Output Feedback/feedforward Control of Discrete Linear RepetitiveProcesses for Performance and Disturbance Rejection

Bartlomiej Sulikowski, Krzysztof Galkowski, Eric Rogers, David H. Owens, Anton Kummert

Abstract— Repetitive processes propagate information in twoindependent directions and are of both systems theoretic andapplications interest. They cannot be controlled by directextension of existing techniques from either standard (termed1D here) or 2D systems theory. Here we give new results on thedesign of physically based feedforward/feedback control lawsto achieve desired performance and disturbance decoupling inthe sense defined in the body of the paper. The new results arefor the case when only output (as opposed to state) informationis used to activate the control law.

I. INTRODUCTION

The essential unique characteristic of a repetitive process is

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

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

To introduce a formal definition, let α < +∞ denote the pass

length (assumed constant). Then in a repetitive process the

pass profile yk(p), 0 ≤ p ≤ α−1, generated on pass k acts as

a forcing function on, and hence contributes to, the dynamics

of the next pass profile yk+1(p), 0 ≤ p ≤ α − 1, k ≥0. 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 (i.e. in the k-direction in the

notation for variables used here).

Physical examples of repetitive processes include long-

wall coal cutting and metal rolling operations (see, for

example, [1]). Also in recent years applications have arisen

where adopting a repetitive process setting for analysis has

distinct advantages over alternatives. For example, they can

be used as an analysis tool iterative learning control (ILC)

schemes, as fully explained in [2], and in the development

of iterative algorithms for solving nonlinear dynamic optimal

control problems based on the maximum principle [3]. More

recently another application has arisen in the context of

self-servo writing in disk drives [4]). Also, there are links

between repetitive processes and the models which arise in

B. Sulikowski and K. Galkowski are with the Institute of Controland Computation Engineering, University of Zielona Gora, Zielona Gora,Poland, {b.sulikowski,k.galkowski}@issi.uz.zgora.pl

E. Rogers is with the School of Electronics and Computer Science,University of Southampton, Southampton, SO17 1BJ, UK.etar@ecs.soton.ac.uk

D.H. Owens is with the Department of Automatic Control and SystemsEngineering, University of Sheffield, Sheffield S1 3JD, UK.

A. Kummert is with Faculty of Electrical, Information and MediaEngineering University of Wuppertal, Rainer-Gruenter-Strasse 21,D-42119 Wuppertal, Germany

This work is partially supported by State /Poland/ Committee for Scien-tific Research, Grant no. 3 T11A 008 26.

K. Galkowski was a Gerhard Mercator Guest Professor (DFG Germany)at The University of Wuppertal from September, 2004 to August, 2005.

the distributed control of so-called spatially interconnected

systems, see e.g. [5] (see the discussion of this point in

Section II).

Attempts to control these processes using standard (or

1D) systems theory/algorithms fail (except in a few very

restrictive special cases) precisely because such an approach

ignores their inherent 2D systems structure, i.e. information

propagation occurs from pass-to-pass and along a given pass

[6]. Moreover, the finite pass length is a key difference

in comparison with other classes of 2D linear systems,

such as those with discrete dynamics described by the well

known and extensively studied Roesser [7] and Fornasini

Marchesini [8] state-space models. Also the initial conditions

are reset before the start of each new pass and can be an

explicit function of points along the previous pass profile

and this alone can cause instability. This latter feature has

no 2D discrete linear systems counterpart and, in general,

many results for the well known and extensively studied

Roesser [7] and Fornasini Marchesini [8] state-space models

of such systems are either not applicable at all or only after

substantial modifications. In particular, the stability of linear

constant pass length repetitive processes (and hence control

law design) can only be based on the abstract model stability

theory of [9].

The structure of linear repetitive processes means that

there is a natural way to write down control laws for them

which can be based on current pass state or output (pass

profile) feedback control and feedforward control from the

previous pass profile. For example, the previous pass profile

(which has already been generated and is hence available for

use) can be combined with state or pass profile information

on the current pass.

The requirement to provide control laws for repetitive

processes which achieve stability and/or performance has

recently been the subject of an increasing level of research

see e.g. [10], [11]. One aspect of this has seen the emergence

of LMI based methods, e.g. [6], as the only currently

available method which allows control laws to be designed

for stability and/or performance as opposed to just obtaining

conditions for stability under control action.

In this paper, we continue the development of control

laws for discrete repetitive processes and, in particular, those

which are only activated by output, i.e. pass profile, as

opposed to the inclusion of state information. The new

results relate to the design of such control laws for stability,

performance and disturbance rejection (as made precise in

the body of the paper). We begin in the next section with a

summary of the relevant background.

Proceedings of the 2006 American Control ConferenceMinneapolis, Minnesota, USA, June 14-16, 2006

FrB06.5

1-4244-0210-7/06/$20.00 ©2006 IEEE 5103

Throughout this paper, the null matrix and the identity

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 all required to be symmetric).

II. BACKGROUND

The state-space model of the discrete linear repetitive

processes considered in this paper has the following form

over 0 ≤ p ≤ α − 1, k ≥ 0

xk+1(p + 1) = Axk+1(p) + Buk+1(p)+ B0yk(p) + Ew(p)

yk+1(p) = Cxk+1(p) + Duk+1(p)+ D0yk(p) + Fw(p) (1)

Here on pass k, xk(p) is the n × 1 state vector, yk(p) is

the m × 1 pass profile vector, uk(p) is the q × 1 vector

of control inputs, and w(p) is a f × 1 disturbance vector

which (see also the conclusions section where we discuss

this assumption again) is the same along each pass.

This state-space model is a natural extension of that first

introduced in [9] to allow for disturbances which influence

both the state and pass profile dynamics on each pass. 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 of the simplest possible form, i.e. xk+1(0) = dk+1, k ≥0, y0(p) = f(p), where the n × 1 vector dk+1 has known

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

are known functions of p over 0 ≤ p ≤ α − 1. For ease

of presentation, we will make no further reference to the

boundary conditions in this paper (with the exception of the

example Section V).

Suppose now that xk(p) �→ x(k, p), uk(p) �→ u(k, p) etc,

r(k, p) := y(k − 1, p). Then, with no disturbance terms for

simplicity, (1) becomes

r(k + 1, p) = D0r(k, p) + Cx(k, p) + Du(k, p)x(k, p + 1) = B0r(k, p) + Ax(k, p) + Bu(k, p)

i.e. a special case of the formulation introduced in [5] (and

subsequent work) for the analysis and control of spatially

distributed systems. In fact, this is a Roesser model inter-

pretation of the process dynamics whose role in the analysis

of the repetitive processes considered here has already been

extensively investigated. Moreover, such a model removes

the key unique feature of these processes, i.e. repeated

sweeps through a set of dynamics defined over the finite pass

length with resetting of part of the boundary conditions (the

pass state initial vector) before the start of each new pass.

Also (as noted in this paper) the boundary conditions can

be an explicit function of each pass and this alone can cause

instability (for a detailed treatment of this area see [12] which

considers so-called differential linear repetitive processes and

differ from the processes considered here only in the fact

that the dynamics along the pass evolve as the solution of a

linear matrix differential equation). Clearly, however, proper

exploitation of the links (at the level of the model struc-

tures and the analysis tools/control law design algorithms)

between these two areas could be highly beneficial (in both

directions).

The stability theory [9] for linear constant pass length

repetitive processes is based on the following abstract model

of the underlying dynamics where Eα is a suitably chosen

Banach space with norm ||.|| and Wα is a linear subspace

of Eα

yk+1 = Lα yk + bk+1, k ≥ 0 (2)

In this model yk ∈ Eα is the pass profile on pass k, bk+1 ∈Wα, and Lα is a bounded linear operator mapping Eα into

itself. The term Lαyk represents the contribution from pass

k to pass k+1 and bk+1 represents known initial conditions,

disturbances and control input effects.

The linear repetitive process (2) is said to be asymptoti-

cally stable if ∃ a real scalar δ > 0 such that, given any initial

profile y0 and any disturbance sequence {bk}k≥1 ∈ Wα

bounded in norm (i.e. ||bk|| ≤ c1 for some constant c1 ≥ 0and ∀k ≥ 1), the output sequence generated by the perturbed

process

yk+1 = (Lα + γ)yk + bk+1, k ≥ 0 (3)

is bounded in norm whenever ||γ|| ≤ δ. This definition is

easily shown to be equivalent to the requirement that ∃ finite

real scalars Mα > 0 and λα ∈ (0, 1) such that

||Lkα|| ≤ Mαλk

α, k ≥ 0 (4)

(where ||.|| is also used to denote the induced operator norm).

Necessary and sufficient conditions for this last condition are

that

r(Lα) < 1 (5)

where r(·) denotes the spectral radius of its argument.

In the case of processes described by (1), stability can

be determined in the absence of the disturbance terms and

it can be shown that asymptotic stability holds if, and only

if, r(D0) < 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 D = 0, E = 0 and F = 0 for ease of

presentation) over 0 ≤ p ≤ α − 1 by

x∞(p + 1) = (A + B0(I − D0)−1C)x∞(p) (6)

+Bu∞(p)y∞(p) = (I − D0)−1Cx∞(p), (7)

x∞(0) = d∞

where d∞ is the strong limit of the sequence {dk}k.In effect, this result states that if a process is asymp-

totically 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(A+B0(I−D0)−1C) < 1

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— a point which is easily illustrated by the case when

A = −0.5, B = 0, B0 = 0.5 + β, C = 1, D = 0, D0

and β > 0 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 abstract model (2), this requires that (4) holds uniformly

with respect to the pass length α. One of several equivalent

statements of this is the requirement that ∃ finite real scalars

M∞ > 0 and λ∞ ∈ (0, 1) independent of α and satisfy

||Lkα|| ≤ M∞λk

∞, ∀ α > 0, ∀ k ≥ 0 (8)

For the processes considered here, there are a wide range

of stability along the pass tests which could be employed.

Here, however, we use the following LMI based condition

since, see also below, it leads immediately to algorithms

for control law design — a feature which is not present in

alternatives. (Note also that r(D0) < 1 and r(A) < 1 are

only necessary conditions (see [9]) for stability along the

pass of processes described by (1).)

Theorem 1: [13] 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⎡⎣ Z − Y (∗) (∗)0 −Z (∗)

A1Y A2Y −Y

⎤⎦ < 0

where

A1 =[

A B0

0 0

], A2 =

[0 0C D0

](9)

III. CONTROL LAW DESIGN FOR STABILITY ALONG THE

PASS

In general, the essential first requirement of any control

scheme or law for repetitive processes will be stability along

the pass. There are, however, applications where asymptotic

stability is all that can be achieved or required, e.g. the

optimal control application [3] where the matrix A in the

state space model arising can never satisfy r(A) < 1. Also

it is routine to conclude that, despite their computational sim-

plicity (in relative terms) control laws which only explicitly

use information from the current pass can only guarantee

stability along the pass in a very few (and restrictive) special

cases.

Recognizing that a control law for these processes must

be activated by a combination of current and previous pass

information, we consider one of the following form over

0 ≤ p ≤ α − 1, k ≥ 0

uk+1(p) = K1yk+1(p) + K2yk(p) (10)

where K1 and K2 are appropriately dimensioned matrices

to be designed. Note that, by definition, the pass profile is

Fig. 1. Block diagram of the controlled process

the process output and is hence always available for control

purposes, unlike the case where yk+1(p) is replaced by the

current pass state vector when it would be necessary to

assume that all elements of this vector are available for

measurement or can be estimated using an observer structure.

As such, this control law assumes that the previous pass

profile has been stored and that there is no detrimental

corruption by noise etc.

A block diagram schematic of the process dynamics with

this control law applied is shown in Figure 1, where z−11 and

z−12 denote backward shift operators in the along the pass

and pass-to-pass directions respectively.

Previous work [14] has shown how to design a control

law of the form (10) to ensure that the controlled process is

stable along the pass by re-formulating it in the form

uk+1(p) = K1xk+1(p) + K2yk(p) (11)

i.e. where the current pass state vector replaces the current

pass profile in the control law actuation. The mechanism here

is c to write

K1 = L1(I + DL1)−1 (12)

K2 = [I − L1(I + DL1)−1D]K2

−L1(I + DL1)−1D0

for any L1 such that I + DL1 is nonsingular, where

K1 = L1C (13)

(This has a parallel in 1D linear systems control problems

where pole placement output feedback control can be written

as a state feedback control problem where the choice of

the corresponding matrix is partially constrained by the

uncontrolled system state space model.)

Now we have the following result.

Theorem 2: [14] Suppose that a control law of the

form (10) is applied to a discrete linear repetitive process

described by (1) and that (13) holds. Then the resulting

process is stable along the pass if there exist matrices Y >0, Z > 0, X > 0 and N such that the following LMI holds

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⎡⎣ Z − Y (∗) (∗)0 −Z (∗)

A1Y + B1NC A2Y + B2NC −Y

⎤⎦<0 (14)

XC = CY

where

B1 =[

B0

], B2 =

[0D

]and C = diag(C, I). Also if this condition holds, the control

law matrices K1 and K2 can be obtained from (12), where

[L1 K2] = NX−1 (15)

and it is required that I + DL1 is nonsingular.

See also [14] for a detailed treatment of the numerical

implementation of this result.

IV. PASS PROFILE BASED CONTROL LAWS FOR

PERFORMANCE AND DISTURBANCE REJECTION

In most cases, control law design will be required for both

stability along the pass and performance. If stability along

the pass holds then the resulting limit profile is guaranteed

to be a stable 1D discrete linear system. Hence its along the

pass dynamics can be quantified using appropriate 1D linear

systems concepts. Also, if as here, disturbances are present

then ideally these should have no influence on the resulting

limit profile dynamics. Moreover, the transient performance

in the pass-to-pass direction should be acceptable.

The above general requirements can be summarized as

follows (and will clearly require refinement for a given

application).

• The controlled process should be stable along the pass

with a limit profile which is (if possible) free from the

effects of the disturbance terms and satisfies such 1D

discrete linear systems performance requires as deemed

appropriate.

• The output sequence of pass profiles produced by the

controlled process should converge to the limit profile

at an acceptable rate in the pass-to-pass direction and

the along the pass dynamics on any pass should also be

acceptable.

Clearly the above is a very general specification and also

control law design to satisfy its requirements is very much

an open question which is almost certain to have no unique

answer. Here we consider the practically motivated case

where an end user specifies that the limit profile must be

some given vector yref(p) which has been chosen to meet

such 1D transient and steady state performance specifications

as deemed appropriate by the particular application under

consideration. (Some similarities with 1D model following

control can be seen here. The design task is to design a

control law of the form (10) that the limit profile is yref(p).

Suppose now that as k → ∞, xk(p) → x∞(p), yk(p) →z∞(p) = yref(p), and uk(p) → u∞(p). Then

x∞(p + 1) = Ax∞(p) + Bu∞(p)+ B0yref(p) + Ew(p)

yref(p) = Cx∞(p) + Du∞(p)+ D0yref + Fw(p) (16)

Now introduce the so-called incremental vectors

yk(p) = yk(p) − yref(p)xk(p) = xk(p) − x∞(p)uk(p) = uk(p) − u∞(p)

Then using (16) and (1) we obtain

xk+1(p + 1) = Axk+1(p) + Buk+1(p) + B0yk(p)yk+1(p) = Cxk+1(p) + Duk+1(p) + D0yk(p) (17)

and in this so-called residual model the influence of the

disturbance vector has been completely rejected. Note that

stability along the pass of (17) defined in terms of the

incremental vectors also ensures stability along the pass of

the original process. Hence the control objectives in this case

are achieved and the remaining task is to design the control

law to ensure stability along the pass.

To undertake this task we use the following equivalent

formulation to (10)

uk+1(p) = u∞(p) + K1yk+1(p)z + K2yk(p)

− (K1 + K2)yref(p) (18)

Then the result of Theorem 2 can be applied and the

remaining task is to construct x∞(p) and u∞(p) when w(p)and yref(p) are known.

Using the pass profile updating equation of (16) for yref(p)we have that

Du∞(p) = (I − D0)yref(p) − Cx∞(p) − Fw(p)

Hence we immediately see that not all yref(p) are available

under this design. In particular, only those which ensure that

rank[D | (I−D0)yref(p)−Cx∞(p)−Fw(p)

]= rank

[D

](19)

0 ≤ p ≤ α − 1, are allowed.

Suppose now that this condition holds (as is the case when

D is full row rank). Then we have that

u∞(p)=D�((I−D0)yref(p)−Cx∞(p)−Fw(p)

)(20)

0 ≤ p ≤ α−1, where (·)� denotes the pseudo inverse of (·).Also if D is square and nonsingular, the pseudo inverse can

be replaced by the matrix inverse and then

u∞(p)=D−1((I−D0)yref(p)−Cx∞(p)−Fw(p)

)0 ≤ p ≤ α − 1.

In the general case, the dimension of the pass profile vector

could exceed that of the input vector and hence neither D nor

Δ are full row rank matrices. Consequently given w(p) and

x∞(p) only those yref(p) for which (19) holds are achievable.

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The control law (18) with (20) computing u∞(p) also

requires knowledge of the limit profile state dynamics. This

is provided using

x∞(p + 1) = (A − BD�C)x∞(p)+ (B0 + BD�(I − D0))yref(p)+ (E − BD�F )w(p)

x∞(0) = d∞

(which is simply a 1D discrete linear systems state equation

which can be computed given d∞ — the strong limit of the

known pass state initial vector sequence).

V. AN EXAMPLE

In this section, we apply the new results of this paper to

a common example which is the special case of (1) defined

by

A =

⎡⎣ 0.8 0.19 0.6−0.08 0.21 −0.73−0.63 0.56 0.78

⎤⎦ , B =

⎡⎣ 2.5−1.45−0.76

⎤⎦B0 =

⎡⎣ 0.00.75−0.7

⎤⎦ , E =

⎡⎣ −0.230.70.05

⎤⎦C =

[0.29 −0.06 0.01

], D=1.14, D0 =1.09, F =0.61

with pass length α = 1000 and boundary conditions

dk(0) = [1.8005 −1.0754 0.4274]T , k ≥ 1

y0(p) = −1 + sin(−5πp

0.5α+ 10π), 0 ≤ p ≤ 999

The disturbance w(p) is generated via the MATLAB code

w(p)=round((rand()-0.5)*100)/100

As a first case, the required limit profile is taken to be 3 full

periods of the cosine function

yref(p) = cos(−3πp

0.5α+ 6π), 0 ≤ p ≤ 999

Implementing the control law design using Theorem 2 now

gives

K1 = 6.4319, K2 = −4.8532

Figure 2 shows the sequence of pass profiles produced in

the presence of disturbance defined above by the resulting

controlled process dynamics in this case

As a second case, suppose the desired limit profile is

as shown in Figure 3. Executing the control law design

algorithm gives the controlled process pass profile sequence

as shown in Figure 4

As a final case, suppose that the required limit profile is the

triangular (or sawtooth) signal shown in Figure 5. Executing

the control law design algorithm gives the controlled process

pass profile sequence as shown in Figure 6

Overall, these cases illustrate that the design method

developed here can work well. Also demonstrate that con-

vergence to the limit profile effectively takes place within

a relatively small number of passes. In context, previous

Fig. 2. Controlled process pass profile sequence for the first disturbance

Fig. 3. Desired limit profile

Fig. 4. Controlled process pass profile sequence for the second disturbance

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Fig. 5. Desired limit profile

Fig. 6. Controlled process pass profile sequence for the third disturbance

work [9] developed computational measures of the error

yk(p) − y∞(p)x for stable along the pass processes which

can be readily applied here.

VI. CONCLUSIONS

This paper has developed new results on the control of

discrete linear repetitive processes. Here the focus is on

output (or pass profile) control law design for stability along

the pass and performance as opposed to the former alone in

previous work. The design algorithms obtained have been

illustrated by the numerical examples. Work is currently

in progress on both the in depth development/evaluation of

these results and their extension. Output from this research

will be reported in due course.

Note again that this paper only deals with disturbances

which are constant from pass-to-pass and this reduces the

general applicability of the new results developed. At present,

it is not clear how (if at all) complete decoupling of distur-

bances which do not satisfy this assumption can be achieved.

One practically relevant alternative is to seek to attenuate the

effects of such disturbances to a prescribed degree using, for

example, H∞, H2, or mixed H2/H∞ techniques. In which

context some significant first results in this direction can be

found in [15].

Finally, it is important to place these results in context,

especially with respect to eventual practical applicability. In

particular, the basic model is practically relevant as it does

arise in approximating the dynamics of physical examples.

It is eventually aimed to apply the control laws of this work

to these problem areas, for which the results in this paper

provide part of the basic foundation.

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[11] ——, “Proportional plus integral control and disturbance rejection fordifferential linear repetitive processes,” in American Control Confer-ence, ACC 2005, 2005, pp. 1782–1787.

[12] D. H. Owens and E. Rogers, “Stability analysis for a class of 2Dcontinuous-discrete linear systems with dynamic boundary condi-tions,” Systems and Control Letters, vol. 37, pp. 55–60, 1999.

[13] K. Gałkowski, J. Lam, E. Rogers, S. Xu, B. Sulikowski, W. Paszke,and D. H. Owens, “LMI based stability analysis and robust controllerdesign for discrete linear repetitive processes,” International Journalof Robust and Nonlinear Control, vol. 13, no. 13, pp. 1195–1211,2003.

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