DIRECTIONAL CHANGE AND WINDUP PHENOMENON

Dariusz Horla

Poznan University of Technology

Institute of Control and Information Engineering

Division of Control and Robotics

Keywords:

Windup phenomenon, Directional change, Control limits, Multivariable systems.

Abstract:

The paper addresses two inherently connected problems, namely: windup phenomenon and directional change

in controls problem for multivariable systems. By comparing two ways of performing anti-windup compen-

sation and two different saturation modes a new deﬁnition of windup phenomenon for multivariable systems

has been obtained, changing deﬁnitions present in the literature. It has been shown that avoiding directional

change does not have necessarily to mean that windup phenomenon has been avoided too.

1 INTRODUCTION

Consideration of control limits is crucial for achieving

high control performance (Peng et al., 1998). There

are two ways in which one can consider possible con-

straints during synthesis of controllers, e.g. imposing

constraints during the design procedure, what leads

to difﬁculties with obtaining explicit forms of control

laws. The other way is to assume the system is linear

and, subsequently, having designed the controller for

unconstrained system – impose constraints, what re-

quires then additional changes in control system due

to presence of constraints.

A situation when because of, e.g., constraints (or,

in general, nonlinearities) internal controller states do

not correspond to the actual signals present in the con-

trol systems is referred as windup phenomenon (Wal-

gama and Sternby, 1993; Horla, 2004). It is obvious

that due to control signal constraints not taken into

account during a controller design stage, one can ex-

pect inferior performance because of infeasibility of

computed control signals.

There are many methods of compensating the

windup phenomenon (Peng et al., 1998; Walgama and

Sternby, 1993), but a few work well enough in the

case of multivariable systems. In such a case, apart

from the windup phenomenon itself, one can also

observe directional change in the control vector due

to, say, different implementation of constraints, what

could affect direction of the original, i.e. computed,

control vector.

The paper aims to compare two strands in con-

troller design subject to constraints, as mentioned be-

fore, and two ways of anti-windup compensation with

respect to directional change in controls.

As a result, a new deﬁnition of windup phe-

nomenon will be obtained with respect to directional

change in controls, which in the case of multivariable

systems cannot be omitted.

2 ANTI-WINDUP

COMPENSATION

There are two general schemes in anti-windup com-

pensation (AWC) connected with controller design.

If the controller has been designed for the case of a

linear plant, i.e. with no constraints, introducing them

would require certain (most often) heuristic modiﬁ-

cations in the control law that usually feed back the

difference in between computed v

t

and constrained

control vector u

t

. This is referred in the literature as a

posteriori AWC (Horla, 2006a; Horla, 2006b).

The second AWC is incorporated implicitly into

the controller, i.e. when controller generates feasible

control vector only (belonging to the domain D of

all control vectors for which a certain control perfor-

369

Horla D. (2007).

DIRECTIONAL CHANGE AND WINDUP PHENOMENON.

In Proceedings of the Fourth International Conference on Informatics in Control, Automation and Robotics, pages 369-374

DOI: 10.5220/0001622903690374

Copyright

c

SciTePress

mance index J

t

is of ﬁnite value), what is addressed

as a priori AWC.

3 A POSTERIORI AWC

One of the most popular AWCs (Peng et al., 1998) are

those based on the RST equation, which in the case of

multivariable systems is of the form (Horla, 2004)

R(q

−1

)v

t

= −S(q

−1

)y

t

+ T(q

−1

)r

t

, (1)

where

R(q

−1

) = I + R

1

q

−1

+ ··· + R

nR

q

−nR

,

S(q

−1

) = S

0

+ S

1

q

−1

+ ··· + S

nS

q

−nS

,

T(q

−1

) = T

0

+ T

1

q

−1

+ ··· + T

nT

q

−nT

are controller polynomial matrices of appropriate

sizes, designed for the unconstrained case, y

t

∈ R

p

is

the output vector, v

t

∈ R

m

is the control vector, d > 0

is a dead-time.

When the nonlinearities, such as control limits,

are taken into consideration the computed vector v

t

is different from the constrained, i.e. applied, control

vector u

t

. In such a case one can modify the control

law according to AWC schemes given below (Horla,

2004; Peng et al., 1998).

• Deadbeat AWC (DB)

v

t

= (I −R(q

−1

))u

t

−S(q

−1

)y

t

+T(q

−1

)r

t

. (2)

The controller is fed back with the constrained

control vector, thus no lack of consistency occurs.

• Generalised AWC (G)

A matrix A

o

(q

−1

) of observer polynomials with

nA

o

≤ nR is added

A

o

(q

−1

)v

t

= (A

o

(q

−1

) − R(q

−1

))u

t

−

−S(q

−1

)y

t

+ T(q

−1

)r

t

. (3)

• Conditioning technique AWC (CT)

The control vector and reference signal are com-

puted as

v

t

= (I − R(q

−1

))u

t

− S(q

−1

)y

t

+

+(T(q

−1

) − T

0

)r

r

t

+ T

0

r

t

, (4)

r

r

t

= r

t

+ T

−1

0

(u

t

− v

t

). (5)

A special case of CT is Modiﬁed conditioning

technique AWC (MCT) where instead of T

−1

0

there is an inversion of matrix (T

0

+ϒ)

−1

which is

responsible for the rate of modiﬁcation of the ref-

erence vector subject to constraints.

In the CT case, often outputs that were intended

to me unmodiﬁed, are modiﬁed due to condition-

ing technique. The latter is a result of directional

change, that is given rise by anti-windup compen-

sation. The two issues are therefore connected.

• Generalised conditioning technique AWC (GCT)

The restoration of the consistency is performed by

modifying the ﬁltered reference vector, i.e. com-

puting the so-called feasible ﬁltered reference

vector,

v

t

= (I − Q(q

−1

)R(q

−1

))u

t

+

+T

2,0

r

f,t

+ (T

2

(q

−1

)L(q

−1

) − T

2,0

)r

r

f,t

+

−Q(q

−1

)S(q

−1

)y

t

. (6)

r

f,t

= Q(q

−1

)L(q

−1

)

−1

T

1

(q

−1

)r

t

, (7)

r

r

f,t

= r

f,t

+ T

−1

2,0

(u

t

− v

t

), (8)

Where T = T

2

T

1

with monic T

1

, and nonsingular

T

2,0

.

4 DEFINITION OF WINDUP

PHENOMENON IN

MULTIVARIABLE SYSTEMS

Currently, one can meet the following deﬁnition of

windup phenomenon in multivariable systems with

its connections to directional change (Walgama and

Sternby, 1993):

Solving the windup phenomenon problem does not

mean that constrained control vector is of the same

direction as computed control vector.

On the other hand, avoiding directional change in

control enables one to avoid windup phenomenon.

In further parts of this paper, it has been shown

where the latter deﬁnition holds, and in what cases it

is invalid.

5 DIRECTIONAL CHANGE

PHENOMENON, AN EXAMPLE

Let us suppose that two-input two-output system is

not coupled and both loops are driven by separate con-

trollers (with no cross-coupling). The system output

y

t

is to track reference vector comprising two sinusoid

waves. It corresponds in the (y

1

,y

2

) plane to drawing

a circular shape.

As it can be seen in the Fig. 1a, the unconstrained

system performs best, whereas in the case of cut-off

saturation of both elements of control vector (Fig. 1b)

the tracking performance is poor. In the applica-

tion for, e.g., shape-cutting performance of the system

from Fig. 1c is superior. Nevertheless, it is to be borne

in mind that the system is always perfectly decoupled.

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

370

6 PLANT MODEL, CONTROL

PROBLEM

The following multivariable CARMA plant model

will be of interest

A(q

−1

)y

t

= B(q

−1

)u

t−d

, (9)

with left co-prime polynomial matrices

A(q

−1

)=I+

−1.4 −0.1

0.1 −1.0

q

−1

+

0.49 0

0 0.25

q

−2

,

B(q

−1

) = I + diag

{

0.5, 0.5

}

q

−1

and d = 1.

The plant is cross-coupled and comprises fourth-

order matrices in the transfer matrix representation

(being stable and minimumphase).

6.1 RST Controller (a Posteriori AWC)

It is assumed that the plant is controlled by a multi-

variable pole-placement controller with characteristic

polynomial matrix

A

M

(q

−1

) = I + diag

{

−0.5, −0.5

}

q

−1

.

The controller is given in RST structure with polyno-

mial matrices R(q

−1

) and S(q

−1

) resulting from Dio-

phantine equation

A(q

−1

)R(q

−1

) + q

−d

B(q

−1

)S(q

−1

) = A

M

(q

−1

)A

o

(q

−1

),

(10)

with A

o

(q

−1

) = I − 0.2Iq

−1

, T(q

−1

) = KA

o

(q

−1

),

A

M

(1) = B(1)K.

Having imposed control limits upon the RST con-

troller requires implementing a posteriori AWC tech-

niques in order to restore good performance quality.

y

1,t

y

2,t

a)

y

1,t

y

2,t

b)

y

1,t

y

2,t

c)

20 40

0

1

−1

t

y

1,t

20 40

0

1

−1

t

y

1,t

20 40

0

1

−1

t

y

1,t

20 40

0

1

−1

t

y

2,t

20 40

0

1

−1

t

y

2,t

20 40

0

1

−1

t

y

2,t

Figure 1: a) unconstrained system, b) cut-off saturation,

c) direction-preserving saturation.

6.2 Optimised Controller (a Priori

AWC)

In comparison, the RST pole-placement controller

performance will be compared with a priori AWC

controller, namely multivariable pole-placement con-

troller utilising the theory of predictive control and

convex optimisation techniques.

In order to enable such a comparison, the pre-

dictive controller has been deprived of all its advan-

tages – the prediction horizon has been chosen as one

step, thus the optimal constrained control vector is

searched (Horla, 2006a; Horla, 2006b)

u

⋆

t

: J

t

(u

⋆

t

) = inf

u

t

∈D(J

t

)

{

J

t

(u

t

)

}

,

where its jth component has been symmetrically con-

strained

|u

j,t

| ≤ α

j

,

where

u

t

=

u

1,t

u

2,t

··· u

m,t

T

.

The performance index has been chosen as a sum

of squared tracking errors resulting from a reference

model output

J

t

=

r

M,t+d

− ˆy

t+d

−

ˆ

ˆy

t+d

2

2

, (11)

where one-step (d = 1) prediction of system output

comprises as in (11) forced and free-response output

vectors.

The performance index can be rewritten into

quadratic form

J

t

=

Gu

t

+

ˆ

ˆy

t+d

− r

M,t+d

T

Gu

t

+

ˆ

ˆy

t+d

− r

M,t+d

,

(12)

and the optimisation can be performed with the use of

its linear matrix inequality (LMI) form with the last

two LMIs responsible for control constraints, as be-

low

min γ

s.t.

I ⋆

u

T

t

(G

T

G)

1/2

γ− (r

M,t+d

−

ˆ

ˆy

t+d

)

T

×

×(r

M,t+d

−

ˆ

ˆy

t+d

)+

+2(r

M,t+d

−

ˆ

ˆy

t+d

)

T

Gu

t

≥ 0,

diag

{

α

1

− u

1,t

,. . ., α

m

− u

m,t

}

≥ 0,

diag

{

α

1

+ u

1,t

,. . ., α

m

+ u

m,t

}

≥ 0,

(13)

where ⋆ detones a symmetrical entry, and G is an

impulse-response matrix.

DIRECTIONAL CHANGE AND WINDUP PHENOMENON

371

7 SIMULATION STUDIES

The simulations have been performed for two con-

trollers: for a pole-placement controller with a group

of a posteriori AWCs and LMI-based predictive pole-

placement controller.

In order to evaluate control performance con-

nected with anti-windup compensation performance

the following performance indices have been intro-

duced

J =

1

N

2

∑

i=1

N

∑

t=1

|r

i,t

− y

i,t

|, (14)

ϕ =

1

N

N

∑

t=1

|ϕ(v

t

) − ϕ(u

t

)| [

◦

], (15)

where (14) corresponds to mean absolute tracking er-

ror on both outputs and (15) is a mean absolute di-

rection change in between computed and constrained

control vector.

The control vector has been constrained in all

cases to α

1

= ±0.2 on the ﬁrst input and α

2

= ±0.3

for the second output. The reference vector is a

square-wave signal of amplitude ±1 and simulation

horizon N = 150.

Performance indices have been given in the Tab. 1.

Table 1: Performance indices for a) cut-off saturation,

b) direction-preserving saturation.

a)

− DB G CT MCT GCT

J 1.1539 1.1539 1.1539 1.1539 1.1536 1.1516

ϕ 0.9003 0.9004 1.1861 1.1862 1.1093 1.5495

b)

− DB G CT MCT GCT

J 1.1772 1.1672 1.1677 1.1677 1.1670 1.1655

ϕ 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

As it can be seen from Tab. 1a, and Figs. 4, 5,

cut-off saturation causes directional change, which is

visible during reference vector changes. It is neces-

sary to alter the decoupling of the plant, in order to

restore high control performance. As it can be seen,

the greater the mean absolute direction change, the

lesser the performance index is. Thus, it can be said,

that directional change supports anti-windup compen-

sation.

On the other hand, as in Tab. 1b, and Figs. 6, 7,

direction-preserving saturation does not cause direc-

tional change. Preserving a constant direction causes

performance indices to increase, dependless of the

method of anti-windup compensation. The coupling

is clearly visible during tracking when reference vec-

tor changes. Constant direction prevents the con-

troller from decoupling the plant – performance is in-

ferior.

In order to stipulate the differences in between

saturation methods, two GCT-AWC cases have been

chosen and compared with LMI-based approach.

50 100

0

1

−1

t

y

1,t

50 100

0

1

−1

t

y

2,t

50 100

0

0.5

−0.5

t

u

1,t

50 100

0

0.5

−0.5

t

u

2,t

50 100 150

0

10

−10

t

ϕ(v

t

) − ϕ(u

t

)

Figure 2: Overall performance for cut-off saturation with

GCT-AWC.

1516

5495

DB G CT MCT GCT

1655

0000

50 100

0

1

−1

t

y

1,t

50 100

0

1

−1

t

y

2,t

50 100

0

0.5

−0.5

t

u

1,t

50 100

0

0.5

−0.5

t

u

2,t

50 100 150

0

10

−10

t

ϕ(v

t

) − ϕ(u

t

)

Figure 3: Overall performance for direction-preserving sat-

uration with GCT-AWC.

In the Fig. 8 one can see a priori anti-windup

compensator performance, where only feasible con-

trol actions are generated. The performance indices

in such a case are of the best values, i.e. J = 1.0450,

ϕ = 64.7063

◦

.

Having compared Figs. 2–8 it can be said, that in

order to achieve the best performance one has to al-

ter the direction of a computed control vector. The

greater the directional change is, the better the con-

trol performance.

For a priori AWC, visible changes in control di-

rection result from decoupling phase, i.e. whenever

control vector encounters constraints it has to be con-

strained in such a way as to achieve the high control

performance (the last plot corresponds to the angle

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

372

difference in between control vector computed in the

unconstrained case using optimisation algorithm, and

constrained a priori AWC control vector).

–

50 100

0

1

−1

t

y

1,t

50 100

0

1

−1

t

y

2,t

DB

50 100

0

1

−1

t

y

1,t

50 100

0

1

−1

t

y

2,t

G

50 100

0

1

−1

t

y

1,t

50 100

0

1

−1

t

y

2,t

CT

50 100

0

1

−1

t

y

1,t

50 100

0

1

−1

t

y

2,t

MCT

50 100

0

1

−1

t

y

1,t

50 100

0

1

−1

t

y

2,t

GCT

50 100

0

1

−1

t

y

1,t

50 100

0

1

−1

t

y

2,t

Figure 4: Tracking performance for cut-off saturation.

–

50 100 150

0

10

−10

t

ϕ

t

DB

50 100 150

0

10

−10

t

ϕ

t

G

50 100 150

0

10

−10

t

ϕ

t

CT

50 100 150

0

10

−10

t

ϕ

t

MCT

50 100 150

0

10

−10

t

ϕ

t

GCT

50 100 150

0

10

−10

t

ϕ

t

Figure 5: Directional change for cut-off saturation.

On the other hand, having constrained the control

vector alters decoupling, thus its direction has to be

additionally altered, what is mostly visible in Fig. 8.

–

50 100

0

1

−1

t

y

1,t

50 100

0

1

−1

t

y

2,t

DB

50 100

0

1

−1

t

y

1,t

50 100

0

1

−1

t

y

2,t

G

50 100

0

1

−1

t

y

1,t

50 100

0

1

−1

t

y

2,t

CT

50 100

0

1

−1

t

y

1,t

50 100

0

1

−1

t

y

2,t

MCT

50 100

0

1

−1

t

y

1,t

50 100

0

1

−1

t

y

2,t

GCT

50 100

0

1

−1

t

y

1,t

50 100

0

1

−1

t

y

2,t

Figure 6: Tracking performance for direction-preserving

saturation.

–

50 100 150

0

10

−10

t

ϕ

t

DB

50 100 150

0

10

−10

t

ϕ

t

G

50 100 150

0

10

−10

t

ϕ

t

CT

50 100 150

0

10

−10

t

ϕ

t

MCT

50 100 150

0

10

−10

t

ϕ

t

GCT

50 100 150

0

10

−10

t

ϕ

t

Figure 7: Directional change for direction-preserving satu-

ration.

Finally, in the Fig. 9 it has been shown that both

a priori and a posteriori AWCs need to alter direction

of controls in order to restore high quality of tracking

performance. In addition, a priori AWC has no ﬁxed

structure, nor decoupling compensator, thus changes

DIRECTIONAL CHANGE AND WINDUP PHENOMENON

373

50 100

0

1

−1

t

y

1,t

50 100

0

1

−1

t

y

2,t

50 100

0

0.5

−0.5

t

u

1,t

50 100

0

0.5

−0.5

t

u

2,t

50 100 150

0

50

100

t

ϕ(v

t

) − ϕ(u

t

)

Figure 8: Overall performance for LMI-based control with

a priori AWC, J = 1.0450,

ϕ = 64.7063

◦

.

50 100 150

0

50

100

t

ϕ(v

t

) − ϕ(u

t

)

a)

50 100 150

0

50

100

t

ϕ(v

t

) − ϕ(u

t

)

b)

50 100 150

0

100

200

ϕ(v

GCT,t

) − ϕ(u

LMI,t

)

c)

t

Figure 9: A comparison of direction changes a) a priori

AWC, b) a posteriori GCT-AWC with cut-off saturation, c) a

priori AWC vs. GCT-AWC with cut-off saturation.

in control direction must be greater than in the case of

GCT-AWC with cut-off saturation.

The comparison of angle difference in between

unconstrained control vector computed for GCT case

and constrained control vector computed by a priori

AWC, shows that approximately a priori AWC acts in

the direction of unconstrained controller with GCT-

AWC, i.e. both of them try to get the system’s perfor-

mance as close as possible to the performance of ideal

pole-placement subject to no constraints.

Nevertheless, the simulations have shown that in

order to obtain a good performance one has to change

direction of control vector. Without the latter one will

observe coupling.

8 SUMMARY – NEW

DEFINITION OF WINDUP

PHENOMENON IN

MULTIVARIABLE SYSTEMS

One can formulate a new deﬁnition:

Solving the windup phenomenon problem does not

have to mean that constrained control vector is of the

same direction as computed control vector if cross-

coupling is present in the control system.

On the other hand, avoiding directional change in

control enables one to avoid windup phenomenon if

and only in the plant is perfectly decoupled or is not

coupled at all. (what due to the constraints is hardly

ever met)

Such a deﬁnition deﬁnitely changes the way one

should look at windup phenomenon and its connec-

tion with directional change problem.

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Horla, D. (2006b). Standard vs. LMI approach to a con-

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Peng, Y., Vran

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´

c, D., Hanus, R., and Weller, S. (1998).

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