n, P

ra d

for

inuor cotureomeith

ments connect the nonlinear geometric and constructive control design techniques with conventional-like schemes employed in industrialpolymer reactor control. The proposed approach is illustrated and tested with a representative example through simulations.

bility decrease, the reactor exhibits strongly nonlinearbehavior, with asymmetric input-output coupling, multi-

sory control schemes. If the reactor is operated with orclose to gel eect, a free monomer controller driven by

studies, the related state of the art can be seen elsewhere[4], and here it suces to mention that: (i) only parts of

able insight and understanding have been gained, theconsideration of these highly nonlinear interactive andmodel-dependent controllers raises serious complexityand reliability concerns among industrial practitioners.Recently, a combined PI-inventory approach was used todraw a control scheme with linear components and reduced

* Corresponding author. Tel.: +52 55 5804 4958; fax: +52 55 5804 4900.E-mail address: jac@xanum.uam.mx (J. Alvarez).

Journal of Process Control 17plicity of steady-states, and parametric sensitivity [1,2].At the cost of less productivity, the reactors can be oper-ated in open-loop stable regime by choosing the conversionsuciently low. In industrial practice, these reactors arecontrolled with volume and cascade temperature linear PIloops, and the monomer content and molecular weight(MW) are regulated by adjusting the monomer and initia-tor and/or transfer agent dosages via supervisory or advi-

the multi-input multi-output (MIMO) control problemhave been addressed, (ii) a diversity of techniques has beenemployed, including linear PI decoupling [2] and modelpredictive (MPC) [5], as well as nonlinear geometric [3,69], MPC [10], and calorimetric [11,12] control techniques,and (iii) the controllers have been implemented withopen-loop [7], extended Kalman lter (EKF) [10], andLuenberger [8,9] nonlinear observers. Even though valu- 2006 Elsevier Ltd. All rights reserved.

Keywords: Nonlinear control; Constructive control; Polymer reactor control

1. Introduction

A wide variety of materials and products are manufac-tured in continuous free-radical polymer reactors. Due tothe strong exothermicity of the reaction and the presenceof the gel eect, which causes reaction autoaccelerationaccompanied by viscosity increase and heat removal capa-

an on-line measurement must be used [3]. Thus, the objec-tive of an advanced joint process-control design scheme isto attain an operation with the best compromise betweensafety, operability, productivity, and quality in the lightof investment and operation costs.

The polymer reactor control problem has been the sub-ject of extensive theoretical, simulation and experimentalConstructive control of co

Jesus Alvarez *

Universidad Autonoma Metropolitana-Iztapalapa, Depto. de Ingenie

Received 16 November 2005; received in revised

Abstract

The problem of controlling (possibly open-loop unstable) contand ow measurements is addressed. The application of a nonlinealinearity, decentralization, robustness and model independency feaand cascade temperature loops, a ratio-type feedforward free montroller, and (ii) a closed-loop nonlocal stability criterion coupled w0959-1524/$ - see front matter 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.jprocont.2006.09.007tinuous polymer reactors

ablo Gonzalez

e Procesos e Hidraulica, Apdo. 55534, 09340 Mexico, DF, Mexico

m 9 August 2006; accepted 26 September 2006

us free-radical solution polymer reactors with temperature, levelnstructive control procedure, with emphasis on the attainment ofs, yields: (i) a measurement-driven control scheme with PI volumer controller, and a material balance molecular weight (MW) con-conventional-like tuning guidelines. The methodological develop-

www.elsevier.com/locate/jprocont

(2007) 463476

ef Ploop unstable) continuous free-radical solution polymerreactors with temperature, level and ow measurements isaddressed. First, geometric control is applied to character-ize the solvability of the problem. Then, the application ofa constructive control procedure yields a MD controlscheme with: (i) decentralized linear PI volume and cascadetemperature loops, a ratio-type free monomer controller,and a material balance MW controller, and (ii) reducedmodel dependency, especially on the components that per-form the stabilization task. An input-to-state (IS) stabilityframework is employed to draw a closed-loop nonlocalrobust stability criterion coupled with conventional-liketuning guidelines. The approach is illustrated and testedwith a representative example through simulations.

2. Control problem

Consider the class of continuous stirred-tank reactorswhere an exothermic free-radical solution homopolymerreaction takes place. Monomer, solvent and initiator arefed to the tank, and heat exchange is enabled by a coolingjacket. Due to the gel eect [20], the reactor can presentsteady-state multiplicity [1,2]. From standard free-radicalpolymerization kinetics [1], and viscous heat exchange con-siderations [2,3], the reactor dynamics are described by thefollowing mass and energy balances (the model functionsare given in Appendix A):

_v qm qs em=qmr q : fv; zv yv v 1a_m r qmqm qm=v : fm; zm m 1b_T Dr qmcm qscsT T e UT T j=C : fT;zT yT T 1c

_T j UT T j qjcjT j T je=Cj fj; yj T j 1d_i r=pi it cdE=pI : fi; zi i 1emodel dependency [13]. However, the design is highlydependent on process insight, and lacks formal connec-tions with previous geometric control studies for polymerreactors.

From the nonlinear control theory we know that [1418]: (i) geometric controllers may be poorly robust, (ii)optimal stabilizing state-feedback (SF) controllers areinherently robust and passive (i.e., have relative degrees lessor equal to 1, and stable zero-dynamics) with respect tosome regulated output, and their measurement-driven(MD) implementation requires closed-loop detectabilitywith respect to the measured output, and (iii) the robustclosed-loop nonlocal stability assessment can be handledwithin an input-to-state (IS) stability framework [19]. Sincethe analytic construction of an optimal controller requiresthe solution of a HamiltonJacobiBellman nonlinear par-tial dierential equation, the direct optimality approach isnot a tractable procedure, and the removal of this obstacleis a central aim of the constructive control method [1618].

In this work, the problem of controlling (possibly open-

464 J. Alvarez, P. Gonzalez / Journal o_I ri wi qI=v : fi 1f_s qsqs qs=v : fs 1g_p r=pf2 r0=r=ip 2r=r0 mwig : fp 1hwhere the polymerization rate (r), initiation rate (ri), free-radical generation rate (r0) heat generation rate (Q), heattransfer coecient (U), heat capacity (C), and polymermass (p), are set by the nonlinear functions

r frv;m; T ; I ; s; Q Dr 2abU fUv;m; T ; T j; s; C fCv;m; s 2cdp fpv;m; s 2eri ET I : friT ; I 2fr0 cdET I itT ;m; s; r : f0v;m; T ; I ; s 2gFor the sake of simplicity, the function dependencies willbe occasionally omitted, and /(x1, . . . ,xn) will be simplywritten as /.

The states (x) are: the reactor (T) and jacket (Tj) temper-atures, the volume (v), the unreacted monomer (m), solvent(s) and initiator (I) masses, as well as the (number-average)MW inverse i M1n and its polydispersity (p). The mea-sured exogenous inputs (d) are: the reactor (Te) and jacket(Tje) feed temperatures, and the solvent (qs) volumetricowrate. The regulated outputs (z) are: the temperature(T), the volume (V), the monomer content (m), and theMW inverse (i). The measured outputs (y) are: the temper-ature (yT), the volume (yv), and the jacket temperature (yj).The control inputs (u) are: the coolant volumetric owrate(qj) through the jacket circuit, the exit owrate (q), themonomer owrate (qm) and the initiator mass feedrate(wi). The (number-average) MW inverse (i) state, with lin-ear dynamics (1e) in i, is used for control simplication andstability analysis purposes. In practice, one is interested inthe conversion (c) and the solid fraction (r) (q is the mix-ture density):

zc c p=vq s; zr r p=vqIn vector notation, the reactor control system (1) is writ-

ten as follows:

_x f x; d; u; pr; y Cyx; z Czx 3where

x v;m; T ; T j; i; I ; s; p0; u q; qm; qj;wi0;y yv; yT; yj0; z zv; zm; zT; zi0

f fv; fm; fT; fj; fi; fs; fp0; d T e; T je; qs0;f x; d; u; pr 0; y Cyx; z Czx

pr is the vector of model parameters, denotes the steady-state nominal value of (), and x can be nonunique, eitherstable or unstable. As it can be seen in (1), this system ishighly nonlinear and interactive. The denition of nonlocalstability that underlies the present study is stated next.

Consider a nonlinear system with time-varying exoge-nous input d (t):

rocess Control 17 (2007) 463476_e fee; de; e0 e0; f e0; 0 0 4a

f Pthere is a KL-class (increasing-decreasing) function a anda K-class (increasing) function c so that

jetj 6 aje0j; t t0 ckdetk; tP 0 4bkdetk sup

tjdetj

where a (or c) bounds the transient (or asymptotic) re-sponse. The (necessary and sucient) Lyapunov character-ization of the IS stability property is given by [19]

a1jej 6 V e 6 a2jej; _V a3jej a4kdek 4cwhere V is a positive denite radially unbounded functionand ai is a K-class function.

Our problem consists in designing a feedback controllerthat, driven by the measurements (y and d), regulates thereactor operation about a (possibly open-loop unstable)nominal steady state x. We are interested in: (i) solvabilityconditions with physical meaning, (ii) closed-loop stablefunctioning, (iii) a systematic construction-tuning proce-dure, and (iv) the attainment, as much as possible, oflinearity, decentralization, robustness and modeling inde-pendency features.

3. Solvability assessment

Assuming that the detailed reactor model (1), (2) and itsstate (x) are known, in this section the related nonlinear(feedforward) FF-(state-feedback) SF control problem isaddressed with geometric [14] and constructive methods[18]. The purposes are the identication of the underlyingsolvability conditions and of the behavior attainable withrobust FF-SF control.

3.1. Relative degrees (RD) and zero-dynamics (ZD)

Regarded individually, the coolant ow-temperatureinputoutput pair qj zT has relative degree (RD) equalto 2, the associated controller (qj) requires the monomerfeed (qm) control time derivative, and consequently, the4-input (u) 4-output (z) reactor system (1) does not haveRDs, meaning that the FF-SF problem cannot be solvedwith static control [14]. From the application of thedynamic extension procedure [14], the next proposition fol-lows [the function f(x1, . . . ,xn) is said to be xi-monotonic ifoxi f is of one sign]:

Proposition 1 (Proof in Appendix B). With the dynamicextension (5a), the augmented reactor system (1, 5a) hasrelative degree vector j (5b)

_q; _qm tq; tqm 5aj jv; jm; jT; ji 2; 2; 2; 2 5b

if and only if, in a compact set about the nominal operation:

m=v 6 q ; T je 6 T j 6abThe steady-state e = 0 is input-to-state (IS) stable [19] if

J. Alvarez, P. Gonzalez / Journal om

fT is T j-monotonic; fi is I-monotonic: r 6cdPhysically speaking, the preceding conditions are alwaysmet because: the free monomer is part of the reacting mix-ture [m < qmv) (6a)], (ii) there is heat exchange betweenthe reacting mixture and the cooling jacket uid[Tje < Tj ) (6b)], (iii) the heat exchange rate U(Tj)(T Tj)is uniquely determined by the jacket temperatureT joT jUT T j < 0) 6c, and (iv) the MW decreaseswith the initiator content [oIfi0) (6d)].

The corresponding zero-dynamics (ZD) are given by

_s skoqs; z; qs; pr qsqs 7a_p kor s; z; prp 2 mwi 7bwhere fi is dened in (1), and fr and fp are given in Appen-dix A

koq qo=v; kor ro=po; ro frv; m; T ; loI s; spo fpv; m; s

qo 1 emro=qm qs=1 m=pof iv; m; T ;i; I ; s 0 ) I loI s

and loI s denotes the solution for I of the algebraic equa-tion fi = 0.

Physically speaking, the ZD (7) represent the reactorbehavior [13] in perfect material balance control [21], withthe mass and the energy (u) delivered to the system exactlybalanced against the load demand z; d without gel eect,or equivalently, without the sole potentially destabilizingmechanism at play [3]. The ZD stability property is for-mally stated in the next proposition.

Proposition 2 (Proof in Appendix B). The ZD (7) are ISstable, with respect to the exogenous input dz = (d,z,pr), if in

a compact set about the nominal operation:

skoq is s-monotonic; koq > 0; kor > 0: r 8acIn a practical reactor, the preceding conditions are alwaysmet because: the residence time is strictly positive (8b), kor issimilar to koqr=p q=v, and the presence of the solvent ismeant to attenuate or dominate the destabilization by geleect, or equivalently, the solvent per se does not inducesteady-state multiplicity (8a).

For its use in subsequent stability analysis, let us recallfrom the proof of the last proposition the ZD IS Lyapunovfunction (LF) characterization:

V o e2s e2p=2 es s s; ep p p 9a_V o aoes; ep; do boes; doP 0 8jesj 6 cos kdokjepjP copkdok 9b

where the stabilizing (ao) and potentially destabilizing (bo)dissipation rates, as well as the asymptotic gains cos ; cop aregiven in Appendix B (B6), do is the exogenous input for theZD, and ea denotes deviation of a from its nominal value:

rocess Control 17 (2007) 463476 465do do0p ; dqs; _dqs0; dop ev; em; eT; ei; e0pr0 ea a a

3.2. FF-SF geometric control

From the enforcement of the linear, noninteractive, poleassignable (LNPA) output regulation dynamics

ea fcaxca _ea xca2ea 0 a v;m; T ; i 10the FF-SF nonlinear geometric dynamic controller follows[the functions are given in (B4) of Appendix B]

_q lgqv;m; T ; T j; i; I ; s; q; qm; T e; qs; _qs 11a_qm lgqmv;m; T ; T j; i; I ; s; q; qm; T e; qs; _qs 11bwi lgwiv;m; T ; i; I ; s; q; qm; T e; qs 11cqj lgqjv;m; T ; T j; i; I ; s; q; qm; T e; T je; qs; _T e; _qs 11d

From well-known results in geometric control [14,15],the IS stability of the resulting closed-loop system is a con-sequence of the IS stability of the LNPA (10) and ZD (7).This controller: (i) has two dynamic components (11ab)and two static ones (11cd), (ii) has a dynamic-to-static

466 J. Alvarez, P. Gonzalez / Journal of Pcomponent cascade interconnection, (iii) requires the deriv-atives of two exogenous inputs (Te and qs), and (ii) the fourcomponents depend on the entire state-input pair xu.These dynamic, interaction and model dependency featuresare depicted in Fig. 1.

3.3. Passive FF-SF control

From a constructive control perspective [16,18], the lackof robustness is a major drawback of the geometric con-troller (11), because it is not underlain by a structure withall the relative degrees equal or less than 1. Following theconstructive control approach, this high RD obstacle forrobustness is then removed by passivation via backstepping[16,18]. For this aim, let us rewrite the four-state subsystemFig. 1. Dependency diagram of the nonlinear geometric (11) and passive(16) controllers.(1cf), associated with the inputoutput pair (qj,wi) (yT,yi), in the following way:

_T aTv;m;T ;T j; sT j bTv;m;T ;T j; r; s;qm;T e;qs 12a_T j ajT j;T jeqj bjv;m;T ;T j; s 12b_i aiv;m;T ; sI biv;m;T ; r; i; s 12c_I kiv;T ;qI wi 12d_p kpv;m;T ; r; s; I ; ip cpv;m;T ; r; i; I ; s 12ewhere

aT U=C; aj cjT je T j=Cj 13abki kq E; ai cdE=p; kq q=v 13cebT Dr qmcm qscsT T e UT =C 13fbj UT T j=Cj; bi kri it 13ghkp kr2 r0=r=i 13icp kr2r=r0 mwi; kr r=p 13jk

Introduce the LF [Vo has been dened in (9)]

V p V pv V pm V pT V pt V o; V pT e2T e2j =2ej T j T j ; ei I I 14a

V pv e2v=2; V pm e2m=2; V pi e2i e2i =2V o cvse2s cvpe2p=2 14b

where T j (or I*) is the jacket temperature (or initiator con-

tent) virtual control (i.e., setpoint).From the enforcement of the dissipation rates

_V pv kve2v; _V pm kme2m 15ab_V pT kTe2T kje2j ; _V pi kie2i kie2i 15cdupon the reactor dynamics (1ab, 12), the nonlinear staticFF-SF passive controller follows:

q r kmemqmkvev qs emr=qm=qmm=v 16aqm r kmemmkvev qs emr=qm=v=qmm=v

16bwi _I kiei kiI aiei; I bi kiei=ai 16cdqj _T j kjej bj aTeT=aj 16eT j bT kTeT=aT 16for equivalently,

q lqv;m; T ; I ; s; qs; qm lqmv;m; T ; I ; s; qs 17abqj lqjv;m; T ; T j; i; I ; s; T e; T je; qs; _T e; _qs 17cwi lwiv; T ;m; i; I ; s; T e; qs 17din a form obtained after performing the time derivations of(16d,f). Take the derivative of the overall LF (14), recall theZD dissipation rate _V o (9b), and apply standard Lyapunovstability arguments [19,22] to draw the dissipation rate

_V p _V p _V p _V p _V p _V o < 0

rocess Control 17 (2007) 463476v m T i

8jejP apkdk; d d0d; _d0d; e0pr0 18

f Pimplying the IS stability of the closed-loop reactor (1) withthe passive controller (17).

Comparing with the geometric controller [(11) andFig. 1], the passive controller (17): (i) has only static ele-ments, (ii) depends on the derivatives of the exogenousinputs Te and qs, (iii) is considerably less interactive, and(iv) is less model dependent, in the sense that the volume(17a) and monomer (17b) controllers do not need partialderivatives of the polymerization rate (fr) and heat transfer(fU) functions. These dynamic and interaction features ofthe passive controller are depicted in Fig. 1.

3.4. Optimality and connection with MPC

From standard arguments in constructive nonlinearcontrol [16,18], the passive controller (17) is an optimal sta-bilizing controller with respect to the objective function

J0;1 Z 10

Lpx; d; up Lsx; d; up; usdt 19

up q; qm; T j ; I; us qj;wi0

Lp Lv Lm LT Li; Ls Lj Liwith primary (Lp) and secondary (Ls) Lagrangian terms

Lv e2vk2v fv; Lm e2m k2m fm; LT e2Tk2T fT;Li e2i k2i fi; Lj e2j k2j fj; Li e2i k2i fi; f T aTT j bTaTej;fi aiI biaiei;fv;fm;fT;fi0 : fpx ; fj;fi0 : fsx

The objective function J is meaningful, in the sense that theregulated outputs and control deviations are eectivelypenalized [16]. This is so because the primary (or second-ary) control up (or us) and the function fpx (or fsx ) are inone-to-one correspondence, and consequently, penalizingfpx (or fsx ) is equivalent to penalizing up (or us). This opti-mality property veries the robustness property of the pas-sive controller (17), and above all reveals a key connectionwith MPC: the analytically constructed passive controller(17) is equivalent to a particular choice of input-outputweighting scheme in an unconstrained model-based MPCdesign over an innite receding horizon [23]. That optimalcontrollers are robust with respect to the particular choiceof weighting function, is a well known fact in linear [24]and nonlinear [16,17] control theory.

3.5. Concluding remarks

The solvability of the robust FF-SF reactor controlproblem has been established with conditions (6, 8) thatbear physical meaning, and these conditions: (i) are genericin the sense that they are met by the entire class of ade-quately designed solution homopolymer reactors (1),regardless of the particular model employed, and (ii) arein agreement with well-known facts and arguments in poly-mer kinetics [25], reaction engineering [3,26], and material

J. Alvarez, P. Gonzalez / Journal obalance control [13,21]. Moreover, the quantitative assess-ment of the control solvability conditions in the light ofequipment characteristics and operating conditions pro-vides a fundamental connection between reactor processand control designs [12]. The a posteriori verication ofthe control robustness property yielded a connection bet-ween passive and MPC polymer reactor control designs.

Finally, it must be pointed out that the IS stable reactordynamics (7), of the (exact model-based) nonlinear passivecontroller (17) represents: (i) the behavior attainable withany robust controller, and (ii) the recovery target for theMD control design of the next section.

4. Measurement-driven control

In this section, the closed-loop behavior of the exactmodel-based passive nonlinear controller (17) is recoveredwith: (i) an interlaced control-observer design on the basisof control model with suitable detectability structure, and(ii) emphasis on the attainment of linearity, decentraliza-tion, and model independency features.

4.1. Control model

From previous polymer reactor nonlinear estimationand control studies [12,27,28], we know that the estimationmodel and its structure are design degrees of freedom thatcan be exploited to attain maximum estimator robustnessin the light of a specic estimation and objective. Followingthese ideas, recall the reactor subsystem (1) and the subsys-tem form (12), replace the nonlinear function pair bT bj(13) by its image value bT bj, introduce the map bv = bv(21c) in the volume dynamics (1a), regard the derivativesof the jacket temperature and initiator setpoints as dynam-ical states (bj and b

i ) under assumption (20j) (which is

standard in signal processing to draw derivative estimates[13]), and obtain the control model

_v q bv; yv v; _T aTT j bT; yT T20ab

_T j ajqj bj; yj T j 20c_m kqm qmqm r; _s kqs qsqs 20de_I kiI wi; bj ; bi _T j ; _I 20fg_i aiI bi kri ci fi 20h_p kpp cp fp; _bv; _bT; _bj; _bj ; _bi 0 20ijwhere the function set {aT,aj,ai, bi,cp,ki,kq,kp} has beendened in (13), and

aT ^U= ^C aT; aj c^jT je T j= ^Cj aj 21ab bv; bT; bi0 bv; bT; bj0 : b 21bbv qm qs emr=qm : bvr; qm; qs; ci aiI krit

21cdr fCjbj CbT U=CCT j cjq T je T je T j T j

rocess Control 17 (2007) 463476 467j

cmqm csqsT e T g=D : bry; b;m; s; u; d 21e

ven by (y,b) [12], provided the subsystem is closed-loop

_^V v xv~b2v _bv~bv; _^V m kq ~m2 ~m~wm ~r 25bc

f PIS stable. Accordingly, the reaction rate pair rU (2a, c)can be on-line reconstructed without needing the reactionrate-heat exchange function pair (fr, fU), and this is modelindependency feature agrees with the calorimetric estima-tion-based control approach for exothermic reactors[12,30].

4.2. State estimator

Recall the auxiliary estimator (22), replace its algebraic-dierential subsystem (22ad) by the battery (23ae) oflinear reduced-order single-input observers [30,31], andobtain the state estimator for the reactor-control modelpair (1,22):

_vv xvvv xvxvyv q; b^v vv xvyv 23a_vT xTvT xTxT yT ajy j; b^T vt xT yT 23b_vj xjvj xjxjy j ajqj; b^j vj xjyj 23c_vj xvj x2j T j ; b^j vj xj T j 23d_vi xvi x2i I; b^i vi xi I 23e_^m qmqm qm^=yv r^; _^s qs^=yv qsqs 23fg~r bryr; b^; m^; s^; u; d 23h_^I q=yv EyTI^ wi; _^i fiyv; yT; r^; m^; s^; I^ ; t^

23ij_^p fpyv; yT; r^; m^; s^; I^ ; i^; p^ 23kwhere xv;xT;xj;xj and x

i are adjustable gains, and theaT (or aj) is an approximation of the steady-state value ofthe function aT (or aj). Knowing that the passive controller(17) does not depend on the polydispersity state (p), thisstate has been introduced for monitoring and model cali-bration purposes.

From the control model (20) the auxiliary estimatorfollows

bv q _yv; bT _yT aT yj 22abbj _yj ajqj; b : bj ; bi _T j ; _I : y 22cd_m qm=v qmqm r; _s qs=v qsqs 22efr bry; b;m; s; u; d 22g_I q=v EyTI wi 22h_i fiv; T ; r;m; s; I ; i 22i_p fpv; T ; r;m; s; I ; i; p 22j

This dierential-algebraic system characterizes the controlmodel detectability structure [27,28]: (i) the unknown input(b,b*) is instantaneously observable [29] because it is time-wise determined by y; _y; y, and (ii) the states (m, s, I, i,p)can be reconstructed by integrating subsystem (22ej) dri-

468 J. Alvarez, P. Gonzalez / Journal osetpoint value T j (or I*) acts as a measurement in

(23d) [or (23e)]._^V T xT~b2T xj~b2j xj ~b2j _bT~bT _bj~bj _bj ~bj 25d_^V i kr~i2 ki~I2 x~b2i i^~i~kr _bi ~bi 25e_^V o kq~s2 kp~p2 ~s~ws ~p~cp p^~kp 25f

wm qmqm; ws qsqs; _bT T aT _T j; _bj T j aj _qj;_bv v _q; _bj T j _bi I

4.3. Measurement-driven (MD) controller

Regard the components of the control (14) and estima-tor (24) LF, set the next four LFs, one for each regulatedoutput:

V v V pv V^ v; V T V pT V^ T 26abV m V pm V^ m; V i V pi V^ i 26cd

and proceed to execute a component wise control con-struction.

Volume controller. Take the time-derivative of the LF(26a), substitute (20a) and (25b), and enforce the controlexpression (27a) to obtain the dissipation rate (27b):

b^v q kvyv v 27a_V v kve2v xv~b2v sve;~xe; d; _d; _u 27b

sv _bv ev~bv~xe x^e xe; xe bv; bT; bj; bj ; bi ;m; s; I ; i; p0Since the detectability property is not invariant underfeedback control, the estimator convergence will beassessed in closed-loop regime. For this aim, let is intro-duce the estimator Lyapunov function V^ , with a passivestructure that is compatible with the one (14) of the nonlin-ear passive controller (17) [~ ^ denotes the esti-mation error of ()]:

V^ V^ v V^ m V^ T V^ t V^ o 24aV^ v ~b2v=2; V^ m ~m2=2 24bcV^ T ~b2T ~b2j ~b2j =2 24d

V^i ~i2 ~I2 ~b2i =2; V^ o ~s2 ~p2=2 24ef

The derivation of V^ followed by the substitution of thecontrol model-estimator pair (22)(23) yields the dissipa-tion rate

_^V _^V v _^V m _^V T _^V i _^V o 25a

rocess Control 17 (2007) 463476d d0d ; d0pc0; dd d d; dpc pc pc

f PTemperature controller. Take the time-derivative of theLF (26b), substitute (22bc) and (25d), and enforce the pri-mary (28a) and secondary (28b) control expressions toobtain the dissipation rate (28c):

aT T j b^T kTyT T 28aajqj b^j aTyT T b^j kjT j T j 28b_V T kT e2T xT~b2T kje2j xj~b2j xj ~b2j sTe;~xe; d; _dsT _bT eT~bT _bj ej~bj _bj ej~bj 28c

Monomer controller. Take the time-derivative of the LF(26c), substitute (20d) and (25c), enforce the control expres-sion (29a) to obtain the dissipation rate (29b) (-m is anadjustable control gain):

qm^=yv qmqm r^ kmm^ m; km -mq=yv29a

_V m kqhem; ~m;-m sme;~xe; d; _d 29bsm ~m em~wm ~rwhere h is a parameterized ellipsoidal form

hx; y;- : -x2 - 1xy y2 > 0- 2 -;-; - 3 2

2

p29c

MW controller. Take the time-derivative of the LF(26d), substitute (20f, h) and (25e), and enforce the primary(30a) and secondary (30b) control expressions to get thedissipation rate (30c):

aiyv; yT; m^; s^I biyv; yT; r^; m^; s^; i^ ki^ii 30a q=yv EyTI^ wi aiyv; yT; m^; s^^ii b^i kiI^ I 30b_V i krhei;~i;-i kihei;~I ;-i sie;~xe; d; _d; 30c-i ki=kr;-i ki=kisi kiei a^iei~I krei a^iei~i ei ~ii~i~kr

_bi ei~bi ~i ei~ciFinally, solve Eqs. (27a), (28ab), (29a), and (30ab) for

the controls q; T j qj; qm and I* wi, respectively, incor-porate the estimator (23), and obtain the measurement-dri-ven (MD) controller:

Volume controller 31aq kvyv v vv xvyv_vv xvvv xvxvyv q Temperature controller 31bT j b^T kTyT T =aTb^T vT xT yT; _vT xTvT xTxT yT ajy jb^j vj xj T j ; _v xj vj x2j T jqj b^j kjy j T j b^j aTyT T =aj

J. Alvarez, P. Gonzalez / Journal ob^j vj xjyj; _vj xjvj xjxjy j ajqj Monomer controller 31c_^m qm^=yv qmqm r^; _^s qs^=yv qsqs;r^ bry; b^; m^; s^; u; dqm ^r qm^=yv kmm^ m=qmkm -mq=yv; -m 2 -;- Molecular weight controller 31d_^I q=yv EyTI^ wi_^i fiyv; yT; r^; m^; s^; I^ ; i^I biyv; yT; r^; m^; s^; i^ ki^ii=aiyv; yT; m^; s^ki -ikryv; r^; m^; s^b^i vi xi I; _vi xi vi x2i I; -i;-i 2 -;-wi b^i kiI^ I q=yv EyTI^

aiyv; yT; m^; s^^ii; ki -ikiyv; yT; q Polydispersity estimator 31e_^p fpyv; yT; r^; m^; s^; I^ ; i^; p^Comparing with the detailed model-based nonlinear geo-metric (11) and passive (17) FF-SF control implementa-tions with (say geometric [3,79]) nonlinear estimators,the above MD controller is considerably less model-depen-dent. Specically: the volume controller (31a) is modelindependent, the temperature controller (31b) needs twostatic parameter approximations (aT and aj) (21), themonomer controller needs calorimetric parameters (pc)(i.e., densities and specic heats), and the MW controller(31d) and polydispersity estimator (31e) need the initia-tion-transfer kinetics parameters. In addition, the propaga-tion-termination kinetic (fr) and heat transfer (fU) modelsare not needed. The MD control interaction, decentraliza-tion, and model dependency characteristics are presented inFig. 2.

In classical PI form, the volume controller (31a) is givenby

q jv ev s1vZ t0

evsds

; ev yv v 32

ja aaxa ka1; sa x1a k1a ; a v;T ; jand the primary (33ab) and secondary (33c) temperaturecontrollers can be written as follows

_^bj xj b^j xj T j 33a

T j jT eT s1TZ t0

eTsds

; eT yT T 33b

qj b^j jj ej s1jZ t0

ejsds

; ej y j T j 33c

where (33b) is a primary-to-secondary feedforward lag ele-ment that performs the setpoint dierentiation.

Thus, from an industrial perspective: (i) the monomer(31c) and MW (31d) SISO components are inventory

rocess Control 17 (2007) 463476 469controllers driven by information generated in the temper-ature controller (31b), (ii) the monomer controller has a

pos

f Pratio-type feedforward component r^=qm that sets a feed-ow qm contribution, (iii) the MW controller has a cascadestructure, with a primary (or secondary) element that setsthe initiator setpoint (or initiator feedrate), including aratio-type correction by chemical reaction r^=p^, and (iv)

Fig. 2. Dependency diagram of the pro

470 J. Alvarez, P. Gonzalez / Journal othe functioning of the volume (31a), temperature (31b),and monomer (31c) controllers is independent of the oneof the MW controller.

4.4. Closed-loop dynamics and tuning

Having as point of departure the LFs (14) of the passivenonlinear controller, the open-loop estimator (24), and theclosed-loop component-wise design (26), the application ofLyapunovs direct method within the IS stability frame-work yields the closed-loop robust stability criteria statedin the next proposition.

Proposition 3 (Proof in Appendix B). Consider the polymerreactor (1) with the MD controller (31). The closed-loopreactor is IS-stable if: (i) the control gains of the unmeasured

outputs (m and I) are chosen as follows:

-m;-i;-i 2 -;- - 3 22

p34a

and (ii) the gains of the primary (kv,kT) and secondary (kj)

control and estimator xv;xT;xj;xj ;xi components aretuned so that the following dynamic separation conditions

are met:

kp < kp < kp cp kj; kj < kj cj x; x < x

34bdwhere kp minkv; kT; x minxv;xT;xj;xj ;xi rConditions (34a) ensure the stability of the state (m, i, I)subsystem associated with the unmeasured-regulated out-puts. In fact, the value that maximizes the area of the sta-bilizing ellipsoidal function h (29c) is -max 3, andtherefore -m;-i;-i should be chosen between 1 (without

ed measurement-driven controller (31).

rocess Control 17 (2007) 463476cross terms in h) and 3. In other words, the unmeasuredoutput gains should at most be three times faster thanthe related dilution rate (q/v or r/p). In Conditions (34bd): (i) x+ is an upper limit imposed by the high frequencyunmodeled dynamics, (ii) kp is a lower primary gain limitset by the open-loop reactor instability, and (iii) cj (orcp ) is an isotonic function that sets an upper limit k

j (or

kp ) for the secondary (or primary) gain, depending onthe faster estimation (x) [or secondary (kj)] gain. Thus,the choice of gains aects and is aected by the sizesof the prescribed (or to be compromised) initial state, exog-enous input, model parameter disturbances, and this is inagreement with a practical stability framework [17,32].

From the preceding stability assessment (34) in conjunc-tion with conventional tuning rules for linear PI controllers[33] and lters [34] the next tuning guidelines follow. (i) Set:the unmeasured output gains at their nominal values oftheir associated dilution rates -m -i -i 1, the con-trol gains at the nominal inverse residence time kv kT kj q=v, and the estimator gains about three times fasterx xv xT xj xi xj 3q=v. (ii) Increase x upto its ultimate value xu (with oscillatory response), andback o so that the behavior is satisfactory (x 6 xu/3).(iii) Increase the volume gain kv up to its ultimate valuekuv, and back o kv 6 kuv=3 for adequate satisfactoryresponse. Repeat the procedure for the secondary temper-ature gain kj kj 6 kuj =3. (iv) Increase the temperature

gain kT up to its ultimate value kuT, and back o until an

adequate response is attained. (v) Apply the sameincrease-back o procedure to the unmeasured outputgains -m;-i;-i, in the understanding that they cannotbe larger than two to four times the related dilution rates(-a 6 2-to-4, a = m, i, i). (vi) If necessary, adjust the estima-tor gains xv;xT;xj;xj ;xi .

4.5. Concluding remarks

Comparing with a previous PI-inventory design withsimilar structure [13], here the temperature (or MW) sec-ondary controller (31b) [or (31d)] has an additional term(aTeT) [or (aie)] that compensates the modeling error asso-ciated with the primary controller design within a conven-tional cascade framework: the intermediate state is at its

ibration aims.

ues (listed in Appendix A) were taken and/or adaptedfrom Alvarez et al. [3], and the nominal inputs wereadapted from a previous solution copolymer reactor study[28]. The reactor has three steady-states (Table 1), with twoof them corresponding to extinction and ignition open-loop stable operations, and one of them being open-loopunstable. In the spirit of the nonlocal IS stability frame-work (4) employed in the control design developments,the unstable steady-state is chosen as the nominal operat-ing point, and the resulting closed-loop system will be sub-jected to initial state, input (persistent model parameterand step/sinusoidal exogenous input) disturbances, andthe kind of transient, a symptotic and combined tran-sient-asymptotic responses will be analyzed.

The adequate fulllment of the relative degree (6) andZD uniqueness with IS stability (8) solvability conditions

closed-loop system was subjected to: an initial state devia-

mete

Tj (K) 329.53

, d

J. Alvarez, P. Gonzalez / Journal of Process Control 17 (2007) 463476 471Mn (kg/kmol) 399149.03I (kg) 1.685s (kg) 501.283p 1.9997c 0.1072r 0.08066

Nominal inputs

u qj; q; qm; wi0 = (30 L/min, 9.1 L/min, 7.34 L/min, 0.0078526 kg/min)05. Application example

To subject the controller to a severe test, let us consideran extreme case of an industrial situation: the operation ofa reactor at high-solid fraction with the potentially destabi-lizing gel-eect at play, about a nominal steady-state whichis open-loop unstable. The monomer is methyl methacry-late, with ethyl acetate (solvent) and AIBN (initiator).The residence time is v=q 220 minutes and nominal vol-ume v 2000 L. The model functions and parameter val-

Table 1Steady-states, nominal inputs, outputs and reactor states, and jacket para

States and outputs Steady-states

Stable (extinction)

m (kg) 1361.089T (K) 329.72setpoint, meaning T j T j (or I = I*). This feature impliesthe relaxation of the dynamic gain separation condition ofthe standard cascade designs, or equivalently, a betterbehavior with an improved performance-robustness trade-o. Moreover, the use of the molecular weight inverse stateled to a simpler MW controller and facilitated the stabilityanalysis, and with the incorporation of the calorimetric-based polydispersity estimates the information containedin the temperature and volume measurements can be fur-ther exploited for monitoring and MW control model cal-Jacket parameters

aT = 4.71 102 min1, aj = 3.78 102 K/Ltion about the open-loop unstable steady-state in conjunc-tion with the reactor and jacket feed temperatures stepchanges shown in Fig. 3a (at t = 500 min: Te from 315 to320 K, and Tje from 328 to 330 K). The closed-loop behav-ior is presented in Fig. 4, showing that: (i) as expected,the MD controller recovers the behavior of the nonlinearFF-SF passive controller, (ii) the volume, temperature

r approximations

Unstable Stable (ignition)

660.082 312.756351.62 373.88341.23 345.61

110384.75 29395.151.3087 0.3513

500.871 498.5611.999 1.99660.5672 0.79540.4269 0.5997

T e; T je; qs0=(315 K, 328 K, 2.54 L/min) 0were veried with combined analytic-numerical testing.The application of the tuning guidelines (given in Section4.4), yielded the following gains

xv xj xT xi xj 1=5min1

: x;

kv kj x=8; kT 2q=v; -m -i -i 1:5

5.1. Behavior recovery

The reactor was controlled with the proposed MD con-trol (31) and its behavior compared with the one of itsexact model-based nonlinear counterpart (16). The modelparameters were xed at their actual values, and the

and monomer outputs reach their set points in about oneresidence time, in the understanding that these controllersensure the stability, the production rate level, and to a goodextent the product quality, (iii) the number-average MW

(Mn) converges in about 2 residence times, meaning twicefaster than the open-loop MW response, (iv) the controlactions eectively cancel out the eect of the input distur-bances on the regulated outputs, and (v) the control actionsoccur in a smooth and coordinated manner, reasonablyaway from saturation.

The preceding test was repeated with sinusoidal inputs(shown in Fig. 3b):

T e 315 K 5 sin6:2832t=110T je 328 K 2 sin6:2832t=110; tP 500min

and the resulting closed-loop behavior is presented inFig. 5, showing that: (i) again, the MD controller recoversthe behavior of its exact SF model-based counterpart, (ii)the volume, temperature and monomer outputs convergeto their setpoints in about one residence time, and the(number-average) MW (Mn) reaches its set point in abouttwo residence times, (iii) the outputs exhibit sustained oscil-

0 500 1000 1500

310

320

330310

320

330

Rea

ctor

(Te)

and

jack

et (T

je) f

eed

tem

pera

ture

s, (K

)

t (min)

Te

Tje

Te

Tje

Rea

ctor

(Te)

and

jack

et (T

je) f

eed

tem

pera

ture

s, (K

) a

b

Fig. 3. Time-varying exogenous inputs: (a) step and (b) sinusoidal.

340

350

360

Tj

T

Reac

tor (

T)an

d ja

cket

(Tj)

tem

pera

ture

s (K

)

600620640660680

m

V

Mon

omer

mas

s, m

(kg)

1.90

1.95

2.00 Volume x 10

-3, V (L)

1.10

1.15

1.20

1.25

Mn

I

ber-

aver

age

MW

x 1

0-5 ,

n (kg

/km

ol)

1.0

1.2

1.4 Initiator mass, I (

340

350

360

Tj

T

Reac

tor (

T)an

d ja

cket

(Tj)

tem

pera

ture

s (K

)

600620640660680

m

V

Mon

omer

mas

s, m

(kg)

1.90

1.95

2.00 Volume x 10 -3, V (L)

1.10

1.15

1.20

1.25

M

I

umbe

r-ave

rage

MW

x 1

0-5 ,

Mn (

kg/k

mol

)

0.8

1.0

1.2

1.4

Initiator mass, I (kg

472 J. Alvarez, P. Gonzalez / Journal of Process Control 17 (2007) 4634760.5

0.6

1.05

Num M 0.8 kg)

c

nver

sion

(c),

frac

tion

(),

(-)

0.4

1.9

2.0

2.1

Co

solid

Poly

disp

ersi

ty,

(-)

0 500 1000 1500

10

20

30

40

t (min)

Coo

lant

(qj),

exi t

(q),

and

mon

omer

(qm

)flo

wra

tes,

(L/m

in)

qmq

wi

qj

5

10

15

Initiator feedrate x 10 3,w

i (kg/min)

Fig. 4. Closed-loop reactor behavior under step input disturbances, withexact model-based SF passive (16) (- - -) and MD (31) (A) controllers, andnominal operation values ( ).0.4

0.5

0.6

1.9

2.0

2.1

1.05 nN )

c

Con

vers

ion

(c),

solid

frac

tion

(),

(-)Po

lydi

sper

sity,

(-)

0 500 1000 1500

10

20

30

40

t (min)

Coo

lant

(qj),

exit

(q),

and

mon

omer

(qm

)flo

wra

tes,

(L/m

in)

qmq

wi

qj

5

10

15

Initiator feedrate x 10 3,w

i (kg/min)

Fig. 5. Closed-loop reactor behavior under sinusoidal input disturbances,

with exact model-based SF passive (16) (- - -) and MD (31) (A) controllers,and nominal operation values ( ).

lations with rather small amplitudes (0.01,0.12,0.1, 0.25)%for (v: volume, T: temperature, m: monomer, Mn) with re-spect to industrial reactor operations.

These tests verify the IS stability property of the closed-loop reactor system with strongly nonlinear FF-SF and themildly nonlinear MD control: after combined initial-statestep-input disturbances, the closed-loop system undergoesa linear-like vanishing transient response, with asymptoticconvergence to the prescribed steady-state. In the case ofpersistent sinusoidal input disturbances, the same transientbehavior is exhibited, and the system converges to a sus-tained oscillatory regime within an acceptable-size compactset about the nominal steady-state.

5.2. Behavior with parameter errors

Finally, the MD control was tested with the followingtypical parameter errors: 14% error in the chain transferparameters (am,as) (Appendix A), and 5% in the initiator

eciency factor (fd), in the understanding that the highlynonlinear and uncertain polymerization rate-heat exchangefunction pair fr fU is not needed by the MD controlscheme. I the closed-loop system was subjected to theabove stated initial state deviation with step (Fig. 3a) andsinusoidal (Fig. 3b) exogenous input disturbances, andthe behavior results are presented in Fig. 6 (with stepinputs) Fig. 7 (with sinusoidal inputs). Comparing withthe corresponding errorless test (Fig. 4), in the case of ini-tial state and step input disturbances, the parameter errorsdo not appreciably aect the control behavior (Fig. 6): (i)the volume, temperature and monomer responses areimperceptibly aected, and (ii) the MW weight responseexhibits 3% asymptotic oset (i.e., smaller than experi-mental measurement uncertainty). Should this oset beunacceptably larger, the initiator and/or transfer constantsshould be occasionally recalibrated on the basis of free-monomer, solid fraction and Mn measurements which areroutinely taken for operation monitoring and product

340

350

360

Tj

T

Reac

tor (

T)an

d ja

cket

(Tj)

tem

pera

ture

s (K

)

600620640660680

m

V

Mon

omer

mas

s, m

(kg)

1.90

1.95

2.00 Volume x 10

-3, V (L)

1.10

1.15

1.20

1.25

I

ber-a

vera

geW

x 1

0-5 ,

n (kg

/km

ol)

1.0

1.2

1.4 Initiator mass, I (

Reac

tor (

T)an

d ja

cket

(Tj)

tem

pera

ture

s (K

)

Mon

omer

mas

s, m

(kg)

Volume x 10

-3, V (L)

ber-a

vera

geW

x 1

0-5 ,

(kg/

kmol

)

Initiator mass, I

340

350

360

Tj

T

600620640660680

m

V

1.90

1.95

2.00

1.10

1.15

1.20

1.25

I

1.0

1.2

1.4

J. Alvarez, P. Gonzalez / Journal of Process Control 17 (2007) 463476 4730.4

0.5

0.6

1.9

2.0

2.1

1.05 MnNum M M 0.8 kg)

c

Conv

ersi

on (c

),so

lid fr

actio

n (

), (-)

Poly

disp

ersi

ty,

0 500 1000 1500

10

20

30

40

t (min)

Coo

lant

(qj),

exi

t (q)

,an

d m

onom

er (q

m)

flow

rate

s, (L

/min

)

qmq

wi

qj

5

10

15

Initiator feedrate x 103,

wi (kg/m

in)

(-

)

Fig. 6. Closed-loop reactor behavior under step input disturbances and

model parameter errors, with MD (A) controllers (31), and nominaloperation values ( ).Num M M

n (kg)

Conv

ersi

on (c

),so

lid fr

actio

n (

), (-)

Poly

disp

ersi

ty,

Coo

lant

(qj),

exi

t (q)

,an

d m

onom

er (q

m)

flow

rate

s, (L

/min

) Initiator feedrate x 103,

wi (kg/m

in)

(-

)

0.4

0.5

0.6

1.9

2.0

2.1

1.05 Mn 0.8

c

0 500 1000 1500

10

20

30

40

t (min)

qmq

wi

qj

5

10

15

Fig. 7. Closed-loop reactor behavior under sinusoidal input disturbances

and model parameter errors, with MD (A) controllers (31), and nominaloperation values ( ).

lines. The proposed controller: is considerably simpler and

r0 cdfriT ; I itT ;m;sfrv;m;T ; I ;s : f0v;m;T ; I;s

f Pless model dependent than previous nonlinear geometriccontrollers, recovers the behavior of its exact model-basedFF-SF counterpart with optimality-based robustness char-acteristic, is equivalent to an unconstrained MPC with in-nite receding horizon, and amounts to a set of coordinateddecentralized and cascade components that resemble theones employed in industrial polymer reactors.

The polymerization of MMA in an open-loop unstableindustrial size reactor was considered as representative case6. Conclusions

A constructive robust MD control design methodologyfor continuous solution homopolymerization reactors withow and temperature measurements has been presented.Structural (relative degree, ZD and detectability) solvabil-ity conditions that bear physical meaning were identiedand employed to set a simplied model for control design.The application of a Lyapunov interlaced estimator-con-trol design yield led to a MD control scheme with: (i) max-imum linearity, decentralization, and model independencyfeatures, (ii) linear-decentralized PI volume and controlcomponents, (iii) material balance monomer and MWcontrollers which exploit the information contained in theintegral actions of the PI controllers, (iv) a systematicconstruction procedure, and (v) a closed-loop nonlinear-nonlocal stability criteria coupled with simple tuning guide-quality assessment purposes. The behavior with sinusoidalinput disturbances (Fig. 3b) is shown in Fig. 7: (i) the tran-sient responses are similar to the ones of the case withoutparameter errors (Fig. 4 with step inputs, and Fig. 5 withsinusoidal inputs), and to the case with parameter errorsand step inputs (Fig. 6), and (ii) the asymptotic responseexhibits an oscillatory behavior with a amplitudes of(0.01, 0.12, 0.1, and 0.25)% in (v,T,m,Mn) about impercep-tible osets in (v,m,T) and an 3% oset in MW.

5.3. Concluding remarks

The responses to initial state deviations, exogenousinput disturbances, and model parameter errors verify: (i)the (nonlinear passive control) behavior recovery as wellas the IS robust stability property of the MD controller,and (ii) the simplicity and eectiveness of the tuning guide-lines drawn from the closed-loop stability assessment. Theclosed-loop testing illustrated and made quantitative theassessment of IS stability features, like transient overshoot,settling time, and asymptotic (constant or oscillatory)behavior. From the comparison of responses with andwithout parameter errors, it follows that the closed-loopoutputs exhibit a nearly linear (i.e., superposition-like)behavior, in accordance with the nearly closed-loop outputbehavior associated with the MD control design.

474 J. Alvarez, P. Gonzalez / Journal oexample with numerical simulations. The closed-loop test-ing veried the control behavior robustness and recoverycd 2f d=mwiET eadbd=T ; kpoT eapbp=T ; ktoT eatbt=Tkpv;m;T ;I ;s kpoT =1 kpoT hpT Efv;m;T ;sk0v;m;T ; I;sk0v;m;T ; I;s k0v;T ; Ik2gv;m;T ; I ;s1=2kgv;m;T ; I;sk0v;T ;I 2f dfriT ; I=vmwikto T kgv;m;T ; I;s fdhtv;T ; IEfv;m;T ;sfriT ; I=vmwiEfv;m;T ;s e2:3m=vqms=vqs=fAT BT m=vqms=vqsghpT eahpbhp=T ; htv;T ; I eahtbhtI=vmwidht=TAT aAbA1T=T g cA1T=T g2;BT aBbB1T=T gitT ;m;s jmT s=mjsT ; jmT 1=mwmeambm=T ;jsT 1=mwseasbs=T

A.2. Heat capacity and transfer [2,3]

cm qmcpm ; cs qscps ; cj qjcpj ;Cmcpm scps fpv;m;scpp : fCV ;m;sq qm1 emm=vqm ess=vqs=1 em : fqv;m;sp Vfqv;m;sm s : fpv;m;sU AfhT ;T j;v;m;s : fUT ;T j;v;m;sh ahk=DLNq=lbhcl=kch l=lwdh : fhT ;T j;v;m;sl alT 273:15blEclf T ;v;m;s : flT ;v;m;s; lw flT j;v;m;s

A.3. Numerical values [3,20]

ad; bd; fd 35:811324; 14896:127; 0:52ap; bp; at; bt 18:39228; 2609:199; 22:49482; 352:758am; bm; as; bs 8:396; 6472:83131; 5:664; 4570:25734aA; bA; cA; aB; bB; T g 0:1678; 0; 1:23517; 0:03; 0; 387:15ahp ; bhp ; aht ; bht ; cht ; dht

35:11094; 13964; 47:03; 48:85; 637:19; 17956The authors gratefully acknowledge the support fromthe Mexican National Council for Research and Technol-ogy (CONACyT Scholarship 118632) for P. Gonzalez.

Appendix A. Polymerization reactor model

A.1. Kinetics [13,20]

riET I : friT ; Ir kpv;m;T ; I ;sk0v;m;T ; I ;sm : frv;m;T ;I ;sfeatures, and illustrated and made quantitative the assess-ment of the closed-loop IS stability property.

The proposed polymer reactor control design methodol-ogy suggests the pursuit of a general-purpose design frame-work that can fruitfully blend theoretical nonlinear andapplied chemical process control techniques.

Acknowledgements

rocess Control 17 (2007) 463476k;D; L;N ;D 0:2768; 13; 7; 250; 134:76

f Puxa; ua; d; _d KpCzx z Kdfv; fm; fT; fi0xa; dKp diagxc2v ;xc2m ;xc2T ;xc2i Kd diagfcvxcv; fcmxcm; fcTxcT; fcixci;recall the ua-invertibility of u (B1b), and solve to last equa-Compare these equations with the ones of the LNPA out-put error dynamics (10), obtain the controller (in v m at-tening coordinates)

ma xc2a ea fcaxcafa; a v;m; T ; i B3substitute these equations into (B1a), obtain the equational; bl; cl 2:484 1029; 9:6973; 1:99ah; bh; ch; dh 0:74; 2=3; 1=3; 0:14mw;mws ;mwi 100:11; 88:1; 164:21qm; qs; qp; em; es 950; 901; 1170; 0:188; 0:23

Appendix B. Solvability assessment

Proof of Proposition 1 (relative degree existence). Recallthe reactor system (3), incorporate the dynamic extension(5a), and obtain the augmented system

_xa faxa; d; u; xa x0; x0a0; xa q; qm0ua _q; _qm; qj;wi0

This system has relative degree j (5b) i the the maps / andu of the coordinate change

v v; fv;m; fm; T ; fT; i; fi0 : /xa; d; oua/ 0/ is xa-invertible B1a

m oxafv; fm; fT; fi0 odfv; fm; fT; fi0 _d : uxa; ua; d; _d/ is ua-invertible B1b

are xa and ua-invertible (inv) [14], respectively, meaningthat (B1a) [or (B1b)] has a unique solution for xa (or ua).Since the inverse (v,m,T, i) 0 = (v1,v3, v5,v7) 0 is trivially gi-ven, / (or u) is xa or (or xu)-invertible i Condition (B2a)[or (B2b)] is met:

fv; fm0 is q; qm-inv () 6afT is T j-inv () 6c; f i is I-inv () 6d B2auv;um0 is _q; _qm-inv () 6auT is qj-inv () 6bc; ui is wi-inv ) 6ad B2b

Consequently, (/,u) is (xa,ua)-invertible i the conditionsof Proposition 1 are met. h

Geometric controller derivation (11). Recall the state-input pair v m (B1), write the Brunovskys controllabilityform [14] associated with the relative degree j (5b)

_v1 v2; _v2 m1; _v3 v4; _v4 m2; _v5 v6; _v6 m1_v7 v8; _v8 m2

J. Alvarez, P. Gonzalez / Journal otion to obtain the dynamic nonlinear FF-SF controller (11)with the following maps [mi is dened in (B3)]:s p

perturbed steady-state solutions, this is,

loqes; dos es bos es; do ) jesj 6 cos jdos j B7alopes; dpep bopes; dop ) jepj 6 cpjesj; jdopj B7b) jepj 6 cpjesj; jdopj 6 cpcos jdos j; jdopj : copkdokThis leads us to conclude the existence of the asymptoticgain functions (cos and c

op) that make non-positive the asso-

ciated dissipation rate (9b), or equivalently, the ZD (7) areIS stable. h

Proof of Proposition 3 (Closed-loop stability with MDcontroller). Introduce the LF V,

V V p V^ ) _V ae;~xe; d e;~xe; d; _d B8abae;~xe; dP 0; s0; 0; 0; 0 0where Vp (or V^ ) is the LF (14) [or (24)] of the nonlinearWrite the steady-state solution (B5) in perturbed form (i.e.,loq bos ; lop bop), and recall condition (8a), to establish theexistence of K-class functions (co and co) that bound thelgqx; q; qm; qs; _qs; T e mm qmmv 1 emoxwi fr # qm _qs=qm m=v

lgqmx; q; qm; qs; _qs; T e fmm m=vvv 1 emm=qmvoxwi fr # m=v _qsg=qm m=v

lgwix; q; qm; qs; T e q=v ET mi hoxwi/i; fwii=oIfilgqjx; q; qm; qs; T e; T je; _qs; _T e mT /j hoxqj/j; fqji

hodwi/j; _qm; _qs; _T e0i=fUoTjfT

B4where f wi fT;fv;fm;fs

0; xwi T ;v;m;s0

fqj f 0wi ;fI0; xqj x0wi ;I

0

#f1em=qmm=vfrqm v=mqmm=vqsgq=v;dwi qm;qs;T e0:

Proof of Proposition 2 (ZD stability). In nominal steady-state regime the ZD (7) become

p 2 mwi 2; sros=v ws B5abimplying that there is a unique steady-state s; p0 i Condi-tion (8a) 2 is met. Recall the ZD LF (9a), take its deriva-tive, and obtain the dissipation rate (9b) with

aoes; ep; do loqes; dos lopes; dop B6aboes; do bos es; dos bopes; dop B6bloqes;dos q=v; q 1 emr=qmqs=1 m=pr frv;m;T ;loI s;slopes;dop r=p; bos es;dos esfws1 loqes;dos =q q^s~qsqs~qsgesp fpv;m;sbopes;dop ~cop p~kop; dos d0z;dqs ;dpr0; dop d0z;dpr0;dz zzdqs qqs; dpr prpr

rocess Control 17 (2007) 463476 475passive FF-SF controller (17) [or estimator (23)]. Takethe derivative of V along the motion of the reactor-MD

control pair (1)(31), substitute the dissipation rates _^V (25),

476 J. Alvarez, P. Gonzalez / Journal of Process Control 17 (2007) 463476_V v (27b), _V T (28c), _V m (29b), and _V i (30c), recall Vo (9a),

determine its dissipation rate component _V o, and obtainthe closed-loop dissipation rate (B8b) with

a kve2v xv~b2v kTe2T xT~b2T kje2j xj~b2j xj ~b2j kqe2s ~s2 e2p ~p2 kqhem; ~m;-m krhei;~i;-i kihei;~I ;-i

d dd; _dd; dpes sve;~xe; d; _d; _u sTe;~xe; d; _d sie;~xe; d; _d

es ~s~ws ~cp ~kpep ~cp p ~p~kp~pwhere {kq,kr,ki,kp} (13) is the dilution rate set, h is theellipsoidal function (29c), and the functions sv (27b), sT(28c), and si (30c) have been already dened. Set the Eq.(B9a), recall the closed-loop IS stability (18) with the pas-sive controller (17), conclude the existence of a localasymptotic gain c (B9b), draw the dissipation rate inequal-ity (B9c):

ae;~xe; d se;~x; d; _d B9a) je0;~x0e0j ckd0; _d00k B9b_V 6 08je0;~x0e0jP ckdtk B9cwith c(0) = 0, and conclude (4), the IS stability of theclosed-loop reactor (1) with the MD driven controller(31), provided the gains are tuned according to Conditions(34) of Proposition 3, to ensure that the stabilizing term (a)dominates the potentially destabilizing one (s)[13]. h

References

[1] J.W. Hamer, T.A. Akramov, W.H. Ray, The dynamic behavior ofcontinuous polymerization reactors II. Nonisothermal solutionhomopolymerization and copolymerization in a CSTR, Chem. Eng.Sci. 36 (12) (1981) 1897.

[2] S. Padilla, J. Alvarez, Control of continuous copolymerizationreactors, AIChE J. 43 (2) (1997) 448.

[3] J. Alvarez, R. Suarez, A. Sanchez, Semiglobal nonlinear controlbased on complete input-output linearization and its application tothe start-up of a continuous polymerization reactor, Chem. Eng. Sci.49 (21) (1994) 3617.

[4] J.P. Congalidis, J.R. Richards, Process control of polymerizationreactors: an industrial perspective, Polym. React. Eng. 6 (2) (1998) 71.

[5] B.R. Maner, F.J. Doyle III, Polymerization reactor control usingautoregressive-plus Volterra-based MPC, AIChE J. 43 (1997) 1763.

[6] P. Daoutidis, M. Soroush, C. Kravaris, Feedforward/feedbackcontrol of multivariable nonlinear processes, AIChE J. 36 (10)(1990) 1471.

[7] M. Soroush, C. Kravaris, Multivariable nonlinear control of acontinuous polymerization reactor: an experimental study, AIChE J.39 (12) (1993) 1920.

[8] J. Alvarez, Output-feedback control of nonlinear plants, AIChE J. 42(9) (1996) 2540.

[9] J.P. Gauthier, I. Kupka, Deterministic Observation Theory andApplications, Cambridge University Press, UK, 2001.[10] R.K. Mutha, W.R. Cluett, A. Penlidis, On-line nonlinear model-based estimation and control of a polymer reactor, AIChE J. 43 (11)(1997) 3042.

[11] I. Saenz de Buruaga, P.D. Armitage, J.R. Leiza, J.M. Asua,Nonlinear control for maximum production rate of latexes of well-dened polymer composition, Ind. Eng. Chem. Res. 36 (1997)4243.

[12] J. Alvarez, F. Zaldo, G. Oaxaca, Towards a joint process and controldesign framework for batch processes: application to semibatchpolymer reactors, in: P. Seferlis, M.C. Georgiadis (Eds.), TheIntegration of Process Design and Control, Elsevier, Amsterdam,The Netherlands, 2004, pp. 604634.

[13] P. Gonzalez, J. Alvarez, Combined proportional/integral-inventorycontrol of solution homopolymerization reactors, Ind. Eng. Chem.Res. 44 (2005) 7147.

[14] A. Isidori, Nonlinear Control Systems, third ed., Springer-Verlag,New York, 1995.

[15] A. Isidori, Nonlinear Control Systems II, Springer-Verlag, London,1999.

[16] R. Sepulchre, M. Jankovic, P. Kokotovic, Constructive NonlinearControl, Springer-Verlag, NY, 1997.

[17] R.A. Freeman, P.V. Kokotovic, Robust Nonlinear Control Design:State-Space and Lyapunov Techniques, Birkhauser, Boston, 1996.

[18] M. Krstic, I. Kanellakopoulos, P. Kokotovic, Nonlinear and Adap-tive Control Design, Wiley, New York, 1995.

[19] E.D. Sontag, The ISS philosophy as a unifying framework forstability-like behavior, in: A. Isidori, F. Lamnabhi-Lagarrigue, W.Respondek (Eds.), Nonlinear Control in the Year 2000, LectureNotes in Control and Information Sciences, vol. 2, Springer-Verlag,Berlin, 2000, pp. 443468.

[20] W.Y. Chiu, G.M. Carratt, D.S. Soong, A computer model for the geleect in free-radical polymerization, Macromolecules 16 (3) (1983)348.

[21] F.G. Shinskey, Process Control Systems, third ed., McGraw-Hill,New York, 1988.

[22] H.K. Khalil, Nonlinear Systems, third ed., Prentice-Hall, New Jersey,2002.

[23] F. Allgower, A. Zheng (Eds.), Nonlinear Model Predictive Control,Birkhauser, Germany, 2000.

[24] R.E. Kalman, When is a linear control system optimal? Trans. ASMESer. D: J. Basic Eng. 86 (1964) 1.

[25] P.J. Flory, Principles of Polymer Chemistry, Cornell University Press,New York, 1953.

[26] J.A. Biesenberger, D.H. Sebastian, Principles of PolymerizationEngineering, Wiley, New York, 1983.

[27] J. Alvarez, Nonlinear state estimation with robust convergence, J.Proc. Cont. 10 (2000) 59.

[28] T. Lopez, J. Alvarez, On the eect of the estimation structure in thefunctioning of a nonlinear copolymer reactor estimator, J. Proc.Cont. 14 (2004) 99.

[29] R. Hermann, A.J. Krener, Nonlinear controllability and observabil-ity, IEEE Trans. Auto. Control. AC-22 5 (1977) 728.

[30] J.J. Alvarez-Ram rez, J. Alvarez, A. Morales, An adaptive cascadecontrol for a class of chemical reactors, Int. J. Adapt. Cont. SignalProcess. 16 (2002) 681.

[31] R.T. Stefani, C.J. Savant Jr., B. Shahian, G.H. Hostetter, Design ofFeedback Control Systems, third ed., Saunders College Publishing,Florida, 1994.

[32] J. LaSalle, S. Lefschetz, Stability by Lyapunovs Direct Method,Academic, New York, 1961.

[33] W.L. Luyben, Process Modeling, second ed.Simulation and Controlfor Chemical Engineers, McGraw-Hill, Singapore, 1990.

[34] J.J. Dazzo, C.H. Houpis, Linear Control System Analysis andDesign, McGraw-Hill, New York, 1981.

Constructive control of continuous polymer reactorsIntroductionControl problemSolvability assessmentRelative degrees (RD) and zero-dynamics (ZD)FF-SF geometric controlPassive FF-SF controlOptimality and connection with MPCConcluding remarks

Measurement-driven controlControl modelState estimatorMeasurement-driven (MD) controllerClosed-loop dynamics and tuningConcluding remarks

Application exampleBehavior recoveryBehavior with parameter errorsConcluding remarks

ConclusionsAcknowledgementsPolymerization reactor modelKinetics [1-3,20]Heat capacity and transfer [2,3]Numerical values [3,20]

Solvability assessmentReferences