Constructive Molecular Weight Control of Continuous Homopolymer

Ind. Eng. Chem. Res. 2008, 47, 9971–9982

9971

Constructive Molecular Weight Control of Continuous Homopolymer Reactors Pablo Gonza´lez and Jesu´s Alvarez/ UniVersidad Autońoma MetropolitanasIztapalapa, Depto. de Ingenierıá de Procesos e Hidraúlica, Apdo. 55534, 09340 Me´xico D.F, Me´xico

In this work, the problem of controlling (possibly open-loop unstable) continuous free-radical solution polymer reactors with continuous measurements of temperature, level and flows and discrete-delayed (DD) measurements of molecular weight (MW) is addressed. On the basis of physical insight and the relative degrees and detectability structure of the reactor system, a muti-input-multi-output (MIMO) interlaced observer-control design is performed within a constructive framework, yielding a linear-decentralized control scheme with volume, temperature, and monomer components driven by continuous-instantaneous measurements, and a MW component driven by DD measurements. The scheme has reduced modeling requirements and conventional-like tuning guidelines drawn from stability considerations. The proposed approach is illustrated and tested with a representative example through simulations. 1. Introduction An important class of materials is produced in continuous freeradical polymer reactors. Due to their strong exothermicity and potential gel effect (reaction autoacceleration with increase of viscosity and decrease of heat removal capability), these reactors exhibit a complex behavior with strong and asymmetric input-output coupling, steady-state multiplicity, and parametric sensitivity,1,2 and consequently, they offer an interesting and challenging multiinput-multi-output (MIMO) nonlinear control problem. In industrial practice,3,4 the production rate, stability, safety, and quality indicators are met by controlling the volume and temperature with conventional linear PI loops,5,6 the conversion with feedback and/ or feedforward control7–10 driven by indirect free-monomer content measurements, and the molecular weight (MW) with supervisory control and offline MW determinations. In particular, the control of the MW is motivated by the need to adequately meet product quality specifications, which are related to the mechanical properties of the polymeric materials.11 The polymer reactor control problem has been the subject of theoretical, simulation, and experimental studies, the related literature can be seen elsewhere,3,4 and here, it suffices to mention that, despite the multi-input (monomer feed, initiator and/or transfer agent dosages, exit flow rate, and heat exchange) multi-output (volume, temperature, conversion, MW) nature of the industrial control problem, only parts of it have been addressed in previous studies. Regarding the MW component: (i) size exclusion chromatography12,13 (SEC) or gel permeation chromatography14 (GPC) discrete-delayed (DD) MW measurements, as well as indirect-continuous viscosity measurements15–17 have been employed, (ii) a diversity of control approaches has been applied, including linear proportional integral (PI) controllers,13 as well as geometric,7,14,18 model predictive16,17 (MPC), and calorimetric19,20 nonlinear controllers, and (iii) the output-feedback (OF) MW controllers have been implemented with open-loop,7,14 extended Kalman filter13,16 (EKF), and Luenberger-type21 nonlinear observers. The resulting OF controllers are strongly nonlinear, interactive, and model-dependent systems with complexity, reliability, and cost drawbacks for industrial applicability. It must be pointed out that these MW control schemes have been tested, via simulations or experiments, without the gel effect at play, or equivalently, with open-loop stable reactors. / To whom correspondence should be addressed. E-mail: jac@ xanum.uam.mx. Phone: +52 (55) 5804 4648, 51, ext 212. Fax: +52 (55) 5804 4900.

Due to its inherent time-ahead prediction capability, the MPC approach seems to be the best suited technique to handle chemical processes with DD measurements;22,23 however, despite its successful implementations to polymer reactors, the nonlinear MPC bears the aforementioned drawbacks for industrial applicability.22 Given that the natural settling time unit has been used to compare closed-loop behaviors of processes with different scales,5,24 and that the Sampling Theorem provides criteria for closed-loop behavior with sampled measurements,25,26 the residence time (τr) to sampling period (∆) quotient (n∆ ) τr/∆) offers a means to compare closed-loop behaviors of continuous processes with different residence times and measurement sampling periods. Regarding continuous polymer reactors, n∆ from two14 to twenty16 have been reported. In the case with the smallest quotient (n∆ ) 2), the MW was regulated within 1.5 residence times (about 3 sampling periods) in a pilot scale experimental polymer reactor. The above-discussed OF MW control studies have established the feasibility of implementing advanced control schemes, and provide valuable insight on the nature of the control problem. However, due to the complexity and model dependency features of the existing advanced controllers, the industrial MW regulation problem is still regarded as a subject that requires further research, with emphasis on simplicity and multiple-loop coordination. Recently,6,27 the MIMO polymer reactor control problem with only continuous instantaneous volume and temperature measurements has been addressed, drawing a scheme with decentralized PI volume and temperature components as well as monomer and MW inventory controllers. In particular, the MW component achieved output regulation within 2 residence times, with an offset that depends on the initiation and chain-transfer kinetics model parameter accuracy. Thus, to preclude a large asymptotic MW offset, the industrial implementation of this MW control component requires periodic calibration of the initiation-transfer model on the basis of MW determinations. These considerations motivate the present study. In this work, the problem of controlling (possibly open-loop unstable) continuous free-radical solution polymer reactors with continuous (temperature, level, and flows) and DD (MW) measurements is addressed, with emphasis on the attainment of a MIMO control scheme with (i) linearity, decentralization, and reducedmodel dependency features, (ii) a systematic construction-tuning scheme, and (iii) conventional-like tuning guidelines drawn from stability considerations. The central issues are the design of a lineardecentralized MW controller driven by DD measurements and its

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9972 Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008

appropriate coordination with the volume, temperature, and monomer components, driven by continuous measurements, of the abovediscussed previous studies without MW measurements.6,27 The proposed approach is illustrated and tested with a representative open-loop unstable reactor through simulations, and the results are put in perspective with the ones obtained with previous modelbased control schemes for open-loop stable reactors. 2. Control Problem 2.1. Polymer Reactor. Consider the class of continuous stirred tank reactors (depicted in Figure 1) where an exothermic freeradical 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. Due to the gel effect,8,28 the reactor can present steady-state multiplicity.1,9 From standard polymerization kinetics1,29 and viscous heat-transfer arguments,9,30 the reactor dynamics are described by the following mass and energy balances: T˙ ) [∆Hr-U(T-Tj) + (Fmqmcm + Fsqscs)(Te - T)]/C:)fT, zT ) T, yT(t) ) T (1a) T˙j ) [U(T-Tj) + Fjqjcj(Tje - Tj)]/CJ:)fj, uj ) qj, V˙)qm + qs - (εm/Fm)r - q:)fv, m ˙ )-r + Fmqm - (q/V)m:)fm,

uv)q, zv)V,

zι ) ι,

yj(t) ) Tj (1b)

described by a Bernoulli equation with quadratic dependency on Mn, in order to draw system (1e) with linear dynamics with respect to ι.27 That a nonlinear coordinate change applied to a nonlinear process leads to a more linear input-output response with a more robust linear OF controller is a well-known fact in high-purity distillation column control.31,32 In compact vector notation, the reactor control system (1) is written as follows: x˙)f(x, d, u),

x(0) ) x0;

y ) Cyx,

z ) Czx

where x ) (T, Tj, V, m, ι,I, s, Q)′,

yv(t) ) V (1c)

um ) qm,

zm ) m

yι(tk) ) ι(tk-1), ∆)tk - tk-1 (1e)

I˙ ) -ri + wi - (q/V)I:)fi,

˙ n ) -(r0/p)Mn2 + (r/p)Mn M

(3)

(1d) ˙ι ) -(r/p)ι + r0/p:)fι,

In the reactor model [(1) and (2)], Fm (or cm), Fs (or cs), and Fj (or cj) are respectively the monomer, solvent, and coolant fluid densities (or specific heat capacities), εm is the monomer contraction factor, CJ is the heat capacity of the cooling system, ∆H is the heat of polymerization per unit monomer mass, and mw is the molecular weight. For the sake of simplicity, the function dependencies will be occasionally omitted, and φ(x) will be simply written as φ. To gain insight into the MW dynamics and facilitate the MW control construction, the MW inverse coordinate change (ι ) 1/Mn) has been applied to the number-average MW dynamics6

uι ) wi

s˙)Fsqs - (q/V)s:)fs

(1f)

f(x¯, d¯, u¯) ) 0 f ) (fT,fj,fv,fm,fι,fi,fs,fQ)′, d ) (Te,Tje,qs)′, u ) (qj,q, qm,wi)′, y ) (yc′,yd)′, z ) (zT,zV,zm,zι)′, yc(t) ) (yT,yj,yv)′(t) ) (T, Tj,V)′, yd(tk) ) yι(tk) ) ι(tk-1) and xj is the (possibly open-loop unstable and/or nonunique) j, u steady-state associated to the nominal values (d j ). 2.2. PI-Inventory Control. From a previous study,27 we know that (i) with the dynamic extension (q˙ ) uq, q˙m ) uqm), the reactor system (3) has relative degree (RD) vector

(1g)

˙ ) -(r/p){[2 - (r0/r)/ι]Q - [2(r0/r) + mw]ι}:)fQ Q (1h) The states (x) are: the reactor (T) and jacket (Tj) temperatures, the volume (V), the free (i.e., unreacted) monomer (m), solvent (s), and initiator (I) masses, the (number-average) MW inverse (ι ) 1/Mn) and its polydispersity (Q). The measured exogenous inputs (d) are the following: the reactor (Te) and jacket (Tje) feed temperatures, and the solvent volumetric flowrate (qs). The regulated outputs (z) are: the temperature (T), volume (V), monomer mass (m), and MW inverse (ι). The continuous measured outputs (yc) are: the reactor (yT) and jacket (yj) temperatures, and the volume (yV). The DD measured output (yd) is the MW inverse (yι), with sampling-delay period (∆) due to the actual measurement device (e.g., GPC), as well as sampling transport delay. The control inputs (u) are: the coolant (qj), exit (q), and monomer (qm) volumetric flowrates and the initiator mass feedrate (wi). The polymerization rate (r), initiation rate (ri), free-radical generation rate (r0), heat generation rate (Qr), heat transfer coefficient (U), heat capacity (C), and polymer mass (p) are set by the nonlinear functions:27 r ) fr(T, V, m, I, s), Qr ) ∆Hr U ) fU(T, Tj, V, m, s), C ) fC(V, m, s), p ) fp(V, m, s) ri)E(T)I:)fri(T, I), r0)cdri+ιtr:)f0(T, V, m, I, s) (2)

K ) (κV, κm, κT, κι)′ ) (2, 2, 2, 2)′

(4)

for the input-output pair (u, z) because the following conditions are generally met by the reactor class (1), m/V * Fm,

Tje * Tj,

Figure 1. Polymer reactor.

fT is Tj-monotonic,

fι is I-monotonic (5a-d)

Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008 9973

and (ii) the corresponding solvent-polydispersity (s, Q) zerodynamics are input-to-state (IS) stable,33 in the sense that admissible-size disturbances produce admissible-size state deviations. Consequently, the feedforward-state feedback (FF-SF) MIMO control problem is solvable with dynamic control, yielding IS-stable closed-loop dynamics and linear, noninteractive pole assignable (LNPA) output regulation dynamics. Moreover, when the MW is not measured, and the initiator dosage (wi) is fixed at its nominal value w j i, the reactor can be controlled with closed-loop IS stability by means of the PIinVentory controller27 [see detailed form in eq (44a–c)] χ˙ c ) Fc(χc, yc, d, uc),

uc ) µc(χc, yc), χc ) (χV, χT, χj, χj, m ˆ , sˆ)′ (6)

made of (i) linear-decentralized PI volume (44a) and cascade temperature (44b) loops and (ii) a near-linear material balance monomer controller (44c) driven by the integral action states (χT, χj) of the temperature controller. It must be pointed out that this (mass-heat balance) monomer controller does not need the polymerization reaction rate function fr (2), as the polymerization rate value (r) is quickly online estimated (rˆ) on the basis of the temperature measurements, according to a calorimetric control design idea.20 On the basis of the preceding RD-based structural characterization, let us rewrite the reactor (3) in the following two-subsystem interconnection form, with subsystem (7a) [or (7b)] associated to the continuous (or discrete) measurements: x˙c ) fc(xc,xd,d, uc),

xc(0) ) xco, yc)(yT,yj,yv)′ ) (T, Tj, V)′ (7a)

x˙d ) fd(xc,xd,q, wi),

xd(0) ) xdo, yd(tk) ) yι(tk) ) ι(tk-1) (7b)

where xc ) (T, Tj, V, m, s, Q)′ uc ) (qj, q, qm)′, zc ) (zT, zV, zm)′ Kc ) RD(uc, zc), Kc ) (κT, κV, κm) ) (2, 2, 2)′ xd ) (ι, I)′, κι ) RD(wi, zι) ) 2 The application of the above-mentioned volumetemperature-monomer PI-inventory controller27 [eq (6) or (44a–c)] yields the following IS-stable partially closed-loop system (the maps Oe and Od are given in the Appendix) x˙e ) φe(xe, I, d), x˙d ) Od(xe, xd, wi),

xe ) (χc′, xc′)′

(8a)

I ) (0, 1)xd

(8b)

with a MW response that settles within three-to-four residence times and has an asymptotic offset depending on the disturbance size.6,27 2.3. Control Problem. In the polymer reaction engineering field,8,14,16,18 it is well-known that (i) the MW-initiator subsystem (8b) exhibits an open-loop stable behavior that is critically damped or oscillatory, and rather sensitive to state (temperature, monomer) changes, and control and exogenous input changes, and (ii) these features manifest themselves as a closed-loop behavior that is quite dependent on the control model and technique, as well on the tuning parameters, and with propensity toward, over-response or saturation of the initiator dosage input. The MIMO reactor problem consists in designing a control scheme to obtain closed-loop IS-stability with regulation of the output z (volume, temperature, monomer, and MW inverse) on the basis of three continuous measurements yc (volume, reactor and jacket temperatures), one DD measurement yι (MW inverse)

with sampling-delay period ∆ ) tk - tk-1[yι(tk) ) ι(tk-1)], three continuous control inputs uc (coolant flow rate, exit flow rate, and monomer feedrate), and one piecewise constant (over ∆) input wi (initiator dosage). Given that the MIMO control component with continuous measurements has been already resolved6 with a threeinput three-output PI-inventory controller (6), our present control problem consists of designing, with minimum steady-state modeling information (to be determined), a linear-decentralized MW control component driven by DD measurements so that its combination with the three-input three-output PI-inventory controller (6) yields a four-input four-output hybrid control scheme with (i) a systematic construction procedure, (ii) conventional-like tuning guidelines, and (iii) MW regulation without offset, with damped response and nonwasteful initiator dosage control action. 3. MW Control with Continuous Measurements In this section, the continuous measurements version of the MW control problem is addressed. The purpose is twofold: (i) the characterization of the control structure and closed-loop behavior for the limiting case, when the sampling-delay period (∆) vanishes and (ii) the setting of conceptual and constructive points of departure for the tackling of the MW control problem with DD measurements. 3.1. Nonlinear Geometric FF-SF MW Control. Assuming that the detailed reactor model (3) is given and the exogenous input (d) and states (x) are known, in this subsection the standard geometric approach34 is applied to address the FF-SF MW control problem. For this aim, recall the MW-initiator dynamics (1e–f) as well as its RD condition (5d) introduce the coordinate change ν ) fι(ι, I, xc)

(9)

to take MW-initiator subsystem (1e–f) into the normal form (the maps R and γ are defined in the Appendix) ˙ι ) ν,

ν˙ ) R(ι, ν, xc)wi + γ(ι, ν, xc, d, uc),

yι ) ι (10)

and from the enforcement of the prescribed closed-loop output regulation error dynamics (see the Appendix) upon system (10), the nonlinear geometric FF-SF controller follows (the maps γg and Rg are defined in the Appendix) wi ) -[k1(ι - ι) + k2fι(ι, I, xc) + γg(ι, I, xc, d, uc)]/Rg(ι, I, xc):)µg(ι, I, xc, d, uc, kg), kg ) (k1, k2)′ (11) where the gains pair (k1, k2) is either chosen by trial-and-error or assigned via pole placement, in the understanding that the gain choice is guided more by closed-loop simulation tests than by open-loop system behavior characteristics. In view of the aforementioned critically damped or oscillatory open-loop MW behavior and closed-loop response that is sensitive to the initiator dosage (not accounted for by the geometric pole placement scheme) and from constructive control perspective,35 the preceding MW geometric controller has a fundamental drawback: since the closed-loop linear dynamics are prescribed without regarding the natural characteristics of the open-loop stable MWinitiator subsystem (10), the controller may unduly cancel or modify those beneficial open-loop dynamical features, yielding malfunctioning with poor robustness and wasteful control action, as it will be illustrated in the section on the Application Example. These comments motivate the study, in the next subsections, of the MW constructive control problem. 3.2. Nonlinear Passive FF-SF MW Control. Following the constructive control approach, in this section the FF-SF MW control


problem is addressed according to the idea that optimal controllers:35 (i) are inherently robust (with respect to modeling errors) and nonwasteful, (ii) are passive (with RD less or equal to 1 and minimum phaseness) with respect to a certain output, (iii) yield stable closed-loop behavior if the open-loop system is detectable with respect to the passive output, and (iv) are robust also with respect to the choice of the objective function, provided it effectively penalizes the state deviation versus the control action. Since the controller construction via direct optimality is a difficult or intractable task because the analytic solution of a Hamiltonian problem is needed, in the constructive control approach an analytic control construction is pursued via indirect optimality, according to the following rationale: on the basis of particular system features, its passivity and detectability structures are characterized, yielding a candidate passive controller whose optimality is a posteriori assessed. Depending on the result, the controller is accepted or redesigned. To gain insight into its dynamical characteristics, let us express the MW-initiator stable subsystem (10) in a nonlinear dampingspring (phase canonical) form36 ¨ι ) R(ι, ˙ι, xc)wi + γ(ι, ˙ι, xc, d, uc),

yι ) ι

(12)

where Rwi and γ represent the “external force” and the combined “damping-restoration” mechanism, respectively. In terms of its linear (local) damping (cδ) and spring (cσ) constants, the preceding system can be written as follows (the map γn is defined in the Appendix) ¨ι + cδ˙ι + cσι ) υ

(13a)

υ ) R(ι, ˙ι, xc)wi + γn(ι, ˙ι, xc, d, uc)

(13b)

where cδ ) λ¯ i + λ¯ ι > 0,

cσ ) λ¯ iλ¯ ι > 0

j, V j, m j, m λ¯ ι ) fr(T j , jI, js)/fp(V j , js) > 0,

j λ¯ ι ≈ qj/V

j ) + E(T j) > 0 λ¯ i ) (qj/V

(14a-b) (14c) (14d)

Here, λjι (or λji) is the real eigenvalue of the MW inverse (or initiator) open-loop stable local dynamics, and υ is a new control input. In deviation variables, eq 13a is written as follows ¨ι˜ + c ˙ι˜ + c ι˜ ) υ˜ , δ σ υ˜ ) υ - υ¯ ,

ι˜ ) ι - ¯ι υ˜ ) cσι

(15a) (15b)

In a mass-damper-spring system analogy, ˜ι is the position deviation, υ˜ is the force acting upon mass, and cδ (or cσ) is the friction coefficient (or spring constant); the corresponding (kinetics plus potential) energy (16a) and its rate of change (16b) are given by e ) ˙ι˜2/2 + cσι˜2/2,

e˙ ) υ˜ ˙ι˜ - cδ˙ι˜2

(16a-b)

The last equation says that (i) when υ˜ ) 0, the natural openloop dissipation rate is -cδ˙ι˜ 2, and (ii) the controller υ˜ induces the adjustable dissipation. These considerations suggest setting the feedback control component (17a) in order to speed-up the dissipation rate with minimum energy injection (17b): υ˜ ) -kd˙ι˜ ⇒ e˙ ) -(kd + cδ)ι˜ < 0 ˙2

(17a-b)

The application of the controller (17a) to the open-loop dynamics (15a) yields the closed-loop output regulation dynamics ¨ι˜ + (c + k )ι˙˜ + c ι˜ ) 0 δ d σ

(18)

Recall the “velocity-feedback” controller (17a) and substitute (13b) and (15b) to obtain the nonlinear passiVe FF-SF MW controller in phase canonical form wi ) -[kd˙ι - cσ¯ι + γn(ι, ˙ι, xc, d, uc)]/R(ι, ˙ι, xc) :) µπ(ι, ˙ι, xc, d, uc, kd) (19) The substitution of ˙ι by fι (9) yields the nonlinear passive MW controller in state-space form wi ) µπ[ι, fι(ι, I, xc), xc, d, uc, kd]:) µp(ι, I, xc, d, uc, kd) (20) with only one adjustable gain (kd) which represents the friction coefficient increasing by control. The preceding passive controller is related to its geometric counterpart (11) via the following expression: wi ) µp(ι, I, xc, d, uc, kd) ) µg(ι, I, xc, d, uc, kg), kg ) (k1, k2)′, k1 ) cσ, k2 ) cδ + kd (21) According to these relationships, the single-gain passive controller (20) is a particular case of the two-gain geometric controller (11), with gains that are not arbitrarily chosen but assigned as follows: the gain k1 is set at the value of the linear (local) springlike restoration constant (k1 ) cσ), and the gain k2 is set at the linear (local) friction-like constant (cδ) plus a feedback contribution (kd) (k2 ) cδ + kd). This equivalence signifies that the passive controller exploits the open-loop dynamical characteristics and, from a geometric control perspective, this means a more effective gain pair choice. In terms of the local damping factor (ζn) and natural frequency (ωn) ωn ) (λ¯ iλ¯ ι)1/2 ) cσ1/2, ζn ) (λ¯ i + λ¯ ι)/[2(λ¯ iλ¯ ι)1/2] ) cδ/(2cσ1/2) (22) associated to the MW-initiator local dynamics (13), the openloop dynamics (15a), the feedback component (17a), and the closed-loop dynamics (18) are given by ¨ι˜ + 2ζ ω ˙ι˜ + ω 2ι˜ ) υ˜ , n n n

υ˜ ) -(2ζ˜ ωn)ι˙˜

⇒ ¨ι˜ + 2(ζn + ζ˜ )ωn¨ι˜ + ωn2ι˜ ) 0

(23a-b) (23c)

These expressions say that the application of the nonlinear passive FF-SF controller (20) to the nonlinear MW-initiator system (13) yields a linear closed-loop output error dynamics (23c) with (i) frequency equal to the natural local one (ωn), and (ii) damping factor equal to the natural one (ζn) plus a component (ζ˜ n) set by the control gain kd ) 2ζ˜ ωn. Next the passivity property of the MW-initiator subsystem (15a) and the optimality property of the passive controller (20) are characterized. Regard the MW inverse derivative (ι˙˜) as a new output ψ, rewrite the open-loop dissipation equation (16b) in terms of ψ υ˜ ψ ) e˙ + cδψ2,

ψ ) ˙ι˜

(24)

and conclude that the initiator-MW inverse subsystem (15) is (i) output strictly passiVe with respect to the input-output storage function triplet (υ˜ , ψ, e), and (ii) dissipatiVe because of the presence of the positive term cδψ2 > 0 for ψ * 0.35–37 On the other hand, the straightforward application a theorem that relates optimality and stability35 yields that the nonlinear passive FF-SF controller (20) is optimal with respect to the infinitetime horizon objective function


J)

∫

∞

0

(cδ + kd/2)[fι(ι, I, xc)]2+(2kd)-1[γg(ι, I, xc, d, uc) +

χ˙ ν)-(cδ + koν)χν + χb + [kob - koν(cδ + koν) - cσ]yι + aιwi, νˆ )χν + koνyι (31a)

cδfι(ι, I, xc) + cσ(ι - ¯ι) + Rg(ι, I, xc)wi]2 dt (25) which is meaningful in the sense that state deviations and control actions are effectively penalized. 3.3. Linear OF MW Control. The direct implementation of the geometric (11) or passive (20) nonlinear FF-SF controller, with a suitable nonlinear (EKF or geometric) observer, requires the detailed initiator-MW inverse model, and this leads to a strongly nonlinear, interactive, and model-dependent dynamical MW control component of the form (with geometric observer) xˆ˙d ) fd(xˆc, xˆd, q, wi) + Gy(xˆc, xˆd, d, u)(yι - Cyxˆ), xˆ(0) ) xˆ0 (26a) wi ) µp(xˆc, xˆd, d, u, kd) (passive) or wi ) µg(xˆc, xˆd, d, u, kg) (geometric) (26b) corresponding to the one employed in most polymer reactor studies, which is interactive in the sense that the MW controller depends on the state xc and the control input uc of the subsystem (7a) associated to the continuous measurements. Since these features signify complexity, fragility, and cost concerns among industrial practitioners, a linear and decentralized passive MW control component with reduced modeling requirements will be designed next, by exploiting the system RD-observability structure. In a way that is analogous to parametric representation of plane curves,38 let us express the open-loop nonlinear dynamics (13) in the linear-dynamical (27a) plus nonlinear-static (27b) form (the map βι is given in the Appendix): ¨ι + cδ˙ι + cσι ) aιwi + bι,

yι ) ι

(27a)

bι)βι(ι, ˙ι, xc, d, uc, wi)

(27b)

where the constant aι is an approximation of steady-state initiator decomposition parameters j )/fp(V j, m j , js) aι ) cdE(T

(28)

In state-space form, this system is written as follows ˙ι ) ν,

ν˙ ) -cσι - cδν + aιwi + bι,

yι ) ι (29a)

bι)βι(ι, ν, xc,d, uc, wi)

(29b)

and the corresponding nonlinear passive controller (19) is given by wi ) -(kdν - cσ¯ι + bι)/aι

(30a)

bι ) βι(ι, ν, xc,d, uc, wi)

(30b)

On the other hand, in terms of (yι, y˙ι, y¨ι) the equation triplet (29a) is expressed as follows yι ) ι,

y˙ι ) ν,

y¨ι ) -cσι - cδν + aιwi + bι

and the solution for (ι, ν, bι) of this algebraic equation triplet yields (ι, ν, bι) ) (yι, y˙ι, y¨ι + cδy˙ι + cσyι - aιwi) meaning that for system (27), the state-input (ι, ν) - bι is instantaneously observable,39 or equivalently, (bι, ν) can be quickly reconstructed via a reduced-order linear observer40 (yι ) ι):

χ˙ b ) -kobχν - koνkobyι,

bˆι ) χb + kobyι

(31b)

where the gains (koν, kob), in terms of the observer frequency (ωo) and damping factor (ζo), are given by koν ) 2ζoωo,

kob ) ωo2

(31c)

The combination of the reduced-order observer (31) with the linear part (30a) of the nonlinear passive FF-SF controller yields the linear passiVe OF MW controller χ˙ ν)-(cδ + koν)χν + χb + [kob - koν(cδ + koν) - cσ]yι + aιwi, νˆ ) χν + koνyι (32a) χ˙ b ) -kobχν - koνkobyι,


wi ) -(kdνˆ - cσ¯ι + bˆι)/aι

(32b) (32c)

The control part (32c) has two components, a feedforward-like one [(cσjι - bˆι)/aι] that performs most of the disturbance rejection task, and a “velocity-feedback” component (-kdνˆ /aι) that ensures the MW (i.e., position) regulation. Moreover, the preceding linear controller (32) (i) recovers, with fast observer convergence rate (≈ ζoωo), the behavior of the exact model-based nonlinear passive FF-SF controller (20), and (ii) constitutes the limiting behavior (as the sampling period vanishes) of the DD OF MW controller to be designed in the next section. 3.4. Concluding Remarks. While the implementation (26) of the nonlinear passive FF-SF controller (20) with a (geometric or Luenberger) nonlinear observer requires the detailed reactor model, the proposed linear passive OF controller (32) requires only three steady-state approximated parameters (cδ, cσ, and aι) of the MWinitiator subsystem (27). For structure-oriented comparison purposes, let us inherit the linearity, decentralization and reduced model dependency features of the proposed passive controller to the geometric OF control design (26), yielding the linear-decentralized geometric controller χ˙ ν ) - (cδ + koν)χν + χb + [kob - koν(cδ + koν) - cσ]yι + aιwi, νˆ ) χν + koνyι (33a) χ˙ b ) -kobχν - koνkobyι,


(33b)

wi ) -[k1(yι - ¯ι) + k2νˆ - cσyι - cδνˆ + bˆι)/aι (33c) which quickly recovers the behavior of the exact model-based nonlinear geometric FF-SF controller (11) and yields the proposed OF passive controller (32) by setting the gains (k1, k2) ) (cσ, cδ + kd). It must be pointed out that the preceding linear realization of the nonlinear geometric OF MW controller (26) constitutes: (i) an improvement over its counterparts employed in previous polymer reactor studies, and (ii) a means to fairly compare the passive and nonpassive control behavior, under the same modeling and observer construction basis. Summarizing, the linearity, decentralization, and reducedmodel features of the preceding linear OF MW controller were attained on the basis of an interlaced observer-controller design that exploited the passivity, dissipativity, and detectability properties of the initiator-MW inverse subsystem. Specifically, passivity led to optimality and robustness, detectability with respect to MW inverse derivative led to stability with FF-SF control, and observability with respect to the measured output (MW inverse) led to OF control linearity, decentralization, and reduced-model dependency.


The combination of the continuous measurements-driven linear OF MW controller with the above-discussed volumetemperature-monomer controllers yields a closed-loop MIMO stable system, provided that the MW control gains are set according to a suitable pole placement scheme, such that the related dynamical separation condition is met with an adequate compromise between performance and robustness.8

In this section, a linear-decentralized OF MW controller is constructed on the basis of DD MW measurements and a suitable discrete model with reduced-model dependency. 4.1. MW Discrete Model. Due to the large sampling periods and delays associated to the industrial operation of polymer reactors, in comparison to (1/2π) times the natural period (Sampling Theorem25), Euler’s method is not appropriate to design a discrete control with sampled-delayed measurements and reduced modeling requirements. Instead, let us perform the analytic solution of the linear part (27a) of the MW-initiator nonlinear dynamics. For this aim, require the control input wi to be constant over the sampling-delay period ∆ on the basis of a standard zero-order hold argument,25,41,42 assume that the exogenous input bι is approximated by its constant (mean) value over ∆, and obtain the second-order nonhomogeneous differential equation ∆ ) tk - tk-1, bι(t) ≈ bk-1 (34)

Recall the definition (14) of the pair (cδ, cσ) in terms of the initiator (λji) and MW (λjι) eigenvalues, and draw the analytic solution for eq (34) according to standard methods41,42 to obtain the MW discrete model ιk ) γ11ιk-1 + γ12νk-1 + β1(aιwk-1 + bk-1),

kp ) ωc2 - cσ,

kd ) 2ζcωc - cδ

(37c)

In terms of the adjustable pair (ωc, ζc), the continuous closed-loop dynamics and its associated sampled version42 are given by (38) and (39), respectively (the maps σc and πc are given in the Appendix): ¨ι˜ + 2ζ ω ˙ι˜ + ω 2ι˜ ) 0 c c c

4. MW Control with DD Measurements

¨ι + cδ˙ι + cσι ) aιwk-1 + bk-1, ι(tk-1) ) ιk-1, ˙ι(tk-1) ) νk-1 t ∈ [tk-1, tk]: wi(t) ) wk-1,

where the gain pair (kp, kd) is set by the frequency-damping pair (ωc, ζc):

yk ) tk-1 (35a)

νk ) γ21ιk-1 + γ22νk-1 + β2(aιwk-1 + bk-1)

(35b)

where the coefficient set {γ11, γ12, γ21, γ22, β1, β2} (Appendix) is determined by the model constant set (cδ, cσ, aι, ∆). In inputoutput form, this model can be realized by the difference equation: ιk+1 - τιk + διk-1 ) β1(aιwk + bk) + βp(aιwk-1 + bk-1) (36) where τ (or δ) is the trace (or determinant) of the coefficient matrix of system (35a) (Appendix). 4.2. Linear FF-SF MW Control. Comparing with its continuous measurements counterpart (32), a controller with DD MW measurements runs with less information and, consequently, exhibits a more sluggish response. This consideration, in conjunction with the employment of a piecewise-constant initiator dosage over the sampling period, suggests us to inherit the (minimum control effort-oriented) damping derivative action of the continuous design and incorporate proportional action (with intensity to be determined) to enable the possibility of speeding-up the closed-loop response. Accordingly, recall the velocity-feedback component (17a) of the passive controller, introduce a proportional component (-kp˜ι) to set the controller (37a), and apply it to the continuous system (15a) to obtain the closed-loop stable system (37b) υ˜ ) -kd˙ι˜ - kpι˜ ⇒ ¨ι˜+(cδ + kd)ι˙˜ + (cσ + kp)ι˜ ) 0 (37a-b)

(38)

ι˜k+1 - σc(∆, ωc, ζc)ι˜k + πc(∆, ωc, ζc)ι˜k-1 ) 0,

ι˜k ) ιk - ¯ι (39)

In terms of a control gain pair (k1c, k2c) and the trace (τ) and determinant (δ) of the open-loop discrete system (36), system (39) is expressed as follows (the maps κc1 and κ2c are given in the Appendix) ι˜k+1 - (τ + kc1)ι˜k + (δ + kc2)ι˜k-1 ) 0, kc1 ) κc1(∆, ωc, ζc),

kc2 ) κc2(∆, ωc, ζc) (40)

Observe that the control gain pair (kc1, kc2) is uniquely determined by the frequency-damping pair (ωc, ζc) of the associated continuous closed loop dynamics (38). From the enforcement of the closed-loop difference equation (40) upon the open-loop difference equation (36), the discrete linear FF-SF MW controller follows wi(tk) ) -(βp/βl)wi(tk-l) - [bk + (βp/βl)bk-1]/al + [(1 τ + δ)/β1](ι¯/aι) + [kcl (ιk - ¯ι) - kc2(ιk-1 - ¯ι)]/(aιβ1) kc1 ) κc1(∆, ωc, ζc),

kc2 ) κc2(∆, ωc, ζc) (41)

In the right side of controller (41), (i) the first term is an integral-like action due to discretization, (ii) the first three terms correspond to feedforward action driven by present (bk) and past (bk-1) load inputs, and (iii) the last term is a feedback correction driven by present (ιk) and past (ιk-1) measurements. As the sampling-delay period (∆) vanishes (∆ f 0), the behavior of the DD measurement-driven controller (41) approaches the one of the continuous measurement-driven passive controller (32), by setting ωc ) ωn [i.e., kp ) 0 in eq (37c)]. 4.3. Linear OF MW Control. Differently from the case with continuous measurements, where the estimation task of the loadstate pair bι-ν can be efficiently performed with a reducedorder observer (31), in the DD measurement case, with less information available, a full-order observer must be designed to effectively tackle the one-step-ahead prediction problem, or equivalently, the delay compensation capability. Thus, the application of the full-order discrete observer technique41 on the input-output discrete model (35) yields the third-order discrete linear state observer (the maps κ1o, κ2o, and κ3o are given in the Appendix): ˆιk ) γ11ˆιk-1 + γ12νˆ k-1 + β1[aιwi(tk-1) + bˆk-1] + kol [yι(tk) - ˆιk-1],

yι(tk) ) ιk-1 (42a)

νˆ k ) γ21ˆιk-1 + γ22νˆ k-1 + β2[aιwi(tk-1) + bˆk-1] + ko2[yl(tk) - ˆιk-1] (42b) bˆk ) bˆk-1 + ko3[yι(tk) - ˆιk-1] ko1 ) κo1(∆, ωo, ζo),

(42c)

ko2 ) κo2(∆, ωo, ζo), ko3 ) κo3(∆, ωo, ζo) (42d)

whose gain triplet (k1o, k2o, k3o) is uniquely determined by the frequency-damping pair (ωo, ζo). The combination of the


observer (42) with the FF-SF controller (41) yields the discrete linear OF MW controller ˆιk ) γ11ˆιk-1 + γ12νˆ k-1 + β1[aιwi(tk-1) + bˆk-1] + ko1[yι(tk) - ˆιk-1],

yι(tk) ) tk-1 (43a)

νˆ k ) γ21ˆιk-1 + γ22νˆ k-1 + β2[aιwi(tk-1) + bˆk-1] + ko2[yι(tk) - ˆιk-1], bˆk ) bˆk-1 + κo3[yι(tk) - ˆιk-1] (43b-c) wi(tk) ) -(βp/β1)wi(tk-1) - [bˆk + (βp/β1)bˆk-1]/a1 + [(1 τ + δ)/β1]ι¯/aι + [kc1(ιˆk - ¯ι) - kc2(ιˆk-1 - ¯ι)]/(aιβ1) (43d) 4.4. MIMO Reactor Control. The combination of the OF MW controller (43) with the volume, temperature, and monomer controllers presented in a previous study27 yields the MIMO polymer reactor control scheme: • Volume controller [yV ) V]

(44a)

χ˙ V ) -ωVχV - ωV(ωVyV - q),

bˆV ) χv + ωVyV

j) q ) bˆV + kV(yV - V • Temperature controller [yT(t) ) T, yj(t) ) Tj]

(44b)

χ˙ T ) -ωTχT - ωT(ωTyT + aTyj), bˆT)χT + ωTyT / j )]/aT Tj ) -[bˆT + kT(yT - T χ˙ j ) -ωjχj - ωj(ωjyj + ajqj) bˆj ) χj + ωjyj / / / / / / / χ˙ j ) -ωj χj - ωj (ωj Tj ) bˆj ) χ/j + ω/j T/j qj ) [bˆ/j - bˆj - kj(yj - T/j )]/aj • Monomer controller

(44c)

m ˆ˙ ) -rˆ + Fmqm - (m ˆ /yV)q, sˆ˙ ) Fsqs - (sˆ/yV)q qm ) [rˆ + (m ˆ /yV)q - km(m ˆ -m j )]/Fm ˆ rˆ ) [fC(yV, m ˆ , sˆ)(bT + aTyj) + CJbˆj]/∆ • MW controller [yι(tk) ) ιk-1]

(44d)

ˆιk ) γ11ˆιk-1 + γ12νˆ k-1 + β1[aιwi(tk-1) + bˆk-1] + ko1[yι(tk) ˆιk-1] νˆ k ) γ21ˆιk-1 + γ22νˆ k-1 + β2[aιwi(tk-1) + bˆk-1] + ko2[yι(tk) ˆιk-1] ˆbk ) bˆk-1 + ko3[yι(tk) - ˆιk-1] wi(tk) ) -(βp/β1)wi(tk-1) - [bˆk + (βp/β1)bˆk-1]/aι + [(1 τ + δ)/β1]ι/aι + [kc1(ιˆk - ι) - kc2(ιˆk-1 - ι)]/(aιβ1) Structurally speaking, the linear OF MW component (44d) amounts to an interlaced estimator-control design, with a secondorder FF-SF controller and a third-order discrete observer, built on the basis of a discrete model (35) with sampling-delay handling capability. The reconstructible terms (bV, bT, bj, bj/) represent the effect of the modeling and unknown disturbance errors; the nonlinear term bι from the continuous representation (27b) is piecewise reconstructed (bˆk) on the basis of the DD MW measurements; this estimated load (bˆk), in conjunction with the estimated loads (bˆV, bˆT, bˆj, bˆj/) from the volume and temperature controllers, counterbalances the (measured or unknown) disturbances and plant/model parameters mismatch, in the same way that PI volume and temperature controllers recover the behavior of a nonlinear SF controller.6,32 The volume (44a) and cascade temperature (44b) components amount to linear-decentralized PI-controllers, and the monomer component (44c) is an inventory controller driven by the information (bˆT,

bˆj) [generated in the temperature controller (44b)] which is exploited in a feedforward component (rˆ/Fm) that sets a feed flowrate qm contribution. The functioning of the volume, temperature, and monomer controllers is independent of the one of the MW controller. Regarding the modeling requirements, the MW control implementation (4.5) needs only the approximated static constants (cσ, cδ, aι) [(14a-b) and (28)] or, equivalently, the nominal residence time (τr) and steady-state approximations of the initiation constant (E) and polymer mass (fp) functions. The modeling requirements of the volume, temperature, and monomer loops are the following:27 two steady-state approximated constants (aT, aj) for the temperature loop and calorimetric parameters (densities and heat capacities) for the monomer loop. These modeling requirements are fewer than the ones of previous polymer reactor control studies with MW measurements.13,14,16 4.5. Tuning Guidelines and Stability Considerations. Given that the principle of separation holds for linear systems but not for nonlinear ones, the assurance of closed-loop stability amounts to the derivation of dynamic separation conditions on the controller and observer gains. In our reactor control problem, this task can be tractably executed on the basis of (i) the Small Gain Theorem (SGT)-based closed-loop stability conditions for the reactor with initiator dosage fixed at its nominal value and PI-inventory controller (driven by continuous temperatures and volume measurements) for the regulation of temperature, volume, and free monomer6,27 and (ii) the derivation of convergence conditions for a geometric estimator with sampleddelayed measurements and a continuous-discrete hybrid estimation error dynamics.39 However, the execution of this technical task goes beyond the scope of the work, and here we circumscribe ourselves to (i) regard these closed-loop stability characterization ideas, (ii) assume the existence of the associated SGT-like conditions for the closed-loop stability of our reactor with continuous and DD measurements, and (iii) numerically assess, in the next section on the Application Example, the closed-loop stability property through representative case-study simulations. Tuning guidelines. For the purpose at hand, recall the residence time (τr) to sampling period (∆) quotient (n∆ ) τr/∆) and introduce the adjustable parameters {no, nc}: no)τo/∆,

nc ) τc/τo,

τo ) 1/ωo,

τc ) 1/ωc

where no (or nc) represents the times that the observer (or controller) characteristic time τo (or τc) is larger than the measurement delay ∆ (or the observer characteristic time). The tuning guidelines for the present case are the following: 1. Set the initiator dosage (wi) at its nominal value (MW in open-loop mode) and apply the tuning guidelines previously presented27 for the volume, temperature, and monomer controllers (44a-c). 2. For the MW controller, (a) In the case of a large quotient n∆ (say greater than 15, meaning close to the continuous measurements case): (i) set the observer frequency (ωo) equal to the one of the continuous controllers (volume, temperature) and the controller frequency (ωc) equal to the MW natural one (ωc ) ωn ) cσ1/2), and (ii) choose the damping factors greater than 1, say (ζo, ζc) ∈ (1, 3], to preclude the amplification of high-frequency unmodeled dynamics.43 (b) In the case of a smaller quotient n∆ (2-15): (i) set the observer characteristic time (τo) three times slower than the measurement delay (no ) 3) and the controller characteristic time (τc) two times slower than the observer

9978 Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008 Table 1. Steady States, Nominal Inputs, Reactor States, and Jacket and MW Approximated Parameters

Table 2. Observer and Controller Gains Volume, Temperature, and Monomer Controllers27

steady states states and outputs

stable (extinction)

unstable

observer gains

stable (ignition)

controller gains j, kV ) kj ) ω/8, kT ) 2qj/V j km ) 1.5qj/V

/

T (K) Tj (K) m (kg) Mn (kg/kmol) I (kg) s (kg) Q

329.72 329.53 1361.089 399149.03 1.685 501.283 1.9997

351.62 341.23 660.082 110384.75 1.3087 500.871 1.999

373.88 345.61 312.756 29395.15 0.3513 498.561 1.9966

nominal inputs uj ) (qjj, qj, qjm, w j i)′ ) (30 L/min, 9.1 L/min, 7.34 L/min, 0.0078526 kg/min)′ j je, qjs)′ ) (315 K, 328 K, 2.54 L/min)′ dj ) (Tj e, T

ωV ) ωj ) ωj ) ωT ) ω ) 1/5 min-1

MW Continuous Controller observer gains -1

controller

ωo (min )

geometric passive

1/10 1/10

controller gains ζo

1.25 1.25

aT ) 4.71 × 10-2 min-1, aj ) - 3.78 × 10-2 K/L MW parameters cδ ) 1.06 × 10-2 min-1, cσ ) 2.76 × 10-5 min-2 aι ) 1.0426 × 10-11 kmol kg-2 min-1 ωn ) cσ1/2 ) 1/190 min-1, ζn ) cδ/(2cσ1/2) ) 1.0088

characteristic time: nc ) 2, (ii) choose the damping factors greater than 1, and (iii) decrease the observer parameter no up to its ultimate value nυo , where the response becomes oscillatory and backoff until a satisfactory response is attained, say at no g 2noυ. 3. If necessary, adjust the damping (ζo, ζc) factors and/or controller parameter (nc). In this manner, the MW controller gains are set according to prescribed root locus-based pole patterns (inside the unit circle) determined by the damping factors (ζc, ζo) and the characteristic frequencies (ωc, ωo) [or equivalently, the pair (nc, no)]. 4.6. Concluding Remarks. The use of the MW inverse (ι) state led to a measurement-driven linear MW control component that is simpler and more robust than previous nonlinear controllers that depend on the detailed polymerization model. The controller is based on a discrete model that is not based on Euler discretization, but on the analytic solution of an approximate continuous linear model whose nonlinear load (bι) is estimated from DD MW measurements; this estimated load (bˆι), in conjunction with the estimated loads (bˆV, bˆT, bˆj, bˆ/j ) from the volume and temperature controllers, accounts for the (measured or unknown) disturbances and plant/model parameters mismatch. This yields a MIMO control scheme with more robust behavior and a better prediction capability than Euler-based schemes, and an initiator feedrate input whose nonwasteful behavior is a result of the design specification (control input is piecewise constant). 5. Application Example To subject the proposed MIMO OF controller (44) to a severe test, the operation of an open-loop unstable reactor was considered (via numerical simulations), at high-solid fraction with the potentially destabilizing gel effect at play. The system has methyl methacrylate (MMA) as monomer, ethyl acetate as solvent, and azobis-isobutyronitrile (AIBN) as initiator. The j ≈ 2000 nominal residence time (τr) is 220 min with a volume V L, and the operating conditions and particular model were previously employed.6 The reactor has three steady states (Table 1), with the unstable steady-state being the control setpoint, and the closed-loop system was subjected to step and sinusoidal exogenous input disturbances. Since the emphasis is put on the

ζc

1/100 1/190

1.2 1.8

MW Discrete Controller observer gains

jacket parameters

ωc (min-1)

controller gains

sampling-period delay ∆

ωo (min-1)

ζo

ωc (min-1)

ζc

5 30 90 180

1/10 1/30 1/90 1/280

3 1.5 1 0.9

1/190 1/120 1/140 1/300

1.8 1 0.7 0.5

design of the MW control component and its appropriate coordination in a MIMO control scheme, in all runs the reactor was controlled with the volume, temperature and monomer with continuous measurements, and among different runs continuous or DD measurements-based MW controllers were tested. In the DD MW measurements case, different sampling periods (delays) were tested. Following the tuning guidelines (subsection 4.5), first the initiator dosage was fixed at its nominal value, and the volume, temperature, and monomer components (44b-c) were tuned accordingly;27 the resulting gains, which were kept fixed in subsequent runs, are listed in Table 2. Then, the MW component was tuned, and the corresponding gains are listed in Table 2 for various sampling periods. Passive and Geometric MW Controllers with Continuous Measurements. On the basis of the open-loop dynamical characteristics (ζn, ωn) ≈ [1, (1/190) min-1] (22) in conjunction with the application of the tuning guidelines, the control gain (kd) of the passive OF MW controller (32) (with continuous measurements) was set with the damping-frequency pair as follows: (ζc, ωc) ≈ (1.8, ωn), meaning a control gain kd with a damping increase of ζ˜ ) 0.8 (23b). On the other hand, the geometric counterpart (33) of the linear passive OF MW controller (32) was implemented with the pole-placement control gain pair (k1, k2) ) (ωc2, 2ζcωc), and the damping-frequency pair value (ζc, ωc) ) (1.2, 10-2 min-1) drawn from closedloop simulation test assisted by standard heuristic considerations (say, the closed-loop dynamics must be overdamped and the controller must be between 5-15 times slower than the observer), in the understanding that further closed-loop testing should lead to a damping-frequency pair value (ζc, ωc) close to the one obtained more directly with the proposed passive control design. The closed-loop reactor was subjected to step changes (shown in Figure 2) in the reactor and jacket feed temperatures (at t ) 100 min: Te from 315 to 320 K, and Tje from 328 to 330 K). The closed-loop behavior is presented in the same figure, showing that (i) the volume, temperature, and monomer reach their set points in about one residence time, with control actions (q, qj, qm) that occur in a smooth and coordinated manner, reasonably away from saturation, (ii) also the MW converges in about one residence time, and (iii) the initiator feedrate input (wi) of the passive MW controller (32) presents a more damped


Figure 2. Closed-loop behavior with passive (s) and geometric (- - -) MW controllers driven by continuous measurements.

Figure 3. Closed-loop behavior with continuous (s) and DD (- - -) (∆ ) 5 min) measurement-driven MW controllers.

response than the one of the geometric OF controller (33). This is so because the passive MW controller only modifies the (local) damping mechanism of the natural open-loop dynamics and not the natural frequency, and this shows the importance of the passivity property that underlies the proposed discrete control scheme for a large residence time (τr) to sampling period (∆) quotient (i.e., close to the continuous measurements case). Behavior Recovery. The reactor was controlled with the proposed MW controller (44d) with DD measurements and a sampling-delay period (∆) of 5 min [meaning a quotient n∆ ) τr/∆ ) 44], and its behavior was compared to the one of the passive controller with continuous measurements (32). The frequencydamping factor pair (ζc, ωc) ) (1.8, ωn) of the discrete MW controller (44d) was set with the same values (Table 2) of its continuous-measurement passive counterpart (32). The corresponding closed-loop behavior with the above stated input step changes is presented in Figure 3. As expected, when the sampling-delay period is small (n∆ ) 44) the discrete and continuous OF MW controllers basically yield the same behavior, and the presence of the MW controllers does not affect the functioning of the volume, temperature, and monomer components. In other words, the discrete MW component does not affect the reactor closed-loop stability and the production rate level. Behavior Dependency on Sampling-Delay Period. The behavior of the proposed OF control scheme (44) for various MW sampling-delay period values is presented in Figure 4: (i) ∆ ) 30 min (n∆ ≈ 7), as the typical delay of industrial GPC measurements,14 (ii) ∆ ) 90 min (n∆ ≈ 2), where the ratio n∆ is similar to the one employed in a previous polymer reactor

control study,14 with the largest MW measurement delay, and (iii) ∆ ) 180 min (n∆ ≈ 1). It must be pointed out that the last two cases must be regarded as extreme case situations with inherent instrument and/or transport delay. The corresponding control gains are listed in Table 2, showing that (i) in the passage from the continuous-like case (n∆ > 15) to the DD one, the control frequency should be greater than the natural frequency (ωc > ωn) to speed up the MW response, (ii) the control frequency decreases as the sampling-delay period increases, and (iii) for typical situations (n∆ > 2), the damping factor should be greater or equal to 1, signifying a control design with overdamped passive-like behavior. The closed-loop responses to temperature step changes are presented in Figure 4, showing that, in all cases, the responses are stable and the MW settling time increases with the sampling-delay period. In particular, the controller with ∆ ) 90 min regulates the MW in about 1.7τr, which is similar (≈ from 1 to 2τr) to the ones obtained with a full model-based and appropriately tuned nonlinear MW control scheme.14 In other words, the proposed MW control scheme can perform the same task with less modeling requirements [(14a-b) and (28)], more robustness with respect to model uncertainty, and under conditions which are close to gel effect. Moreover, in comparison to previous studies without MW measurements,6,27 the proposed MW controller regulates this output with shorter settling time, fewer modeling requirements, and without offset. For all sampling periods tested, the regulation of the monomer, temperature, and volume is not affected by the MW loop, meaning that the control of this output is performed in a coordinated way with linear decentralized volume, temperature, and conversion components.


Figure 4. Closed-loop behavior with DD measurement-driven MW controller for different sampling periods. ∆ ) 5 (s), 30 (- - -), 90 (- · - · -), and 180 ( · · · ) min.

Figure 5. Closed-loop behavior with continuous (volume, reactor and jacket temperatures) and DD (MW) noisy measurements (∆ ) 30 min): response to feed reactor and jacket temperature sinusoidal disturbances.

Behavior with Noisy Measurements. The closed-loop reactor was subjected to Gaussian measurement noises with standard deviations that correspond to common online instruments: 5 L for the volume (yV), 0.5 K for the temperatures (yT, yj), 1000 kg/kmol for the molecular weight [yι(tk) ) 1/Mn(tk-1)]; for the continuous measurements (yV, yT, yj), the noise was injected every minute, and for the DD measurement (yι), the noise was injected each sampling time. The sampling-delay period of the MW measurements was ∆ ) 30 min (as the typical delay of industrial GPC measurements), and the application of the tuning guidelines yielded the following settings:

is a slight deacceleration in the MW settling time, but basically the same qualitative behavior is obtained. Input-to-State Stability Property. Finally, to test the inputto-state stability property of the closed-loop reactor with the proposed control scheme (44), the system was subjected to sinusoidal disturbances in the monomer feed concentration and in the reactor and jacket feed temperatures (shown in Figure 6):

• (V, T, M)-controller gains

Te ) 315 K + 5 sin(2πt/110 min), Tje ) 328 K + 2 sin(2πt/110 min), t g 100 min

ωV ) ωj )

-1

ω/j

) ωT ) ω ) 1/8 min , j, kV ) kj ) 1/45 min-1, kT ) 2qj/V

j km ) 1.5qj/V

• MW controller gains ωo ) 1/35 min-1,

ζo ) 2.2, ωc ) 1/140 min-1,

ζo ) 1.3

As expected, comparing with the gains of the noiseless case (Table 2), the noisy measurements led to gain detuning (10-15% slower with more damping). The corresponding closed-loop behavior is presented in Figure 5; as it can be seen in the figure, the presence of noise slightly affects the output responses, as they remain arbitrarily close to the nominal values. Comparing with the noiseless MW measurements case, there

me ) Fm(qm + q˜m)/(Fmqm + Fsqjs) ) Fmqm/(Fmqm + Fsqjs) + 0.02 sin(2πt/220 min),

q/m ) qm + qjm

/ Thus, the actual monomer feedrate input (qm ) is the one calculated from the monomer controller (44c) plus a deviation term (q˜m) calculated from the sinusoidal change in the monomer feed concentration (me), meaning an actuator error (unknown disturbance) not accounted for by the monomer controller (44c). The sampling-delay period of the MW measurements was ∆ ) 30 min, the control gains are the ones presented in Table 2, and the resulting closed-loop behavior is presented in Figure 6. The outputs exhibit sustained oscillations with the following amplitudes: (0.008, 0.1, 1.2, and 0.45)% for (V: volume, T: temperature, m: monomer, and Mn: number-average molecular weight), which are rather small with respect to industrial operations and instrumentations; the control inputs act in an effective coordinated way to compensate for the sinusoidal


closed-loop stable dynamics for an ample interval of sampling periods, (ii) shows a MW response with settling time that is similar to the ones obtained earlier with full model-based nonlinear MW controllers, and (iii) recovers the behavior of a controller with continuous measurements, as the sampling-delay period vanishes. Acknowledgment The authors gratefully acknowledge the support from the Mexican National Council for Research and Technology (CONACyT Scholarship 118632) for P. Gonza´lez. Appendix. Nonlinear maps • Closed-loop reactor with PI-inventory control and continuous measurements (8) Oe(xe, I, d) ) {Fc[χc, Ccχc, d, µc(χc, Ccxc)], fc[xc, xd, d, µc(χc, Ccxc)]}′, I ) (0, 1)xd Od(xe,xd,wi) ) fd[xd, xc, µcq(χc, Ccxc), wi], q ) µcq(χc, Ccxc),

yc ) Ccxc

• Geometric controller (11) Prescribed closed-loop output regulation error dynamics ¨ι˜ + k ˙ι˜ + k ι˜ ) 0, 2 1

k1, k2 > 0

ι˜ ) ι -ι¯,

Maps Figure 6. Closed-loop behavior with DD measurement-driven MW controller (∆ ) 30 min): response to monomer concentration and feed reactor and jacket temperature sinusoidal disturbances.

disturbances, and the presence of the unknown disturbance does not affect the stability of the closed-loop system. This closedloop behavior corroborates the IS stability property of the reactor system with the proposed OF controller: with persistent sinusoidal input disturbances, the system converges to a sustained oscillatory regime within an acceptable-size compact set about the nominal steady-state. Conclusions The control of continuous free-radical solution polymer reactors with continuous measurements of temperature, level, and flows and discrete-delayed measurements of molecular weight (MW) has been addressed within a constructive control framework, with emphasis on the design of a MW component with applicability-oriented features (linearity, decentralization, and reduced-model dependency). The MW component was drawn by combining physical insight, passivity, discrete model realization, and controllability-detectability considerations. The resulting MIMO control scheme has linear decentralized PItype volume and temperature components, a material balance monomer controller, and a discrete linear decentralized MW component that amounts to a second-order discrete input interlaced to a third-order discrete observer. The MIMO control scheme has a systematic construction, simple tuning guidelines, and fewer modeling requirements than previous polymer reactor control studies with MW measurements. The implementation (via numerical simulations) in an open-loop unstable industrialsize case study showed that the proposed controller (i) yields

R(ι, ν, xc) ) Rg[ι, fι -1(ι, ν, xc), xc],

Rg(ι, I, xc) ) ∂Ifι

γ(ι, ν, xc, d, uc) ) γg[ι, fι -1(ι, ν, xc), xc, d, uc] γg(ι, I, xc, d, uc) ) -(∂Ifι)λi(xc, uc)I - λι(I, xc)fι + (∂xcfι)fc(xc, I, d, uc) λi(xc, uc) ) q/V + E(T), λι(I, xc) ) fr(T, V, m, I, s)/fp(V, m, s),

I ) fι -l(ι, ν, xc)

• Continuous nonlinear passive FF-SF controller (19) and linear OF control (30a-b) γn(ι, ˙ι, xc, d, uc) ) γ(ι, ˙ι, xc, d, uc) + cδ˙ι + cσι βι(ι, ˙ι, xc, d, uc,wi) ) γn(ι, ˙ι, xc, d, uc) + [R(ι, ˙ι, xc) - aι]wi • MW discrete model (34) ¯ ¯ γ11 ) (λ¯ ie-λι∆ - λ¯ ιe-λi∆)/(λ¯ i - λ¯ ι),

¯

γ12 ) β2 ) (e-λι∆ ¯ e-λi∆)/(λ¯ i - λ¯ ι) ¯ γ22 ) (λ¯ ie-λi∆ -

¯ ¯ γ21 ) λ¯ iλ¯ ι(e-λi∆-e-λι∆)/(λ¯ i - λ¯ ι),

¯ λ¯ ιe-λι∆)/(λ¯ i - λ¯ ι) ¯ ¯ β1)[(1 - e-λι∆)/λ¯ ι-(1 - e-λi∆)/λ¯ i]/(λ¯ i - λ¯ ι) j

j

τ)e-∆λi + e-∆λι,

j

j

δ ) e-∆(λi+λι),

• Discrete closed-loop dynamics (39)

βp ) γ12β2 - γ22β1


σc(∆, ωc, ζc) ) γc1(∆, ωc, ζc) + γc2(∆, ωc, ζc) ) 2e-∆ζcωc cosh[∆ωc(ζc2 - 1)1/2], (for ζc g 1) ) 2e-∆ζcωc cos[∆ωc(1 - ζc2)1/2], (for ζc < 1) πc(∆, ωc, ζc) ) γc1(∆, ωc, ζc)γc2(∆, ωc, ζc) ) e-2∆ζcωc γc1(∆, ωc, ζo) ) e-∆ωc[ζc+(ζc -1) ], 2

1/2

γc2(∆, ωc, ζc) )

e-∆ωc[ζc-(ζc -1) • Discrete control (41) and observer (42) gains 2

1/2]

κc1(∆, ωc, ζc) ) -τ + σc(∆, ωc, ζc) κc2(∆, ωc, ζc) ) -δ + πc(∆, ωc, ζc) κo1(∆, ωo, ζo) ) 3 - [σo(∆, ωo, ζo) + γo3] κo2(∆, ωo, ζo) ) {5 - 3[σo(∆, ωo, ζo) + γo3] + (1 + γo3)e-2∆ζoωo + σo(∆, ωo, ζo)γ03}/(2∆) κo3(∆, ωo, ζo) ) {1 - [σo(∆, ωo, ζo) + γo3] + (1 γo3)e-2∆ζoωo + σo(∆, ωo, ζo)γo3}/∆2 σo(∆, ωo, ζo) ) γo1(∆, ωo, ζo) + γo2(∆, ωo, ζo) ) 2e-∆ζoωo cosh [∆ωo(ζo2 - 1)1/2], (for ζo g 1) ) 2e-∆ζoωo cos[∆ωo(1 - ζo2)1/2], γo1(∆, ωo, ζo) ) e-∆ωo[ζo+(ζo

(for ζo < 1)

2-1)1/2]

γo2(∆,

,

ωo, ζo) ) e-∆ωo[ζo-(ζo

2

- 1)1/2]

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ReceiVed for reView October 10, 2007 ReVised manuscript receiVed May 13, 2008 Accepted May 21, 2008 IE071361R

Constructive Molecular Weight Control of Continuous Homopolymer

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