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.
REFERENCES
[1] S. Arimoto, S. Kawamura, and F. Miyazaki, “Bettering operations of
robots by learning,” Journal of Robotic Systems, vol. 1, no. 2, pp.
123–140, 1984.
[2] D. A. Bristow, M. Tharayil, and A. G. Alleyne, “A survey of iterative
learning control,” IEEE Control Systems Magazine, vol. 26, no. 3, pp.
96–114, 2006.
[3] H.-S. Ahn, Y. Chen, and K. L. Moore, “Iterative learning control: brief
survey and categorization,” IEEE Transactions on Systems, Man and
Cybernetics, Part C, vol. 37, no. 6, pp. 1109–1121, 2007.
[4] J. E. Kurek and M. B. Zaremba, “Iterative learning control synthe-
sis based on 2-D system theory,” IEEE Transactions on Automatic
Control, vol. 38, no. 1, pp. 121–125, 1993.
[5] E. Rogers, K. Galkowski, and D. H. Owens, Control Systems Theory
and Applications for Linear Repetitive Processes, ser. Lecture Notes
in Control and Information Sciences.
349.
[6] P. P. Khargonekar, I. R. Petersen, and K. Zhou, “Robust stabilization
of uncertain linear systems: quadratic stabilizability and H∞ control
theory,” IEEE Transactions on Automatic Control, vol. 35, no. 3, pp.
356–361, 1990.
[7] L. Hladowski, K. Galkowski, Z. Cai, E. Rogers, C. T. Freeman, and
P. L. Lewin, “A 2D systems approach to iterative learning control with
experimental validation,” in Proc. of the 17th IFAC World Congress,
Soeul, Korea, July 2008, pp. 2832–2837.
[8] L. Hladowski, Z. Cai, K. Galkowski, E. Rogers, C. Freeman, and
P. L. Lewin, “Using 2D systems theory to design output signal
based iterative learning control laws with experimental verification,”
in Proceedings of the 47th IEEE Conference on Decision and Control,
December 2008, pp. 3026–3031.
[9] Y. Wang and R. W. Longman, “Use of non-causal digital signal
processing in learning and repetitive control,” Advances in the As-
tronautical Sciences, vol. 90, no. 1, pp. 649–668, 1996.
[10] J. Wallen, M. Norrlof, and S. Gunnarsson, “Arm-side evaluation of ILC
applied to a six-degrees-of-freedom industrial robot,” in Proceedings of
17th IFAC World Congress 2008, Seoul, Korea, July 2008, pp. 13450–
13455.
[11] M. Norrlof and S. Gunnarsson, “A frequency domain analysis of a
second order iterative learning control algorithm,” in Proceedings of
the 38th IEEE Conference on Decision and Control, Phoenix, Arizona,
USA, December 1999, pp. 1587–1592.
[12] J. D. Ratcliffe, J. J. Hatonen, P. L. Lewin, E. Rogers, T. J. Harte,
and D. H. Owens, “P-type iterative learning control for systems that
contain resonance,” International Journal of Adaptive Control and
Signal Processing, vol. 19, no. 10, pp. 769–796, 2005.
[13] B. Zhang, D. Wang, K. K. Zhou, Y. Ye, and Y. Wang, “Cyclic pseudo-
downsampled iterative learning control for high performance tracking,”
Control Engineering Practice, vol. 17, no. 8, pp. 957–965, 2008.
Springer Verlag, 2007, vol.
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