CONTROL OF DISCRETE LINEAR REPETITIVE PROCESSES

WITH VARIABLE PARAMETER UNCERTAINTY

B. Cichy

∗

∗

Institute of Control and Computation Engineering, University of Zielona G

´

ora

ul. Podg

´

orna 50, 65-246 Zielona G

´

ora, Poland

K. Gałkowski

∗, †

and A. Kummert

‡

†,‡

Bergische Universit

¨

at Wuppertal

Rainer-Gruenter-Strasse 21, D-42119 Wuppertal, Germany

E. Rogers

§

§

School of Electronics and Computer Science

University of Southampton, Southampton SO17 1BJ, UK

Keywords:

Controller design, variable uncertainty, robust control, LMI.

Abstract:

This paper is devoted to solving the problem of stabilising an uncertain discrete linear repetitive process, where

the model uncertainty is a result of the variable along the pass uncertainty of the parameters. The analysis is

applied to the engineering example of the material rolling process, which can be modelled as a repetitive

process (Rogers and Owens, 1992; Gałkowski et al., 2003b). Due to its analytical simplicity and due to

computational effectiveness, the LMI based approach to design a robust state controller for 2D systems has

been used here.

1 INTRODUCTION

Repetitive processes are a distinct class of 2D systems

of both system theoretic and applications interest.

The essential 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 func-

tion on, and hence contributes to, the dynamics of the

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 oscil-

lations that increase in amplitude in the pass-to-pass

direction.

The analysis of linear repetitive processes has re-

ceived considerable attention in the literature — see,

for example, (Rogers et al., 2005; Gałkowski and

Wood, 2001; Roberts, 2000; Rogers and Owens,

1992). Although these processes are well known,

many control design problems for them are still (rel-

∗

This work is partially supported by State /Poland/ Com-

mittee for Scientiﬁc Research, Grant no. 3 T11A 008 26.

†

K. Gałkowski is currently a Gerhard Mercator Guest

Professor in the University of Wuppertal

atively) open. This stems from the fact that control

of these processes using standard (or 1D) systems

theory/algorithms fails (except in a few very restric-

tive special cases) precisely because such an approach

ignores their inherent 2D systems structure, i.e. in-

formation propagation occurs from pass-to-pass and

along a given pass, and also the pass initial conditions

are reset before the start of each new pass.

Material rolling is one of a number of physically

based problems which can be modelled as a linear

repetitive process (Rogers and Owens, 1992). In this

paper, we use material rolling as a basis to illustrate

numerically algorithms the solution we develop to a

currently open robust stability and stabilization prob-

lem for the underlying sub-class of so-called discrete

linear repetitive processes. The design itself can be

executed in terms of a linear matrix inequality (LMI)

which, in turn, can be solved with well established

effective numerical algorithms (Gahinet et al., 1995;

Nesterov and Nemirovskii, 1994).

Inn physical applications, the system or process

parameters are most often not known exactly and

only some nominal values or admissible intervals are

available. Hence, although the nominal process is

most often time invariant, the uncertain process can

be time variant. This will be the case here and to

37

Cichy B., Gałkowski K., Kummert A. and Rogers E. (2005).

CONTROL OF DISCRETE LINEAR REPETITIVE PROCESSES WITH VARIABLE PARAMETER UNCERTAINTY.

In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics - Robotics and Automation, pages 37-42

DOI: 10.5220/0001161500370042

Copyright

c

SciTePress

solve the problem we generalize previously reported

LMI based design algorithms for uncertain repetitive

processes (Gałkowski et al., 2002) and other classes

of systems (Daafouz and Bernussou, 2001) with poly-

topic uncertainty. As the resulting solution requires

the presence of a convex uncertainty admissible set,

the question then arises as to what can be done in

cases where this is not true. Here we propose a so-

lution to this problem by ﬁrst obtaining the convex

hull of the original non-convex uncertainty set.

Throughout this paper, the null matrix and the iden-

tity matrix with appropriate dimensions are denoted

by 0 and I, respectively. Moreover, a matrix, say

M < which is real symmetric and positive deﬁnite

is denoted by M > 0.

2 MATERIAL ROLLING AS

A LINEAR REPETITIVE

PROCESS

Material rolling is an extremely common industrial

process where, in essence, deformation of the work-

piece takes place between two rolls with parallel axes

revolving in opposite directions.

In practice, a number of models of this process can

be developed depending on the assumptions made on

the underlying dynamics and the particular mode of

operation under consideration. Here, we restrict at-

tention to a linearised model of the dynamics. In

particular, following, for example (Gałkowski et al.,

2003b), the model considered is a so-called discrete

linear repetitive process whose state space model is

of the form

x

k+1

(p + 1) = Ax

k+1

(p) + Bu

k+1

(p) + B

0

y

k

(p) (1)

y

k+1

(p) = Cx

k+1

(p) + Du

k+1

(p) + D

0

y

k

(p)

Here on pass k, x

k

(p) ∈ R

n

is the state vector,

y

k

(p) ∈ R

m

is the pass proﬁle vector, u

k

(p) ∈ R

l

is the vector of control inputs. To detail the structure

for our material rolling example, ﬁrst introduce

u

k+1

(p)=F

M

(p)

x

k+1

(p)=

y

k+1

(p−1) y

k+1

(p−2) y

k

(p−1) y

k

(p−2)

T

A=

a

1

a

2

a

4

a

5

1 0 0 0

0 0 0 0

0 0 1 0

B=

b

0

0

0

B

0

=

a

3

0

1

0

(2)

C=

a

1

a

2

a

4

a

5

D= b D

0

= a

3

where

a

1

=

2M

λT

2

+M

a

2

=

−M

λ

2

T +M

a

3

=

λ

λT

2

+M

T

2

+

M

λ

1

a

4

=

−2λM

λ

1

(λT

2

+M)

a

5

=

λM

λ

1

(λT

2

+M)

b =

−λT

2

λ

2

(λT

2

+M)

The systems variables in above expressions are:

y

k+1

(p) and y

k

(p), which denote thickness of the ma-

terial on the current and previous pass respectively,

M is the lumped mass of the roll-gap adjusting mech-

anism, λ

1

is the stiffness of the adjustment mecha-

nism spring, λ

2

is the hardness of the material strip,

λ =

λ

1

λ

2

λ

1

+λ

2

is the composite stiffness of the material

strip and the roll mechanism. Finally, F

M

(p) is the

force developed by the motor and T is the sampling

period.

To complete the process description it is necessary

to specify the pass length and the initial, or boundary,

conditions, i.e. the pass state initial vector sequence

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

of the form

x

k+1

(0) = d

k+1

, k ≥ 0

y

0

(p) = f (p), 0 ≤ p ≤ α − 1 (3)

where d

k+1

is an n × 1 vector with constant entries

and f(p) is an m × 1 vector whose entries are known

functions of p. For ease of presentation, we make

no further explicit reference to the boundary condi-

tions except in Section 5 where a numerical example

is given.

3 STABILITY AND

STABILIZATION OF DISCRETE

LINEAR REPETITIVE

PROCESSES

The stability theory (Rogers and Owens, 1992) for

linear repetitive processes consists of two distinct

concepts but here it is the strongest of these, termed

stability along the pass, which is of interest. In

essence, this property (recall the unique control prob-

lem for these processes) demands bounded-input

bounded-output stability (deﬁned in terms of the norm

on the underlying function space) uniformly, i.e. in-

dependent of the pass length. Several sets of neces-

sary and sufﬁcient conditions for its existence but in

this paper it is the following result which is the start-

ing point. Even though this condition is sufﬁcient but

not necessary, it forms a basis for control law design,

a feature which is not present in currently available

necessary and sufﬁcient conditions.

Theorem 1 (Gałkowski et al., 2003a) A discrete lin-

ear repetitive process described by (1) is stable along

the pass if ∃ matrices W

1

> 0 and W

2

> 0 such that

the Lyapunov inequality

A

T

W A − W < 0 (4)

holds and W = diag {W

1

, W

2

} > 0, where

A =

A B

0

C D

0

(5)

ICINCO 2005 - ROBOTICS AND AUTOMATION

38

To provide a Lyaunov interpretation of this result

(which will be used extensively in the analysis to fol-

low in this paper), introduce the candidate Lyapunov

function as

V(k, p) = x

T

k+1

(p)W

1

x

k+1

(p) + y

T

k

(p)W

2

y

k

(p) (6)

where W

1

> 0, W

2

> 0, with associated increment

∆V(k, p) = x

T

k+1

(p + 1)W

1

x

k+1

(p + 1) + y

T

k+1

(p)W

2

y

k+1

(p) − x

T

k+1

(p)W

1

x

k+1

(p) − y

T

k

(p)W

2

y

k

(p). (7)

Then it is easy to show that

∆V(k, p) < 0 (8)

is equivalent to (4). For this reason, and the quadratic

structure of the Lyapunov function, stability along the

pass is also referred to as quadratic stability.

An extensively analyzed control law for the

processes considered here has the following form over

0 ≤ p ≤ α − 1, k ≥ 0

u

k

(p + 1) =

K

1

K

2

x

k

(p)

y

k−1

(p)

(9)

where K

1

and K

2

are appropriately dimensioned ma-

trices to be designed. In effect, this control law is

composed of the weighted sum of current pass state

feedback and feedforward of the previous pass pro-

ﬁle.

The LMI of (4) extends in a natural manner to the

design of (9) for stability along the pass (or quadratic

stability), but here we will use the approach based

on (Peaucelle et al., 2000) and ﬁrst adopted for repet-

itive processes in (Gałkowski et al., 2003a). This will

prove to be of particular use in the analysis of the case

when there is uncertainty in the model.

Theorem 2 (Gałkowski et al., 2003a) Suppose that

a control law of the form (9) is applied to a discrete

linear repetitive process of the form described by (1).

Then the resulting process is stable along the pass if

∃ matrices W = diag {W

1

, W

2

}, W

1

> 0, W

2

> 0,

G = diag {G

1

, G

2

}, and N = diag {N

1

, N

2

}, such

that

−G − G

T

+ W (AG + BN )

T

AG + BN −W

< 0 (10)

If this condition holds, stabilizing K

1

and K

2

in the

control law (9) are given by K = N G

−1

, where K =

diag{K

1

, K

2

} and

b

B is given by

B = diag {B, D} (11)

4 ROBUST STABILITY AND

STABILIZATION OF DISCRETE

LINEAR REPETITIVE

PROCESSES

The design of control laws for discrete linear repeti-

tive processes has been the subject of much research

effort (Gałkowski et al., 2003b; Gałkowski et al.,

2003a; Gałkowski et al., 2002) and here we continue

the development of this general area by giving new re-

sults relating to the practical case when there is uncer-

tainty associated with the process (state space model)

description. In particular, we consider the case when

the model matrices

b

A,

b

B

are not precisely known,

but belong to a convex bounded (polytope type) un-

certain domain D. This, in turn, means that any un-

certain matrix can be written as a convex combination

of the vertices of the polytope D deﬁned as follows

D =

A(ξ(k, p)), B(ξ(k, p)) : A(ξ(k, p)),

B(ξ(k, p)) =

v

i=1

ξ

i

(k , p)

A

i

, B

i

;

v

i=1

ξ

i

(k , p ) = 1;

ξ

i

(k , p) ≥ 0; k ≥ 0; 0 ≤ p ≤ α − 1

(12)

where v denotes the number of vertices. Note also

that the uncertainty here is variable in both indepen-

dent directions of information propagation, i.e. along

the pass (depends on p) and pass-to-pass (depends

on k).

Now we can write the following linear parameter

dependent system describing the process dynamics

x

k+1

(p + 1) = A

ξ(k, p) x

k+1

(p) + B ξ(k, p) u

k+1

(p)

+ B

0

ξ(k, p) y

k

(p)

y

k+1

(p) = C ξ(k, p) x

k+1

(p) + D ξ(k, p) u

k+1

(p)

+ D

0

ξ(k, p) y

k

(p) (13)

and also the parameterized candidate Lyapunov func-

tion

V(k, p, ξ(k, p)) = x

T

k+1

(p)

T

W

1

(ξ(k, p))x

k+1

(p)

+ y

T

k

(p)W

2

(ξ(k, p))y

k

(p) (14)

with

W

1

(ξ(k, p)) =

v

i=1

ξ

i

(k , p )W

1i

W

2

(ξ(k, p)) =

v

i=1

ξ

i

(k , p )W

2i

(15)

Also V(0, 0, ξ(0, 0)) < ∞ and the Lyapunov func-

tion increment is given by

∆V(k, p, ξ(k, p)) = x

T

k+1

(p + 1)W

1

ξ(k, p + 1)

x

k+1

(p+1)+y

T

k+1

(p)W

2

ξ(k+1, p) y

k+1

(p)−x

T

k+1

(p)

W

1

ξ(k, p) x

k+1

(p) − y

T

k

(p)W

2

ξ(k, p) y

k

(p) (16)

Hence we can deﬁne the so-called poly-quadratic

stability for the repetitive processes considered here

(see (Daafouz and Bernussou, 2001) for the 1D sys-

tems case).

Deﬁnition 1 A discrete linear repetitive process de-

scribed by (1) with uncertainty deﬁned by (12) is said

to be poly-quadratically stable provided

∆V

k, p, ξ(k, p) < 0 (17)

∀ k ≥ 0, 0 ≤ p ≤ α − 1

CONTROL OF DISCRETE LINEAR REPETITIVE PROCESSES WITH VARIABLE PARAMETER UNCERTAINTY

39

The requirement of (17) is easy seen to be equiva-

lent to

A ξ(k, p )

T

diag

W

1

ξ(k, p + 1) , W

2

ξ(k + 1, p)

A ξ(k, p) − diag W

1

ξ(k, p) , W

2

ξ(k, p) < 0

(18)

where

b

A of (5) now becomes

A ξ(k, p) =

A ξ(k, p ) B

0

ξ(k, p)

C ξ(k, p) D

0

ξ(k, p)

=

v

i=1

ξ

i

(k , p )

A

i

(19)

and

b

A

i

is a polytope vertex, see (12).

Remark 1 When diag{W

1

ξ(k, p + 1) , W

2

ξ(k +

1, p)

} = diag{W

1

ξ(k, p) , W

2

ξ(k, p) } = W then

poly-quadratic stability reduces to quadratic stability

as in Corollary 1.

Now we have the following result from (Cichy

et al., 2005) which (drawing on the work in (Daafouz

and Bernussou, 2001)) aims to minimize the conser-

vativeness present from the use of a sufﬁcient but not

necessary stability condition.

Theorem 3 A discrete linear repetitive process of the

form described by (1) with uncertainty deﬁned by (12)

is poly-quadratically stable if ∃ block diagonal matri-

ces

b

S

i

, i = 1, 2, . . . , v, i.e.

b

S

i

= diag {S

i1

, S

i2

}, and

matrices

b

G = diag {G

1

, G

2

} such that

G + G

T

− S

i

G

T

A

T

i

A

i

G S

j

> 0 (20)

for all i, j = 1, 2, . . . , v.

With the control law (9) applied, (20) becomes

G + G

T

− S

i

G

T

(A

i

+ B

i

K)

T

(

A

i

+ B

i

K)G S

j

> 0 (21)

where K = diag {K

1

, K

2

}. The following re-

sult now gives a sufﬁcient condition for the existence

of a poly-quadratically stabilizing control law of the

form (9) for the case under consideration.

Theorem 4 Suppose that a control law of the

form (9) is applied to a discrete linear repeti-

tive process described by (1) with uncertainty de-

ﬁned by (12). Then the resulting process is poly-

quadratically stabilizable if ∃ symmetric matrices

b

S

i

> 0, i = 1, 2, . . . , v, i.e.

b

S

i

= diag {S

i1

, S

i2

}

and

b

G = diag {G

1

, G

2

},

b

N = diag {N

1

, N

2

}, such

that the following LMI is feasible

G + G

T

− S

i

G

T

A

T

i

+ N

T

B

T

i

A

i

G + B

i

N S

j

> 0 (22)

for all i, j = 1, . . . , v. If this condition holds then

stabilizing K

1

and K

2

in the control law are given by

(9) with

K =

NG

−1

(23)

where K = diag {K

1

, K

2

} .

Proof: Follows immediately from (21) on setting

K

b

G =

b

N.

5 APPLICATION TO THE

MATERIAL ROLLING

EXAMPLE

In this section we apply our new design to the material

rolling model of Section 2 when the model parameters

T, M, λ

1

, λ

2

are uncertain. In order to avoid a control

law with very large entries in the deﬁning matrices,

we limit attention to solutions of the LMI (22) where

the matrix N is diagonal. The boundary conditions

are x

k+1

(p) = 0, k ≥ 0 and y

0

(p) = 1, 0 ≤ p ≤

14, and in the simulations given below x

1

k

(p) denotes

the ﬁrst entry of the state vector x

k

(p), the number of

passes is 26, and the number of points along the pass

is 15.

Consider ﬁrst the case when the discretization pe-

riod belongs to the interval

T ∈ [T

, T ] = [0.21, 0.25] (24)

and the rest of parameters satisfy

M ∈ [M , M ] = [90, 110], λ

1

∈ [λ

1

, λ

1

] = [430, 600]

λ

2

∈ [λ

2

, λ

2

] = [1970, 2070]. (25)

Here the uncertainty domain has 16 vertices, but is

not convex and hence the new design procedure de-

veloped in this paper cannot be applied. To overcome

this difﬁculty, we ﬁrst use the Geometric Bounding

Toolbox (GBT) to numerically estimate the minimum

convex domain (i.e. convex hull) which covers the

original non-convex one. (This, of course, introduces

extra conservativeness.)

The resulting convex domain which now be

used to execute our new design algorithm for the

example considered has the following 6 vertices

Vertex 1

A =

1.7502 −0.0041151 −1.4491 0.72457

1 0 0 0

0 0 0 0

0 0 1 0

C =

1.7502 −0.0041151 −1.4491 0.72457

B =

−6.0343e − 5

0

0

0

B

0

=

0.84948

0

1

0

D = −6.0343e − 5 D

0

= 0.84948

Vertex 2

A =

1.5159 −0.001699 −1.162 0.58098

∗

C =

1.5159 −0.001699 −1.162 0.58098

B =

−0.00012288

∗

B

0

=

0.82305

∗

D = −0.00012288 D

0

= 0.82305

ICINCO 2005 - ROBOTICS AND AUTOMATION

40

Vertex 3

A =

1.6886 −0.0024702 −1.2944 0.6472

∗

C =

1.6886 −0.0024702 −1.2944 0.6472

B =

−7.9026e − 5

∗

B

0

=

0.80288

∗

D = −7.9026e − 5 D

0

= 0.80288

Vertex 4

A =

1.6035 −0.0028319 −1.3277 0.66386

∗

C =

1.6035 −0.0028319 −1.3277 0.66386

B =

−9.5766e − 5

∗

B

0

=

0.8621

∗

D = −9.5766e − 5 D

0

= 0.8621

Vertex 5

A =

1.5117 −0.001661 −1.172 0.58599

∗

C =

1.5117 −0.001661 −1.172 0.58599

B =

−0.00011795

∗

B

0

=

0.83015

∗

D = −0.00011795 D

0

= 0.83015

Vertex 6

A =

1.5159 −0.001699 −1.162 0.58098

∗

C =

1.5159 −0.001699 −1.162 0.58098

B =

−0.00012288

∗

B

0

=

0.82305

∗

D = −0.00012288 D

0

= 0.82305

where ∗ denotes an entry equal to that of the corre-

sponding value for Vertex 1.

The parameters T, M, λ

1

, λ

2

vary on each

pass k stochastically with p within the con-

stant intervals (24) and (25) respectively and

are denoted by T (k, p), M (k, p), λ

1

(k, p), and

λ

2

(k, p) respectively. Note also that the functions

T (k, p), M (k, p), λ

1

(k, p), and λ

2

(k, p) can be

different on each pass k.

Applying Theorem 4 now gives the stabilizing con-

trol law matrices

K

1

=

18161.073 −31.212 −10010.633 7335.353

K

2

= 9425.674. (26)

Figures 1 and 2 show the pass proﬁle se-

quences generated by the uncontrolled and controlled

processes respectively (with zero input sequence).

To conclude this section, it is instructive to give

some comments on the steps necessary to apply the

new design algorithm developed in this paper. The

ﬁrst point is that we have 4 parameters which can

vary between given maximum and minimum values

0

10

20

0

5

10

0

1

2

3

4

x 10

5

pass to pass

along the pass

Figure 1: The process pass proﬁle sequence with no control

action applied (plot is x

1

k

(p)).

0

10

20

0

5

10

−0.5

0

0.5

1

pass to pass

along the pass

Figure 2: The process pass proﬁle sequence with control

action applied (plot is x

1

k

(p)).

and hence there are 16 combinations for the uncer-

tainty domain vertices, and the uncertainty domain is

required to be convex. If, as in the numerical example

above, convexity is not present, then the most obvi-

ous idea is to numerically estimate the convex hull of

the vertices where here we have used (GBT). This,

of course, introduces extra conservativeness into the

design and how to reduce this is clearly a subject for

further research.

Suppose now that a control law has been success-

fully designed and we wish to simulate the response

of the process under control action. Then we ﬁrst have

to determine ξ

i

(p) ∀i = 1, 2, . . . , v and 0 ≤ p ≤

α−1, where v denotes number of vertices. We can do

this by applying Matlab function fmincon to solve

the following problem: determine ξ

i

(k, p) ∈ R

+

,

i = 1, 2, . . . , v such that

v

i=1

ξ

i

(k , p)V

i

= P (k, p) (27)

CONTROL OF DISCRETE LINEAR REPETITIVE PROCESSES WITH VARIABLE PARAMETER UNCERTAINTY

41

where

v

i=1

ξ

i

(k , p) = 1; ξ

i

(k , p) ≥ 0;

k ≥ 0; 0 ≤ p ≤ α − 1

Here the V

i

denote the minimal convex domain ver-

tices and P (k, p) denotes a matrix within the obtained

polytope for point p on pass k, i.e the corresponding

process state space model matrix computed from the

corresponding T (k, p), M(k, p), λ

1

(k, p), λ

2

(k, p).

Repeating this procedure ∀ k ≥ 0 and ∀ p,

0 ≤ p ≤ α − 1, enables the process response with or

without control action applied

6 CONCLUSION

In this paper, we have extended previous results on

the stability and control of discrete linear repeti-

tive processes to the case when the deﬁning state

space model matrices are subject polytopic uncer-

tainty. This has led to a design algorithm which can

be implemented using well tested software. Also we

have attempted to minimize the conservativeness in-

troduced by the use of sufﬁcient only conditions for

stability.

REFERENCES

Cichy, B., Gałkowski, K., and Rogers, E. (2005). Control

of uncertain discrete linear repetitive processes based

on the physical example. Proceedings of 4th Interna-

tional Workshop On Multidimensional (ND) Systems,

Wuppertal, Germany (in press).

Daafouz, J. and Bernussou, J. (2001). Parameter depen-

dent lyapunov functions for discrete time systems with

time varying parametric uncertainties. Systems &

Control Letters, 43:355–359.

Gahinet, P., Nemirowski, A., Laub, A. J., and Chilali, M.

(1995). LMI Control Toolbox for use with MATLAB.

The Mathworks Partner Series. The MathWorks Inc.

Gałkowski, K., Lam, J., Rogers, E., Xu, S., Sulikowski, B.,

Paszke, W., and Owens, D. H. (2003a). LMI based

stability analisys and robust controller design for dis-

crete linear repetetive processes. International Jour-

nal of Robust and Nonlinear Control, 13:1195–1211.

Gałkowski, K., Rogers, E., Paszke, W., and Owens, D.

(2003b). Linear repetitive process control theory ap-

plied to a physical example. Applied Mathematics and

Computer Science, 13(1):87–99.

Gałkowski, K., Rogers, E., Xu, S., Lam, J., and Owens, D.

(2002). LMIs -a fundamental tool in analysis and con-

troller design for discrete linear repetitive processes.

IEEE Transactions on Circuits and Systems I, Funda-

mental Theory and Applications, 49(6):768–778.

Gałkowski, K. and Wood, J. (2001). Multidimensional Sig-

nals, Circuits and Systems. Taylor & Francis.

Nesterov, Y. and Nemirovskii, A. (1994). Interior-point

Polynomial Algorithms in Convex Programing, vol-

ume 13 of SIAM Studies in Applied Mathematics.

SIAM, Philadelphia.

Peaucelle, D., Arzelier, D., Bachelier, O., and Bernussou, J.

(2000). A new robust D-stability condition for poly-

topic uncertainty. Systems & Control Letters, 40:21–

30.

Roberts, P. D. (2000). Numerical investigations of a stabil-

ity theorem arising from 2-dimensional analysis of an

iterative optimal control algorithm. Multidimensional

Systems and Signal Processing, 11 (1/2):109–124.

Rogers, E., Gałkowski, K., and Owens, D. (2005). Control

Systems Theory and Applications for Linear Repeti-

tive Processes. Lecture Notes in Control and Informa-

tion Sciences. Springer-Verlag. (in print).

Rogers, E. and Owens, D. (1992). Stability Analysis for

Linear Repetitive Processes, volume 175 of Lecture

Notes in Control and Information Sciences. Springer-

Verlag.

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