables z= [z1 z2]T is given by

n

r

PII

W

PWWW

W

e

z

u

eee

a

00

00

(10)

2452

H∞ strategies are generally applied to find a sub-optimal

controller for the linear fractional transformation problem. e

is the tracking error between the reference command and the

displacement output, and K1(s) is the designed H∞ feedback

controller.

+

u

y

r

z1

z2

Wu Wa

∆

P -

We e

K1 n

Fig. 3 The entire-connection of the robustly -controlled system.

The optimization problem is stated as

1

TW

RW

SW

a

u

e

(11)

Then the unique H∞ controller can be found by solving

two Riccati equations [4]. The controller K1 stabilizes the

plant with uncertainties, such that the H∞ norm of the closed

loop transfer matrix is minimized providing robust

performance.

In order to reduce the high frequency content of the

control signal, the following weighting function Wu is

selected [4]

bc

ubc

us

MsW

1

/ (12)

where ωbc is the bandwidth of the controller, Mu is the

maximum value of the control signal, and α1 is a small

number.

The following performance weight formula is selected

first as a simple high pass filter to develop the closed loop

controlled system [4].

b

bs

es

MsW

/ (13)

where Ms is the maximum value of the sensitivity function in

all frequencies. α is a small number to approximate the

integral part of the filter with a pole near the origin, and ωb

the system bandwidth is to be selected using optimization

procedure as will be described next

B. Optimized Performance Weighting Function

The selection of the weighting function We is optimized in

this paper using a Constrained Optimization (Inequalities

and Bounded) technique; The aim is to ensure having a good

robust performance by satisfying the condition of that the

norm of the nominal plant should be less than 1 on all

frequencies. A constrained nonlinear optimization is used to

optimize the selection of Ms and ωb order to improve the

sensitivity function of the closed loop controlled system. 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.

C. Integral Robust Control Scheme

In practical application of the designed controller for a

servo system, it will be useful to add an integrator to the

closed loop controlled system in order to guarantee zero

tracking error [16]. In this work, the integrator is added to

the closed loop controlled system in a modified scheme, as

shown Fig. 4. The effective control signal will be:

]))(())((

[)( 12s

Kssy

s

KssrKsu II

(14)

where KI>0. The integral action improves the performance at

low frequencies, and the phase advance term s+KI maintains

the gained robustness and wide bandwidth from the H∞

robust controller.

e u2 y

ei

rs

K I1K

rP

+

+

--

Fig. 4 The Integral -Robust Controller scheme.

V. EXPERIMENTAL WORK AND RESULTS

A. Device Description

Experimental setup of the overall motion control scheme

for the motor-table direct drive system is shown in Fig.5.

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 1μm

resolution for measurements. In the system, position

feedback signal is the only sensing available, which is

obtained via an incremental encoder. The proposed control

algorithm is implemented as Simulink blocks in MATLAB/

Simulink/xPC. The controller is compiled and downloaded

to the card to carry out the real-time control. The sampling

interval for the real-time experiment is 1ms. The desired

control signal is generated by the designed H∞ controller

then it is sent to the servo power amplifier to regulate the

actuator’s position.

B. Intelligent Robust Control Implementation

The following procedure is applied to develop the

intelligent robust controller:

1. By examining the system nominal model and design

requirement, proper We and Wu can be evaluated using

(12) and (13).

2. The structure of ANFIS must be carefully chosen to be

applied. Followed by offline training within MEM

framework of Fig.2 (to identify the uncertainty bounds)

using experimental closed loop data and ANFIS of four

rules. And since the order of the H∞ controller is

directly related to the order of the evaluated weighting

2453

function, a second order transfer function is substituted

for the intelligent bound.

3. Test the validity of Wa by designing the H∞ controller,

and evaluating the v-gap and bG,C. If the achieved γ

value is < 1.0 and bG,C > v-gap δv, then the robust

stability is achieved and continues to step 4.

4. Optimize the performance weighting function We for

better performance specifications, taking into account

some specified constraints, as explained in Section

IV.B.

Power

Amplifier

Encoder

Interfacing

Board

PC

MATLAB

/Simulink/

XPc

Target

DC servo

motot

Fig. 5 Experimental Setup of the Single Axis Positioning System.

C. Experimental Results

To obtain the nominal model, offline identification

procedure using experimental input/output data is applied.

Different types of test signals were used to select the best

possible model parameters, as given in Table 1. The model

that achieves the minimum Final Prediction Error (FPE) will

be selected for the controller design. The resulted difference

between the two transfer functions is in the gain of the

identified transfer function. That is due mainly to the effect

of friction nonlinearity. Then the associated uncertainty is

identified as explained in Section II.C and Section II.D. Fig.

6 shows the membership function plots of both input

variables |Ge(jω)| and |FRE(jω)| to ANFIS after training.

Fig. 7 shows the evaluated second order weighting function

for the identified uncertainty bounds.

TABLE 1

Comparison between three different identified nominal models of the

positioning system

The type of input test

signal

The Identified Transfer Function of the

Nominal Model (μm/V)

PRBS

ssG

9401.29

0928.13920

Unit impulse

ssG

0698.29

0669.13120

For the purpose of illustration, some performance

requirements were considered; Settling time Ts<0.2 seconds,

Rise time Tr<0.1 seconds and steady state error ess≤1.0μm.

Table 2 gives some experimental results of applying the

designed robust controller to the positioning system. It is

clear that using the intelligent identified uncertainty

weighting function Wa in the H∞ controller synthesis, robust

stability and larger range of stable controllers is guaranteed,

as indicated by the bG,C and v-gap values, where δv is much

less than bG,C indicating guaranteed large stability region.

Then for further improving the performance of the

controlled system, the optimized performance weighting

function We is utilized to repeat the robust controller design.

The corresponding experimental result is shown in Fig. 8. It

is also, obvious that all the performance requirements are

achieved.

Fig. 6 Membership function plots of inputs of ANFIS; |Ge(jω)| and |F(jω)|.

Fig.7 ANFIS uncertainty bound and evaluated uncertainty function Wa.

TABLE 2

Validation , robust stability test of Wa

and transient characteristics of controlled system.

γ value v gap δv B b-𝛿v

0.9811 0.0027 0.2681 0.2654

Tr sec. Ts sec. Max. Control signal (V.) ess (𝜇m)

0.089 0.163 1.5410 0.0

Fig. 8 The transient response of the closed loop controlled system, using optimized performance weighting function in the controller design.

A conventional PID controller is utilized for comparison.

Although, PID controller can be easily tuned for better

performance, still the reference tracking and speed of

response can’t be achieved, as clearly shown in Fig. 9, where

the PID closed-loop controlled system is compared with that

of optimized intelligent Integral-H∞ closed-loop controlled

system. Each experiment was repeated 20 times, the average

error and standard deviation were calculated as well. The

reference used in the experiments is a step of 500 µm. The

average errors and standard deviations of 20 similar

experiments are shown in Table 3. The differences in

performance between the two controllers are very large,

proving the suitability of the proposed controller scheme for

precise positioning.

20 40 60 80 100

0

0.2

0.4

0.6

0.8

1

input1

Deg

ree

of m

embe

rshi

p

in1mf1 in1mf2

100 200 300

0

0.2

0.4

0.6

0.8

1

input2

Deg

ree

of m

embe

rshi

p

in2mf1 in2mf2

10-2

-140

-120

-100

-80

-60

-40

-20

Ma

gn

itud

e (

dB

)

Bode Diagram

Frequency (rad/sec)

ANFIS Unc.Bound

|Wa(jw)|

0 0.2 0.4 0.6 0.8 10

100

200

300

400

500

600

Time (sec.)

Pos

ition

(Mm

)

Wa ANFIS

We opt. added

0.1 0.15 0.2

420

440

460

480

500

520

Time (sec.)

Pos

ition

(Mm

)

Wa ANFIS

We opt. added

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Fig. 9 The transient response of the closed loop controlled system,

compared with a conventional PID controlled system.

TABLE 3

Position control performance of a conventional PID and the proposed Intelligent Integral H∞ controllers

Controller Average (μm) Standard deviation (μm)

PID 7.55 4.2472

proposed- H∞ 0.40 0.5026

Finally, it is necessary to investigate the robustness of the

intelligent H∞ controller to load changes. Therefore a mass of

2.5 kg was mounted on the single axis positioning table. Fig.

10 illustrates closed loop tracking results for the system

under added load. The performance of the closed loop

controlled system is slightly affected as shown in Fig. 11,

where the control signal rest at less than 0.025 Volt higher

than that when there is no load effect.

Fig. 10. The transient response of the closed loop controlled system,

Subject to load variations.

Fig. 11 The control signals of the closed loop controlled system subject to

load variations.

VI. CONCLUSION

This paper has described an intelligent robust controller

design and implementation for a DC servo positioning

system, which simultaneously achieves high resolution, high

bandwidth, and robustness to load variations. To achieve

these multiple objectives, an adaptive neuro fuzzy inference

system is trained to identify uncertainty bounds necessary to

estimate uncertainty weighting function; optimized

performance weighting function is formulated; and a special

integral configuration for further illumination of tracking

error in the practical implementation is developed.

Experimental demonstrations validate the benefits of the

proposed methodology; enhanced performance as compared

with a conventional PID is provided, better tracking is

accomplished, and faster response and robust stability are

achieved.

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0 0.2 0.4 0.6 0.8 1-200

-100

0

100

200

300

400

500

600

Time(sec.)

Po

sitio

n (

Mm

)

Reference

PID

I.Robust

0 0.2 0.4 0.6 0.8 10

50

100

150

200

250

300

350

400

450

500

Time (sec.)

Posit

ion (M

m)

Reference

With Load

No Load

0 0.2 0.4 0.6 0.8 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time (sec.)

Con

trol S

igna

l (V

.)

No load

with load

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