# Experimentally verified 2D systems theory based robust iterative learning control

**ABSTRACT** This paper uses 2D control systems theory to develop robust iterative learning control laws for linear plants with experimental validation on a gantry robot used for `pick and place' operations commonly found in industries such as food processing. In particular, the stability theory for linear repetitive processes provides the setting for analysis and this allows design to take account of trial-to-trial error convergence, transient response along the trials and robustness. The mechanism for this is the use of a strong form of stability for repetitive processes/2D linear systems known as stability along the pass (or trial) with the added requirement for maintaining this property in the presence of model uncertainty. The resulting design computations are in terms of Linear Matrix Inequalities (LMIs) and the control laws can be implemented without the need to estimate state vector entries.

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**ABSTRACT:**An algorithm is presented for iterative learning of the control input for a linear discrete-time multivariable system. Necessary and sufficient conditions are stated for convergence of the proposed algorithm. The algorithm synthesis and analysis are based on two-dimensional (2-D) system theory. A numerical example is givenIEEE Transactions on Automatic Control 02/1993; · 2.72 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, a multirate cyclic pseudo-downsampled iterative learning control (ILC) scheme is proposed. The scheme has the ability to produce a good learning transient for trajectories with high frequency components with/without initial state errors. The proposed scheme downsamples the feedback error and input signals every m samples to arrive at slower rate. Then, the downsampled slow rate signals are applied to an ILC algorithm, whose output is then interpolated and applied to an actuator. The main feature of the proposed scheme is that, for two successive iterations, the signal is downsampled with the same m but the downsampling points are time shifted along the time axis. This shifting process makes the ILC scheme cyclic along the iteration axis with a period of m cycles. Experimental results show significant improvement in tracking accuracy. Additional advantages are that the proposed scheme does not need a filter design and also reduces the computation and memory size substantially.Control Engineering Practice 01/2009; 17(8):957-965. · 1.67 Impact Factor - SourceAvailable from: Mikael Norrlof[Show abstract] [Hide abstract]

**ABSTRACT:**Experimental results from a first order P-type ILC algorithm applied to a large size six degrees of freedom commercial industrial robot are presented. The ILC algorithm is based on measurements of the motor angles, but in addition to the conventional evaluation of the ILC algorithm based on the control error on the motor side, the tool path error on the arm side is evaluated using a laser tracker. Experiments have been carried out in three dierent operating points using movements that represent typical paths in a laser cutting application, and dierent choices of algorithm design parameters have been studied. The motor angle error is reduced substantially in all experiments, and the tool path error is reduced in most of the cases. In one operating point, however, the error does not decrease as much and an oscillatory tool behaviour is observed. Changed filter variables can give worse error reduction in all operating points. To achieve even better performance, especially in dicult operating points, it is concluded that an arm side measurement, from for example an accelerometer, needs to be included in the learning.

Page 1

Experimentally Verified 2D Systems Theory based Robust Iterative

Learning Control

Lukasz Hladowski, Krzysztof Galkowski, Zhonglun Cai, Eric Rogers, Chris T. Freeman, Paul L. Lewin

Abstract—This paper uses 2D control systems theory to

develop robust iterative learning control laws for linear plants

with experimental validation on a gantry robot used for ‘pick

and place’ operations commonly found in industries such as

food processing. In particular, the stability theory for linear

repetitive processes provides the setting for analysis and this

allows design to take account of trial-to-trial error conver-

gence, transient response along the trials and robustness. The

mechanism for this is the use of a strong form of stability for

repetitive processes/2D linear systems known as stability along

the pass (or trial) with the added requirement for maintaining

this property in the presence of model uncertainty. The resulting

design computations are in terms of Linear Matrix Inequalities

(LMIs) and the control laws can be implemented without the

need to estimate state vector entries.

I. INTRODUCTION

Iterative learning control (ILC) is a technique for con-

trolling systems operating in a repetitive (or pass-to-pass)

mode with the requirement that a reference trajectory yref(t)

defined over a finite interval 0 ≤ t ≤ α is followed to a

high precision. Examples of such systems include robotic

manipulators that are required to repeat a given task to high

precision, chemical batch processes or, more generally, the

class of tracking systems.

Since the original work [1], the general area of ILC has

been the subject of intense research effort. An initial source

for the literature here are the survey papers

The analysis of ILC schemes is firmly outside standard (or

1D) control theory, although it is still has a significant role

to play in certain cases of practical interest. Such schemes

do, however, have a natural 2D systems structure and hence

we make use of the links with repetitive processes where

information propagation in one of the two directions only

occurs over a finite duration.

In ILC, a major objective is to achieve convergence of

the trial-to-trial error and often this has been treated as

the only objective. In fact, it is possible that enforcing

fast convergence could lead to unsatisfactory along the trial

performance. Moreover, in many applications there will be

uncertainty associated with the model used for design and

hence a robust control theory that takes account of this trade-

off is required.

[2] and [3].

L.Hladowski

and

andK.Galkowski

Engineering,

L.Hladowski@issi.uz.zgora.pl,

arewith

University

theInstitute

of

of

Control

Gora,

K.Galkowski@issi.uz.zgora.pl

Z. Cai, E. Rogers, C. T. Freeman and P. L. Lewin are with the

School of Electronics and Computer Science, University of Southampton,

Southampton SO17 1BJ, UK

This work has been partially supported by the Ministry of Science and

Higher Education in Poland under the project N N514 293235.

Computation

Poland

Zielona

This paper shows how robust ILC, with due regard to

both trial-to-trial error convergence and along the trial per-

formance, can be achieved by extending robust control law

design algorithms for discrete linear repetitive processes. The

key feature here is the use of the strong concept of stability

along the pass (or trial) for these processes in an ILC setting.

The analysis results in control law design algorithms that can

be computed using Linear Matrix Inequalities (LMIs) and

the laws themselves have a well defined physical structure.

Finally, representative results from the experimental applica-

tion of the algorithms developed to a gantry robot facility

that executes a pick and place operation commonly found in

the process industries are given, where the plant modeling

starts from frequency response data.

In this paper the null and identity matrices with the

required dimensions are denoted by 0 and I respectively.

Also Γ ≻ 0 and Γ ≺ 0 are used to denote symmetric matrices

that are positive definite and negative definite, respectively.

The symbol r(·) is used to denote the spectral radius of a

given matrix. In particular, if M is a p × p matrix with

eigenvalues λi, 1 ≤ i ≤ p, then r(M) = max1≤i≤p|λi|.

Finally, ∗ is used to denote block entries in symmetric LMIs.

II. BACKGROUND

Consider the case when the plant to be controlled can be

modeled as a differential linear time-invariant system with

state-space model defined (state, input and output matrices

respectively) by {A,B,C}. Suppose also that this model is

sampled using a zero-order hold. Then in an ILC setting the

discrete dynamics are written as

xk(p + 1)

yk(p)

=

=

Axk(p) + Buk(p),

Cxk(p), p = 0,1,...,α − 1,

(1)

where on trial k, xk(p) ∈ Rnis the state vector, yk(p) ∈ Rm

is the output vector, uk(p) ∈ Rpis the vector of control

inputs, and the trial length α < ∞. If the signal to be

tracked is denoted by yref(p) then ek(p) = yref(p)−yk(p)

is the error on trial k, and the most basic requirement is to

force the error to converge in k. In particular, the objective

of constructing a sequence of input functions such that the

performance is gradually improving with each successive

trial can be refined to a convergence condition on the input

and error

lim

k→∞||ek|| = 0, lim

k→∞||uk− u∞|| = 0,

where || · || is a signal norm in a suitably chosen function

space with a norm-based topology.

2010 American Control Conference

Marriott Waterfront, Baltimore, MD, USA

June 30-July 02, 2010

ThB10.3

978-1-4244-7425-7/10/$26.00 ©2010 AACC3475

Page 2

It is, however, possible that trial-to-trial error convergence

will occur but produce along the trial performance that is

far from satisfactory for many practical applications, e.g.

a gantry robot whose task is to collect an object from a

location, place it on a moving conveyor, and then return for

the next one and so on. If, for example, the object has an

open top and is filled with liquid, and/or is fragile in nature,

then unwanted vibrations during the transfer time could have

very detrimental effects. Hence in such cases there is also a

need to control the along the trial dynamics and in this paper

the method is to use a stronger form of stability theory for

linear repetitive processes.

One approach to the analysis of ILC schemes with the

potential to address the dynamics in both directions, i.e. trial-

to-trial and along the trial respectively is to use a 2D systems

setting. For example, in [4] it was shown how trial-to-trial

error convergence of linear ILC schemes in the discrete

domain could be examined as a stability problem in terms of

a 2D discrete linear systems state-space model interpretation

of the dynamics. To-date, however, relatively little attention

has been directed towards control law design in a 2D systems

setting for both trial-to-trial convergence and along the trial

dynamics.

Given that the trial length is finite by definition, it follows

that ILC fits naturally into the class of so-called repetitive

processes [5]. The unique characteristic of such a process is

a series of sweeps, termed passes, through a set of dynamics

defined over a fixed finite duration known as the pass length.

On each pass an output, termed the pass profile, is produced

which acts as a forcing function on, and hence contributes

to, the dynamics of the next pass profile. This, in turn, leads

to the unique control problem that the output sequence of

pass profiles generated can contain oscillations that increase

in amplitude in the pass-to-pass direction.

Attempts to control these processes using standard systems

theory and algorithms fail (except in a few very restrictive

special cases) precisely because such an approach ignores

their inherent 2D systems structure, i.e. information propaga-

tion occurs from pass-to-pass (k direction) and along a given

pass (t direction) and also the initial conditions are reset be-

fore the start of each new pass. To remove these deficiencies,

a rigorous stability theory has been developed [5] based on an

abstract model of the dynamics in a Banach space setting that

includes a very large class of processes with linear dynamics

and a constant pass length as special cases, including those

described by (2) below. In terms of their dynamics, it is

the pass-to-pass coupling (noting again their unique feature)

which is critical. This is of the form yk+1= Lαyk, where

yk∈ Eα(Eαa Banach space with norm || · ||) and Lαis a

bounded linear operator mapping Eαinto itself.

Consider now discrete linear repetitive processes described

by the following state-space model over p = 0,1,...,α −

1,k ≥ 0,

xk+1(p + 1) =

yk+1(p) =

ˆAxk+1(p) +ˆBuk+1(p) +ˆB0yk(p),

ˆCxk+1(p) +ˆDuk+1(p) +ˆD0yk(p),

(2)

where on pass, or trial in an ILC setting, k, xk(p) ∈ Rn

is the state vector, yk(p) ∈ Rmis the pass profile vector

and uk(p) ∈ Rris the control input vector. To complete the

process description, it is necessary to specify the initial, or

boundary, conditions, i.e. the state initial vector on each pass

and the initial pass profile. Here these are taken to be zero. In

the next section, it is shown how a repetitive process setting

can be used to analyze ILC schemes and, in particular, how

the stability theory of these processes can be employed to

develop algorithms for control law design for trial-to-trial

error convergence and along the trial performance.

The stability theory [5] for linear repetitive processes

consists of two distinct concepts. Asymptotic stability for (1)

holds if, and only if, r(ˆD0) < 1. Also if this property holds

and the control input sequence applied {uk}k converges

strongly to u∞ as k → ∞ then the resulting output

pass profile sequence {yk}k converges strongly to y∞, the

so-called limit profile defined (with ˆD = 0 for ease of

presentation) over p = 0,1,...,α − 1 by

x∞(p + 1)=

+

(ˆA +ˆB0(I −ˆD0)−1ˆC)x∞(p)

Bu∞(p),

(I −ˆD0)−1ˆCˆ x∞(p),

ˆd∞,

ˆ y∞(p)

ˆ x∞(0)

=

=

(3)

whereˆd∞ is the strong limit of the state initial vector

(xk+1(0), k ≥ 0) sequence. In effect, this result states

that if a process is asymptotically stable then its repetitive

dynamics can, after a sufficiently large number of passes,

be replaced by those of a 1D differential linear system.

Note, however, that this property does not guarantee that

the limit profile is stable as a 1D discrete linear system, i.e.

r(ˆA+ˆB0(I−ˆD0)−1ˆC) < 1, a point which is easily illustrated

by the case whenˆA = −0.5,ˆB = 0,ˆB0= 0.5 + β,ˆC = 1,

ˆD = 0,ˆD0= 0 and β is a real scalar such that |β| ≥ 1.

The reason why asymptotic stability does not guarantee

a limit profile which is stable along the pass is due to the

finite pass length. In particular, asymptotic stability is easily

shown to be bounded-input bounded-output (BIBO) stability

with respect to the finite and fixed pass length. Also in cases

where this feature is not acceptable, the stronger concept

of stability along the pass must be used. In effect, for the

model (1), this requires that the BIBO stability property holds

uniformly with respect to the pass length α.

For the processes considered here, there are a wide range

of stability along the pass tests but in this work it is the

following LMI based condition that is the starting point for

control law design in the ILC case.

Theorem 1: [5] A discrete linear repetitive process de-

scribed by (2) is stable along the pass if there exist matrices

Y ≻ 0 and Z ≻ 0 such that the following LMI holds

Y − Z

0

ˆA1Y

∗∗

∗−Z

ˆA2Y

−Y

≺ 0,

(4)

3476

Page 3

where

ˆA1=

?

ˆA

0

ˆB0

0

?

,ˆA2=

?

0

ˆC

0

ˆD0

?

.

(5)

We also make use of the following result.

Theorem 2: [6] For any FTF ≤ I and a scalar ǫ > 0

the following holds

Σ1FΣ2+ ΣT

2FΣT

1≤ ǫ−1Σ1ΣT

1+ ǫΣT

2Σ2.

(6)

III. ILC DESIGN

Consider again (1). In many applications there will al-

ways be uncertainty associated with the plant dynamics and

hence there is a need for robust design algorithms. Then to

undertake robust ILC design in a repetitive process setting

we consider the case when this is modeled as additive

uncertainty in the state-space matrices of the model (1) of

the form

A

=

˜A + ∆A,

B

=

˜B + ∆B,

(7)

where˜A and˜B represent the nominal versions of A and

B respectively and ∆A and ∆B are uncertainties that by

assumption satisfy

∆A

∆B

=

=

H1F1E1,

H2F2E2,

where F1 = FT

F1 = Rn×n, F2 = Rn×n, H1, E1, H2, E2 are matrices

of appropriate dimensions. Here H1, H2, E1 and E2 are

assumed constant but F1 and F2 can vary both from trial-

to-trial and from point-to-point along the trial, but under the

requirement that they are norm bounded. Here we aim to

control the ILC dynamics by a linear control law which only

uses plant output information.

Let yref(p) denote the reference signal and ek(p) =

yref(p) − yk(p) the error on trial k. Then it is easy to see

that in this case

1, FT

1F1 ? I, F2 = FT

2, FT

2F2 ? I and

ek+1(p) − ek(p)=

CA(−xk+1(p − 1) + xk(p − 1))

CB(−uk+1(p − 1) + uk(p − 1)),

+

(8)

and introduce

ηk+1(p + 1)

∆uk+1(p)

=

=

xk+1(p) − xk(p),

uk+1(p) − uk(p).

(9)

Then in [7] it has been shown how to design the control law

∆uk+1(p) = K1ηk+1(p + 1) + K3ek(p + 1),

(10)

using an LMI setting to ensure stability along the trial. The

only difficulty with this analysis is that in many cases the

state vector xk(p) may not be available or, at best, only some

of its entries are and hence, in general, an observer will be

required. An alternative is to use a control law of the form

∆uk+1(p) = K1µk+1(p + 1) + K2µk+1(p) + K3ek(p + 1),

(11)

where

µk(p) = yk(p − 1) − yk−1(p − 1) = Cηk(p).

(12)

Note that in comparison with (10) the vector variable

ηk+1(p + 1) has been replaced by µk+1(p + 1). The extra

term in the control law considered here has been added as

a means, if necessary, of compensating for the effects of

replacing pure state vector information.

Now introduce

˜ ηk+1(p + 1)

ek+1(p)

=

=

? A˜ ηk+1(p) +?B0ek(p),

?C˜ ηk+1(p) +?D0ek(p),

k+1(p + 1)

ηT

(13)

where ˜ η(p + 1) =?

ηT

k+1(p)

?Tand

? A

?B0

?C

=

?

?

?

(I − CBK3),

˜ A + ∆A + (˜B + ∆B)K1C

I

(˜B + ∆B)K3

0

−CA − CBK1C

(˜B + ∆B)K2C

0

?

,

=

?

,

=

=

−CBK2C

?,

?D0

which is of the form (2) and hence the repetitive process

stability theory can be applied to this ILC control scheme. In

particular, stability along the pass (or trial in the ILC setting)

is equivalent to uniform bounded input bounded output

stability (defined in terms of the norm on the underlying

function space), i.e. independent of the trial length, and hence

we can (potentially) achieve trial-trial error convergence with

acceptable along the trial dynamics. (See also [8] where

the same control law was developed for the case when no

uncertainty is present in the process model.)

Note 1: A necessary condition for stability along the trial

is r(?D0) = r(I − CBK3) < 1 which is the necessary

This last condition is precisely that obtained by applying

2D discrete linear systems stability theory to (13) as first

proposed in [4] to ensure trial-to-trial error convergence only.

The case where this could lead to unacceptable along the trial

dynamics is illustrated by any example where the resulting

limit profile is unstable as a standard linear system, i.e.,

r(? A) < 1.

Now we have the following result.

Theorem 3: An ILC scheme described by (13) is stable

along the trial if there exist matrices Y ≻ 0, Z ≻ 0, N1, N2,

N3, P, Q and real scalars ˜ e1> 0 and ˜ e2> 0 such that the

following LMI with linear constraints holds

(14)

and sufficient condition for asymptotic stability in this case.

CY1= PC, CY2= QC,

Z − Y

0

Ω31

0

Ω51

Ω61

0ΩT

ΩT

Ω33

Ω43

Ω53

0

31

0

0

ΩT

0

ΩT

0

Ω55

0

51

ΩT

ΩT

0

0

0

Ω66

61

−Z

Ω32

0

0

Ω62

3262

ΩT

Ω44

0

0

43

53

≺ 0,

(15)

3477

Page 4

where

Y

=

Y1

0

0

˜ AY1+˜BN1C

Y1

0

00

0

Y2

0

Y3

,

Ω31

=

˜BN2C

0

0

˜BN3

0

0

0

0

,

Ω32

=

0

0

0

0

−CAY1− CBN1C

−Y1

0

0

−Y2

00

˜ e2HT

2

0˜ e2HT

0

−˜ e2I

0

−˜ e2I

0

0

0

E2N1C

˜ e1HT

1

0

00

00

−˜ e1I

0

−˜ e1I

0

00

E1Y1

0

00

E2N1C

0

E1Y1

−˜ e2I

0

−˜ e1I

0

−CBN2CY3− CBN3

,

Ω33

=

0

0

−Y3

,

Ω43

=

00

0

2

0

−˜ e2HT

0

0

−˜ e2I

2CT

,

0

0

E2N3

,

Ω44

=

0

0

Ω51

=

0

0

E2N2C

,

Ω53

=

0

−˜ e1HT

1CT

0

0

0

−˜ e2I

Ω55

=

0

0

,

Ω61

=

0

0

0

,

Ω62

=

E2N2C

0

0

0

E2N3

0

0

.

,

Ω66

=

0

0

0

−˜ e1I

(16)

If this LMI with equality constraints is feasible, control law

matrices can be calculated using

K1= N1P−1, K2= N2Q−1, K3= N3Y−1

Proof: The first step is to interpret Theorem 1 in terms

of the controlled dynamics and then use Theorem 2 in a

manner that directly follows repetitive process analysis and

the details can be found in [5].

The result of Theorem 3 provides infinitely many solutions

for the control law matrices where in the example here these

are scalars. To select these, we use the following objective

function

3

.

(17)

f(N2,N3,Y3) =

?

N2·h1−

?

N3·h2+

?

Y3·h3, (18)

where hk, k = 1,2,3, are appropriately chosen constants and

sums involved are performed for all elements of the appro-

priate matrices. Tweaking this function (e.g. by modifying

hk) allows us to obtain a set of control law matrices which

satisfy (15) and this can also be used to tune performance.

To apply the control law, note that, after simple algebraic

manipulations, we obtain

uk(p) =

uk−1(p) + K1(yk(p) − yk−1(p))

+K2(yk(p − 1) − yk−1(p − 1))

+K3(yref(p + 1) − yk−1(p + 1)).

(19)

Here the last term is phase advance on the previous trial error,

where in ILC such a term is well known in simple structure

algorithms. Such an advance appears in the discrete-time

implementation of the derivative ILC algorithm, where it was

used to extend the applicability of the original ILC algorithm

[1]. Since then, a variable advance has been considered [9]

and found to lead to accurate training (or learning) in practice

on a range of systems [10]. The second and third terms are

proportional in nature acting on the error between the current

and previous pass trials at p and p − 1 respectively. Whilst

use of current trial data has appeared in many approaches

to manipulate the plant dynamics along the trial [11] and

has been found to increase initial tracking and disturbance

rejection [12], the coupling of previous and current trial data

is a novel addition to this class of updates.

Implementation of this control law does not require a state

observer (in repetitive process terms y is the process output

and hence available for measurement by definition) but does

assume in this first work that the level of noise and other

disturbances on the measurements is negligible. Note also

that this control law design method covers both the multiple-

input multiple-output (MIMO) and single-input single-output

(SISO) cases.

IV. EXPERIMENTAL VERIFICATION

The new ILC control law design algorithms developed in

this paper have been experimentally validated using a multi-

axis gantry robot, see Fig. 1, previously used for testing

and comparing the performance of other ILC algorithms,

see, for example, [12] and this section gives (some of) the

results obtained together with supporting discussion. Each

axis of the gantry robot is modeled from frequency response

data where, since the axes are orthogonal, it is assumed

that there is minimal interaction between them. Consider

the X-axis (the one parallel to the conveyor in Fig. 1), for

which frequency response tests (using the Bode gain and

phase plots in Figure 2) result in a 7th order continuous

time transfer-function given in [12] as an adequate model of

the dynamics to use for control law design (this reference

also gives the models for the Y and Z axes). The transfer-

function is discretized with a sampling time of Ts = 0.05

seconds to develop a discrete linear state-space model. The

required reference trajectory is designed to simulate a “pick

and place” process of duration 2 seconds, and the signal

is used in all algorithm tests in order to make all results

comparable. The X-axis component of the 3D reference

trajectory is shown in Figure 3 (and for brevity we only

consider this axis from this point onwards).

We also consider uncertainty defined by the following

matrices, which move at least one eigenvalue of the matrix

A outside the unit circle and hence a necessary condition for

stability along the trial is violated.

3478

Page 5

H1=

0.0019

0.0038

0.0042

0.0016

0.0011

0.0050

0.0004

0.0003

0.0003

0.0033

0.0050

0.0010

0.0020

0.0047

0.0042

0.0047

0.0008

0.0009

0.0010

0.0049

0.0005

0.0011

0.0017

0.0048

0.0021

0.0032

0.0013

0.0024

0.0001

0.0045

0.0048

0.0001

0.0027

0.0041

0.0046

0.0016

0.0045

0.0047

0.0046

0.0006

0.0000

0.0010

0.0046

0.0005

0.0026

0.0018

0.0000

0.0018

0.0034

0.0021

0.0017

0.0017

0.0017

0.0040

0.0010

0.0006

,

H2=

0.0050

0.0004

0.0047

0.0047

0.0035

0.0036

0.0031

0.0031

0.0001

0.0001

0.0019

0.0047

0.0032

0.0039

0.015

0.0003

0.0021

0.0044

0.0043

0.0004

0.0046

0.0010

0.0032

0.0033

0.0028

0.0045

0.0039

0.0047

0.0048

0.0009

0.0010

0.0035

0.0035

0.0015

0.0002

0.0027

0.0037

0.0014

0.0020

0.0028

0.0018

0.0030

0.0037

0.092

0.032

0.042

0.055

0.028

0.094

0.082

,

E1=

0.061

0.006

0.006

0.030

0.041

0.027

0.024

0.062

0.031

0.074

0.100

0.000

0.062

0.019

0.039

0.099

0.089

0.088

0.098

0.006

0.060

0.0342

0.0794

0.0585

0.0304

0.0336

0.0946

0.0835

0.049

0.095

0.092

0.019

0.025

0.035

0.079

0.027

0.043

0.030

0.036

0.042

0.085

0.079

,E2=

0.024

0.093

0.039

0.060

0.014

0.050

,

and both F1and F2are taken as a 7×7 diagonal matrix with

each non-zero entry set at 0.99. For this data r(A) = 1.0017

and applying Theorem 3 (with h1= 0.001, h2= −1, h3=

25 in (18)) gives

K1= −121.53, K2= −13.88, K3= 57.46.

(20)

This control law was applied to the robot where hardware

constraints necessitated the use of a multi-sampling approach

(for a similar technique see, for example, [13]). The basic

idea of a multi-sampling algorithm is that we sample the

system with relatively large sampling period Ts,max and

replace a single low-frequency (with sampling frequency

fs.min=

Ts,max) controller by ˆ n low-frequency controllers

working independently on different parts of the system

frequency response. This task is accomplished by simply

dividing Ts,maxinto ˆ n equal parts, i.e. Ts,c=

controller k (k = 0,1,..., ˆ n − 1) is designed for a plant

model sampled with period Ts,maxand works also with this

period but starts from k·Ts,c,

controllers can be regarded as one super-controller operating

with sampling period Ts,c.

Figure 4 shows the progression of the input, output and

error as the trials are completed and Fig 5 the mean squared

error against trial number. These show that the objectives

of trial-to-trial error convergence and along the trial perfor-

mance can both be well controlled and further refinement

to, for example, place stronger emphasis on the mean square

error reduction is possible.

1

Ts,max

ˆ n

. Each

k = 0,1,..., ˆ n−1. These ˆ n

Fig. 1.The gantry robot with the axes marked.

10−1

100

101

102

103

−80

−60

−40

−20

Magnitude (dB)

X−axis Frequency Response

10−1

100

101

102

103

−180

−135

−90

Frequency (rad/s)

Phase (degree)

Experiment 1

Experiment 2

Fitted model

Fig. 2.Bode gain and phase plots of the X-axis frequency response.

V. CONCLUSIONS

This paper has shown how 2D systems theory, in the form

of stability analysis for linear repetitive processes can be

used to design ILC laws in the presence of uncertainty. The

resulting design has been experimentally verified for a case

where in the absence of control action the uncertainty results

in an unstable along the trial process. Moreover, the final

control law has a well defined physical structure and can

3479

Page 6

Fig. 4.Experimentally measured input (left), output (center) and error (right) progression with trial number.

020406080100120140160 180200

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Sample Number

X−axis Reference Trajectory

Fig. 3.The X-axis reference trajectory.

020 40 6080 100120140160180200

10−4

10−3

10−2

10−1

100

101

102

103

Trial Number

Mean Squared Error (mm2)

X−axis MSE

Fig. 5.

number.

Experimentally measured mean square error plotted against trial

be implemented without the need for state vector estimation.

These results are a significant advance in terms of ILC design

using a 2D systems setting. Of course, there are many other

possible ways of studying the robustness problem for ILC

and the experimental results, in particular, also provide a

basis for comparison against alternatives.

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[10] J. Wallen, M. Norrlof, and S. Gunnarsson, “Arm-side evaluation of ILC

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[11] M. Norrlof and S. Gunnarsson, “A frequency domain analysis of a

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[12] J. D. Ratcliffe, J. J. Hatonen, P. L. Lewin, E. Rogers, T. J. Harte,

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