ROBUST ILC DESIGN USING MÖBIUS TRANSFORMATIONS

C. T. Freeman, P. L. Lewin and E. Rogers

University of Southampton

School of Electronics and Computer Science, University Road, Southampton, SO17 1BJ, United Kingdom

Keywords:

Learning control, non-minimum phase systems, optimisation.

Abstract:

In this paper a general ILC algorithm is examined and it is found that the ﬁlters involved can be selected to

satisfy frequency-wise uncertainty limits on the plant model. The probability of the plant model being at a

given point in the uncertainty space is speciﬁed, and the ﬁlters are then chosen to maximise the convergence

rate that can be expected in practice. The magnitude of the change in input over successive trials and the

residual error have also been incorporated into the cost function. Experimental results are presented using a

non-minimum phase test facility to show the effectiveness of the design method.

1 INTRODUCTION

Iterative Learning Control (ILC) is a control method

that is applicable to systems which perform the same

action repeatedly. Operating in this way it is able to

use past control information such as input signals and

tracking errors in the construction of the present con-

trol action. This sets ILC apart from most other con-

trol techniques and has allowed it to provide improved

performance with reduced knowledge of the plant

when compared with other control approaches. A lit-

erature survey of ILC can be found in (Moore, 1998)

and there exist textbooks on the subject (Moore, 1993;

Bien and Xu, 1998). One such ILC update law is

given by

k+1

= F (z)u

+ S(z)e

(1)

where e

= y

− y

and F (z) and S(z) are ﬁlters

which may be non-causal. This has been analysed in

the frequency-domain in, for example, (Norrlöf and

Gunnarsson, 1999) and (Norrlöf, 2000). Conversion

of (1) to the frequency-domain using the sampling pe-

riod, T

, gives

k+1

= F (e

jωT

+ S(e

jωT

(2)

Let the uncertain plant be described by G(e

jωT

) =

jωT

)U(e

jωT

) where U (e

jωT

) is a multiplica-

tive uncertainty and G

the nominal plant model. In

this case the error evolution is

k+1

F (e

jωT

) − S(e

jωT

)U(e

jωT

) E

1 − F (e

jωT

) Y

(3)

so that



F (e

jωT

) − S(e

jωT

)U(e

jωT

)



< 1

(4)

is a necessary condition for stability since, for each

frequency, (3) represents a state-space system in the

iteration domain having a state transition matrix with

eigenvalues less than 1. In particular, if F (e

jωT

) =

1 this becomes a necessary condition for monotonic

convergence which can be shown to be essentially a

sufﬁcient condition as well (Longman, 2000). More

generally, the same argument means



F (e

jωT

) − S(e

jωT

)U(e

jωT

)



< l

(5)

is a sufﬁcient condition for the eigenvalues of the state

transition matrix to be less than l, or a reduction of

in the magnitude of the error over consecutive cycles

if F (e

jωT

) = 1. To accommodate both these cases,

satisfying (5) will be said to produce a convergence

rate of

Remark 1 Adding a ﬁlter, T (z), on the (k + 1)

cy-

cle error to the algorithm given in (1) produces

k+1

= F (z)u

+ S(z)e

+ T (z)e

k+1

(6)

and we can then write

k+1

G(z)T (z)

1+G(z)T (z)

G(z)

1+G(z)T (z)

(F (z)u

+ S(z)e

)

(7)

141

T. Freeman C., L. Lewin P. and Rogers E. (2005).

ROBUST ILC DESIGN USING MÖBIUS TRANSFORMATIONS.

In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics, pages 141-146

DOI: 10.5220/0001172601410146

 SciTePress

to give

k+1

T (z)

1+G(z)T (z)

(F (z)u

+ S(z)e

)

(8)

The substitutions ˆy



T (z)

S(z)

+ 1



and

G(z) =

G(z)

1+G(z)T (z)

, or alternatively ˆy



S(z)

T (z)



and

G(z) =

G(z)T (z)

1+G(z)T (z)

, allow this to be written as

(1). Any design method for S(z) and F (z) can there-

fore be implemented as (6) using these substitutions.

Any robustness properties, however, will only apply to

G(z) instead of G(z).

2 DFT

An alternative expression for (2) is found by taking

the Discrete Fourier Transform (DFT) of both sides

of the update law (1) which results in

ˆu

k+1

F ⊙ ˆu

S ⊙ ˆe

(9)

which is possible since e

and u

are known prior to

the k + 1

trial. Here ⊙ denotes component-wise

multiplication and ˆu is the DFT of u. Similarly, the

DFT of (3) results in an alternative description

ˆe

k+1

= (

F −

⊙

U ⊙

S) ⊙ ˆe

+(I −

F )⊙ ˆy

(10)

Note that now the design is conducted in the

frequency-domain, the designer can select each ele-

ment

and

individually and build the new in-

put u

k+1

using (9) combined with expressions for the

DFT and IDFT of a signal. Similar transparency can-

not be found in the time domain. Since (4) is a steady-

state requirement which assumes a reasonable tran-

sient response, the use of a steady-state update of the

input is not disadvantageous. It just remains to choose

the value of the ﬁlters S and F at every frequency.

3 APPROACH TO FILTER

DESIGN

By introducing the variable v = U (e

jωT

) it is possi-

ble to write (5) as

sup

ω ∈[0,2π]

|f(v)| < l (11)

where

f(v) = F (e

jωT

) − S(e

jωT

)v v ∈ C

(12)

Let the open disc of radius l centred at the origin,

and its boundary be deﬁned as

D = {r e

jθ

| θ ∈ [−π π), r ∈ [0 l)},

δD = {le

jθ

| θ ∈ [−π π)}

(13)

so that the inverse function

−1

(v) =

F (e

jωT

) + v

S(e

jωT

)

v ∈ C (14)

applied to D gives the range of values taken by

U(e

jωT

) for (5) to be satisﬁed.

The domain and co-domain can be enlarged to

equal the extended complex plane

C (Jones and

Singerman, 2004) which is the union of C and the

point at inﬁnity; thus

C = C ∪ {∞}. In this case

−1

(v) is an extended Möbius transformation when

ω and T

are ﬁxed, and maps

•

C one-one onto

• generalised circles onto generalised circles

A generalised circle is deﬁned as either a circle or an

extended line (see (Jones and Singerman, 2004) for

details).

It is therefore possible to apply f

−1

(v) to δD and

ﬁnd the boundary of the region U(e

jωT

) must oc-

cupy to ensure a convergence rate of

, or monotonic

convergence if l ≤ 1. The image of a Möbius trans-

formation can be found by simply applying f

−1

(v) to

three points of δD and ﬁnding the unique generalised

circle which passes through the resulting points (see

(Jones and Singerman, 2004)). Since {±l, lj} ∈ δD,

−1

(+l) =

F (e

jωT

)+l

S(e

jωT

)

−1

(−l) =

F (e

jωT

)−l

S(e

jωT

)

−1

(lj) =

F (e

jωT

)+lj

S(e

jωT

)

(15)

belong to this extended circle. However a simpler

method involves noting that the inverse points of δD

are 0 and ∞ so that the inverse points of the image are

given by

−1

(0) =

F (e

jωT

)

S(e

jωT

)

, f

−1

(∞) = ∞

(16)

and the mapping has the form



z −

F (e

jωT

)

S(e

jωT

)



= k(e

jωT

) (17)

Since f

−1

(l) =

F (e

jωT

)+l

S(e

jωT

)

∈ δD then



F (e

jωT

) + l

S(e

jωT

)

−

F (e

jωT

)

S(e

jωT

)



= k(e

jωT

) =

|S(e

jωT

(18)

which is the equation for a circle with centre, λ, and

radius, r, where

λ =

F (e

jωT

)

S(e

jωT

)

, r =

|S(e

jωT

(19)

ICINCO 2005 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

142

This circle can then be drawn for every frequency of

interest and hence the region U(e

jωT

) must occupy

in order to guarantee monotonic convergence. Note

that a Möbius transformation, t(z), maps a region R

in its domain to a region t(R), and the boundary of

R to the boundary of t(R). Figure 1 shows the ac-

tion of f

−1

(v) on δD and the region U(e

jωT

) must

occupy for a convergence rate of

. Any uncertainty

Uncertaintyspace

f (v)

-1

Figure 1: Geometry of uncertainty space

that is contained within the right-hand circle for all

frequencies will satisfy (5). At a general point, p, in

the uncertainty space we can equate the distance from

p to λ with r



p −

F (e

jωT

)

S(e

jωT

)



|S(e

jωT

(20)

in order to ﬁnd the convergence rate at that frequency



p −

F (e

jωT

)

S(e

jωT

)



|S(e

jωT

|S(e

jωT

)p − F (e

jωT

(21)

This provides a useful means of selecting S and F :

Let us choose to maximize the convergence rate that

can be expected given that the probability that the

plant is at the point p at a given frequency is known

or can be estimated. It will be assumed that this prob-

ability is symmetrical about the nominal value, that

is +1 in the uncertainty space. This is a situation

which is realistic given the methods of obtaining a

plant model commonly used in practice. In this case it

will be a function of the distance to the nominal plant

model which equals

|p−1|

in the uncertainty space.

The optimisation will therefore take the form

max

S,F

J(S, F ) (22)

with the cost function

J(S, F ) =



p −



|SG



|p − 1|



δp

|SG



p −





|p − 1|



δp

(23)

where P (·) is the probability that the plant is at p,

and A is a region of uncertainty over which P (·) is

valid. The frequency dependence of the ﬁlters has

been dropped for conciseness. Note that a small circle

around the singularity must be removed if the equa-

tion is solved numerically. Figure 2 shows three cases

using the probability function P (x) = x

−α

and α

equal to 2, 1 and 0.5. The probability function is able

0 0.5 1

5000

10000

P(x) = x

−2

0 0.5 1

100

P(x) = x

−1

0 0.5 1

P(x) = x

−0.5

Figure 2: Examples of P (x)

to shift focus between robustness and convergence

speed by varying the weighting of the optimisation.

In the ﬁrst case P (x) is only large for x ≈ 0 so the

maximisation will lead to a solution which is tailored

to only values close to the nominal plant model. As α

is reduced it becomes more important to choose S so

that the whole of A has a value of l < 1. These cases

are shown in Figure 3. In a)

≈ 1 to give a high

convergence rate for the nominal plant, whilst in b) it

is smaller but most of the region A has the property of

satisfying (5). This choice of P (x) produces the cost

f (1)

-1

f (1)

-1

Figure 3: Optimal solutions for a) α > 1 and b) α < 1

J(S, F ) =

|SG

|p − 1|



p −



δp (24)

ROBUST ILC DESIGN USING MÖBIUS TRANSFORMATIONS

143

and the values of S and F are found by solving

∂

∂S

J(S, F ) = 0 and

∂

∂F

J(S, F ) = 0 (25)

at each frequency and ensuring that it is a global max-

imum. If F is ﬁxed at 1 then any solution, S

∗

which

solves (25), so that

dJ(S

∗

, 1)

= 0 (26)

will also solve the optimisation using





|F |

|S||G

|p − 1|



p −



δp (27)

since only a substitution of variables has been applied.

S and F can assume any values as long as

= S

∗

If F is now ﬁxed at

F , the corresponding value of

F S

∗

. This must also solve the optimisation using

(24) as this is simply (27) multiplied by a constant.

Therefore it is enough to solve the optimisation using

(24) with F = 1 and simply substitute

S =

F S

∗

to obtain the solution for every other value of F . The

resulting cost is

J = |

F |J

∗

where J

∗

is the cost using

F = 1. In order to incorporate other considerations in

the cost, let us consider the change in input from one

trial to the next. Since this is given by

k+1

− u

= (F − 1)u

+ Se

(28)

which, with the repeated application of (3), becomes

k+1

− u

= S (F − SG

S(1 − F )

k−1

i=0

(F − SG

+ (F − 1)u

(29)

and the residual error is given by

= (F − SG

+(1−F )

k−1

i=0

(F − SG

(30)

It can be seen that reducing F from 1 therefore has

the effect of reducing the cost (24) with the compro-

mise of a likelihood of increased residual error and

input change. To tackle these effects directly for an

arbitrary e

and y

, it is required that each term in

(29) and (30) has a small modulus for each frequency

considered. Assuming that (4) is satisﬁed it remains

to reduce |S| and also the bound, λ, on the remaining

term which is given by

(1 − F )

k−1

i=0

(F − SG

1−F

1−(F −SG

= λ

(31)

This can be achieved using the mapping technique

that has already been described. Using this, we ﬁnd

that at a point in the uncertainty space, p, the bound λ

equals

1 − F



p +

1−F



1−F

p + 1



(32)

Therefore the functions Q(|S|

−1

) and





1−F

p + 1





can be incorporated into the

cost to limit the upper bound of the residual error

and change in successive inputs. Since these are

dependent on the plant and choice of y

, they will be

neglected in order to maintain focus on the general

case.

3.1 Experimental Test Facility

The experimental non-minimum phase test-bed has

previously been used to evaluate a number of Repet-

itive Control and ILC schemes (see (Freeman et al.,

2005) for details) and consists of a rotary mechanical

system of inertias, dampers, torsional springs, a tim-

ing belt, pulleys and gears. An encoder records the

ouput shaft position and a standard squirrel cage in-

duction motor drives the load. The system has been

modelled using a LMS algorithm to ﬁt a linear model

to a great number of frequency response test results.

The resulting continuous time plant transfer function

has thus been established as

(s) =

1.202(4 − s)

s(s + 9)(s

+ 12s + 56.25)

(33)

A PID loop around the plant is used in order to act as

a pre-stabiliser and provide greater stability. The PID

gains used are K

= 137, K

= 5 and K

= 3. The

resulting closed-loop system constitutes the system to

be controlled.

4 EXPERIMENTAL RESULTS

In polar co-ordinates let us deﬁne the region

A = {re

jθ

| θ ∈ [θ

], r ∈ [r

]} (34)

in the uncertainty space over which the probabil-

ity function is valid. The parameters T = 6 and

n = 1024 are chosen for convenience to give fs =

1024/6. Let us solve the optimisation using (24) with

F = 1 and α = 0.2. Figure 4 shows a Bode plot of the

resulting S vectors when using θ

= −

, θ

= 0 and r

T ω

2πλ

with λ = 4, 6, 8, 10.

These have been chosen using previous experience of

the plant uncertainty. As λ increases, r

decreases,

which, in turn, increases the magnitude of S at each

frequency as the plant effectively becomes less un-

certain as A diminishes in size. Since θ

= −θ

ICINCO 2005 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

144

−150

−100

−50

Gain (dB)

Frequency (rad/s)

−2000

−1000

1000

2000

Phase (degrees)

λ = 4

λ = 6

λ = 8

λ = 10

λ = 4

λ = 6

λ = 8

λ = 10

Figure 4: Bode plot using various r

functions and α =

0.2

A is symmetrical about the real axis which results in

∠S = −∠G

. Figure 5 shows error results using the

sinewave demand (shown in ﬁgure 11). The normal-

ized error ‘NE’ is the cumulative error incurred over

an iteration divided by the integral of the reference de-

mand. Data has been recorded over the course of 200

iterations. It can be seen that smaller λ values lead to

reduced ﬂuctuations in the cycle error. Figure 6 shows

0 40 80 120 160 200

0.2

0.4

0.6

0.8

Cycle No.

λ = 4

λ = 6

λ = 8

λ = 10

Figure 5: Cycle error for sinewave demand

error results using the sinewave demand (shown in

ﬁgure 7). Due to the higher frequencies present in

the demand, the convergence is slower and the resid-

ual error greater. Higher frequency properties of the

optimisation play a greater role and so the effect of

λ on the learning transients is more pronounced than

previously. The convergence of the output to the de-

mand is shown in ﬁgure 7 using λ = 6. The repeating

sequence demand is also drawn.

To investigate the effect of variation in α, the op-

timisation using (24) with F = 1 has again been

solved but using α = 0.1, 0.3, 0.5, 0.7. Figure 8

shows a Bode plot of the resulting S vectors when

0 40 80 120 160 200

0.2

0.4

0.6

0.8

Cycle No.

λ = 4

λ = 6

λ = 8

λ = 10

Figure 6: Cycle error for repeating sequence demand

Figure 7: Output tracking using repeating sequence demand

and λ = 6

using θ

= −

, θ

, r

= 0 and r

T ω

2πλ

with λ = 6. As α increases in magnitude more em-

phasis is put on fast convergence for the nominal plant

and the gain is larger for a greater range of frequen-

cies. Figure 9 shows error results using the sinewave

demand and it can be seen that higher α values pro-

duce greater learning transients, and, ultimately, di-

vergence. They do, however, produce faster initial

convergence. Figure 10 shows error results using the

repeating sequence demand which conﬁrm the previ-

ous ﬁndings. The convergence of the output to the

demand is shown in ﬁgure 11 using α = −0.5 and

the sinewave demand which is also shown.

5 FUTURE WORK

Since the uncertainty probability function can only be

an approximation, an obvious method to increase al-

gorithm robustness is to use additional plant data and

attempt to locate the plant model with greater accu-

ROBUST ILC DESIGN USING MÖBIUS TRANSFORMATIONS

145

−150

−100

−50

Gain (dB)

Frequency (rad/s)

−1000

−500

500

1000

Phase (degrees)

α =−0.7

α =−0.3

α =−0.1

α =−0.5

α =−0.7

α =−0.3

α = −0.1

α =−0.5

Figure 8: Bode plot using various α, λ = 6

0 40 80 120 160 200

0.2

0.4

0.6

0.8

Cycle No.

α = −0.1

α = −0.3

α = −0.5

α = −0.7

Figure 9: Cycle error for sinewave demand

racy as the iterations progress. At the same time, the

added conﬁdence in the nominal plant value could al-

low the focus of the optimisation to be shifted from

robustness to convergence speed. This would be

achieved by adaptively varying the probability func-

tion from trial to trial and seeking to reduce the radii

of circles corresponding to convergence rates for each

frequency around the nominal plant model. This use

of adaptive parameter tuning will be investigated in

conjunction with different plant models to conﬁrm its

effectiveness.

The terms that have been derived in order to tackle

the magnitude of the residual error and the change in

input over successive cycles must be added to the cost

function and experiments conducted to examine their

performance in practice.

REFERENCES

Bien, Z. and Xu, J. (1998). Iterative Learning Con-

trol, Analysis, Design, Integration and Applications.

Kluwer Academic Publishers.

0 40 80 120 160 200

0.2

0.4

0.6

0.8

Cycle No.

α = −0.1

α = −0.3

α = −0.5

α = −0.7

Figure 10: Cycle error for repeating sequence demand

Figure 11: Output tracking using sinewave demand and

α = 0.5

Freeman, C. T., Lewin, P. L., and Rogers, E. (2005). Exper-

imental evaluation of iterative control algorithms for

non-minimum phase plants. International Journal of

Control, 78(11):806–826.

Jones, G. A. and Singerman, D. (2004). Complex Func-

tions: An Algebraic and Geometric Viewpoint. Cam-

bridge University Press.

Longman, R. W. (2000). Iterative learning control and

repetitive control for engineering practice. Interna-

tional Journal of Control, 73(10):930–954.

Moore, K. L. (1993). Iterative Learning Control for Deter-

ministic Systems. Springer-Verlag.

Moore, K. L. (1998). Iterative learning control - an expos-

itory overview. Applied and Computational Controls,

Signal Processing and Circuits, 1(1):151 – 214.

Norrlöf, M. (2000). Comparative study on ﬁrst and second

order ilc- frequency domain analysis and experiments.

In Proceedings of the 39th Conference on Decision

and Control, pages 3415–3420.

Norrlöf, M. and Gunnarsson, S. (1999). A frequency do-

main analysis of a second order iterative learning con-

trol algorithm. In Proceedings of the 38th Conference

on Decision and Control, pages 1587–1592.

ICINCO 2005 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

146