Robust estimation of free-radical homopolymer reactors

Jesu s Alvarez *, Teresa Lo pez 1, Eduardo Herna ndez

Universidad AutoÂnoma Metropolitana-Iztapalapa, Depto. de IngenierõÂa de Procesos e HidraÂulica, Apdo. 55534, 09340 Mexico D.F., Mexico

Abstract

In this work is presented a practical (i.e. nonlocal) robust estimation design to infer the main safety, production, and qualityvariables of batch, semibatch, or continuous free-radical homopolymer reactors with on-line measurements of free monomer,

temperature, and volume. The solvability of the problem is established a priori, and the design features a robust nonlocal uniformconvergence criterion coupled with a systematic construction-tuning procedure. The nonlocal stability framework exhibits the per-formance-robustness tradeo�, as well as the relationship between the sizes of the estimate error, initial state, exogenous input, and

model parameter errors. The tuning of gains of the estimator can be executed with conventional ®ltering techniques for linearsingle-output ®lters. The polymerization of methyl methacrylate with AIBN initiator is considered as an application example, andthe robust nonlocal functioning of the estimator is corroborated with numerical simulations. # 2000 Elsevier Science Ltd. All rightsreserved.

Keywords: Chemical industry; Polymerization reactors; Robust nonlinear estimation; Nonlocal state estimation; Semiglobal estimation

1. Introduction

The problem of estimating the states of a nonlinearplant from its model in conjunction with secondarymeasurements is an important problem in polymerreactor engineering, with applicability and implicationsin conventional and advanced feedback control, super-visory control, product quality monitoring, data recon-ciliation and failure detection, software sensors, theassistance of experimental designs intended for modelvalidation or scale-up procedures, etc. In fact, theimplementation of a geometric [1] or model predictive[2] controller depends on the functioning of a robustnonlinear estimator.There are four main approaches to the design of

nonlinear observers: (i) the extended Kalman ®lter(EKF) [3], whose design is simple but lacks convergencecriteria and systematic tuning procedures, and bears theburden of having to solve a set of nonlinear di�erentialRiccati equations; (ii) the geometric observer (GO)design [4], which guarantees convergence with linearoutput error dynamics but applies to an extremelyrestrictive class of plants; (iii) the high-gain (HG)

approach [5], which guarantees convergence but has acomplex tuning procedure; and (iv) the sliding-mode(SM) approach [6], which guarantees robust stabilitybut has an elaborated design. These techniques arerestricted to nonlinear open-loop, observable plants,and their extensions to detectable plants depends on theidenti®cation by inspection of the observable andunobservable parts of the plant. In polymer reactorengineering the EKF is by far the most widely used stateestimation technique (see [7,8]), while the use of thelocal GO, HG, and SD approaches is a recent and lesswidespread development (see [8,9], and referencestherein). Since a global approach may be neither justi-®ed nor required in applied systems engineering, thenonlocal notions of practical [10] and semiglobal [11]stability have been introduced in state-feedback andobserver-based nonlocal nonlinear geometric control.However, as they stand, these designs cannot be appliedto open-loop estimators or to closed-loop plants withnon-geometric controllers. By combining features of theHG and GO observer designs, in [12] and [8] was pre-sented an (either observer or detector) estimator designthat featured a robust local convergence criterion, and asystematic construction-tuning procedure. This estima-tion methodology was applied to establish the solva-bility of the local nonlinear estimation problem of free-radical homopolymer reactors with measurements offree monomer, temperature, and volume. The local sol-vability conditions were interpreted with properties

0959-1524/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved.

PI I : S0959-1524(00 )00014-7

Journal of Process Control 10 (2000) 389±398

www.elsevier.com/locate/jprocont

* Corresponding author. Tel.: +52-5-724-4648-51; fax: +52-5-724-

4900.

E-mail address: jac@xanum.uam.mx (J. Alvarez).1 Present address: Centro de Investigacion en Polimeros, M. Achar

2, Tepexpan, Edo. Mex 55855, Mexico.

bearing physical meaning, and the numerical testing ofthe estimator exhibited an adequate degree of toleranceto model parameter and measurement noise errors. Thisnonlocal like robust functioning feature is in agreementwith the successful experimental implementationsreported earlier (see [8] and references therein) in massand solution free-radical homopolymer reactors. Thisclass of reactors, which captures the essence of a wideclass of industrial polymer reactors, has been the subjectof extensive earlier theoretical and experimental non-linear estimation and control studies (see [8] and refer-ences therein). Thus, the aim of the present work is thedevelopment of a rationale to establish the nonlocalrobust solvability of the nonlinear estimation problemof free-radical homopolymer reactors.In this work, is presented, a practical (i.e. nonlocal)

robust estimation design to infer the main safety, pro-duction, and quality variables of batch, semibatch, orcontinuous free-radical homopolymer reactors with on-line measurements of free monomer, temperature, andvolume. The solvability of the problem is established apriori, and the design features a robust nonlocal uni-form state convergence criterion coupled with a sys-tematic construction-tuning procedure. The outputmodel-plant mismatch is asymptotically eliminated withalmost LNPA (linear, noninteractive, and pole-assign-able) dynamics, enabling a gain tuning scheme thatamounts to the independent tuning of single-outputlinear ®lters, one for each measured signal. The non-local stability framework exhibits the nature of theperformance-robustness tradeo�, as well as the relation-ship between the sizes of the estimation, initial state, exo-genous input, and model parameter errors. Di�erentlyfrom the local approach employed in [8], where expo-nential convergence is guaranteed for su�ciently smalldisturbances (of unde®ned size), here the uniform con-vergence of the estimate is guaranteed for prescribedand compatible sets of disturbances and of estimationerrors. The polymerization of methyl methacrylate withAIBN initiator is considered as an application example,and the robust nonlocal functioning of the estimator iscorroborated with numerical simulations.

2. Reactor estimation problem

Consider the stirred tank reactor, where a stronglyexothermic free-radical homopolymerization takesplace, heat exchange being enabled by a heating/coolingjacket. The reactor dynamics are described by the fol-lowing six equations [13,14,15]:

I: � ÿrI I;T� � � "Irp I;m;T� � � ie ÿ qeI� �=V :� fI

m: � ÿ 1ÿ "m� �rp I;m;T� � � qe me ÿm� �=V :� fm; y1 � m

T: � � m� �rp I;m;T� � ÿ m;T;V� � Tc ÿ T� �

� qe 1ÿ "me� � Te ÿ T� �= 1ÿ "m� �V� � :� fT; y2 � T

V: � ÿ"Vrp I;m;T� � � qe ÿ q :� fV; y3 � V

�:0 � r0 I;m;T� ��"�0rp I;m;T� ��qe �0e

ÿ �0

ÿ �=V :� f�o

�:2 � r2 I;m;T� � � "�2rp I;m;T� � � qe �2e ÿ �2� �=V : f�2

The states of the reactor are: I (concentration of initia-tor), m (dimensionless concentration M=M0 of mono-mer, M and M0 are the molar concentrations ofmonomer and of pure monomer), T (temperature), �0

(zeroth moment of the chain length distribution -CLD-),�2 (second moment of the CLD), and V (volume). Theexogenous inputs are: ie (initiator feedrate), qe (inlet¯owrate), q (exit ¯owrate), Te (inlet temperature), andTc (coolant jacket temperature). The measured outputsare: m (using a densitometer [16] or a refractive indexsensor [17]), temperature T (using a thermocouple), andvolume V (by reactor gravimetry or by using a levelsensor). " is a contraction factor to account for the dif-ferent monomer �m� � and polymer �p

ÿ �densities.

rI; rp; r0, and r2 are smooth and strictly positive scalar®elds that describe the rates of initiator decomposition,polymerization, and of change of the zeroth and secondCLD moments, respectively. � and are ratios of heatgeneration and exchange capabilities to heat capacity,respectively. The above model can represent batch,semibatch, and continuous operations. Due to the pre-sence of the gel e�ect, a continuous reactor can havemultiplicity of steady-states [13], and a batch or semi-batch reactor can have an open-loop unstable statemotion [15].The reactor estimation problem is the following.

Based on the three-measurement vector y t� �, obtainrobustly convergent estimates of conversion c, averagemolecular weight Mn, and polydispersity Q; which arekey variables to monitor the safety level, the productionrate, and the product grade of an industrial reactor:

c � 1ÿm� �= 1ÿ "m� �; Mn � �m 1ÿm� �= 1ÿ "� ��0� �;Q � 1ÿ "� �= 1ÿm� �� �2 �2=�0� �

3. Motion stability

In this section is introduced the notion of nonlocaluniform robust stability that underlies the proposal forthe estimation scheme to reconstruct the state motionsof a nonlinear nonautonomous (i.e. with time-varyingexogenous inputs) dynamical system. In vectornotation, the polymer reactor can be written as follows:

390 J. Alvarez et al. / Journal of Process Control 10 (2000) 389±398

x: � f x; u t� �;r� �; y� h x;r� �; z� g x;r� �; x to� � � xo �1a�

xo2X�o� X�; x2X�; r2R�; u2U�; u t� �2 U� �1b�

where the state x� �, the exogenous input u� �, the mea-sured output y� �, and the output z� � to be estimated aregiven by:

x � I;m;T;V;�0;�2� �0; u � ie;qe;q;Te;Tc;me;�0e;�2e� �0;y � m;T;V� �0; z � c;Mn;Q� �0

r is the vector of model parameters, and U� is a set ofexogenous inputs. The input, the state, and the para-meter take values in the sets U�;X�, and R�, respec-tively, that are compact (i.e. bounded and closed) due tophysical and practical considerations. It must be poin-ted out that the sets X�o;X

�;R�;U�;U�: (i) are compa-tible with the solutions of the nonautonomousdynamical system [Eq. (1)], and (ii) some of them areeither given or to be found.If the entries of u t� � are piecewise continuous func-

tions of time, from the physics of the reactor followsthat the maps f and h are su�ciently smooth (i.e. dif-ferentiable) so that each set xo; u t� �; r�

uniquely deter-mines a (possibly unstable) state motion x�t� which inturn determines an output trajectory pair y t� �; z t� �� �

x t� � � �x t;to;xo; u t� �;r� �; y t� � � h x t� �;r� �; z t� � � g x t� �;r� �

�2�

In the next de®nition is given a uniform version of thenonlocal notion of practical stability [10]. First, somenotation and de®nitions must be introduced. Let X�o;U�,and R� be compact sets of initial state, exogenous input,and parameter perturbations with radii (i.e. sizes) ��o; �

�u,

and ��r , respectively:

X�o � �o j �o ÿ xok k4��o�

;

U� � � t� �j � t� � ÿ u t� � 4��u�

;

R� � �j �ÿ rk k4��r�

In other words, X�o;U�, and R� contain bounded initialstate, exogenous input, and parameter disturbancesabout the corresponding unperturbed values xo; u t� �,and r, respectively, of the nonautonomous dynamicalsystem [Eq. (1)]. Let S� denote the disturbance set

S� � X�o � U� � R�

and let Xo;U;R, and S be compact subsets ofX�o;U�;R�, and S�, respectively:

Xo � �o j �o ÿ xok k4�of g; U � � t� �j � t� � ÿ u t� � 4�u

� ;

R � �j �ÿ rk k4�rf g

S�Xo�U�R � S�; 04�o4��o; 04�u4��u; 04�r4��r

In the next de®nition is stated the notion of nonlocaluniform robust stability that is required to address thereactor estimation problem. Let � a; b� � be a real func-tions with real arguments a; b50� �. � is of class KK if ithas strictly increasing dependencies in a and b, and� 0; 0� � � 0 [18]. � is of class LK if it has strictlydecreasing dependency in a, strictly increasing depen-dency in b; � a; 0� � � 0, and � 0; b� � > 0 for all b > 0.

De®nition 1. The motion x t� � [Eq. (2)] of system 1 is saidto be PU(practically uniformly)-stable if, for each com-pact subset S � S�, there exists a LK-class function� :; :� � and a KK-class functions � :; :� � such that the per-turbed motions

� t� � � �x t; to; �o; � t� �; �� �; �o; � t� �; �� � 2 S

converge to the unperturbed motion x t� � [Eq. (2)],according to the following inequality:

� t� � ÿ x t� � 4� tÿ to; �o ÿ xok k� �� � � t� � ÿ u t� � s; �ÿ rk kÿ �

;

� t� � ÿ u t� � s� supt

� t� � ÿ u t� � In particular, if � t; s� � � celts and � a; b� � � baa� bbb,

the motion x t� � is said to be PE(practically exponen-tially)-stable.^

This de®nition of stability implies the boundedness ofthe perturbed trajectories � t� �, determining the follow-ing compact sets of motions about the unperturbed onex t� �: the ``excursion'' set X� S�� �, subsets X S� � of X� S�� �,the ultimate (i.e. to be reached asymptotically) setsX�F S�� �, and its subsets XF S� �

X S� � � � t� � k� t� � ÿ x t� ��� 4" �o; �u; �r� �� � X� S�� �

� � t� � � t� � ÿ x t� � 4"��� �

XF S� � � � t� �jk� t� � ÿ x t� �k4� �u; �r� �� � X�F S�� �

� � t� �jk� t� � ÿ x t� �k4� ��u; ��r

ÿ �� where " ; ;� � is a KKK-class (i.e. strictly increasing in allits arguments) function

" �o; �u; �r� � � � 0; �o� � � � �u; �r� �; "� � " ��o; ��u; ��rÿ �

which represents the size of the perturbed state motionset. The preceding notion of practical stability isequivalent to nonlocal uniformly ultimately bounded

J. Alvarez et al. / Journal of Process Control 10 (2000) 389±398 391

stability [18], and implies the following nonlocal notionsof stability: uniform asymptotic (i.e. for initial stateerrors: �o 6� 0; �u � �r � 0), uniform BIBO (bounded-input bounded-output) (i.e. for exogenous input errors:�o � 0; �u 6� 0; �r � 0), and structural (i.e. for parametererrors: �o � 0; �u � 0; �r 6� 0). Evidently, practical stabi-lity implies local stability with respect to initial state,exogenous input, and parameter disturbances: the per-turbed trajectories � t� � can stay arbitrarily close to theunperturbed trajectory x t� � by starting su�ciently closeto it and by making the disturbances su�ciently small.There is a fundamental di�erence between local andnonlocal stability approaches: in a local framework [8]the stability property is established for some ``su�-ciently small'' open set of disturbances of unde®ned size,in a nonlocal or practical framework the stability prop-erty is established for an either given or to be foundcompact set of bounded disturbances whose sizes theprocess systems engineer must not merely specify, butabove all compromise against the rate of convergence aswell as the sizes of the excursion X� � and ultimate bound(i.e. o�set) XF� � sets. In our reactor estimation problem,the unperturbed motion x t� � shall stand for the actualreactor motion, and the perturbed motion � t� � shall beregarded as its model-based estimate.

4. Estimability property

In this section the local [12] de®nition of RE-estimability of a state motion is extended to the nonlocalcase, and the nonlocal solvability of the reactor esti-mator problem is established. For this purpose, somede®nitions and notation must be introduced ®rst.The map � x; xu; r� � is said to be Rx-invertible

(robustly invertible for x) at the compact set �� (aboutx t� �; xu t� �; r� �) if � is continuously di�erentiable forx; xu� � and there is an inverse �ÿ1 so that

�ÿ1 � �; �u; �� �; �u; �� � � �; 8 �u; �; �� � 2 ��

� and �ÿ1 are (nonlocally) L(Lipschitz)-continuous ateach compact subset � � ��,

�� � � �; �u; �� �j � t� � ÿ x t� � 4"�;

k�u t� � ÿ xu t� �k4��xu; k�ÿ rk4��r ; t50 �3�

Let the set k � �1; . . . ; �mf g of positive integers (i.e.observability indices, one for each measured output)determine the maps �I and ':

�I x; xu; r� � � h1; . . . ;Lx1ÿ1f h1; . . . ; hm; . . . ;Lxmÿ1

f hm

� �0 �4�

' x; xu; v� � � L�1f h1; . . . ;L�m

f hm

� �0 �5�

where xu and v are vectors made of the inputs u� � andsome of their time-derivatives u i� � � diu=dti

ÿ �,

xu � u1; . . . ; u�1ÿ11 ; . . . ; up; . . . ; u�pÿ1p

h i0t� �;

v � u�l� �1 ; . . . ; u

�p� �p

h i0; �i > 0

and Lif� is the recursive directional derivative of the

scalar � x; t� � along the vector f x; t� �:

Li�1f � � Lf Li

f�ÿ �

; i51; L0f� � �; Lf� � �xf� �t;

�x � @�=@x; �t � @�=@t

The map �I determine the nÿ �� �-dimensional unob-servable dynamics x� t� � in the unobservable surface � t� �:

x: � � f x�; u t� �; r� �; x� 0� � � xo;

x� t� � 2 � t� � � x 2 X j �I x; xu t� �; r� � � ye t� �� �6�

where ye is a vector made of the measured outputs y� �and some of their time-derivatives

ye t� � � y1; . . . ; y �1ÿ1� �1 ; . . . ; ym; . . . ; y��mÿ1�

m

h i0t� �

De®nition 2. The motion x t� � [Eq. (2)] of the reactor [Eq.(1)] is PU(practically uniformly)-estimable at the com-pact set �� [Eq. (3)] if there are m � 3 integers (obser-vability indices) �1; . . . ; �3 �1 � �2 � �3 � �46; �i > 0� �and a map �II x; xu; r� � � ���1; . . . ; �n� �0 such that, at ��:

i. The map � x; xu; r� � � �0I; �0II

� �0�x;xu;r� [�I de®ned in

Eq. (4)] is Rx-invertible.ii. The map ' x; xu; v� � [Eq. (5)] is L-continuous.iii.The unobservable motion x� t� � [Eq. (6)] is PU-

stable.

If � � n [i.e. condition (iii) is trivially met] x t� � is PU-observable. If � < n; x t� � is PU-detectable.^

The next lemma is a straightforward consequence of:(i) the local version of the lemma in [8], and (ii) the ful-®llment of the nonlocal conditions of De®nition 2, onthe basis of the smoothness of the kinetics and heatexchange expressions of the reactor model [15].

Lemma 1. The motions [Eq. (2)] of the (batch, semi-batch or continuous) reactor [Eq. (1)] are PU-detectable

392 J. Alvarez et al. / Journal of Process Control 10 (2000) 389±398

with the following observability structures S kxIIf g [i.e.the pair index(k)-unobservable state xII� �] pairs:

S � 1; 1; 1; 1� �; I; �0; �2� �� ; 2; 1; 1� �; �0; �2� ��

;

1; 2; 1� �; �0; �2� �� ; 1; 1; 2� �; �0; �2� ��

:^

The preceding lemma says that the moment states�0; �2 are not observable from free-monomer, tem-perature, and volume measurements, and this is inagreement with well known facts in polymer reactionengineering [19].

5. Estimator construction

Having established the PU solvability of the reactornonlinear estimation problem for all the admissibleestimability structures, the construction and tuning ofthe corresponding PI(proportional-integral) nonlinearestimators follow from a straightforward application ofthe procedures provided by the nonlinear geometricestimation design methodology [8,20]. Since [20]addressed the e�ect of the observability structure on therobust functioning of the estimator, here we circum-scribe ourselves to consider only the observabilitystructure

S � 2; 1; 1� �; �0; �2� �� which, on the basis of observations made at the imple-mentation-testing stage, has been reported to yield thebest robust functioning or free-radical homopolymerreactors [8]. For the above estimability structure, thegeometric nonlinear estimator is given by

�:

u � Au�u � Ku su� �u t� �; � � �u�u; su > 0 �7a��:

m � KI so� � yÿ h �; �� �� �; so > 0 �7b�

�: � f �;�u�u; �� � � G �; �u; �; so� � yÿ h �; �� �� �

�H �; �u; �� ��m; � h �; �� �; � � g �; �� � �7c�

where � is an approximation of the model parameter r,� is the estimate of the state x, �u is the estimate of theaugmented input vector xu (made of u and some of itstime derivatives), �m is the estimate of the persistentmodeling errors in the input(u)-output(y) estimatorpath, and G and H are nonlinear gain matrices

G �; �u; �;So� � � �I �; �u; �� �Ko so� �;H �; �u; �� � � �I �; �u; �� ��o; �I;�II� � :� �ÿ1x �; �u; �� �

where � is an Rx-invertible nonlinear map,

� x; uI; r� � � x2; f2 x1; x2; x3; x4; uI; r� �; x3; x4; x5; x6� �0;uI � qe;me� �0

and Ku;Ko, and KI are parametrized gain matrices [1]

Ku su� � � diag suku11; suk

u12

ÿ �Ko so� � � bd sok

o11; s

2ok

o21

ÿ �0; sok

o12; sok

o13

h iKI so� � � diag s3ok

I1; s

2ok

I2; s

2ok

I3

ÿ �whose ``reference gain'' vectors (one for each measuredinput or output signal)

kTe � ku11; ku12

ÿ �0feed temperature� �

km � ko11; k

o21; k

I1

ÿ �0free monomer� � �8a�

kT � ko12; k

I2

ÿ �0reactor temperature� �;

kv � ko13; k

I3

ÿ �0reactor volume� � �8b�

are chosen so that the reference LNPA (linear, non-interactive and pole-assignable) output error dynamics

�:i � ku1i�i � 0; i � 1; 2; �i � u

i ÿ ui �9a�

�_�1 � ko11��1 � ko21�:1 � kI

1�1 � 0 �9b�

�� i � ko1i�:i � kI

i�i � 0; i � 2; 3; �i � i ÿ ui �9c�

are stable, according to the noise or robust oriented polepatterns given in Table 2 in [8], for the input and outputsignals, respectively. Accordingly, in terms of referencefrequency factors !u

i ; !1; !2; !3

ÿ �and damping factor

�1; �2; �3� �, the estimator gains are given by:

ku1i �!ui ; i � 1; 2; ko11� 2�1 � 1� �!1; ko21� 2�1 � 1� �!2

1;

kI1 � !3

1; ko1i � 2�i!i; kIi � !2i ; i � 2; 3 �10�

The time-scaling parameters su and so set the ratesof convergence of the input �� � and output � � esti-mates, respectively, and, as we shall see, their adequatetuning enables the possibility of obtaining a PU con-vergent estimate motion � t� � towards the actual reactormotion x t� �. The corresponding PU-stable unobservablemotion x��t� � �x�I �t��0; �x�II�t��0

� 0�t� and two-dimensionaldynamics are given by

x: �

II�t� � f5; f6ÿ �0

�x�I�t�;x�

II�t�;u�t�;r�;x�

II��x�

5;x�

6�0

x�I �t� � f ÿ1I � y1�t�; y2�t�; uI�t�; r�; y1�t�; y2�t�; y3�t��

where x�5 and x�6 are the zeroth �0� � and second �2� �moments of the chain length distribution, respectively

J. Alvarez et al. / Journal of Process Control 10 (2000) 389±398 393

and the nonlinear map f ÿ1I is the Rx1-inverse of thenonlinear map fI in the right-side of the initiator di�er-ential equation.In the next de®nition is introduced the robust and

uniform version of the nonlocal notion of semiglobalstability [11].

De®nition 3. The motion � t� � of the nonlinear estimator[Eq. (7)] PU-converges to the reactor motion x t� � for thedisturbance set S� if, for each disturbance subset S 2 S�

there is a tuning parameter pair su; so� �S (depending onS) such that the estimator motion � t� � PU-converges tothe reactor motion x t� �, for all �o; � t� �; �� � 2 S.

This de®nition says that the tuning parameter pairsu; so� � depends on the size of the disturbance subsetS � S�, and that this subset can be speci®ed or com-promised, depending on the design objectives andrestrictions, as well as on the kind of reactor motion.

6. Convergence

In this section are presented the main results of thepresent work: the proof of the PU-convergence of theestimator motion � t� �, the nature of the correspondingperformance-robustness tradeo�, and the relationshipbetween the size of the estimation error and the sizes ofthe initial state, exogenous input, and parameter dis-turbances.

Theorem 1. (Proof in Appendix B.) The motions � t� � ofthe estimator [Eq. (7)] with k � 2; 1; 1� � PU-converge tothe reactor motion x t� � if the tuning parameter pairso; su� � is set su�ciently large (i.e. fast) so that thefollowing inequalities are met:

sulu > soleo > ae

oMI:^ �11�

As explained in the local case discussed in [8], in thepreceding inequality, lu is determined by the referencegain kTe [Eq. (8a)] that sets the pole pattern [8] of thelinear exo-observer [Eq. (7a)], the pair (aeo; l

eo) is deter-

mined by the reference gains km; kT, and kV [Eq. (8)]that set the pole patterns [8] of the qLNPA output errordynamics, and the set lu; a

eo; l

eo

ÿ �is independent of the

operating conditions. MI=inf> is a state-set dependentLipschitz constant that measures the size of a nonlinearerror term qI� � in the quasi-linear observable errordynamics [Eq. (B1b) in Appendix B], and whose poten-tially source of divergence is dominated by making so

su�ciently large. If the state-set dependent constant MI

is replaced by its state-set independent limit m1 (as theradius " of the set X tends to zero), the nonlocalinequality [Eq. (11)] becomes the one of the local casederived in [8].

In the next corollary is given an explicit characteriza-tion of the PU-convergence features associated to theful®llment of the stability condition of Theorem 1. Forthis purpose, let us introduce ®rst some de®nitions. L�I

x

denotes the state-set dependent L (Lipschitz)-constantof the map � x; xu; r� � with respect to x;Mu;MI, and Mr

are L-constants of the nonlinear error feedback term qI

(de®ned in Appendix A) that is present in the quasi-lin-ear observable error dynamics in a suitable ¯atteningcoordinate system [12] associated to Condition (i) of thePU-estimability property (in De®nition 2);

qI e; v; er; t� � 4Mu euk k �MI eIk k �Mr�r �12�

au (independent of so and the state-set) is the amplitudeconstant set by the choice of pole pattern for the exo-observer; LI is an ``e�ective'' exponential decay constantrelated to the inequality condition of Theorem 1 [Eq. (11)]

LI � soleo ÿ ae

oMI > 0; lu � sulu

and eeo and eIIo are nonlinear transformations of theinitial observable and unobservable estimation errors,respectively.

Corollary 1. (Proof in Appendix B.) Let the PU-con-vergence condition [Eq. (11)] of Theorem 1 be met.Then,

(i) the output error PE-vanishes with (exponential)qLNPA dynamics, according to the followinginequality:

t� � ÿ y t� � 4aeo eeok keÿLL tÿto� � � aeo=LL

ÿ �Mu"u �Mr�r� �;

(ii) the state estimate w t� � and the output estimate z t� �PU-converge as follows:

� t� � ÿ x t� � 4L�ÿ1

z

� t� � ÿ y t� � � � tÿ to; eIIok k� �

� � "u; "e; �r� ��� L�ÿ1

r �r

� t� � ÿ z t� � 4Lgx � t� � ÿ x t� � � Lg

r �r

where:

"e � aeo eeok k � aeo=LI

ÿ �Mu"u �Mr�r� �;

eeok k4L�Ix �o � L�Iu "uo � L�Ir �r;

eIIok k4L�IIx �o; "u � au�uo � au=lu� ��v:^

Statement (i) says that, by making so su�ciently large,the output error vanishes exponentially. Statement (ii)says that one (observable) part of the state � t� �

394 J. Alvarez et al. / Journal of Process Control 10 (2000) 389±398

exponentially converges with arbitrarily fast dynamics,and that another (unobservable) part PU-convergeswith a dynamics imposed by the ones of the unobser-vable motion. From the inequality set of the corollaryfollows that the sizes of the estimation errors "y; "x; "z

ÿ �and of the disturbances �o; �xe; �r� � are related as follows

"y �o; �uo; �r; �v� � � aeo eeok k � ae

o=LI

ÿ �Mu"u �Mr�r� �;

"z �o; �uo; �r; �v� � � Lgx"x � Lg

r �r

"x �o; �uo; �r; �v� � � L�ÿ1

z "y � � t; to; eIIok k� � � � "u; "e; �r� �� �� L�

ÿ1r �r

and the ultimate bounds (as t tends to in®nity) are givenby:

"Fy �o; �uo; �r; �v� � � aeo=LI

ÿ �Mu"u �Mr�r� �;

"Fz �o; �uo; �r; �v� � � Lgx"x � Lg

r �r

"Fx �o; �uo; �r; �v� � � L�ÿ1

z "y � � "u; "e; �r� �� �� L�ÿ1

r �r

From these expressions follows that: (i) the outputo�set "Fy can be made arbitrarily small by tuning theparameter so su�ciently large (fast); and (ii) there willbe state "Fz

ÿ �and output "F

x

ÿ �o�sets, depending on the

quality of the estimator model. Summarizing, Theorem 1and Corollary 1 display the basic factors whose compro-mise determines the robust functioning of the nonlinearestimator and the nature of its underlying robustness-performance tradeo�.

7. Estimator functioning

Let us recall the test motion and the gains employedin Alvarez and Lopez' [8] locally RE-convergent design,for the polymerization of methyl methacrylate withAIBN initiator, in a continuous reactor that holds anominal volume V� of 2000 1 and has the followingnominal inputs: q�e � 40 l=min of pure monomerm� e � 1; �� 0e � �� 2e � 0� � at T� e � 300K are processed,the coolant temperature T� c is 300 K, the initiator feed-rate 1�e is 0.08 gmol/min, and the exit ¯owrate isq�=34.9359 l/min. The reactor has (at least) threesteady-states [13], one stable with high conversion, onestable with low conversion, and one unstable withintermediate conversion. Initially the reactor is incontinuous operation, at an unstable steady-state, witha volume Vo � 1500 l that is 500 l below its nominalvalue of 2000 l. At to � 0, the reactor is subjected to atime-varying vanishing feed temperature and exit ¯owrate

Te t� � � 300� 10eÿ4t=120 sin 2�t=60� �;Fig. 1. Evolutions of the reactor (±Ð), of the reactor with initial-state

and model parameter errors (ÐÐ), and of the estimator run with

model parameter errors (± ± ±).

J. Alvarez et al. / Journal of Process Control 10 (2000) 389±398 395

q t� � � q�e ÿ "Vrp I;m;T� � ÿ 500� � 4=125� �eÿ4t=125

exogenous disturbances so that the reactor reaches itshigh-conversion steady-state, undergoing considerableand abrupt changes over a rather large region of itsstate space. With small initial-state errors, the perturbedmotions reach either its low or a high conversionsteady-state, meaning that the reactor motion isunstable. To test the nonlocal robust estimator design,the model was run with the initial state and the para-meter errors listed in [8]. These errors signify a �16%underestimation of the rate of the heat-producing pro-pagation reaction, a �10% underestimation in the cap-ability of heat removal, and a �5% underestimation ofthe initiator e�ciency. The corresponding unperturbedand perturbed reactor motions are shown in Fig. 1:instead of reaching the terminal high-conversion steady-state of the unperturbed motion, the perturbed motionreaches the low-conversion steady-state. This divergentmotion con®rms the instability of the unperturbedreactor motion, must be seen as the zero-gain case of thenonlinear geometric estimator [Eq. (7)], and displayswhat the robust geometric estimator must achieve: tosteer back the perturbed motion (thin-continuousplots in Fig. 1) towards the ``actual'' reactor motion(thick-continuous plots in Fig. 1).The reference frequencies (minÿ1) and damping factor

[Eq. (10)] for the estimator are [8] !m; !T; !V;!Te� � �4 1=125; 1=75; 1=5; 1=3� � and �m=T=V � 0:71, withsu � 1; so � 10. The discontinuous plots of Fig. 1 showthe functioning of the estimator with initial state andparameter errors. As it can be seen in the ®gure, asexpected: (i) the states and outputs (initiator, monomer,temperature, and volume) which are more linked to theobservable part of the reactor converge fastly and with-out signi®cant o�set, (ii) the unobservable states (zerothand second moment) converge slowly and with someo�set within the error range of measurement devices,(iii) the settling time of the states more linked to theobservable part resembles the one (about 10 times fasterthan the one of the unobservable error dynamics) set bythe so-time scaled reference pole pattern. In [8] can beseen further details on the performance-robustnesstrade-o� of the proposed estimator design, includingnoisy measurements.

8. Conclusions

The (nonlocal) practical uniform solvability of thenonlinear state estimation problem has been establishedfor the class of free-radical homopolymer reactorsoperating in batch, semibatch, and continuos operationregimes, explaining the robust estimator functioningreported earlier in simulation and experimental studiesin the same class of reactors. The nonlocal robust

solvability condition has been established rigorously forany reactor motion. The estimator design featured asystematic construction-tuning procedure coupled to arobust convergence condition. The estimation frame-work: (i) identi®ed the basic components and tuningdecisions that lead to the ful®llment of the nonlocalrobust convergence condition, (ii) the nature of theperformance-robustness tradeo� that underlies the non-local robust functioning of the geometric type nonlinearestimator, and (iii) the relationship between the size ofthe estimation error and the sizes of the initial estimate,exogenous input, and parameter errors. In principle, thepresent qualitative approach could be endowed withquantitative measures of nonlocal estimability, alongthe methodological line presented in [20]; and should bethe point of departure to extend to the nonlocal case theestimator-based nonlinear geometric control presentedin [1] for the same class of reactors. The nonlocal stabi-lity treatment employed here should enable the con-sideration of the nonlocal case for the general class ofnonlinear plants addressed locally in [8], where theobservable and unobservable error dynamics exhibit atwo-way coupling.

Appendix A. Matrices and maps

bdM:=a block-diagonal matrix M

ÿu=o � bd ÿu=o1 ; . . . ;ÿu=o

p=m

h i; �u=o � bd �u=o

1

h i;

�u=o� bd �u=o1 ; . . . ;�u=o

p=m

h i; �u=o

i 1x�i� �= 1x�i� �� 1; 0; . . . ;0� �

Au=o � ÿu=o ÿ Ku=o�u=o; Aeo � ÿe

o ÿ Keo�e

o;

ÿu=oi �ix�i� �= �ix�i� � �

0 1 0 � � 0� �� �0 � � � 0 10 � � � � 0

266664377775;

�u=oi vix1� �= vix1� � �

0...

01

26643775; ÿe

o � bd ÿe1; . . . ;ÿe

m

� �;

�eo � bd �el ; . . . ; �e

m

� �; �e

o � bd �el ; . . . ; �em� �

;

ÿei � ÿo

i �i � 1� �= �i � 1� �; �ei � �oi ; 0ÿ �

; �ei � �oi0; 0

ÿ �0Ke

o � bdh

sokol ; . . . ; s�1o ko�1 ; s

�1�1o kIl

� �0;

. . . ; sokolm; . . . ; s�m

o ko�mm; s�1�1o kIm

� �0i

396 J. Alvarez et al. / Journal of Process Control 10 (2000) 389±398

T e0I; e0q

h i0� el; . . . ; e�1 ; e1; . . . ; e�m�1; . . . ; e�; e

qm

� �0eI � el; . . . ; e�1; . . . ; e�m�1; . . . ; e�

� �0; eq � e1; . . . ; eqm

� �0� z; v; r� � � ' x; xu; x

:u; r� ��

x��ÿ1 z;r� �;xu�zu:xu�ÿuzu��uv� �;

z � z0u; z0I; z0�

ÿ �0qI eu; eI; eII; v; er; t� � � � z� e; v; r� er� � ÿ � z; v; r� �� � z t� �;v t� �� �

w z; v; r� � � �IIxf� �IIux:

u� � x��ÿ1 z;r� �;xu�zu;x:

u��ÿuzu��uv� �! e; er; t� � � w z� e; v; r� er� � ÿ w z; v; r� �� � z t� �;v t� �� �

Appendix B. Proofs of Theorem 1 and Corollary 1

Recall the reactor [Eq. (1)] and its estimator [Eq. (7)],apply the error coordinate change

eu � �u ÿ xu; eI � �I �; �u; �� �;eII � �II �; �u; �� � ÿ �II x; xu; r� �;

eq � KI

�tto

h x; r� � ÿ h �; �� �� �d�

and obtain the estimation error dynamics (the matricesAu;�u;A

eo;�

eo, and T and the nonlinear maps qI and !

are given in Appendix A):

e:u � Aueu ÿ�uv t� �; v t� � 4ev; � � �ueu �B1a�

e:e � Ae

oee ��eoqI eu; ee; v t� �; er; t� �; � � �e

oee �B1b�

e:II � ! eu; eI; eII; er; t� �; e0e � T e0I; e

0q

h i0�B1c�

system (B1c) with eu; eI; er� � � 0 is the unobservabledynamics [Eq. (6)]. The nonlinear map qI is L-boundedin the set � � �� [Eq. (3)]. Integrate (B1a), take norms,and obtain (B2a). Integrate (B1b), take norms, sub-stitute [Eq. (12)], apply Gronwall's Lemma [18] followedby integration by parts, and obtain (B2b)

eu t� � 4au euok keÿ1u tÿto� � � au=lu� �"v; lu � aulu �B2a�

where

ee t� � 4aeo eeok keÿLI tÿto� �

� aeo

�tto

eÿLI tÿ�� � Mu eu �� � �Mr"r

� �d� �B2b�

where

etAu su�1� � 4aueÿlut; etA

eo so�1� � 4aeoeÿl

eot �B3�

According to inequality (B2b), the observable erroree� � is independent of the unobservable one eII� �.Take norms in (B2) and obtain (B4a,b). Recall De®-

nition 1 and write the PU-stability inequality of theunobservable motion eII t� � � 0 of system B1c, and writeinequality (B4c) (� is a LK-class function and � is aKKK-class function)

eu t� � 4au euok keÿ1u tÿto� � � au=lu� �"v �B4a�

ee t� � 4aeo eeok keÿLI tÿto� � � aeo=LI

ÿ �Mu"u �Mr�r� �;

"u � au euok k � au=lu� �"v �B4b�

eII t� � 4� tÿ to; eIIok k� � � � "u; "e; �r� �;

"e � aeo eeok k � aeo=LI

ÿ �Mu"u �Mr�r� � �B4c�

From condition [Eq. (11)] of Theorem 1 in conjunctionwith the inequality pair (B4a,b) follows that eu t� � andee t� � are PE-stable motions, and this in turn implies thatthe motion eII t� � is PU-stable, or equivalently, that themotion � t� � [Eq. (7c)] PU-converges to x t� �. Since thisstabilization can be done for any � � ��, according toDe®nition 3, the motion � t� � PU converges to x t� �. Thiscompletes the proof of Theorem 1. The inequalities ofLemma 1 follow from the inequality set (B4) and theapplication of the inverse of the above coordinatechange. QED

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