FURTHER RESULTS ON OUTPUT FEEDBACK CONTROL OF
DISCRETE LINEAR REPETITIVE PROCESSES
B. Sulikowski, K. Galkowski
Institute of Control and Computation Engineering
University of Zielona Gora, Poland
E. Rogers
School of Electronics and Computer Science
University of Southampton, Southampton SO17 1BJ, UK
D. H. Owens
Department of Automatic Control and Systems Engineering
University of Sheffield, Sheffield S1 3JD, UK
Keywords:
repetitive dynamics, stability, stabilization, output controller design, LMI
Abstract:
Repetitive processes are a distinct class of 2D systems (i.e. information propagation in two independent
directions) of both systems theoretic and applications interest. They cannot be controlled by direct extension
of existing techniques from either standard (termed 1D here) or 2D systems theory. Here we give new results
on the relatively open problem of the design of physically based control laws using an LMI setting. These
results are for the sub-class of so-called discrete linear repetitive processes which arise in applications areas
such as iterative learning control.
1 INTRODUCTION
Repetitive processes are a distinct class of 2D sys-
tems of both system theoretic and applications inter-
est. The essential unique characteristic of such a pro-
cess 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 dy-
namics next pass profile. This, in turn, leads to the
unique control problem for these processes in that the
output sequence of pass profiles generated can contain
oscillations that increase in amplitude in the pass-to-
pass direction.
To introduce a formal definition, let α < +∞ de-
note the pass length (assumed constant). Then in a
repetitive process the pass profile y
k
(p), 0 ≤ p ≤ α,
generated on pass k acts as a forcing function on, and
hence contributes to, the dynamics of next pass profile
y
k+1
(p), 0 ≤ p ≤ α, k ≥ 0.
Physical examples of repetitive processes include
long-wall coal cutting and metal rolling operations
(see, for example, (Benton, 2000)). Also in recent
years applications have arisen where adopting a repet-
itive process setting for analysis has distinct advan-
tages over alternatives. Examples of these so-called
algorithmic applications include classes of iterative
learning control (ILC) schemes (Amann et al., 1998)
and iterative algorithms for solving nonlinear dy-
namic optimal control problems based on the maxi-
mum principle (Roberts, 2000). In the case of ILC for
the linear dynamics case, the stability theory for so-
called differential and discrete linear repetitive pro-
cesses is the essential basis for a rigorous stabil-
ity/convergence theory for such algorithms.
One unique feature of repetitive processes in com-
parison to other classes of 2D linear systems is that
it is possible to define physically meaningful control
laws for their dynamics. For example, in the ILC
application, one such family of control laws is com-
posed of state feedback control action on the current
pass combined with information ‘feedforward’ from
the previous pass (or trial in the ILC context) which,
of course, has already been generated and is therefore
available for use.
In the general case of repetitive processes it is
clearly highly desirable to have an analysis setting
where control laws can be designed for stability
and/or performance. In which context, previous work
has shown that an LMI re-formulation of the stabil-
ity conditions for discrete linear repetitive processes
263
Sulikowski B., Galkowski K., Rogers E. and H. Owens D. (2004).
FURTHER RESULTS ON OUTPUT FEEDBACK CONTROL OF DISCRETE LINEAR REPETITIVE PROCESSES.
In Proceedings of the First International Conference on Informatics in Control, Automation and Robotics, pages 264-269
Copyright
c
 SciTePress

leads naturally to design algorithms to ensure closed
loop stability along the pass under control laws of this
form — see, for example, (Gałkowski et al., 2002).
To implement a control law which uses the cur-
rent pass state vector will, in general, require an ob-
server to estimate the elements in this vector which
are not directly measurable. As an alternative, this
paper shows how to use the LMI setting to design
control laws which only require pass profile informa-
tion (which has already been generated and hence is
available control action) for implementation. Note
here that LMI based methods have also been inves-
tigated as a means of stability analysis and controller
design for 2D discrete linear systems described by the
well known Roesser (Roesser, 1975) and Fornasini
Marchesini (Fornasini and Marchesini, 1978) state
space models, see, for example, (Hinamoto, 1997; Du
and Xie, 2002). Discrete linear repetitive processes
have strong structural links with such systems class
of systems and some results can be exchanged be-
tween these classes of linear systems. The key novelty
in this paper is the use of physically motivated con-
trol schemes which are actuated only by pass profile
information (in previous work the current pass state
feedback vector was used and hence the possible need
for an observer to implement such a scheme) and also
the development of numerically feasible design algo-
rithms for them.
Throughout this paper, the null matrix and the iden-
tity matrix with the required dimensions are denoted
by 0 and I, respectively. Moreover, M > 0 (< 0)
denotes a real symmetric positive (negative) definite
matrix. We use (∗) to denote the transpose of matrix
blocks in some of the LMIs employed (which are re-
quired to be symmetric).
2 BACKGROUND
Following (Rogers and Owens, 1992), the state-space
model of a discrete linear repetitive process has the
following form over 0 ≤ p ≤ α, k ≥ 0,
x
k+1
(p + 1) =Ax
k+1
(p) + Bu
k+1
(p) + B
0
y
k
(p)
y
k+1
(p) =Cx
k+1
(p) + Du
k+1
(p) + D
0
y
k
(p)
(1)
Here on pass k, x
k
(p) ∈ R
n
is the state vector,
y
k
(p) ∈ R
m
is the pass profile vector and u
k
(p) ∈ R
l
is the vector of control inputs.
To complete the process description, it is necessary
to specify the boundary conditions, i.e. the state initial
vector on each pass and the initial pass profile. Here
no loss of generality arises from assuming x
k+1
(0) =
d
k+1
∈ R
n
k ≥ 0, and y
0
(p) = f (p) ∈ R
m
, where
d
k+1
is a vector with known constant entries and f(p)
is a vector whose entries are known functions of p
over [0, α]. (For ease of presentation, we will make no
further explicit reference to the boundary conditions
in this paper.)
The stability theory (Rogers and Owens, 1992)
for linear repetitive processes consists of two distinct
concepts but here it is the stronger of these which
is required. This is termed stability along the pass
and several equivalent sets of necessary and sufficient
conditions for processes described by (1) to have this
property are known, but here the essential starting
point is based on the so-called 2D characteristic poly-
nomial for these processes given next.
Define the delay operators z
1
, z
2
in the along the
pass (p) and pass-to-pass (k) directions respectively
as
x
k
(p) := z
1
x
k
(p + 1), x
k
(p) := z
2
x
k+1
(p) (2)
Then the 2D characteristic polynomial for processes
described by (1) is defined as
C(z
1
, z
2
) = det
·
I − z
1
A −z
1
B
0
−z
2
C I − z
2
D
0
¸
(3)
and it can be shown (Rogers and Owens, 1992) that
stability along the pass holds if, and only if,
C(z
1
, z
2
) 6= 0, ∀ |z
1
| ≤ 1, |z
2
| ≤ 1
Note that stability along the pass can also be ex-
pressed in the form
C(z
1
, z
2
) = det(I − z
1
b
A
1
− z
2
b
A
2
) 6= 0,
∀ |z
1
| ≤ 1, |z
2
| ≤ 1 (4)
where
b
A
1
=
·
A B
0
0 0
¸
,
b
A
2
=
·
0 0
C D
0
¸
(5)
In this work, we use the following LMI based suf-
ficient condition derived from (4) which, unlike all
other existing stability tests, leads immediately (see
below) to systematic methods for control law design.
The proof of this result can be found in (Gałkowski
et al., 2002).
Theorem 1 A discrete linear repetitive process de-
scribed by (1) is stable along the pass if there exist
matrices Y > 0 and Z > 0 such that the following
LMI holds


Y − Z (∗) (∗)
0 −Z (∗)
b
A
1
Y
b
A
2
Y −Y


< 0
The control law considered in previous work has
the following form over 0 ≤ p ≤ α, k ≥ 0
u
k+1
(p) = K
1
x
k+1
(p)+K
2
y
k
(p) := K
·
x
k+1
(p)
y
k
(p)
¸
(6)

where K
1
and K
2
are appropriately dimensioned ma-
trices to be designed. In effect, this control law uses
feedback of the current state vector (which is assumed
to be available for use) and ‘feedforward’ of the pre-
vious pass profile vector. Note that in repetitive pro-
cesses the term ‘feedforward’ is used to describe the
case where state or pass profile information from the
previous pass (or passes) is used as (part of) the input
to a control law applied on the current pass, i.e. to in-
formation which is propagated in the pass-to-pass (k)
direction. The basic result for the design of this con-
trol law for closed loop stability along the pass is as
follows.
Theorem 2 (Gałkowski et al., 2002) Consider a dis-
crete linear repetitive process of the form described
by (1) subject to a control law of the form (6). Then
the resulting closed loop process is stable along the
pass if there exist matrices Y > 0, Z > 0, and N
such that the following LMI holds.


Z − Y (∗) (∗)
0 −Z (∗)
b
A
1
Y +
b
B
1
N
b
A
2
Y +
b
B
2
N −Y


< 0 (7)
where
b
A
1
,
b
A
2
are given in (5) and
b
B
1
=
·
B
0
¸
,
b
B
2
=
·
0
D
¸
(8)
If (7) holds, then a stabilizing K in the control law (6)
is given by
K = NY
−1
(9)
3 OUTPUT FEEDBACK BASED
CONTROLLER DESIGN
In many cases the state vector x
k+1
(p) may not be
available or, at best, only some of its entries are.
Hence, we now consider the use of output based feed-
back based control laws to achieve closed loop stabil-
ity along the pass. The first law considered has the
following form over
0 ≤ p ≤ α, k ≥ 0.
u
k+1
(p) =
e
K
1
y
k+1
(p) +
e
K
2
y
k
(p) (10)
This control law is, in general, weaker than that
of (6) and examples are easily given where stability
along the pass can be achieved using (6) but not (10).
To consider the effect of a controller of the
form (10) on the process dynamics, first substitute the
pass profile (second) equation of (1) into (10) to ob-
tain (assuming the required matrix inverse exists)
u
k+1
(p) =(I
l
−
e
K
1
D)
−1
e
K
1
Cx
k+1
(p)
+(I
l
−
e
K
1
D)
−1
[
e
K
2
+
e
K
1
D
0
]y
k
(p)
(11)
and hence (11) can be treated as a particular case
of (6) with
K
1
=(I
l
−
e
K
1
D)
−1
e
K
1
C
K
2
=(I
l
−
e
K
1
D)
−1
(
e
K
2
+
e
K
1
D
0
)
(12)
This route may, however, encounter serious numerical
difficulties (arising from the fact that they are a set of
matrix nonlinear algebraic equations) and hence we
proceed by rewriting these last equations to obtain
(I
l
−
e
K
1
D)K
1
=
e
K
1
C
(I
l
−
e
K
1
D)K
2
=
e
K
2
+
e
K
1
D
0
(13)
and assume that
K
1
= L
1
C (14)
Then it follows immediately that
e
K
1
= L
1
(I + DL
1
)
−1
e
K
2
= [I − L
1
(I + DL
1
)
−1
D]K
2
−
−L
1
(I + DL
1
)
−1
D
0
(15)
for any L
1
such that I + DL
1
is nonsingular, and we
have the following result.
Theorem 3 Suppose that a discrete linear repetitive
process of the form described by (1) is subject to a
control law of the form (10) and that (14) holds. Then
the resulting closed loop process is stable along the
pass if there exist matrices Y > 0, Z > 0, X > 0
and N such that the following LMI holds


Z − Y (∗) (∗)
0 −Z (∗)
b
A
1
Y +
b
B
1
N
e
C
b
A
2
Y +
b
B
2
N
e
C −Y


< 0
X
e
C =
e
CY (16)
where
b
B
1
,
b
B
2
,
b
A
1
,
b
A
2
, N are defined as in Theo-
rem 2, and
e
C =
·
C 0
0 I
¸
(17)
Also if this condition holds, the controller matrices
e
K
1
and
e
K
2
can be obtained from (15), where
[L
1
K
2
] = N X
−1
(18)
and it is required that I + DL
1
is nonsingular.
Proof: From (18) we have that N = LX, L :=
[
L
1
K
2
] , and substitution into the LMI of (16)
now gives


Z − Y (∗) (∗)
0 −Z (∗)
b
A
1
Y +
b
B
1
LX
e
C
b
A
2
Y +
b
B
2
LX
e
C −Y


< 0
X
e
C =
e
CY

or with the equality constraint applied


Z − Y (∗) (∗)
0 −Z (∗)
b
A
1
Y +
b
B
1
L
e
CY
b
A
2
Y +
b
B
2
L
e
CY −Y


< 0
Finally, set L
e
C = K to obtain the LMI stabilisation
condition (i.e Theorem 1 applied to the closed loop
process)


Z − Y (∗) (∗)
0 −Z (∗)
(
b
A
1
+
b
B
1
K)Y (
b
A
2
+
b
B
2
K)Y −Y


< 0
and the proof is complete.
The design developed above is easily implemented
using LMI toolboxes, such as Scilab or Matlab, but
has one real disadvantage in that it is based on a
sufficient but not necessary stability condition. This
means that there could well be a high degree of con-
servativeness in the sense that in many cases it will
fail to produce a stabilising controller when one actu-
ally exists. To avoid, or lower, the level of conserva-
tiveness present, we next develop an extension of the
control law considered in this section based on the
additional use of the delayed current pass profile and
pass-to-pass profile information.
4 EXTENDED OUTPUT
FEEDBACK BASED
CONTROLLER DESIGN
The control law considered in this section has the fol-
lowing form and is, in effect, (10) augmented at point
p by additive contributions from points p − 1 on the
current pass and p on the previous pass respectively
u
k+1
(p) =
e
K
1
y
k+1
(p) +
e
K
2
y
k
(p)
+
e
K
3
y
k
(p − 1) +
e
K
4
y
k−1
(p)
(19)
Substituting the second equation of (1) into the
control law (19) now yields
u
k+1
(p) =(I −
e
K
1
D)
−1
³
e
K
1
Cx
k+1
(p)
+ [
e
K
2
+
e
K
1
D
0
]y
k
(p)
+
e
K
3
y
k
(p − 1) +
e
K
4
y
k−1
(p)
´
(20)
which is the particular case of the so-called extended,
mixed state, pass profile controller
u
k+1
(p) =K
1
x
k+1
(p) + K
2
y
k
(p)
+K
3
y
k
(p − 1) + K
4
y
k−1
(p)
(21)
This last control law is, in effect, an extension of
that of the previous section but here it is used as an
intermediate step in the computation of the matrices
e
K
i
, i = 1, . . . , 4, through use of the following result.
Theorem 4 Suppose that a discrete linear repeti-
tive process of the form described by (1) is sub-
ject to a control law of the form (21) and that (14)
holds. Then the resulting closed loop process is sta-
ble along the pass if there exist matrices Y > 0,
X = diag(X
1
, X
2
, X
3
, X
4
) > 0, Z > 0 and N such
that


Y − Z (∗) (∗)
0 −Z (∗)
b
A
1
Y +
b
B
1
N
b
C
b
A
2
Y +
b
B
2
N
b
C −Y


< 0 (22)
X
b
C =
b
CY
where
b
A
1
=



A −I 0 B
0
0 0 0 0
0 0 0 0
0 0 0 0



,
b
A
2
=



0 0 0 0
0 0 0 0
0 0 0 0
C 0 −I D
0



,
b
B
1
=



B 0 0 B
0 B 0 0
0 D 0 0
0 0 0 0



,
b
B
2
=



0 0 0 0
0 0 B 0
0 0 D 0
D 0 0 D



,
N =



N
1
0 0 0
0 0 0 −N
3
0 0 0 −N
4
0 0 0 N
2



,
b
C =



C 0 0 0
0 I 0 0
0 0 I 0
0 0 0 I



(23)
with



L
1
0 0 0
0 0 0 K
3
0 0 0 K
4
0 0 0 K
2



= NX
−1
(24)
Also if (22) holds, the controller matrices
e
K
1
and
e
K
2
can be computed using (15) and then
e
K
3
=[I − L
1
(I + DL
1
)
−1
D]K
3
e
K
4
=[I − L
1
(I + DL
1
)
−1
D]K
4
(25)
where it is assumed that I + DL
1
is nonsingular.
Proof. Substitute (21) into (1) and using (14) we
obtain the closed loop state space model
x
k+1
(p + 1) =(A + BL
1
C)x
k+1
(p)
+(B
0
+ BK
2
)y
k
(p) + BK
3
y
k
(p − 1)
+BK
4
y
k−1
(p),
y
k+1
(p) =(C + DL
1
C)x
k+1
(p)
+(D
0
+ DK
2
)y
k
(p) + DK
3
y
k
(p − 1)
+DK
4
y
k−1
(p)
(26)
This last description is not in the form to which
Theorem 1 can be applied but it is possible to obtain
an equivalent state space model for which this is the
case. Here the route is by using the delay operators

of (2) and the 2D characteristic polynomial. To be-
gin, rewrite (26) by introducing the substitutions
l := k + 1 v
l
(p) := y
k+1
(p) (27)
and then apply (2) to obtain
x
l
(p) =z
1
(A + BL
1
C)x
l
(p)
+z
1
(B
0
+ BK
2
)v
l
(p) + z
2
2
BK
3
v
l
(p)
+z
1
z
2
BK
4
v
l
(p)
v
l
(p) =z
2
(C + DL
1
C)x
l
(p)
+z
2
(D
0
+ DK
2
)v
l
(p) + z
1
z
2
DK
4
v
l
(p)+
+z
2
2
DK
4
v
l
(p)
(28)
and introduce
C
c
(z
1
, z
2
) :=
det
·
I − z
1
e
A −z
1
e
B
0
− z
2
1
F
1
− z
1
z
2
F
3
−z
2
e
C I − z
2
e
D
0
− z
1
z
2
F
2
− z
2
2
F
4
¸
Application of appropriate elementary operations
(which leave the determinant invariant) to the right-
hand side of this last expression now yields
det




I − z
1
e
A z
1
I 0 −z
1
e
B
0
0 I 0 z
1
F
1
+ z
2
F
3
0 0 I z
1
F
2
+ z
2
F
4
−z
2
e
C 0 z
2
I I − z
2
e
D
0




(29)
where
e
A = A + BL
1
C,
e
B
0
= B
0
+ BK
2
e
C = C + DL
1
C,
e
D
0
= D
0
+ DK
2
F
1
= BK
3
, F
2
= DK
3
F
3
= BK
4
, F
4
= DK
4
At this stage, the closed loop state space model has
a 2D characteristic polynomial which is of the form
required for use in (4) (and therefore Theorem 1 can
be directly applied) with
e
A
1
=



e
A −I 0
e
B
0
0 0 0 −F
1
0 0 0 −F
2
0 0 0 0



e
A
2
=



0 0 0 0
0 0 0 −F
3
0 0 0 −F
4
e
C 0 −I
e
D
0



(30)
Application of Theorem 1 together with some ob-
vious algebraic operations now yield directly the LMI
of (22) as a sufficient condition for closed loop sta-
bility along the pass. Finally, by comparing (21)
and (20) we have that
e
K
1
and
e
K
2
can be computed us-
ing (15) and
e
K
3
and
e
K
4
using (25), provided I +DL
1
nonsingular, and the proof is complete.
As a numerical example, consider the following
process which can be shown to be unstable along the
pass
A =


−1.36 −1.29 −0.8
0.15 0.34 0
−0.19 0 −1.36


, B
0
=


0.44 0.51
0.93 0.14
0.65 0


B =


0.18 −2.35 0.8
1.07 −2.5 0.5
−0.43 0.8 2.82


, C =
·
−0.38 0 −0.37
0 0 −0.98
¸
D =
·
−2.85 −0.65 −2.5
−0.28 −2.98 1.96
¸
, D
0
=
·
−1.15 0
−0.42 1.13
¸
Then here Theorem 2 (explicit contribution from the
current pass state vector) is successful but Theorem 3
is not. Theorem 4 is, however, successful with the
following solution matrices Y > 0, Z > 0, X =
diag(X
1
, X
2
, X
3
, X
4
)
X
1
=
·
101.1824 208.8786
208.8786 492.3833
¸
X
2
=


2.1312 0 0
0 1.6720 −2.7704
0 −2.7704 17.2027


X
3
=
·
133.4038 173.4733
173.4733 238.0715
¸
X
4
=
·
2619.8248 3243.2668
3243.2668 4033.1097
¸
and N is of the structure defined in (23) where
N
1
=


280.3089 516.6972
84.8 155.6090
−159.7937 −310.9259


N
2
=


−281.4519 −339.9495
898.5330 1115.4205
−678.3573 −847.8786


N
3
= 10
−11
×


−0.2198 −0.2865
0.0085 0.0110
0.1886 0.2458


N
4
= 10
−11
×


0.0478 0.0625
0.0172 0.0225
−0.0156 −0.0204


Hence
L
1
=


4.8613 −1.0129
1.4944 −0.3179
−2.2186 0.3097


and K is of the structure (24) where

K
1
=


−1.8473 0 −0.8061
−0.5679 0 −0.2414
0.8431 0 0.5174


K
2
=


−0.6894 0.4701
0.1328 0.1698
0.2964 −0.4486


K
3
= 10
−14
×


−0.9040 0.7980
0.0351 −0.0309
0.7767 −0.6855


K
4
= 10
−14
×


0.2108 −0.1850
0.0757 −0.0665
−0.0686 0.0602


Finally, the output controller matrices for closed loop
stability along the pass under the application of (19)
computed using (15) and (25) are
e
K
1
=


49.4899 −40.7956
14.2722 −11.7740
−44.4902 36.4625


e
K
2
=


−1.7708 0.9557
−0.1795 0.3063
1.2597 −0.9670


e
K
3
= 10
−12
×


0.3698 −0.3263
0.1098 −0.0968
−0.3293 0.2905


e
K
4
= 10
−13
×


0.6785 −0.5950
0.1968 −0.1726
−0.6067 0.5321


Remark It is of interest to note that in the exam-
ple here the elements of
e
K
3
and
e
K
4
are significantly
smaller in magnitude than those in the other controller
matrices. Also if these matrices are deleted from the
control then it can be verified that the closed loop pro-
cess is still stable along the pass but the design method
of Theorem 3 fails. This feature also appears in nu-
merous other numerical examples computed to date.
Hence it can be conjured that this last design method
can be exploited to reduce the degree of conservative-
ness due to the fact that the underlying LMI control
law design is based on a sufficient but not necessary
condition for stability along the pass.
5 CONCLUSIONS
One unique feature of repetitive processes in compari-
son to other classes of 2D systems is that it is possible
define physically meaningful control laws for them.
It is hence essential to have an analysis setting where
such control laws can be designed for stability and/or
performance.
Previous work has shown that, of the currently
available tools, it is only an LMI based setting can
meet this last specification. In this paper we have con-
tinued the development of control laws based on this
analysis setting which critically remove the need to
use current pass state feedback information.
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