—————

2That said, extensive literature exists on the subject of controllabilityof linear systems with saturated inputs (see, for example, Sontag andSussmann, 1990 or the references cited in Tyan and Bernstein, 1995).

—————

*Corresponding author. Tel.: #44 116 223 1303; fax: #44 116 2522619; e-mail: ce@sun.engg.le.ac.uk.

1This paper was not presented at any IFAC meeting. This paper wasrecommended for publication in revised form by Editor Peter Dorato.

Automatica 35 (1999) 761—765

Technical Communique

An anti-windup scheme with closed-loop stability considerations1

Christopher Edwards*, Ian Postlethwaite

Control Systems Research, Department of Engineering, University of Leicester, Leicester LE1 7RH, UK

Received 4 August 1997; revised 18 May 1998; received in final form 3 November 1998

Abstract

This paper considers the so-called model-based approach to anti-windup compensation and analyses the stability of theclosed-loop system for a class of input nonlinearities. The degrees of freedom in the anti-windup scheme are used to minimise theH

=norm of the transfer function around the nonlinearity and the Small Gain Theorem is used to prove stability. ( 1999 Elsevier

Science Ltd. All rights reserved.

Keywords: Control nonlinearities; Feedback control; Nonlinear systems; Saturation; H= control; Anti-windup; Bumpless transfer

1. Introduction

A good deal of the control systems literature concernsitself with the problem of designing linear controllers forplants which are assumed to be linear. In practice, ofcourse, this assumption is an oversimplification: forexample, the actuators associated with any implementa-tion limit the magnitude of the signal entering the plantand thus behave as nonlinear saturation elements be-tween the controller and the plant. If the controller isunstable or contains integrators a phenomenon referredto as integral windup is prone to occur: characteristicallyif the control signal saturates, the controller continues tointegrate the tracking error thus producing even largercontrol signals keeping the system ‘locked’ in saturationwhich usually results in undesirable performance in theform of large overshoots.

This paper considers anti-windup schemes whichmodify controllers so that if the signal from the controllerdiffers from that which enters the plant, corrective feed-back is employed to reduce the discrepancy. This proce-dure will be referred to as anti-windup compensation.Typically, these schemes are ‘retro-fitted’ to the existingcontroller which is assumed to have been designed in anoptimal fashion for a linear model of the plant. Many‘solutions’ to the problem have been reported in the

literature; however, until recently, there have been rela-tively few attempts to analyse the resulting closed-loopbehaviour from the point of view of stability:2 notableexceptions are Miyamoto and Vinnicombe (1996), Glatt-felder and Schaufelberger (1988), Doyle et al. (1987),Campo and Morari (1990) and more recently Kothareand Morari (1995) and Teel and Kapoor (1997). Kothareand Morari (1995) consider the closed-loop behaviour ofdifferent anti-windup compensation schemes in the caseof saturation. Their approach is to use a fractional trans-formation framework and to represent the saturationelement as a sector/conic nonlinearity. This generalapproach will be adopted in this paper. The state-spacenotation

G s"CA

1B1

C1

D1D , K s"C

A B

C DD (1)

will be used throughout to represent the plant and con-troller, respectively. Furthermore A

1is assumed to be

Hurwitz. For convenience, in the state-space manipula-tions that will be presented it will be assumed that D"0.

A general unifying approach to anti-windup compen-sation using coprime factors has been put forward byKothare et al. (1994). In this approach, the controller isconditioned by two signals affecting the controller output

0005-1098/99/$—see front matter ( 1999 Elsevier Science Ltd. All rights reservedPII: S 0 0 0 5 - 1 0 9 8 ( 9 8 ) 0 0 2 2 5 - 8

—————

3A nonlinear (scalar) function / : x>/(x) is said to belongto the sector [a, b] for some scalars a, b with a'b if (/(x)!ax)(/(x)!bx)40 for all x.

Fig. 1. A model-based conditioning scheme.

and the controller states, which results in the conditionedcontroller

u"Ve#(I!U) u' , (2)

where the pair of transfer functions U and V representa coprime factorization of the controller. All the commonlinear approaches discussed in the literature can be ex-pressed in the form of Eq. (2) for appropriate choices ofU and V (for details see Kothare et al., 1994). This schemehas subsequently been analysed with respect to closed-loop stability using an LMI framework (Kothare andMorari, 1995, 1997) under the assumption that the non-linearity belongs to the conic sector3 [0, 1]. This work,however, focuses solely on closed-loop stability and notthe performance of the anti-windup scheme. The work ofMiyamoto and Vinnicombe (1996) provides an elegantanalysis and synthesis procedure for the class of nonlin-earities in the sector [!1, 1]. However, this representsa conservative choice of sector for saturation nonlineari-ties. The approach described here may be regarded asa special case of the coprime factor framework and theanalysis of Kothare and Morari (1995) but addressesperformance issues as well as stability considerations.

2. Model-based schemes

The so-called model-based scheme presented byHanus (1988) from unpublished work by Irving, is givenin block diagram form in Fig. 1 where N represents thenonlinearity. In this framework, G

.represents a user-

defined transfer function usually referred to as the directmodel. The resulting conditioned controller is given by

u"Qe#QG.u' , (3)

where

Q"(I#KG.)~1 K. (4)

One approach for synthesising G.

is given in Glattfelderand Schaufelberger (1996). Another is proposed inEdwards and Postlethwaite (1997) by considering thefeedback loop associated with the signals u, u' , y

.and e.

This approach whilst effecting anti-windup compensa-tion does not address closed-loop stability issues which isthe contribution of the sequel. Notice that Eq. (3) can bewritten as

u"Qr!Q(Gu'!G.u' ) (5)

and thus if G."G the transfer function between u' and

u is zero. Consequently, the signal u is unaffected by the

nonlinearity and global closed-loop stability is assured.From this perspective the choice of G

."G is appealing,

however, the performance may be poor in the sense thatthe plant output y may be detrimentally affected. Indeedin certain cases steady-state tracking resulting from theuse of integral action in the controller may be lost (see,for example, Campo and Morari, 1990; Edwards andPostlethwaite, 1997).

In the model-based approach there is no inherentrequirement for G

."G. The fundamental idea is thus to

synthesise a direct model to guarantee that in the pres-ence of saturation the closed loop remains globallystable, whilst maintaining an acceptable output responsefrom the plant.

Note that this model-based approach fits into thecoprime framework of Kothare et al. (1994) if a non-minimal realisation of the controller is employed(Edwards and Postlethwaite, 1997).

2.1. Synthesis of the direct model

Suppose the nonlinearity N in Fig. 1 is such thatu'"( (u) where the nonlinear map ( belongs to themultidimensional sector [0, I], i.e.

(T ((!u)40. (6)

Note that this class of nonlinearities includes the com-ponentwise saturation function

u'i"sat(u

i)"G

1 if ui'1,

ui

if!14ui41,

!1 if ui(!1.

(7)

The block diagram from Fig. 1 can be re-drawn as thenonlinear feedback loop shown in Fig. 2 where

F"(I#KG)~1K(G.!G) (8)

and the nonlinearity /"u!u' is the ‘dead-zone’ non-linearity belonging to the sector [0, I]. Because of theprocedure that will be proposed to synthesise the directmodel G

., it is convenient at this stage to apply ‘loop-

shifting’ to the nonlinear feedback loop given in Fig. 2.

762 C. Edwards, I. Postlethwaite/Automatica 35 (1999) 761—765

Fig. 2. Nonlinear feedback configuration.

Fig. 3. Loop shifting equivalence diagram.

Fig. 4. New nonlinear feedback loop.

Consider the ‘loop-shifting’ diagram shown in Fig. 3and define

/I "/!12

u and F4"(I#1

2F)~1 F. (9)

It follows that stability of Fig. 2 is equivalent to stabilityof Fig. 4 where /I belongs to the sector [!1

2I, 1

2I]. From

the Small Gain Theorem (Zames, 1966) it follows that thefeedback loop in Fig. 4 is stable if

E12F4E=(1, (10)

since F4is a finite-dimensional linear operator. One pos-

sibility is therefore to synthesise a direct model G.

toensure good performance whilst ensuring the nonlinearclosed loop is stable via a ‘small gain’ argument. This willbe accomplished using a standard H

=controller design

framework.Define a new output signal uJ "1

2u and notice that as

a result /"/I #uJ . It follows from Fig. 1 that

u"Ky.!KG(u!/) (11)

and thus after loop shifting

uJ "(I#12KG)~1 1

2Ky

.#(I#1

2KG)~1 1

2KG/I . (12)

The ‘exogenous signal’ to the generalised plant will beconsidered to be /I and the ‘control input’ will be y

.. The

outputs that drive the controller will be u'!u"!/"

!uJ !/J and the error signal to be kept small will bedefined to be

z"CuJ

W1y.D , (13)

where W1

represents a high pass filter used to tune theclosed-loop response. Note the transfer function from/I to uJ is 1

2F

4. The ‘generalised plant’ representation of

this problem is: find a stabilising G.

such that

EFl(P, G

.)E

=(1, (14)

where Fl

represents a lower fractional transforma-tion (see, for example, Skogestad and Postlethwaite,1996) and

Remark. This approach does not guarantee that thesynthesised ‘controller’ G

.is stable. In practice, however,

no difficulty has been found in selecting weights to givea stable G

.. An intuitive rationale for this may be that

under the assumption that G is stable, the choiceG

."G represents a feasible solution (for an appropriate

W1

since in this case F4"0). It is not unreasonable

therefore to expect that for a class of W1

the problem hasa solution.

2.2. Analysis with respect to a step input

In this section an analysis of the closed loop will bemade with respect to a step response of magnitude rN ina manner similar to that of Glattfelder and Schaufel-berger (1988). Suppose, in the absence of the nonlinearity,the plant state vector at steady state is xN

1and control

state vector at steady state is xN#with inputs and outputs

uN and yN , respectively. If x1(t) and x

#(t) represent the plant

and controller states, respectively, and e1(t)"x

1(t)!xN

1and e

#(t)"x

#(t)!xN

#then

eR1(t)"A

1x1(t)#B

1(u!/)!(A

1xN1#B

1uN ) (15)

"A1e1(t)#B

1(u!uN )!B

1/ (16)

C. Edwards, I. Postlethwaite/Automatica 35 (1999) 761—765 763

Fig. 6. Response of the constrained system.

Fig. 5. Nonlinear feedback configuration for u%.

and

eR#"Ax

##B (r!y#y

.)!(AxN

##BrN!ByN ) (17)

"Ae#!B(y!yN )#By

., (18)

since r (t)"rN . Furthermore, u (t)!uN "Ce#(t) and

y(t)!yN "C1e1(t)#D

1(u(t)!/!uN ) and thus

eR1"A

1e1#B

1Ce

#!B

1/, (19)

eR#"(A!BD

1C) e

c!BC

1e1#BD

1/#By

.. (20)

The dynamics of the original system in response to thestep input is therefore equivalent to the free system inEqs. (19) and (20) with initial conditions e

1(0)"!xN

1and

e#(0)"!xN

#(assuming the initial conditions x

1(0)"0

and x#(0)"0). If

u%(t)"Ce

#(t), (21)

then u(t)"u%(t)#uN and Eqs. (19)— (21) can be written in

transfer function form as

u%"(I#KG)~1Ky

.#(I#KG)~1KG/. (22)

With respect to u%, componentwise, the nonlinearity / is

the unsymmetrical dead-zone function with /izero in the

interval [!1!uNi, 1!uN

i]. Crucially, each component

/ibelongs to the sector [0, 1] with respect to u

%provided

!1(uNi(1. From Eq. (22), if y

."!G

./ then the

state-space system (19)—(21) is equivalent to the blockdiagram in Fig. 5 where F is given in Eq. (8). This systemis asymptotically stable (Haddad and Bernstein, 1993)and hence e

1(t)P0 and thus steady-state tracking is

retained.

Remarks. The restriction !1(uNi(1 has a very natu-

ral interpretation, namely, the steady-state values of thecontrol action for the nominal linear system must liewithin the control action constraints. The relationshipuN "G (0)~1rN , indicates the range of step inputs for whichthe analysis guarantees that steady-state tracking will bepreserved.

The fact that a specific nonlinearity has been con-sidered but stability has been proven for any sectornonlinearity within [0, 1] suggests that unnecessary con-servatism has been introduced. However, in the analysisabove, step inputs of different magnitudes equated todead-zone nonlinearities shifted by different amountsalong the horizontal axis and thus may be thought of asa class of nonlinearities within the sector [0, 1].

3. An example

The design approach described previously will now beapplied to an example taken from Doyle et al. (1987). Theexample is somewhat artificial but was constructed toexpose the limitations of existing anti-windup schemesand to test new ones. The plant model comprises twosecond-order Butterworth filters in series, and the con-troller is taken to be pure integral action. In Doyle et al.(1987), the effect of limiting the control signal to theinterval $1 is considered. With no anti-windup compen-sation the closed-loop response degenerates to a limitcycle of large amplitude. Fig. 6 shows the performanceobtained from using the optimisation scheme with

W1"C

!0.230 0.220

!0.092 0.100D (23)

which has been obtained through trial and error. It canbe verified that this weight gives a stable ‘controller’ G

.with E1

2F4E=(0.83.

For comparison, the response obtained from usingG

."G is shown in Fig. 7 (as the solid line). The dotted

line is a reprise of the response in Fig. 6 for comparison.

764 C. Edwards, I. Postlethwaite/Automatica 35 (1999) 761—765

Fig. 7. Response of the constrained system.

4. Summary

An optimisation approach has been suggested forutilising the design freedom available in the so-calledmodel-based approach to anti-windup compensation.The direct model has been chosen in an attempt to ensurethat a certain transfer function satisfies the requirementsof the Small Gain Theorem thus guaranteeing the stabil-ity of the closed-loop despite the presence of actuatorsaturation. The procedure is straightforward and utilisesstandard commercially available H

=software.

References

Campo, P., & Morari, M. (1990). Robust control of processes subject tosaturation nonlinearities. Comput. Chem. Engng, 14, 343—358.

Doyle, J., Smith, R., & Enns, D. (1987). Control of plants with inputsaturation nonlinearities. In Proc. American Control Conf.,Minneapolis (pp. 2147—2152).

Edwards, C., & Postlethwaite, I. (1998). Anti-windup and bumpless-transfer schemes. Automatica, 34, 199—210.

Glattfelder, A., & Schaufelberger, W. (1988). Stability of discrete over-ride and cascade-limiter single loop control systems. IEEE ¹rans.Automat. Control, 33, 532—540.

Glattfelder, A., & Schaufelberger, W. (1996). Stability of trajectorygenerator system with anti-windup. In Proc. CESA +98 Multiconf.,Symp. on Control, Optimization and Supervision, Lille, France(pp. 1020—1025).

Haddad, W., & Bernstein, D. (1993). Explict construction of quadraticLyapunov functions for the small-gain, positivity, circle and Popov

theorems and their application to robust stability. Int. J. RobustNonlinear Control, 3, 313—339.

Hanus, R. (1988), Anti-windup and bumpless transfer: a survey. InProc. 12th IMACS ¼orld Congress on Scientific Computing, Paris(pp. 59—65).

Kothare, M., & Morari, M. (1995). Multiplier theory for stabilityanalysis of anti-windup control systems. In Proc. 34th Conf. onDecision and Control, New Orleans (pp. 3767—3772).

Kothare, M., & Morari, M. (1997). Stability analysis of anti-windupcontrol systems: a review and some generalisations. In Proc.European Control Conf. ECC +97, Brussels, Belgium.

Kothare, M., Campo, P., Morari, M., & Nett, C. (1994). A unifiedframework for the study of anti-windup designs. Automatica, 30,1869—1883.

Miyamoto, S., & Vinnicombe, G. (1996). Robust control of plants withsaturation nonlinearity based on coprime factor representations.In Proc. 35th Conf. on Decision and Control, Kobe, Japan(pp. 2838—2840).

Skogestad, S., & Postlethwaite, I. (1996). Multivariable FeedbackControl. New York: Wiley.

Sontag, E., & Sussmann, H. (1990). Nonlinear output feedback designfor linear systems with saturating controls. In Proc. 29th Conf. onDecision and Control (pp. 3414—3416).

Teel, A., & Kapoor, N. (1997). The L2

anti-windup problem: itsdefinition and solution. In Proc. European Control Conf. ECC+97,Brussels, Belgium.

Tyan, F., & Bernstein, D. (1995). Anti-windup compensator synthesisfor systems with saturation actuators. Int. J. Robust NonlinearControl, 5, 521—537.

Zames, G. (1966). On the input—output stability of time-varying non-linear feedback systems. part 1: Conditions derived using conceptsof loop gain, conicity and positivity. IEEE ¹rans. Automat. Control,AC-11, 228—238.

C. Edwards, I. Postlethwaite/Automatica 35 (1999) 761—765 765