Integralâ Inventory Control of Solution

Ind. Eng. Chem. Res. 2005, 44, 7147-7163

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Combined Proportional/Integral-Inventory Control of Solution Homopolymerization Reactors Pablo Gonza´ lez and Jesu ´ s Alvarez* Universidad Auto´ noma MetropolitanasIztapalapa, Departamento de Ingenierıá de Procesos e Hidra´ ulica, Apartado 55534, 09340 Me´ xico DF, Me´ xico

In this work, a proportional/integral (PI) material-balance scheme to control continuous freeradical solution homopolymer (possibly open-loop unstable) reactors with level, temperature, and flow measurements is presented. First, the combination of inventory and constructive control ideas yields (i) a nonlinear feedforward state-feedback static passive controller whose closedloop dynamics are the limiting behavior attainable by robust feedback control and (ii) the related solvability conditions with physical meaning. Then, such a limiting behavior is recovered via a measurement-driven dynamic control system with (i) linear PI-type decentralized volume and cascade temperature controllers and (ii) material-balance monomer and molecular weight (MW) controllers. The temperature controller requires two approximate static parameters, the monomer controller requires the heat capacity function, the MW controller requires initiation-transfer parameters, and the automatic-to-manual switching of the MW control does not affect the functioning of the controllers that perform the stabilization task. The design methodology has a systematic control construction and conventional-like tuning guidelines coupled with a nonlocal stability assessment. The proposed approach is put into perspective with previous polymer reactor model predictive control and geometric control studies and tested with an industrial-size reactor through numerical simulations. Introduction An important class of materials is produced via freeradical exothermic polymerizations in continuous reactors, which are highly nonlinear dynamical systems with complex behavior: strong and asymmetric inputoutput interaction, possibility of steady-state multiplicity, parametric sensitivity, runaway potential, and sometimes operation about an open-loop unstable steady state.1,2 In a typical industrial setting, the volume (or temperature) is controlled with a decentralized single (or cascade) linear proportional/integral (PI) controller by manipulating the exit flow (or heat exchange), and the conversion (or molecular weight, MW) is feedback (FB) and/or feedforward (FF) controlled with inferential, advisory, or supervisory control schemes by manipulating the monomer feed (or initiator and/or chain-transfer agent) dosage rate. If the reactor is open-loop unstable, a conversion FB control is needed to stabilize the reactor.1-4 Motivated by the need of designing processes with better compromises between safety, productivity, and quality, by the availability of reliable online computing capability, and by the development of nonlinear control methods, in the last two decades the polymerization reactor problem has been the subject of extensive theoretical, simulation, and experimental studies. The related state of the art can be seen elsewhere,5-7 and here it suffices to mention that (i) parts of the multiinput multi-output (MIMO) control problem have been addressed,3,4,6-15 (ii) a diversity of control techniques have been employed, including linear PI decoupling2,8 and model predictive9 (MPC) as well as nonlinear * To whom correspondence should be addressed. Tel.: +52 (55) 5804 4648 or 51, ext 212. Fax: +52 (55) 5804 4900. E-mail: [email protected].

geometric,3,4,10-12 MPC,13 and calorimetric14,15 control techniques, and (iii) the nonlinear controllers have been implemented with open-loop,11 extended Kalman filter13 (EKF), and Luenberger3,4,15 (L) nonlinear observers. The resulting nonlinear dynamic controllers are strongly interactive and model-dependent, signifying complexity, reliability, and cost drawbacks for industrial applicability. On the other hand, the fundamental connections between optimality, passivity, and robustness have been central issues in the development of the constructive nonlinear control theory,16,17 the subject has been related14,18 to the inventory balance framework employed in chemical industrial practice to design combined FF-FB control schemes,19,20 and the passivity concept has been applied to design nonlinear robust estimators.21 The inventory control concepts in conjunction with the connections between optimality, passivity, and robustness have been exploited to systematize and simplify (i) the joint motion and control design of batch distillation columns22 and (ii) the calorimetric control design for semibatch emulsion polymer reactors.14 Earlier ideas on the relationship between inventory balance, thermodynamics, and control can be found in (i) the use of extensive variables for process control,23 (ii) the energy balance-based nonlinear geometric control,24 and (iii) the interplay between inventory control, passivity, and thermodynamics.18 The preceding comments motivate the need of developing application-oriented control methodologies to combine the simplicity and reliability of the linear PI and inventory industrial controllers19,20 with the structure and robustness design tools offered by the nonlinear geometric25 and constructive16,17 control approaches. In this work, the problem of controlling continuous free-radical solution homopolymer (possibly open-loop

10.1021/ie040207r CCC: $30.25 © 2005 American Chemical Society Published on Web 08/06/2005

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unstable) reactors with level, temperature, and flow measurements is addressed. The aim is the derivation of a robust MIMO control scheme with maximum simplicity, in the sense of linearity, decentralization, and modeling dependency. First, an inventory-based passive inversion yields an exact model-based nonlinear FF state-FB static controller whose closed-loop dynamics are the limiting behavior attainable by robust FB control. Then, this behavior is recovered via a measurement-driven dynamic control system with (i) linear PItype decentralized volume and cascade temperature controllers and (ii) material-balance monomer and MW controllers. The design methodology has solvability conditions with physical meaning, a systematic controller construction, and conventional-like tuning guidelines coupled with a nonlocal stability26,27 assessment. The approach is put into perspective with previous polymer reactor MPC13 and geometric control studies3,4,10-12 and tested with an industrial-size reactor through numerical simulations. Polymerization Reactor and Control Problem Polymer Reactor. Consider the class of continuous stirred tank reactors (depicted in Figure 1) where an exothermic free-radical solution homopolymer reaction takes place. Monomer, solvent, and initiator are fed to the tank, and heat exchange is enabled by a cooling jacket with a recirculation circuit. Because of the gel effect,3,28 the reactor can have multiple steady states.1,2 From standard polymerization kinetics1,29,30 and viscous heat-transfer considerations,2,12 the reactor dynamics are given by the following energy and material balances:

T˙ ) {∆fr(T,V,m,I,s) - fU(T,Tj,V,m,s) (T - Tj) + (Fmqmcm + Fsqscs)(Te - T)}/ fC(V,m,s) :) fT, zT ) yT ) T (1a) T˙ j ) {fU(T,Tj,V,m,s) (T - Tj) + Fjqjcj(Tje - Tj)}/ CJ :) fj, yj ) Tj, uj ) qj (1b) V˙ ) qm + qs - (m/Fm)fr(T,V,m,I,s) - q :) fV, zV ) yV ) V, uV ) q (1c)

Figure 1. Solution homopolymer reactor.

perature (T), volume (V), monomer content (m), and molecular weight (π). The measured outputs (y) are the temperature (yT), volume (yV), and jacket temperature (yj). The control inputs (u) are the coolant (qj), exit (q), and monomer (qm) volumetric flow rates and the initiator mass feed rate (wi). The reactor model includes the polymerization (fr), initiator decomposition (fri), and zeroth (f0) and second moment (f2) rate functions,1-3,29,30 as well as the heattransfer coefficient (fU), heat capacity (fC), density (F), and polymer mass (mp) functions.2,12,14 In particular, the functions (fri, f0, and fC) and parameters (pc and pit) that will appear in the monomer and MW components of the proposed control scheme are given by

fri(T,I) ) Ade-Ed/RTI, f0(T,V,m,I,s) ) 2fdfri(T,I) + κ(T,m,s) fr(T,V,m,I,s) (2a,b) κ(T,m,s) ) κm(T) + (s/m)κs(T), κm(T) ) ame-Lm/RT, κs(T) ) ase-Ls/RT (2c)

m ˘ ) -fr(T,V,m,I,s) + Fmqm - (m/V)q :) fm, zm ) m, um ) qm (1d)

fC(V,m,s) ) mcm + scs + mp(V,m,s) cp, mp(V,m,s) ) VF(V,m,s) - m - s (2d)

π˘ ) [fr(T,V,m,I,s) - πf0(T,V,m,I,s)]π/ [VF(V,m,s) - m - s] :) fπ, zπ ) π (1e)

F(V,m,s) ) Fm[1 - mm/VFm - ss/VFs]/(1 - m), m ) 1 - Fm/Fp, s ) 1 - Fs/Fp (2e)

I˙ ) -fri(T,I) + wi - (I/V)q :) fi, ui ) wi

(1f)

pc ) (cm, cs, cp, Fm, Fs, Fp, CJ, ∆)′, pit ) (fd, Ad, Ed, am, Lm, as, Ls)′ (2f)

s˘ ) Fsqs - (s/V)q :) fsx

(1g)

µ˘ 2 ) f2(T,V,m,I,s) - (µ2/V)q :) fµ2

(1h)

The states (x) are the reactor (T) and jacket (Tj) temperatures, the volume (V), and the free (i.e., unreacted) monomer (m), solvent (s), and initiator (I) masses, as well as the (number-average) molecular weight (π) and second moment (µ2) of the MW distribution. The measured exogenous inputs (d) are the reactor (Te) and jacket (Tje) feed temperatures and the solvent volumetric flow rate (qs). The regulated outputs (z) are the tem-

where (i) Ad, Ed, and fd are respectively the preexponential, activation energy, and efficiency constants of the initiator decomposition rate, (ii) κm (or κs) is the Arrhenius-type monomer (or solvent) transfer-to-propagation quotient30 with parameter pair (am, Lm) [or (as, Ls)], (iii) Fm (or cm), Fs (or cs), Fp (or cp), and Fj (or cj) are respectively the monomer, solvent, polymer, and coolant fluid densities (or specific heat capacities), (iv) m (or s) is the monomer (or solvent) contraction factor, (v) CJ is the heat capacity of the cooling system, including the cooling fluid as well as the reactor, jacket, and recirculation walls,14 (vi) ∆ is the heat of polymerization

Ind. Eng. Chem. Res., Vol. 44, No. 18, 2005 7149

per unit monomer mass, and (vii) pc (or pit) is the calorimetric (or initiation-transfer) parameter vector. In practice, one is interested in looking and (indirectly) controlling the conversion (c), the solid fraction (ψs), and the MW polydispersity (Q) (Wm is the monomer molecular weight):

c ) mp(V,m,s)/[VF(V,m,s) - s], ψs ) mp(V,m,s)/VF(V,m,s), Q ) Wm µ2/[mp(V,m,s) π] In compact vector notation, the reactor control system (1) is given by

x3 ) f(x,d,u,p), x(0) ) x0; y ) Cyx, z ) Czx (3) where

x ) (T, Tj, V, m, π, I, s, µ2)′, f ) (fT, fj, fV, fm, fπ, fi, fsx, fµ2)′, f(x j ,d h ,u j ,p) ) 0 d ) (Te, Tje, qs)′, u ) (qj, q, qm, wi)′, y ) (yT, yj, yV)′, z ) (zT, zV, zm, zπ)′ p ) (p′r, p′it, p′U, p′c)′, pr ) (pr1, ..., pnr r)′, U pU ) (pU 1 , ..., pnU)′

p is the model parameter vector, including the calorimetric14 (pc), initiation-transfer (pit), polymerization rate (pr), and heat-transfer (pU) parameter vectors,2,12 and x j is the (possibly open-loop unstable and/or nonunique) steady state associated with the input (d h, u j) and parameter p. Stability. Let us recall the input-to-state17,26 (IS) practical-like31 nonlocal stability notion that has been employed in polymer reactor estimation studies.27,32 For this purpose, rewrite system (3) in deviation form:

|x˜ 0| < δ0, ||v˜ || e δ, w |x˜ (t)| < ae-λt|x˜ 0| + γxv||v˜ || < ) aδo + γxvδ, a, λ, γxv > 0 (6) If is below a given limit +, system (4) is said to be practically stable.31 System (4) is ZI stable if its ZI version is stable.[ Control Problem. Our problem consists of controlling the reactor, about a (possibly open-loop unstable and nonunique) nominal steady state x j , by manipulating the four inputs (u) to maintain the four regulated outputs (z) about their nominal setpoint (zj ), on the basis of the three-output (y) and three-input (d) online measurements. For applicability purposes, the design methodology must combine simplicity and robustness attributes, meaning (i) the pursuit of linearity, decentralization, model independency, and loop-failure tolerance features, (ii) the attainment of linear noninteractive pole assignable (LNPA) output regulation dynamics so that conventional-like tuning guidelines for linear single-input single-output systems can be applied, and (iii) the derivation of simple and straightforward tuning rules coupled with a closed-loop nonlocal IS stability assessment. Methodologically speaking, we are interested in (i) identifying the underlying solvability conditions with physical meaning and (ii) putting the proposed approach into perspective with applicationoriented PI and material-balance control schemes,19,20 as well as with the nonlinear MPC,13 geometric,3,4,10-12 and calorimetric14,15 control approaches employed in earlier polymer reactor studies. Inventory Control

|x˜ 0| < δ0, w |x˜ (t)| < ae-λt|x˜ 0| < ) aδ0, a, λ > 0 (5)

According to the inventory control approach employed in industrial practice,19,20 the most effective way to control a process is a combined FF-FB system: the FF component, which is a process inverse that balances the mass and energy delivered to the system against the demand of the load, performs most of the disturbance rejection task, and the FB is dedicated to compensate for the modeling errors of the FF component. From a constructive control perspective,16,17 such a control scheme is robust if the process inverse is passive, that is to say, stable and with relative degrees equal to 0 or 1. On the basis of these ideas, the inventory-based FFFB reactor robust control problem is addressed in this section. The purposes are the identification of the underlying solvability conditions, the delimitation of the behavior attainable with robust state-FB control, and the setting of a point of departure with regard to our output-FB control problem. Reactor Passive Inverse. The passive inversion problem consists of (uniquely) reconstructing the applied control input u(t) on the basis of the initial state (x0), the exogenous input (d), the regulated outputs (z), the derivatives d4 , z3 , and z1 , and the mass and energy balances (1), by means of a (possibly cascade) structure with relative degrees equal to 1.14,17,25 (a) Primary Inverse. Assume the solvent state (s) is given, enforce the regulated outputs (z), their derivatives (z3 ), and the exogenous inputs (d), verify the fulfillment of the conditions (∂xf ) ∂f/∂x)

System (4) is IS stable if, for given (δ0, δ), the motion deviations x˜ (t) are bounded as follows:17,26

fU(T,Tj,V,m,s) > (T - Tj)∂Tj fU(T,Tj,V,m,s) (7a)

x˜3 ) f(x˜ +x j ,d ˜ +d h ,u ˜ +u j ,p ˜ +p) :) f˜[x˜ ,v˜ (t)], x˜ (0) ) x˜ 0, f˜(0,0) ) 0 (4) x˜ ) x - x j , v˜ ) v - v j , v(t) ) [d′, u′(t), p′]′, |x˜ 0| e δ0, ||v˜ || ) sup ˜ (t)| e δ, |x˜ (t)| e t |v where v(t) is an augmented input, δ0 (or δ) is the size of the initial state (or input) disturbance, and is the perturbed motion size. While in the standard local framework the stability property holds for sufficiently small initial state deviations (with size δ0), in a nonlocal practical framework the IS stability property holds for initial condition, model parameter, input, and state deviations with specified or to be determined sizes (δ0, δ, and ). Next, this notion of stability is formally stated. Definition 1. The zero-input (ZI) system x˜3 ) f˜(x˜ ,0) :) g(x˜ ) is stable if, for given δ0, the motion deviations x˜ (t) are bounded as follows:

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m + s < VF(V,m,s), ∂I fr(T,V,m,I,s) > π∂If0(T,V,m,I,s) (7b,c)

us ) σs(z,xs,xI,v*,d,up,p), us ) (qj, wi)′, σs ) (σj, σi)′ (14)

and conclude that the balance set (1a,c-e) is solvable for the control (q, qm) and state (Tj, I) pairs, according to the following expressions (σ/j , σ/i , σV, and σm are defined in appendix B.2):

(c) Passive Inverse. The combination of the primary (10 and 12) and secondary (14) inverses with the solvent-second moment dynamics (1g, 1h), followed by the enforcement of the regulated output at its nominal (time-invariant) value (i.e., z ) zj , zj3 ) zj1 ) 0), yields the cascade (passive) inverse with relative degrees equal to 1, in internal model control (IMC) form:

˘ ,p), Tj ) σ/j (T,V,m,I,s,Te,qs,T˙ ,V˙ , m I ) σ/i (T,V,m,π,s,π˘ ,p) (8a,b) q ) σV(T,V,m,I,s,qs,V˙ ,m ˘ ,p), ˘ ,p) (8c,d) qm ) σm(T,V,m,I,s,qs,V˙ ,m

x3 I ) f ιI(xI,d,up,p), xI(0) ) xI0; up ) ιp(s,d,p), us ) ιs(s,d,d4 ,p) (15a,b) where

Condition (7a) says that, given (T, V, m, s), the heatexchange rate is uniquely determined by the jacket temperature Tj. Condition (7b) means that there is converted monomer in the reactor, and condition (7c) says that the MW balance (1e) is met with a unique initiator content (I), or equivalently, that there is a oneto-one correspondence between the number of live polymer chains and the initiator content, which is a well-known fact in polymerization kinetics.30 The time derivation of the jacket temperature (8a) [or the initiator content (8b)] inverse determines the rate of change in T˙ j (or I˙ ) (ν/j and ν/i are defined in appendix B.3):

T˙ j :) v/j ) ˘ ,T ¨ ,p) (9a) ν/j (T,Tj,V,m,I,s,vi,Te,qs,T˙ e,q˘ s,q,qm,q˘ m,T˙ ,V˙ ,m ˘ ,π˘ ,π¨ ,p) (9b) I˙ :) v/i ) ν/i (T,V,m,π,I,s,qs,q,T˙ ,V˙ ,m In compact notation, the preceding primary inverse (8) is written as follows:

xp ) z, xs ) σ*(z,xI,d,z3 ,p), up ) σp(z,xs,xI,d,z3 ,p) (10) where

xp ) (T, V, m, π)′, xs ) (Tj, I)′, σ* ) (σ/j , σ/i )′, up ) (q, qm)′, σp ) (σV, σm)′, xI ) (s, µ2)′ (11) xp (or xs) is the primary (or secondary) state, up is the primary control, and xI is the internal state. The corresponding secondary-state time derivative is given by

x3 s :) v* ) ν*(z,xs,xI,d,d4 ,up,z3 ,z1 ,p), v* ) (v/j , v/i )′, ν* ) (ν/j , ν/i )′ (12) (b) Secondary Inverse. The substitution of the primary inverse (10) and the secondary-state derivative map (12) into the secondary balance pair (1b, 1f) yields an algebraic equation pair that is solvable for the secondary control pair (qj, wi), according to the following expressions (σj and σi are defined in appendix B.4):

qj ) σj(T,Tj,V,m,s,v/j ,Tje,p), wi ) σi(T,V,I,v/i ,q,p) (13a,b) In compact notation, this secondary inverse is given by

ιp(s,d,p) ) σp[zj ,ι*(s,d,p),xI,d,0,p], ιs(s,d,d4 ,p) ) σs[zj ,ι*(s,d,p),xI,v*,d,up,p] ι*(s,d,p) ) σ*(zj ,xI,d,0,p), v* ) ι/d(s,d,d4 ,p) ) ν*[zj ,ι*(s,d,p),xI,d,d4 ,up,0,0,p] f ιI(xI,d,up,p) ) fI [zj ,ι*(s,d,p),xI,d,up,p], fI ) (fsx, fµ2)′ In physical coordinates, the inverse-state dynamics (15a) are given by

s˘ ) -[ιV(s,qs,p)/V h ]s + Fsqs, s(0) ) s0

(16a)

h ]µ2 + f2[T h ,V h ,m j ,ι/i (s,p),s,p], µ˘ 2 ) -[ιV(s,qs,p)/V µ2(0) ) µ2,0 (16b) where (σ/i and σV are defined in appendix B.2)

h ,V h ,m j ,ι/i (s,p),s,qs,0,0,p], ιV(s,qs,p) ) σV[T h ,V h ,m j ,π j ,s,0,p) ι/i (s,p) ) σ/i (T These solvent-second moment (1g, 1h) internal dynamics, in perfect inventory control mode (i.e., z ) zj ), consist of two mixing-like processes in cascade interconnection with time-varying characteristic time q(t)/V h , without any destabilizing mechanism at play because fixing the reactor temperature, volume, and monomer content at their nominal values implies that the instability by the gel effect is precluded.3 (d) Proposition 1 (Proof in Appendix A). The solvent-second moment internal dynamics (16) are IS stable with respect to (zj , xs, d, p), with amplitude-decay estimate (aI, λq ) q j /V h ).[ In deviation form, the inverse-state dynamics (16) are written as follows (OI and θI are defined in appendix B.5):

j I, e3 I ) OI(p;eI) + θI(eI,p;ed); eI ) xI - x h , OI(p;0) ) 0, θI(eI,p;0) ) 0 (17) ed ) d - d Henceforth, in deviation-form dynamical systems, φ(x,y) signifies that φ vanishes with (x, y) and φ(x;y) means that φ vanishes with y. FF Control. The FF controller

Mι: sˆ˘ ) Fsqs - (sˆ /V)q, sˆ (0) ) sˆ 0

(18a)

Cι: up ) ιp(sˆ ,d,p), us ) ιs(sˆ ,d,d4 ,p)

(18b)


z3 ) -Kp(xp - x j p), Kp ) diag(kT, kV, km, kπ), kT, kV, km, kπ > 0 (19a) x3 s ) x3 /s - Ks(xs - x/s ), x/s ) (T/j , I*)′, Ks ) diag(kj, ki), kj > kT, ki > kπ (19b) where kT, kV, km, and kπ (or kj and ki) are the primary (or secondary) control gains, x/s is the time-varying secondary setpoint vector (determined by the primary controller), and x3 /s is its time derivative, according to the equations [σ* and ν* have been presented in (11) and (12)]

j p),p] :) η*(xp,xI,d,Kp,p) x/s ) σ*[xp,xI,d,-Kp(xp-x j p), x3 /s ) ν*{xp,xs,xI,d,d4 ,up,-Kp(xp-x j p)],p} :) η/d(xp,xs,xI,d,d4 ,up,Kp,p) Kp[Kp(xp-x The substitution of these equations into the static (15b) component of the reactor inverse yields the inventorybased FF-FB controller [σp and σs have been presented in (11) and (14)]

j p),p] :) up ) σp[xp,xs,xI,d,-Kp(xp-x ηp(xp,xs,xI,d,Kp,p) us ) σs[xp,xs,xI,x3 /s -Ks(xs-x/s ),d,up,p] :) ηs(xp,xs,xI,d,d4 ,up,Ks,p) or, equivalently, in vector form

Figure 2. Closed-loop reactor (R) with (a) FF (18), (b) FF-FB (20), and (c) nonlinear estimator-based FF-FB (25) controllers.

simply amounts to the online computation of the stable reactor inverse (15) driven by the nominal (timeinvariant) setpoint19 zj , with sˆ being the computed value of the solvent content (s). The second moment differential equation (16b) has been dropped because its state is not required to compute the control u. As expected,19 the FF controller has static (18b) and dynamic (18a) components. The reactor (3) with the FF controller (18) is depicted in Figure 2. When the nominal steady state (x j ) is open-loop stable and the control model and its initial internal state are exact (sˆ 0 ) s0), the computed and actual states coincide [i.e., xˆ (t) ) x(t)] and, consequently, the application of the FF controller (18) to the reactor (3) keeps its four regulated outputs at their prescribed nominal values [i.e., z(t) ) zj ]. Otherwise, when there is an initial state deviation, the output trajectory z(t) asymptotically reaches its nominal j /V h . As it stands, value zj , with exponential rate λq ≈ q the (stable) FF controller (18) can only be applied to an open-loop stable reactor because its stability property is not modified by FF control. FF-FB Control. Assume that the reactor state x is known, recall the reactor inverse system (15), drop its dynamic component (15a), and run its static component (15b) with x3 p ) z3 ) 0 (or x3 s ) v*) replaced by the righthand side of the enforced primary (or secondary) regulation (or tracking) LNPA first-order output error dynamics (19a) [or (19b)]:

Cη: u ) η(xp,xs,xI,d,d4 ,Kp,Ks,p), u ) (u′p,u′s)′, η ) (η′p,η′s)′ (20) The reactor (3) with the FF-FB controller (20) is depicted in Figure 2, and the corresponding closed-loop system in deviation form is given by

e3 s ) -Kses, e3 p ) -Kpep + Pp(ep,eI,ed,p;es), e3 I ) OI(p;eI) + PI(eI,Kp,p;ep,es,ed) (21a-c) where eI has been defined in (17), (Pp, PI) is defined in appendix B.6, and

jp es ) xs - x/s , ep ) xp - x In compact form, system (21) is written as follows (bd M: block-diagonal matrix M)

e3 ) fe(Kc,p;e,ed), e ) (e′p,e′s,e′I)′, Kc ) bd(Ks,Kp) (22) (a) Proposition 2 (Proof in Appendix A). The closed-loop system (21) or (22) with the exact modelbased FF-FB controller (20) is IS stable, with amplitudedecay estimate

j /V h , λc ≈ min(λp,λs), [ae, λe ≈ min(λc,λq)], λq ) q λp ≈ min(kT,kV,km,kπ), λs ≈ min(kj,ki) where λp (or λs) is the dominant frequency of the enforced primary (19a) [or secondary (19b)] regulation (or tracking) LNPA output error dynamics. The

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regulated outputs (z) converge to their nominal values as follows: ∼kT

∼kV

∼km

∼kp

T(t) 98 T h , V(t) 98 V h , m(t) 98 m j , π(t) 98 π j ∼λ

where (‚)(t) f (‚h) denotes “(‚)(t) tends to (‚h) with approximate rate λ”.[ Connections with Geometric Control and MPC. The direct application of the geometric control technique, employed in previous polymer reactor studies,3,4,10-12 to our polymer reactor problem leads to the following results: (i) because of the fulfillment of the invertibility conditions (7), the reactor (3) augmented with an input dynamic extension (q˘ ) uq, q˘ m ) uqm) has relative degree vector (2, 2, 2, 2)′; (ii) the corresponding stable zero dynamics are given by the ZI version of the inverse-state dynamics [i.e., (15a) or (16) with d ) d h ]; and (iii) consequently,25 the reactor control problem is solvable with a nonlinear dynamic FF-FB controller that is more complex than its passive-static counterpart (20), as can be seen in polymer reactor geometric controllers with a second-order relative degree for the monomer content,3,4,11,12 temperature,10,11 or molecular weight.10 From a geometric control viewpoint, the passive FF-FB controller (20) has a primary (or secondary) component with input-output pair (u′p, x′s)′-z (or u′s-x′s) and relative degree vector (1, 1, 1, 1)′ [or (1, 1)′]. Thus, from a constructive control perspective,16 the FF-FB controller (20) corresponds to a simpler and robustified redesign, via passivation by backstepping, of the aforementioned geometric controllers. From an optimal control viewpoint, the passive FFFB controller (20) is an optimal stabilizing controller with respect to the objective function16 (fp and fs are given in appendix B.6):

J ) Jp(0,∞) + Js(0,∞), Jp(0,∞) )

∫0∞[(xp - xj p)′(xp - xj p) +

f′p(xp,x/s ,xI,d,up,p) Kp-2fp(xp,x/s ,xI,d,up,p)] dτ Js(0,∞) )

∫0∞[(xs - x/s)′(xs - x/s) +

f′s(xp,xs,xI,d,up,us,p) Ks-2fs(xp,xs,xI,d,up,us,p)] dτ that is meaningful in the sense that the outputs and controls are effectively penalized because no cross u-z terms are present. Because of the one-to-one correspondence between the actual (u′p, x′s)′ (or us) and the “new” control input fp (or fs), penalizing fp (or fs) is equivalent to penalizing (u′p, x′s)′ (or us). Consequently, the passive FF-FB controller (20) is (or is nearly) an unconstrained model-based MPC with infinite (or finite) time primary and secondary receding horizons, according to expression (23a) [or (23b)]:

J ) Jp(t,t+∞) + Js(t,t+∞), J ≈ Jp(t,t+4/λp) + Js(t,t+4/λs) (23a,b) It must be pointed out that, to a good extend, the control saturation issue is handled by the nonwasteful control feature inherent to optimal designs. This matter will be further discussed in subsequent sections. Concluding Remarks. The solvability of the FF and FF-FB reactor control problems is equivalent to the

passive invertibility (7) with stability (proposition 1) of the reactor control system, and these conditions are met by the entire class of free-radical polymer reactors, regardless of the polymerization system and reactor dimensions. The construction of the passive FF-FB controller (20) requires the model parameter vector p (3) and the set FSF with the kinetics (fr, fri, f0), heat transfer (fU), and heat capacity (fC) functions as well as some of their partial derivatives:

MSF: {FSF, p} FSF: {fr, fri, f0, fU, fC, ∂xfr, ∂xf0, ∂xfU, ∂xfC} p ) (p′r, p′it, p′U, p′c)′

(24)

with the understanding that the same modeling requirements (MSF) hold for the aforementioned more complex geometric controllers. According to the inversion-based inventory19 and the robustness-oriented constructive16,17 control approaches, the closed-loop reactor dynamics (22) with the passive FF-FB controller (20) constitute the limiting behavior attainable by FB control with maximum robustness, minimum control effort, and LNPA output regulation dynamics that enable the application of conventionallike tuning rules. In the next section, such behavior is regarded as the recovery target for the development of the related measurement-driven controller. Output-FB Control In this section, the behavior of the exact model-based FF-FB passive controller (20) is recovered by means of a measurement-driven controller with linear-decentralized components and reduced model dependency. Estimator-Based FF-FB Control. Along the approach of earlier polymer reactor studies, the FF-FB controller (20) can, in principle, be implemented with an EKF13,33 or a L observer.3,4,15 While the EKF yields a control system with 28 ordinary differential equations (ODEs), including 21 Riccati ones, the standard L observer yields a control system with 7 ODEs. In the case of an L-type PI geometric estimator design,21 the nonlinear passive detectability property is verified, the corresponding estimator (25a,b) is constructed, and its combination with the FF-FB controller (20) yields the dynamic control system

ˆ4 ) Kd(d-d ˆ ), E: xˆ3 m ) Km(y-Cyxˆ ), xˆ m(0) ) xˆ m0; d d ˆ (0) ) d ˆ 0 (25a) ˆ ,u,p ˆ ) xˆ m + xˆ4 ) f(xˆ ,d ˆ ,u,pˆ ) + Gm(xˆ ,d ˆ ,u,pˆ ) (y - Cyxˆ ), xˆ (0) ) xˆ 0 (25b) Gy(xˆ ,d Cηe: u ) η[xˆ ,d ˆ ,Kd(d-d ˆ ),Kc,p ˆ]

(25c)

with 10 ODEs that include three simple integral actions to eliminate the output mismatch due to modeling errors. Here, Gm and Gy (or Km and Kd) are nonlinear (or linear) gain matrices and xˆ m is the integral action state. It must be pointed out that the modeling requirements are the ones MSF (24) of the FF-FB controller (20). The reactor with the nonlinear estimator-based FF-FB controller is depicted in Figure 2. The corresponding closed-loop stability assessment and gain tuning aspects have been discussed before,3,21 and here it suffices to mention that (i) the gains are set according


to a pole placement scheme, (ii) the closed-loop stability is ensured by making the observer output convergence faster (typically 3-10 times faster) than the closed-loop dynamics (21), and (iii) the resulting closed-loop behavior approaches, with the estimation convergence rate, the behavior of the reactor with the exact model-based FF-FB passive controller (20). Even though the geometric estimator-based implementation (25) of the passive FF-FB controller (20) yields a control system that is simpler than the ones employed in the polymer reactor studies,3,4,11 from an industrial control perspective, where the volume and temperature are typically controlled with decentralized PI FB loops,20 the preceding controller (25) is still a complex dynamical system, with strong nonlinear inputoutput interaction and model dependency or, equivalently, with maintenance and reliability drawbacks for applicability. The same kinds of concerns34 hold for the applicability of the nonlinear MPC13 and calorimetric control15,35,36 schemes. These considerations motivate the control simplification and robustification pursuits of the present section. Parametric Control Realization. In a way that is analogous to the parametric representation of surfaces in calculus,37 let us rewrite the reactor model (1) in the parametric form:

V˙ ) bV - q, T˙ ) aTTj + bT, T˙ j ) ajqj + bj, yV ) zV ) V, yT ) zT ) T, yj ) Tj (26a) m ˘ ) Fmqm - bm, s˘ ) Fsqs - bs, I˙ ) wi - bi, π˘ ) bπ - aπI, zm ) m, zπ ) π (26b) b ) β(x,d,u,p), b ) (bV,bT,bj,bm,bs,bi,bπ)′, aπ ) Rπ(V,T,m,s,π,pc,pit) (26c) where aT and aj are approximations of steady-state heatexchange and inflow jacket parameters

hˆ /C hˆ , aj ) Fˆ jcˆ j(T h je - T h j)/C ˆJ aT ) U

(27)

U hˆ and C hˆ are approximations of steady-state values of the heat-transfer coefficient (fU) and heat capacity (fC) functions and aπ and the load disturbances (b) are given by (βV, βT, and βj are defined in appendix B.7)

bm ) [fC(V,m,s) (bT + aTTj) + CJbj]/ ∆ + (m/V)q :) βm(V,bT,Tj,bj,m,s,q,pc) (28a) bπ ) [1 - κ(T,m,s) π][fC(V,m,s) (bT + aTTj)]π/ [VF(V,m,s) - m - s]

decentralized, and driven by the synthetic load disturbance b and a static one (26c) that is nonlinear and interactive. Accordingly, the nonlinear FF-FB controller (20) can be rewritten as follows (µ/di is defined in appendix B.7):

T/j ) -[bT + kT(T - T h )]/aT :) µ/j (T,bT,kT), j )]/aπ :) µ/i (π,aπ,bπ,kπ) (29a,b) I* ) [bπ + kπ(π - π T˙ /j ) -(b˙ T + kTT˙ )/aT :) µ/dj(T˙ ,b˙ T,kT), j ,kπ) (29c,d) I˙ * ) µ/di(π,a˘ π,b˙ π,π q ) bV - kV(V - V h ) :) µV(V,bV,kV), j )]/Fm :) µm(m,bm,km) (30a,b) qm ) [bm - km(m - m qj ) [T˙ /j - bj - kj(Tj - T/j )]/aj :) µj(Tj,bj,T/j ,T˙ /j ,kj), wi ) I˙ * + bi - ki(I - I*) :) µi(I,bi,I*,I˙ *,ki) (30c,d) This representation is unique because it is based on the linear reactor form (26) that results from an input coordinate change.25 Combined PI-Inventory Controller. (a) PI Volume and Temperature Controllers. Let us recall the model subsystem (26a) associated with the measured outputs (y) and express it as follows:

x˘ ι ) aιuι + bι, yι ) xι, ι ) V, T, Tj, uV ) q, uT ) Tj, uj ) qj (31) Because the load bι is determined by the input-output pair (uι, yι), that is, bι ) x˘ ι - aιuι, bι can be quickly reconstructed via a linear observer, say a reduced-order one:38,39

χ˘ ι ) -ωιχι - ωι2yι - ωιaιuι, χι(0) ) χι0 ) -ωιyι0, bˆ ι ) χι + ωιyι (32) where ωι is the observer gain. Assume that bι is known, enforce the output regulation dynamics (19) on system (31), solve the resulting equation for uι, and obtain the linear controller set

uι ) [yj˘ι - kι(yι - yjι) - bι]/aι, ι ) V, T, Tj The combination of these controllers with their corresponding filters (32) yields the linear volume and cascade temperature controllers

(28b)

χ˘ ι ) -ωιχι - ωι2yι - ωιaιuι, χι(0) ) -ωιyιo, uι ) [yj˘ι - kι(yι - yjι) - χι - ωιyι]/aι; ι ) V, T, Tj (33a)

bs ) (s/V)q :) βs(V,s,q) (28c)

ˆ -T h )]/aT, yj˘ι ) 0, ι ) V, T yjj ) T/j ) [-bˆ T - kT(T (33b)

:) βπ(V,T,bT,Tj,m,s,π,pc,pit) bi ) fri(T,I) + (I/V)q :) βi(V,T,I,q,pit),

(bV, bT, bj)′ ) [βV, βT, βj]′(V, T, Tj, m, s, I, Te, Tje, qs, qj, qm, p) (28d) aπ ) (2fdAde-Ed/RTπ2)/[VF(V,m,s) - m - s] :) Rπ(V,T,m,s,π,pc,pit) (28e) In representation (26), the reactor system has two components, a dynamical one (26a,b) that is linear,

which in classical PI form are given by38,39

uι ) yj˘ι/aι - κι[(yι - yjι) + (1/τι)

∫0t(yι - yjι) dτ],

κι ) (ωι + kι)/aι, τι ) ωι-1 + kι-1, ι ) V, T, Tj (34) with κι being the proportional gain and τι the integral time. Because the linear-decentralized control model [(26a) or (31)] is an exact representation of the actual nonlinear volume (1c) and temperature (1a,b) reactor

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dynamics and the filter set (32) quickly reconstructs the load disturbances (bV, bT, bj), the linear-decentralized PI controllers (34) basically behave like their considerably more model-dependent and interactive counterparts in the nonlinear estimator-based FF-FB controller (25). In the case of measurements with a significant noise level, a filter should be added between the measurement and the PI controller or, equivalently, the PI controller (34) should be redesigned on the basis of a linear secondorder filter,40 and the result is the controller set (see Figure 3)

EV: V ˆ˙ ) bˆ V - q + 2ζVωV(yV - V ˆ ), bˆ˙ V ) ωV2(yV - V ˆ) CV: q ) bˆ V + kV(V ˆ -V h)

(35a)

ET: T ˆ˙ ) aTT ˆ j + bˆ T + 2ζTωT(yT - T ˆ ), ˆ ) (35b) bˆ˙ T ) ωT2(yT - T CTp : T ˆ /j ) -[bˆ T + kT(T ˆ -T h )]/aT, ˆ) + vˆ j ) -{ωT2(yT - T kT[aTT ˆ j + bˆ T + 2ζTωT(yT - T ˆ )]}/aT (35c) ˆ˙ j ) ajqj + bˆ j + 2ζjωj(yj - T ˆ j), bˆ˙ j ) ωj2(yj - T ˆ j) Ej: T ˆj - T ˆ /j )]/aj CTs : qj ) [vˆ j - bˆ j - kj(T

(35d)

where ωι and ζι (ι ) V, T, Tj) are the characteristic frequency and damping factor, respectively. Equation 35b-d is a linear cascade temperature control, with primary (CTp , ET) and secondary (CTs , Ej) components. These controllers are PI-type in the sense that they consist of standard decentralized PI controllers coordinated with first-order measurement filters. It must be pointed out that the construction of these controllers needs only approximated values of the steady-state heat-exchange and inflow jacket parameters (aT, aj) (27). (b) Material-Balance Monomer and MW Controllers. The combination of the FF-FB monomer controller (30b) with the monomer and solvent balances yields the following material balance (MB) monomer controller in IMC form (depicted in Figure 3):

Figure 3. Closed-loop reactor (R) with PI-inventory controller (38).

In a similar way, the combination of the FF-FB MW controller (30d) with the initiator and MW balances yields the following MB cascade MW controller (depicted in Figure 3):

Ei: Iˆ˙ ) wi - bˆ i, bˆ i ) βi(V ˆ ,T ˆ ,Iˆ,q,pˆ it)

ˆ˘ ) Fmqm - bˆ m, bˆ m ) βm(V ˆ ,bˆ T,T ˆ j,bˆ j,m ˆ ,sˆ ,q,pˆ c); Em : m ˘sˆ ) F q - bˆ , bˆ ) β (V ˆ ,sˆ ,q) (36a)

ˆ ,T ˆ ,bˆ T,T ˆ j,m ˆ ,sˆ ,πˆ ,pˆ c,pˆ it) Eπ: πˆ˘ ) bˆ π - aˆ πIˆ, bˆ π ) βπ(V

Cm: qm ) [bˆ m - km(m ˆ -m j )]/Fm

j )]/aˆ π, Cπp: Iˆ* ) [bˆ π + kπ(πˆ - π aˆ π ) Rπ(V ˆ ,T ˆ ,m ˆ ,sˆ ,πˆ ,pˆ c,pˆ it)

s s

s

s

s

(36b)

with the estimated loads bˆ T and bˆ j being available from the cascade temperature controller (35b-d). As can be seen in (28a), the computation of bˆ m only needs the static parameter aT (27), the heat capacity function fC(V,m,s) (2d), and the estimated values (in pˆ c) of the calorimetric parameter vector pc (2f). In terms of the polymerization rate fr, the load bm (28a) is given by

bm ) fr(T,V,m,I,s) + (m/V)q meaning that, from an industrial perspective,19 the last controller can be seen as a ratio-type FF component that sets the monomer flow rate qm according to the polymerization rate fr and the exit flow rate q.

E*: χˆ˘ /i ) -ω/i χˆ /i - (ω/i )2 Iˆ*, vˆ i ) χˆ /i + ω/i Iˆ

Cπs : wi ) vˆ i + bˆ i - ki(Iˆ - Iˆ*)

(37a)

(37b)

with the estimated load bˆ T being available from the temperature controller (35b-d). The subsystem Cπp-Eπ is the primary controller that generates the initiator content setpoint Iˆ*, E* is a first-order filter that provides the setpoint derivative estimate vˆ i of I˙ *, and these signals drive the secondary controller Cπs -Ei that sets the initiator feed rate wi. The first-order filter (E*) has been introduced to circumvent the (cumbersome) analytical calculation, via (29d) of the derivative I˙ *, and the need of the derivatives a˘ π and b˙ π. The implementation


of the cascade MW controller needs the static parameter aT (27), the initiator decomposition fri(T,I) (2a) and heat capacity fC(V,m,s) (2d) functions, and the estimated values (in pˆ c and pˆ it) of the calorimetric (pc) and initiation-transfer (pit) parameter vectors (2f). Preferably, the initiator decomposition and transfer rate constants should be offline-calibrated and occasionally recalibrated on the basis of the free monomer, solid fraction, and MW measurements, which are usually taken for monitoring and supervisory control purposes, according to standard experimental procedures.30 (c) PI-Inventory Controller. The combination of the linear-decentralized PI volume and cascade temperature controllers (35) with the MB monomer (36) and cascade MW (37) controllers yields the combined PIinventory controller in IMC form (the matrices Bι, Bu, and Cι and the maps fν and µ are defined in appendix B.8):

where xe is the estimator state, e (or ξ) is the reactor (or estimator) state regulation (or estimation) error, and d ˜ (or p˜ e) is the exogenous input measurement (or estimator parameter) error. Observe that, when the estimator, exogenous input measurement, and estimator parameter errors vanish (i.e., ξ ) 0, d ˜ ) 0, and p˜ e ) 0), the preceding closed-loop dynamics becomes the one (22) of the reactor with the exact model-based FF-FB controller (20). (a) Proposition 3 (Proof in Appendix A). The reactor (1) with the PI-inventory controller (38) yields an IS stable closed-loop system (40) if the (reactor state, estimator state, actual exogenous input, measured exogenous input, and estimator parameter) disturbance sizes (, ξ, δd, δδ, δp), the control (Kc), and the estimator (Kι and Kν) gains meet the next conditions (the bounding + + functions lq, l* , l* , lι , and lι are defined in the proof of proposition 3 in appendix A):

xˆ3 ι ) Bι(pι) xˆ ι + Bu(pι) u + Kι(y - Cιxˆ ι), xˆ3 ν ) fν(xˆ ι,xˆ ν,d ˆ ,u,pι,p ˆ ν),

(ia, b) 0 < kc,

ˆ ,Kc,pι,p ˆ ν) (38a-c) u ) µ(xˆ ι,xˆ ν,d

kc < k+ c ) lq(Kι,Kν,,ξ,δd,δδ,δp)

+ (iia, b) λι < ωo < λ ι

where xι (or xν) is the state of the innovated (or noninnovated) subsystem21 and pι (or pν) is the corresponding parameter vector

/ /+ (iiia, b) ω/i < ωi < ωi

xι ) (V, bV, T, bT, Tj, bj)′, xν ) (m, s, I, π, χ/i )′,

kc ) min(kV, kT, km, kπ), kc ) max(kV, kT, km, kπ, kj, ki), λq ) q j /V h, ωo ) min(ωV/ζV, ωT/ζT, ωj/ζj)

Kι ) bd[(2ζjωj,ωV2)′,(2ζTωT,ωT2)′,(2ζjωj,ωj2)′] pι ) (aT, aj)′, pν ) (p′c, p′it)′ As can be seen in (35)-(37) and Figure 3, the lineardecentralized PI-type volume and temperature controllers drive the MB monomer and MW controllers, and thus the control scheme has a one-way decoupling structure: (volume, temperature) f monomer f MW. The implementation of the controller (38) requires the estimator parameter vector pe and the function set FOF:

+ - + (λι , λι )′ ) [lι , lι ]′(Kc, Kι, Kν, , ξ, δd, δδ, δp), /+ - + (ω/i , ωi )′ ) [l* , l* ]′(Kc, Kι, λq, , ξ, δd, δδ, δp)

If these conditions are met, the regulated outputs (z) converge to their nominal values (zj ) as follows: ∼k

∼k

∼λ

v T I h , T(t) 98 T h , m(t) 98 m j +m j˜ , V(t) 98 V

∼2λI

MOF: {FOF, pe} FOF: {fri, f0, fC}, pe ) (p′ι, p′ν)′

where

j+π j˜ (41) π(t) 98 π (39)

These modeling requirements MOF (39) are significantly smaller than the ones MSF (24) of the estimator-based FF-FB controller (25) and previous estimator-based geometric3,4,11 or MPC13 schemes for polymer reactors. Because the calorimetric parameters in pc (2f) are less uncertain that the ones of the initiation-transfer vector pit (2f) and the temperature, volume, and monomer controllers do not depend on pit, the critical reactor stabilization task can be executed in a rather robust manner. Closed-Loop Dynamics and Tuning. The application of the PI-inventory controller (38) to the reactor [(1) or (26)] yields the closed-loop system (the matrices Aξ and Kν and the maps θξ and θe are defined in appendix B.9)

ξ4 ) -Aξ(Kc,Kι,Kν,pι) ξ + θξ

(Kc,Kι,Kν,p;e,ed,e3 d,e1 d,ξ,d ˜ ,d ˜4 ,p ˜ e), ξ ) xˆ e - xe (40a)

e3 ) fe(Kc,p;e,ed) + θe(e,ed,e3 d,Kc,p;ξ,d ˜ ,p ˜ e), ˜ )d ˆ - d, p ˜e ) p ˆ e - pe (40b) xe ) (x′ι, x′ν)′, d

where m j˜ (or π j˜ ) is a monomer content (or MW) asymptotic offset, depending on the calorimetric (or initiationtransfer) parameter errors p˜ c (or p˜ it).[ Condition (ia) states that FB is needed for closed-loop stability, Condition (ib) says that the control gains are bounded from above, condition (iia) [or (iib)] says that the gains of the innovated estimator subsystem (38a) are bounded from below (or above), and condition (iiia) [or (iiib)] says that the gain of the initiator setpoint filter [in (38b)] is bounded from below (or above). For given state (, ξ), input (δd, δδ), and parameter (δp) disturbance sizes, the closed-loop stability property can be ensured by making the control gains sufficiently small (typically equal or up to 3 times faster than λq) so that + + the control [(kc , kc )], estimator [(λι , λι )], and initiator //+ setpoint filter [(ωi , ωi )] gain intervals are sufficiently ample to enable a robust tuning. Expression (41) states that the PI-inventory controller (38) recovers the behavior of the exact model-based FF-FB controller (20) with (i) the adjustable rate ωo for the innovated dynamics (38a) and (ii) the fixed rate λq for the noninnovated dynamics (38b). Because the monomer and MW are not measured, the corresponding control gains (km and kπ) must be set equal to λq.

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From proposition 3 in conjunction with conventionallike tuning guidelines for linear PI controllers20 and filters,40 the next tuning rules for the proposed PIinventory controller follow: 1. Set all of the control gains at the inverse of the nominal residence time, kV ) kT ) kj ) km ) ki ) kπ ) λq ) q j /V h , and set the estimator frequency parameter (ωo) about 3 times faster, ωo ) ωV ) ωT ) ωj ) ω/i ) 3λq. For the second-order observers (35), choose a damping factor greater than 1, say ζ ) ζV ) ζT ) ζj ∈ (1, 3], to preclude the amplification of the high-frequency (jacket) dynamics. 2. Increase the estimator parameter ωo up to its ultimate value ωυo , where the response becomes oscillatory, and back off until a satisfactory response is attained, say at ωo e ωυo /3. 3. Increase the volume gain kV up to its ultimate value kυV, then back off until a satisfactory response is attained, say kV e kυV/3, and repeat the same procedure for the secondary temperature gain kj: kj e kυj /3. 4. Increase the primary temperature gain kT up to its ultimate value k/T, back off until an adequate response is attained, and repeat the procedure for the secondary MW gain (ki). 5. If necessary, adjust the damping (ζV, ζT, ζj) and/or frequency (ωV, ωT, ωj, ω/i ) estimator parameters. Different from the standard rules for the tuning of volume and temperature PI control loops, which have a single ultimate control gain,20 in the preceding rules there are control and estimator ultimate gains, meaning a simplification of the multiloop tuning task. On the other hand, the time-scale multiplicity feature of the closed-loop dynamics is reflected in the gain boundedness stated in proposition 3 and in the related tuning guidelines. Concluding Remarks. While in previous polymer reactor control studies, only up to three manipulated inputs have been considered,12 and in some cases measurements other than temperature and volume have been used,3,4,11,13 in the proposed scheme, the overall MIMO control problem, with four control inputs, has been addressed. From a nonlinear control perspective,16 the proposed controller (38) is (i) a more robust and less model-dependent version of previous geometric controllers, (ii) a stabilizing controller that is optimal with respect to the meaningful objective function (23), and (iii) a particular form of an estimator-based unconstrained MPC, with the understanding that the control saturation problem is to a good extent handled by the nonwasteful nature of the optimal stabilizing controllers in conjunction with the antireset windup protection inherent to the IMC form of the controller. According to the definitions (appendix B) of the load disturbances (bT and bj) associated with the linear cascade temperature control (35b-d), estimating the pair (bT, bj) is equivalent to estimating the heat generation rate-heat-exchange coefficient pair (∆fr, fU) or, equivalently, the linear PI-type cascade temperature controller (35b-d) performs the same task of their more complex (EKF or L) observer-based calorimetric control counterparts.14,15,35,36 Application Example To subject the proposed PI-inventory controller (38) to a severe test, let us consider an extreme case of an

Table 1. Steady States, Nominal Inputs, Outputs, and Reactor States, and Jacket Parameter Approximations steady states states and outputs

stable (extinction)

unstable

T (K) 329.72 351.62 Tj (K) 329.53 341.23 m (kg) 1 361.089 660.082 I (kg) 1.685 1.3087 s (kg) 501.283 500.871 π (kg/kmol) 399149.03 110384.75 µ2 (kg) 1.907 × 106 1.303 × 106 c 0.1072 0.5672 ψs 0.08066 0.4269 Q 1.9997 1.999 nominal inputs u j ) (q j j, q j, q j m, w j i)′ ) (30 L/min, 9.1 L/min, 7.34 L/ min, 0.007 852 6 kg/min)′ d h ) (T h e, T h je, q j s)′ ) (315 K, 328 K, 2.54 L/min)′ jacket parameters aT ) 4.71 × 10-2 min-1, aj ) -3.78 × 10-2 K/L

stable (ignition) 373.88 345.61 312.756 0.3513 498.561 29395.15 7.127 × 105 0.7954 0.5997 1.9966

industrial situation: the operation of a reactor at high solid fraction with the gel effect at play, about a nominal steady state that is open-loop unstable. The homopolymerization system has methyl methacrylate as the monomer, ethyl acetate as the solvent, and azobis(isobutyronitrile) as the initiator. The reactor residence time is V h /q j ) 220 min with nominal volume zjV ≈ 2000 L. The functionalities and parameter values were taken and/or adapted from Alvarez et al.,12 and the nominal inputs (shown in Table 1) were adapted from a solution copolymer reactor study.21 The reactor has three steady states (listed in Table 1), with two of them corresponding to extinction and ignition open-loop stable operations and one of them being the nominal open-loop unstable operating point. In Figure 4, the open-loop reactor response to an initial state perturbation about the unstable steady state is shown: the reactor reaches its ignition stable steady state after evolving in an ample region of the state space, with an ignition-vs-extinction damped oscillatory response. The nominal values of the open-loop unstable steady state are shown in the same figure, illustrating the control task: to steer-back the states to their nominal values in the open-loop unstable steady-state regime. Nominal Behavior. The application of the tuning guidelines yielded the following estimator and control gains [the parameter approximations (aT, aj) are shown in Table 1]:

ωo ) ωυo ) (U h + Fjq j jcj)/CJ ) 1/5 min-1, ζ ) 0.71 (42a) kV ) kj ) ωo/8, kT ) 2λq, ki ) (4/3)λq, km ) kπ ) λq ) 1/220 min-1 (42b) The closed-loop behavior with the exact model-based FF-FB (20) and the proposed PI-inventory (38) controllers is presented in Figure 5, showing that (i) as expected from the theoretical development, the PIinventory controller basically recovers the behavior of its exact model-based FF-FB counterpart, (ii) the volume, temperature, and monomer content outputs reach their prescribed setpoints in about one residence time, with the understanding that the regulation of these outputs ensures the stability, the production rate level, and an important part of the product quality, (iii) the MW reaches its setpoint in about two residence


Figure 4. Open-loop reactor behavior (s) and nominal operation values (‚‚‚).

Figure 5. Closed-loop reactor behavior with exact model-based FF-FB (- - -) and PI-inventory (s) controllers, and nominal operation values (‚‚‚).

times, (iv) the polydispersity reaches its nominal value in about three residence times, with less than 5% overshoot, and (v) as expected from the nonwasteful control feature inherent to the optimality property (23) that underlies the proposed passive control scheme, the four manipulated inputs evolve in a smooth-coordinated and away-from-saturation manner. The closed-loop behavior with the PI-inventory controller, with the MW control in manual (wi ) w j i) and automatic modes, is presented in Figure 6. In both cases, (i) the volume, temperature, and monomer content are regulated in about one residence time, verifying that the functioning of the corresponding controllers is not affected by the operation mode of the MW input, and (ii) the manipulated inputs evolve in a smooth-coordinated and away-from-saturation fashion. In the same figure, the advantage of manipulating the initiator feed rate to control the MW is shown: the response in automatic mode is twice as fast as the one in manual mode. Robust Behavior. In Figure 7, the closed-loop behavior with the PI-inventory controller is presented,

with and without typical parameter errors: -14% in the monomer (am ) 4430.25) and solvent (as ) 288.46) transfer preexponential parameters and -5% in the initiator efficiency constant (fd ) 0.52). As can be seen in the figure, because of the one-way decoupling control structure, the presence of these errors does not affect the stability and the production rate level and yields only a 3% MW asymptotic offset, which is smaller than the measurement device uncertainty. As mentioned before, in the case of unacceptable large MW offsets, the initiator and/or transfer constants should be occasionally recalibrated on the basis of the free monomer, solid fraction, and MW measurements that are usually taken for operation and product monitoring purposes, within a supervisory control scheme. The corresponding control responses are presented in Figure 7, showing that (i) the above-mentioned parameter errors do not affect significantly the smooth-coordinated and away-fromsaturation behavior of the control inputs and (ii) the rise of the initiator feed rate causes the initiator content to increase and hence the MW offset. On the other hand, the upsurge of the initiator content produces an increase

7158


Figure 6. Closed-loop reactor behavior with PI-inventory controller, with MW control in automatic (s) and manual (- - -) modes, and nominal operation values (‚‚‚).

of the heat generation rate, accompanied by an increase of the coolant flow rate to maintain the reactor temperature at its nominal value. As we shall see (in Figure 8), all of these features are also shown in the case with noisy measurements. Finally, the closed-loop reactor was run with the above-stated parameter errors in conjunction with Gaussian measurement noises injected every 30 s, with standard deviations that correspond to common online instruments: 0.5 K for the temperatures (yT and yj) and 4 L for the volume (yV). In this case, the application of the tuning guidelines yielded the following settings:

ωo ) 1/8 min-1, ζ ) 1.2

(43a)

kV ) kj ) (1.1)ωυo /8, kT ) 2.2λq, ki ) (4.4/3)λq, km ) kπ ) λq (43b) In comparison with the gains (42) of the noiseless case,

Figure 7. Closed-loop reactor behavior with PI-inventory controller, with (- - -) and without (s) parameter errors, and nominal operation values (‚‚‚).

the presence of noise led to gain detuning (10-15% slower with more damping). The corresponding closedloop behavior is presented in Figure 8, including the noiseless case for comparison purposes. As can be seen in the figure, the presence of noise slightly affects the volume, temperature, and monomer responses but not the ones of the MW and polydispersity. As expected from a cascade control scheme, the high-frequency disturbances due to modeling errors and measurement noise are mostly compensated for by the secondary control components and do not appreciably manifest themselves in the primary control loops: while the secondary states (Tj and I) and controls (qj and wi) exhibit responses with some high-frequency components, the regulated outputs (T, V, m, and π; primary states) and the primary controls (q and qm) present rather smooth responses with low-frequency components set by the prescribed primary loop gains and the fixed internal dynamics. Because of the optimality feature that underlies the passive control scheme, the multiple control response


linear PI controllers need two approximate steady-state parameters, the monomer controller needs the heat capacity function, and the MW controller needs the initiation-transfer constants. The control system has a one-way loop decoupling structure that tolerates automatic-to-manual mode switching and failures in the MW control loop. The proposed dynamic controller (i) recovers the behavior of an exact model-based nonlinear passive static controller, with LNPA output error dynamics, maximum robustness, and minimum control effort, (ii) is equivalent to an observer-based unconstrained MPC, (iii) has a nonlocal IS stability criterion coupled with conventional-type tuning guidelines, and (iv) has solvability conditions with physical meaning that are met in any practical situation. An open-loop unstable industrial-size reactor was considered as an application example with numerical simulations, finding that (i) the PI-inventory controller basically behaves like its exact model-based nonlinear counterpart, (ii) the nearly model-independent ratio-type monomer control loop effectively performs the key stabilizing task, (iii) the MW controller halves the MW response of its manual mode counterpart and can be switched between automatic and manual modes without affecting the functioning of the other loops and the closed-loop stability property, and (iv) the closed-loop behavior corroborates the formally established nonlocal robustness, behavior recovery, and nonwasteful features associated with the passivity and optimality properties that underlie the proposed inventory-based output-FB control design. Acknowledgment The authors gratefully acknowledge the support from the Mexican National Council for Research and Technology (CONACyT Scholarship 118632) and from a CIPCOMEX graduate student award for P.G., as well as the CIP’s Faculty award for J.A. Appendix A: Stability Proofs Tools. (a) Theorem 1.26 The system Figure 8. Closed-loop reactor behavior with a PI-inventory controller with parameter errors [noisy (- - -) and noiseless (s) measurements], and nominal operation values (‚‚‚).

exhibits a smooth-coordinated and away-from-saturation behavior. In summary, the closed-loop nominal and robust behavior tests corroborate the nonlocal stability and behavior recovery features stated in proposition 3 as well as the related tuning rules and manifest the robustness and nonwasteful control features associated with the passivity and optimality properties that underlie the proposed inventory inversion-based control design. Conclusions On the basis of inventory and constructive control ideas, a combined PI-inventory control scheme for continuous (possibly open-loop unstable) polymer reactors has been derived. The resulting control system has a subsystem made by linear-decentralized PI-type volume and cascade temperature controllers and a subsystem made by monomer and MW controllers. The

x3 ) f[x,v(t)], x(0) ) x0; f(0,0) ) 0

(A1)

is IS stable if it (i) is ZI stable and (ii) has an asymptotic gain γxv [i.e., ∃ γxv 3 limtf∞ |x(t)| < γxv||v||].[ (b) Lemma 1.32 Let system x3 ) f(x) be (ax, λx) stable. The perturbed system

x3 ) f(x) + p(x,v), |p(x,v)| e Lpx(,δ)|x| + Lpv(,δ)|v|, |x| e , |v| e δ (A2) is IS stable if (ax/λx)Lpx(,δ) < 1. The resulting decay parameter is λx - axLpx(,δ), and Lpx is the (nonlocal) Lipschitz constant of p(x,v) with respect to x: Lpx(,δ) ) max(x,v) |f(xˆ ,v) - f(x,v)|/|xˆ - x|.[ (c) (Small-Gain) Theorem 2.17 Let system (A3a) [or (A3b)] be IS stable with asymptotic gain γxy (or γyx) with respect to y (or x). The two-system interconnection (A3) is IS stable if γxy γyx < 1.

x3 ) f(x,y,v), f(0,0,0) ) 0; y3 ) g(x,y,v), g(0,0,0) ) 0 [ (A3a,b)

7160


(d) Corollary 1. The cascade interconnection [(A3a) with γxy ) 0] of two IS stable systems is IS stable.[ The combination of lemma 1 with theorem 2 yields the following lemma. (e) Lemma 2.27,32 The two-system interconnection

x3 ) f(x) + p(x,y,v), |p(x,y,v)| e Lpx(x,y,δ)|x| + Lpy(x,y,δ)|y| + Lpv(x,y,δ)|v|, |x| e x (A4a) y3 ) g(y) + q(x,y,v), |q(x,y,v)| e Lqx(x,y,δ)|x| + Lqy(x,y,δ)|y| + Lqv(x,y,δ)|v|, |y| e y, |v| e δ (A4b) is IS stable if

and (aI, λq) stable, respectively, with the unity amplitude factors being a consequence of the passive structure of the primary and secondary controllers (20). The IS stability of subsystems (21a) and (21b) with (ac, λc) ≈ [1, min(λp, λs)] follows from corollary 1 and the Lipschitz boundedness of Pp. The IS stability of system (21) with [ae, λe ≈ min(λc, λq)] follows from corollary 1 and the Lipschitz boundedness of PI. QED (c) Proof of Proposition 3. Rewrite system (40a) in terms of its innovated-noninnovated state partition (the matrices Aι and Aν and the maps θι and θν are defined in appendix B.9):

ξ4 ι ) -Aι(Kι,pι) ξι + θι

(Kc,Kι,Kν,p;e,ed,e3 d,e1 d,ξι,ξν,d ˜ ,d ˜4 ,p ˜ e) (A5a)

ξ4 ν ) -Aν(Kc,Kν) ξν +

(ia) x3 ) f(x) is (ax, λx) stable

θν(Kc,Kι,Kν,p;e,ed,e3 d,e1 d,ξι,ξν,d ˜ ,d ˜4 ,p ˜ e) (A5b)

(ib) y3 ) g(y) is (ay, λy) stable (iia) γxx(λx,x,y,δ) < 1

where the last equation in (A5b) is the initiator setpoint filter error dynamics (the map θ/i is given in appendix B.9):

(iib) γyy(λy,x,y,δ) < 1

˜ ,d ˜4 ,p˜ e), ξ˙ χ ) -ω/i ξχ + θ/i (Kc,Kι,Kν,p;e,ed,e3 d,e1 d,ξι,ξν,d

(iii) γxy(λx,x,y,δ) [1 - γxx(λx,x,y,δ)]-1γyx(λy,x,y,δ) [1 - γyy(λy,x,y,δ)]-1 < 1 where

γxR(λx,x,y,δ) ) (ax/λx)LpR(x,y,δ), γyR(λy,x,y,δ)

)

(ay/λy)LqR(x,y,δ),

R ) x, y [

Proposition Proofs. (a) Proof of Proposition 1. Consider the Lyapunov function Vs (or Vµ2), write its j s [or (16b) time derivative along system (16a) with qs ) q j s)], observe that ιV(s,q j s,p) > 0 and with (s, qs) ) (sj, q j s,p) > 0, conclude that the ZI solvent (or second ∂sιV(s,q moment) dynamics are stable (esx ) s - sj and eµ2 ) µ2 µ j 2):

Vs ) esx2/2, j s,p)/V h ]esx2 - sj[ιV(esx+sj,q j s,p) V˙ s ) -[ιV(esx+sj,q ιV(sj,q j s,p)/V h ]esx < 0 2

2

j s,p)/V h ]eµ2 < 0 Vµ2 ) eµ2 /2, V˙ µ2 ) -[ιV(esx+sj,q and observe that ιV(esx+sj,q j s,p)/V h ≈ λq. Write the asymptotic behavior of system (16) as

h Fsqs :) Rs(s,qs,p), 0 ) sιV(s,qs,p) - V ∂sRs ) ιV(s,qs,p) + s∂sιV(s,qs,p) > 0 h f2[T h ,V h ,m j ,ι/i (s,p),s] :) 0 ) µ2ιV(s,qs,p) - V Rµ2(µ2,s,qs,p), ∂µ2Rµ2 ) ιV(s,qs,p) > 0 meaning that bounded disturbances produce asymptotically bounded deviations (esx, eµ2). From theorem 1, (16a) [or (16b)] is IS stable. From corollary 1, the system (16) is IS stable with (aI, λq). QED (b) Proof of Proposition 2. Subsystems (21a), (21b) with Pp ) 0 and (21c) with PI ) 0 are (1, λs), (1, λp),

ξχ ) χˆ /i - χ/i (A6) Subsystem (A5a) with θι ) 0 [or (A5b) with θν ) 0] is (aι, λι) stable [or (aν, λq) stable]. According to lemma 1, the innovated (A5a) [or initiator setpoint filter (A6)] estimation error dynamics are stable if

ωo > aιLθξιι(Kc,Kι,Kν,,ξ,δd,δδ,δp), ω/i > Lθξχi (Kc,Kι,Kν,,ξ,δd,δδ,δp) (A7a,b) where , ξ, δd, δδ, and δp are respectively the reactor state, estimator state, actual exogenous input, measured exogenous input, and estimator parameter disturbance sizes. Because the right-hand side of inequality (A7a) [or (A7b)] grows nonlinearly with ωo (or ω/i ) and the equality version of inequality (A7a) [or (A7b)] is met + /and ω/+ with two values, ωo and ωo (or ωi i ) of ωo (or / ωi ), this is, (A8) [or (A9)] is met: + // /+ ωo < ωo < ωo ; ωi < ωi < ωi

(A8a,b)

+ + (ωo , ωo )′ ) [Lι , Lι ]′(Kc, Kν, , ξ, δd, δδ, δp), /+ - + (ω/i , ωi )′ ) [l* , l* ]′(Kc, Kι, λq, , ξ, δd, δδ, δp) (A9a,b)

These from above and below bounds resemble the ones drawn in the stability proof of a nonlinear estimator with sampled measurements.41 Inequalities (A8) [or (A9)] ensure the IS stability of the innovated (or initiator setpoint filter) error dynamics (A5a) [or (A6)], and (A9) yields condition (iii) of proposition 3. According to lemma 1, the noninnovated error dynamics (A5b) are IS stable if

λq > aνLθξνν(Kc,Kι,Kν,,ξ,δd,δδ,δp) w kc < lν(Kι,Kν,,ξ,δd,δδ,δp) (A10) In virtue of (A8)-(A10), the application of lemma 2 to (A5) yields that the estimation error dynamics (A5) are


IS stable if θι θν θν -1 ωo > ωo + aιaν[Lξν Lξι (λq - aνLξν ) ]

(Kc,Kι,Kν,,ξ,δd,δδ,δp) :) lι (Kc,Kι,Kν,,ξ,δd,δδ,δp) (A11a) θι θν θν -1 ω o < ω+ o - aιaν[Lξν Lξι (λq - aνLξν ) ]

(Kc,Kι,Kν,,ξ,δd,δδ,δp) :) l+ ι (Kc,Kι,Kν,,ξ,δd,δδ,δp) (A11b) or, equivalently, if condition (ii) of proposition 3 is met. In other words, the closed-loop estimation error dynamics (40a) are IS stable. The closed-loop regulation error dynamics (40b) are IS stable because of Lθe e ) 0 and lemma 1. Regarding the two-IS stable system interconnection (40), apply lemma 2 and conclude that the closed-loop system (40) is IS stable if

λq > aξae[Lθe ξ Lθξ e(λq - aξLθξ ξ)-1] (Kc,Kι,Kν,,ξ,δd,δδ,δp) w kc < leξ(Kι,Kν,,ξ,δd,δδ,δp) (A12) or, equivalently, in light of (A10), if condition (ib) of proposition 3 is met with

ν/j (T,Tj,V,m,I,s,vi,Te,qs,T˙ e,q˘ s,q,qm,q˘ m,T˙ ,V˙ ,m ˘ ,T ¨ ,p) ) ˘ - (∂IφT)I˙ [T ¨ - (∂TφT)T˙ - (∂VφT)V˙ - (∂mφT)m (∂sφT)φsx - (∂qsφT)q˘ s - (∂TeφT)T˙ e - (∂qmφT)q˘ m]/∂TjφT B.4. Secondary Inverse Maps (13).

σj(T,Tj,V,m,s,vj,Tje,p) ) {CJT˙ j - φU(T - Tj)}/[Fjcj(Tje - Tj)] σi(T,V,I,vi,q,p) ) I˙ + fri(T,I) + (I/V)q B.5. Inverse State Dynamics (17).

OI(p;eI) ) fuI (0,0,eI,0,Kp,p), θI(eI,p;ed) ) PI(eI,Kp,p;0,0,ed) B.6. Closed-Loop Reactor with FF-FB Controller (21) and Optimal Objective Function (23).

Pp(ep,eI,ed,p;es) ) fup(ep,es,eI,ed,Kp,p) fup(ep,0,eI,ed,Kp,p)

lq(Kι,Kν,,ξ,δd,δδ,δp) ) min[lν(Kι,Kν,,ξ,δd,δδ,δp), leξ(Kι,Kν,,ξ,δd,δδ,δp)]

PI(eI,Kp,p;ep,es,ed) ) fuI (ep,es,eI,ed,Kp,p) -

In summary, the closed-loop system (40) with the PIinventory controller (38) is IS stable if conditions (i)(iii) of proposition 3 are met. QED

j p,es+η/e f up(ep,es,eI,ed,Kp,p) ) fp[ep+x

Appendix B: Nonlinear Maps

j p,es+η/e f uI (ep,es,eI,ed,Kp,p) ) fI[ep+x

B.1. Values Oa of the Nonlinear Function fr in (B.2)-(B.4) and (B.7).

φr ) fr(T,V,m,I,s), φU ) fU(T,Tj,V,m,s) φsx ) fsx(V,s,qs,q), φπ ) fπ(T,V,m,π,I,s), φT ) fT(T,Tj,V,m,I,s,Te,qs,qm) B.2. Primary Inverse Maps (8).

˘ ,p) ) σ/j (T,V,m,I,s,Te,qs,T˙ ,V˙ ,m ˘ ),T˙ ] fT-1[T,V,m,I,s,Te,qs,σm(T,V,m,I,s,qs,V˙ ,m ˘ ,p) σ/i (T,V,m,π,s,π˘ ,p) ) f-1 π (T,V,m,π,s,π

fuI (0,0,eI,0,Kp,p)

(ep,eI,ed,Kp,p),eI+x j I,ed+d h ,ηep(ep,es,eI,ed,Kp,p),p]

(ep,eI,ed,Kp,p),eI+x j I,ed+d h ,ηep(ep,es,eI,ed,Kp,p),p] fp(xp,xs,xI,d,up,p) ) [fT, fV, fm, fπ]′(xp, xs, xI, d, up, p) fs(xp,xs,xI,d,up,us,p) ) [fj, fi]′(xp,xs,xI,d,up,us,p), fI(xp,xs,xI,d,up,p) ) [fsx, fµ2]′(xp,xs,xI,d,up,p) ηep(ep,es,eI,ed,Kp,p) ) j p,es+η/e(ep,eI,ed,Kp,p),eI+x j I,ed+d h ,Kp,p] ηp[ep+x j p,eI+x j I,ed+d h ,Kp,p) η/e(ep,eI,ed,Kp,p) ) η*(ep+x B.7. Parametric Control Realization (26).

˘ ,p) ) σV(T,V,m,I,s,qs,V˙ ,m [m ˘ - FmV˙ + (1 - m)φr + Fmqs]/(Fm - m/V)

β ) (βV, βT, βj, βm, βs, βi, βπ)′, βV(V,T,m,s,I,qs,qm,p) ) qm + qs - (m/Fm)φr

˘ ,p) ) σm(T,V,m,I,s,qs,V˙ ,m ˘ ,p)]/Fm [m ˘ + φr + (m/V)σV(T,V,m,I,s,qs,V˙ ,m

βT(V,T,Tj,m,s,I,Te,qs,qm,p) ) {∆φr - φU(T - Tj) + (Fmqmcm + Fsqscs)(Te - T)}/fC(V,m,s) - aTTj

B.3. Secondary State Time Derivatives (9).

ν/i (T,V,m,π,I,s,qs,q,T˙ ,V˙ ,m ˘ ,π˘ ,π¨ ,p) ) [π¨ - (∂Tφπ)T˙ - (∂Vφπ)V˙ - (∂mφπ)m ˘ - (∂πφπ)π˘ (∂sφπ)φsx]/∂Iφπ

βj(V,T,Tj,m,s,Tje,qj,p) ) [φU(T - Tj) + Fjqjcj(Tje - Tj)]/CJ - ajqj µ/di(π,a˘ π,b˙ π,π j ,kπ) ) (∂πµ/i )[-kπ(π - π j )] + (∂aπµ/i )a˘ π + (∂bπµ/i )b˙ π

7162


θ/i ) -(ω/i )2µ˜ /i - I¨*,

B.8. PI-Inventory Controller (38).

(

)

( )

˜ e+pe) µ˜ /i (e,ed,Kc,pe;ξ,p˜ e) ) µ/i (ξ+xe,Kc,p

Γ 0 0 -Γ 0 0 , Bι(pι) ) 0 Γ ΛT(pι) , Bu(pι) ) 0 Λ (p ) 0 0 Γ j ι 0

( )

( )

µ/i (xe,Kc,pe) ˜4 ,d ˜ ,p ˜ e), I¨* ) µ¨ /i (Kc,Kι,Kν,pe;e,ed,e3 d,e1 d,ξ,d Bβ ) (Bβ1, Bβ2, Bβ3), Bβ1 ) (0, 1, 0, 0, 0, 0)′,

( )

a 0 a 0 0 1 , Λj(pι) ) j , , ΛT(pι) ) T Γ) 0 0 0 0 0 0 Cι ) bd(cι,cι,cι) cι ) (1, 0), fν(xι,xν,d,u,pι,pν) ) -(q/V)D(xι,xν,u,pν) xν + Bν(xι,xν,d,u,pι,pν) D(xι,xν,u,pν) ) bd[I2,Mν(xι,xν,u,pν)], I2 ) diag(1, 1) Mν(xι,xν,u,pν) )

[

1 0 0 (V/q)Rπ(V,T,m,s,π,pc,pit) 0 0 0 0 (V/q)ω/i

]

Bν(xι,xν,d,u,pι,pν) ) [-{fC(V,m,s)(bT+aTTj)+ CJbj}/∆+Fmqm,Fsqs,-fri(T,I) + wi,βπ(V,T,bT,Tj,m,s,π,pc,pit),-(ω/i )2µ/i (xι,xν,pι,pν)]′, µ) (µj, µV, µm, µi)′ B.9. Closed-Loop Reactor (40) with PI-Inventory Controller. B.9.a. Regulation Error Dynamics (40b).

˜ ,p ˜ e) ) f(e+x j ,ed+d h ,u,p) θe(e,ed,e3 d,Kc,p;ξ,d f(e+x j ,ed+d h ,usf,p) θe ) (θeT, θej , θeV, θem, 0, θei , θes, θµe 2)′ u ) usf + u ˜ ) η(e+x j ,ed+d h ,e3 d+d h4 ,Kc,p) + ˜ ,p ˜ e), usf ) η(x,d,d4 ,Kc,p) µ˜ (e,ed,Kc,pe;ξ,d xe ) (x′ι, x′ν)′ ) (x′y, b′ι, x′ν)′, xy ) (V, T, Tj)′, j ,ed+d h ,u,p) bι ) (bV, bT, bj)′ :) βι(e+x ˜ ,p˜ e) ) µ(ξ+xe,d ˜ +d,Kc,p ˜ e+pe) µ˜ (e,ed,Kc,pe;ξ,d µ(xe,d,Kc,pe)

Bβ2 ) (0, 0, 0, 1, 0, 0)′, Bβ3 ) (0, 0, 0, 0, 0, 1)′ Notation Ad ) preexponential constant of the initiator decomposition rate, min-1 aj ) steady-state inflow jacket parameter, K/L am, as ) preexponential constants of the monomer and solvent transfer-to-propagation quotients aT ) steady-state heat-exchange jacket parameter, min-1 b ) load disturbance vector c ) conversion CJ ) heat capacity of the cooling system, J/K cm, cs, cp, cj ) monomer, solvent, polymer, and coolant fluid specific heat capacities, J/kg‚K d ) exogenous input vector Ed ) activation energy of the initiator decomposition rate, J/mol fC ) heat capacity function, J/K fd ) initiator efficiency constant fr, fri, f0, f2 ) polymerization, initiator decomposition, and zeroth and second moment rate functions, kg/min fU ) heat-transfer coefficient function, J/min‚K I ) initiator mass, kg kT, kV, km, kπ, kj, ki ) temperature, volume, monomer, MW, jacket, and initiator control gains, min-1 Lm, Ls ) activation energy constants of monomer and solvent transfer-to-propagation quotients, J/mol m, mp ) monomer and polymer masses, kg p, pc, pit, pr, pU, pe ) reactor, calorimetric, initiationtransfer, polymerization, heat-transfer, and estimator parameter vectors Q ) MW polydispersity qm, qs, qj, q ) monomer, solvent, coolant, and exit volumetric flow rates, L/min s ) solvent mass, kg T, Tj ) reactor and jacket temperatures, K Te, Tje ) reactor and jacket feed temperatures, K u ) control input vector up, us ) primary and secondary control vectors V ) volume, L wi ) initiator mass feed rate, kg/min x ) reactor state vector xp, xs, xI ) primary, secondary, and internal state vectors y, z ) measured and regulated output vectors Greek Symbols

B.9.b. Estimation Error Dynamics (40a) or (A5,6).

Aξ ) diag(Aι, Aν), Aι(Kι,pι) ) KιCι - Bι(pι), j /V h , ki, kπ, ω/i ), Aν(Kc,Kν) ) diag(km, q j /V h , ω/i )′ Kν ) (q ˜ ,d ˜4 ,p˜ e) ) θξ ) (θ′ι, θ′ν)′, θι(Kc,Kι,Kν,p;e,ed,e3 d,e1 d,ξ,d -Bββ4 ι(Kc,Kι,Kν,p;e,ed,e3 d,e1 d,ξ,d ˜ ,d ˜4 ,p ˜ e) j /V h )ξs + θes, - θei + ξχ + θν ) [-θem, (q/V - q (ki + ω/i )µ˜ /i , aπei + aˆ π(- ξi - ei + µ˜ /i ), θ/i ]′

∆ ) heat of polymerization, J/kg m, s ) monomer and solvent contraction factors referred to the polymer density κm, κs ) monomer and solvent transfer-to-propagation quotients µ2 ) second moment of the molecular weight distribution, kg π ) number-average molecular weight, kg/kmol F ) density function, kg/L Fm, Fs, Fp, Fj ) monomer, solvent, polymer, and coolant fluid densities, kg/L ω/i , ωV, ωT, ωj ) initiator setpoint derivative, volume, temperature, and jacket estimator frequency parameters, min-1 ψs ) solid fraction

Ind. Eng. Chem. Res., Vol. 44, No. 18, 2005 7163 ζV, ζT, ζj ) volume, temperature, and jacket estimator damping parameters

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Received for review July 27, 2004 Revised manuscript received March 24, 2005 Accepted May 4, 2005 IE040207R

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