Nonlinear State Estimation in a Polymerization Reactor - American

Ind. Eng. Chem. Res. 1997, 36, 2679-2690

2679

Nonlinear State Estimation in a Polymerization Reactor Srinivas Tatiraju and Masoud Soroush* Chemical Engineering Department, Drexel University, Philadelphia, Pennsylvania 19104

This paper concerns nonlinear state estimation in a continuous polymerization reactor where free-radical solution polymerization of methyl methacrylate takes place. Initiator and solvent concentrations and the leading moments of the molecular weight distribution of the polymer are estimated in three measurement cases. For one of the measurement cases, the global convergence of the nonlinear observer is proved. The convergence of the nonlinear observers is shown by numerical simulations, for different values of observer gains, for different observer initial conditions, and in the presence of a measurable disturbance. Furthermore, in each case, the implementation and performance of the nonlinear observer are compared with those of a deterministic extended Kalman filter. 1. Introduction Polymer products are used in a vast number of applications. The quality, processability, and utility of these products greatly depend on the molecular weight distribution (MWD) of the polymer, which is specified at the synthesis stage. Thus, monitoring and controlling MWD in polymerization reactors are of considerable importance. In many cases, the leading moments of MWD are adequate to characterize the polymer product quality. In practice, the leading moments are, however, not measured on-line or measured on-line with significant time delays. To address this measurement problem, several research directions have been pursued. These include the following: (a) development of faster and more reliable sensors [see Chien and Penlidis (1990) for a review of available sensors], (b) study of the quantitative relations between easily available measurements and product quality indices (for example, between viscosity and average molecular weights), and (c) reliable estimation of the unmeasurable state variables from readily available measurements by using state observers/estimators. State estimators are deterministic/stochastic dynamic systems that are used to reconstruct the inaccessible but important process state variables, from easily measured variables. It is generally agreed that linear observers/estimators are inadequate for many severely nonlinear processes (Kantor, 1989; Ogunnaike, 1995; Valluri and Soroush, 1996), motivating the use of nonlinear estimators/observers. The problem of state estimation in chemical/petrochemical processes has been studied extensively since the mid 1970’s. In particular, the extended Kalman filter (EKF) has been used widely for state estimation in polymerization reactors (Adebekun and Schork, 1989; Ellis et al., 1988; Jo and Bankoff, 1976; Kim and Choi, 1991; Kozub and MacGregor, 1992; Ogunnaike, 1994; Quintero-Marmol et al, 1991; Robertson et al., 1995; Schuler and Suzhen, 1985). The estimator is designed on the basis of a linear approximation of process model (as in the case of polymerization reactors), and thus, it is inadequate for severely nonlinear processes (Gudi et al., 1995). Since the early 1980’s, in parallel to the advances in nonlinear control, many attempts have been made to solve the problem of nonlinear state observer design. These have led to the development of several methods * To whom correspondence should be addressed. Fax: (215) 895-5837. E-mail: [email protected]. S0888-5885(96)00905-0 CCC: $14.00

such as output injection (Krener and Isidori, 1983), high gain observer (van Dootingh et. al., 1992), open-loop reduced-order observer (Daoutidis and Kravaris, 1992; Soroush and Kravaris, 1994), extended Luenberger observers (Ciccarella et al., 1993; Kazantzis and Kravaris, 1995; Kurtz and Henson, 1995; Zeitz, 1987), moving horizon state estimator (Robertson et al., 1996), and closed-loop reduced-order observer (Soroush, 1997). In these methods, a nonlinear process model, without any linear approximation, is used directly for observer design. In this paper, the nonlinear reduced-order observer design method presented in Soroush (1997) is applied to a polymerization reactor where solution polymerization of methyl methacrylate takes place. In particular, initiator and solvent concentrations and the leading moments of the MWD of the polymer are estimated in several measurement scenarios. This work is the first application of the nonlinear observer design method to estimate polymer product quality indices from easily available measurements in a polymerization reactor. The global convergence of the nonlinear observer for a class of polymerization reactors is proved in one of the measurement scenarios. Furthermore, the implementation and performance of the nonlinear observer design method are compared to those of a deterministic EKF. This paper begins with a brief review of the nonlinear observer design method presented in Soroush (1997) and a deterministic EKF (Smith and Roberts, 1979). The polymerization system and its mathematical model are then described, followed by the application of the nonlinear observer and the EKF to the reactor. Finally, the performance of the nonlinear observer and the EKF are shown and compared through numerical simulations. 2. Nonlinear Observer Design Method Consider a nonlinear process with a dynamic model of the form

{

x˘ (t) ) f(x(t), u(t)), x(0) ) x0 y(t) ) h(x(t))

(1)

where x ) [x1 ... xn]T, u ) [u1 ... um]T, and y ) [y1 ... yq]T represent the vectors of state variables, measurable inputs (measurable disturbances and manipulated inputs), and measurable outputs, respectively. Unlike the measurable outputs, the measurable inputs do not include any information on any of the state variables. It is assumed that (a) the vector functions h(x) and f(x, u) are smooth; (b) the process is at least locally detect© 1997 American Chemical Society

2680 Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997

able; and (c) the q × n matrix

∂h(x) ∂x

(2)

has q linearly independent rows or columns; that is, the measurable outputs are not “redundant”. The assumption (c) ensures the existence of a locally one-to-one change of variables of the form

[]

[ ]

Px η ) u (x) ) h(x) y

where η ) [η1, ..., ηn-q]T and P is a constant (n - q) × n matrix, which for the sake of simplicity is chosen such that (i) each row of the matrix P has only one nonzero term equal to 1, and (ii) locally the determinant of the n × n matrix

[ ] P ∂h(x) ∂x

η˘ ) Fη(η, y, u) y˘ ) Fy(η, y, u)

(3)

where

|

{

ηˆ˙ ) Fη(ηˆ , y, u) + L[y˘ - Fy(ηˆ , y, u)] xˆ ) u

-1

(ηˆ , y)

(4)

where the [(n - q) × q] matrix L is the observer gain. If the observer gain is chosen to be constant, by using the new variable z ) ηˆ - Ly, we can recast the observer of (4) in the form

{

z˘ ) Fη(z + Ly, y, u) - LFy(z + Ly, y, u) xˆ ) u

-1

(z + Ly, y)

-1

(8)

(ηˆ , y)

e˘ ) Fη(e + η, y, u) - Fη(η, y, u)

(9)

which has a universal equilibrium point at e ) 0. Thus, a necessary condition for the local asymptotic convergence of this open-loop observer is that locally all the eigenvalues of the (n - q) × (n - q) Jacobian matrix

∂Fη(η, y, u) ∂η

(η, y), u]

Once an estimate of η, denoted by ηˆ , is obtained by using an “open-loop” or “closed-loop”, reduced-order observer designed for the first (n - q) differential equations of (3), the vector of estimated state variables, xˆ , is calculated by using the relation x ) u -1(ηˆ , y). 2.1. Reduced-Order Observer. If a process of the form of (1) is locally observable, then one can design a state observer with locally asymptotically stable error dynamics and with a completely adjustable rate of convergence. These can be achieved by using a closedloop, reduced-order observer of the form (Soroush, 1997)

xˆ ) u

that is, integration of the η subsystem in (3), which is driven by the measurable inputs and outputs. In this case, the observer error is governed by the system

-1

-1

(7)

lie in the left half of the complex plane. This condition is, of course, a necessary condition for local asymptotic stability of the equilibrium point e ) 0. In many practical cases, it is sufficient to check the preceding condition. Remark 1: In the case that the observer gain L ) 0, then the closed-loop observer of (4) and (5) will be an open-loop reduced order observer

ηˆ˙ ) Fη(ηˆ , y, u)

{

(η, y), u]; ∂h(x) Fy(η, y, u) ) f[u ∂x x)u -1(η,y)

∂Fη(η, y, u) ∂Fy (η, y, u) -L ∂η ∂η

{

is nonzero. The new variables η1, ..., ηn-q are simply (n - q) state variables of the original model of (1), which satisfy the condition of ii. The system of (1), in terms of the new state variables η1, ..., ηn-q, and y, takes the form

Fη(η, y, u) ) Pf[u

(Soroush, 1997). Because e ) 0 is the universal equilibrium point of the error dynamics of (6), if the observer error e ) 0 at t ) t0, then e(t) ≡ 0, ∀t g t0, regardless of any variation in the measurable inputs and outputs. The constant observer gain, L, should be chosen such that locally all the eigenvalues of the [(n - q) × (n q)] matrix

(10)

lie in the left half of the complex plane. The rate of convergence of this open-loop observer is, of course, not adjustable and is dictated by the process model itself. 3. Deterministic Extended Kalman Filter If the Kalman filter (KF), as described by Kalman and Bucy (1961), is implemented for deterministic systems, the algorithm gives rise to a filter with infinite gain. This failure of KF is attributed to the optimal gain of the filter, and thus, the use of a suboptimal gain should alleviate this problem (Smith and Roberts, 1979). A nonlinear extension of the KF of Smith and Roberts (1979) to nonlinear deterministic systems of the form of (1) is

{

xˆ˙ ) F(xˆ )xˆ + cP[H(xˆ )]T[y - H(xˆ )xˆ ], xˆ (0) ) xˆ 0 P˙ ) F(xˆ )P + P[F(xˆ )]T - 2cP[H(xˆ )]T H(xˆ ) P, P(0) ) P0

(11) where

(5)

whose implementation does not need the time-derivative of the measurable outputs. The corresponding observererror e, where e ) ηˆ - η, is governed by

e˘ ) Fη(e + η, y, u) - Fη(η, y, u) L{Fy(e + η, y, u) - Fy(η, y, u)} (6) which has a universal equilibrium point at e ) 0. The origin is a universal equilibrium point in the sense that it is an equilibrium point for every triplet (η, y, u)

F(x) )

∂f(x) , ∂x

H(x) )

∂h(x) ∂x

the expression cPHT is the time-varying filter gain, c is an adjustable positive scalar constant, and P is the error covariance matrix. The EKF of (11) consists of [(n + 1)n] ordinary differential equations (ODEs), whereas the nonlinear reduced-order observer consists of (n - q) ODEs. Even when the off-diagonal elements of the matrix P are set to zero (Pij(t) ≡ 0, i * j, i, j ) 1, ..., n), the number of ODEs to be integrated will be higher in the case of the EKF. Furthermore, the EKF requires

Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997 2681

F(x) and H(x), which should be calculated symbolically or numerically. 4. Free-Radical Polymerization Reactor The dynamic model described in Schmidt and Ray (1981) is used to represent the solution polymerization of methyl methacrylate in a continuous stirred tank reactor (CSTR)

where

Ciin

f1 )

τ Csin

f2 )

τ

1 f3(Ci, Cs, Cm, T) ) [(kfmCm + ktdP + kfsCs)]RP + ktcP2 2 f4(Ci, Cs, Cm, T) ) Mm[(kfmCm + P ktdP + kfsCs)(2R - R2) + ktcP] (1 - R) f5(Ci, Cs, Cm, T) ) Mm2[(kfmCm + ktdP + P kfsCs)(R3 - 3R2 + 4R) + ktcP(R + 2)] (1 - R)2 f6(Ci, Cm, T) ) -kpCmP + f7(Ci, Cm, T, Tj) ) γkpCmP + f8(T, Tj) )

Cmin - Cm τ

US(Tj - T) Tin - T + FcpV τ

Twin - Tj US (T - Tj) + FwcpwVj τj kpCm

R)

(kp + kfm)Cm + kfsCs + ktP P)

x

2f*Ciki kt

Table 1. Operating Conditions of the Reactor Cm(0) Ci(0) Cs(0) T(0) Tj(0) λ0(0) λ1(0) λ2(0) Cmin Ciin Csin Tin Twin

) ) ) ) ) ) ) ) ) ) ) ) )

5.162 kmol‚m-3 0.032 kmol‚m-3 4.580 kmol‚m-3 3.460 × 102 K 3.260 × 102 K 0.020 kmol‚m-3 1.000 kg‚m-3 5.000 × 102 kg2‚kmol-1‚m-3 5.162 kmol‚m-3 0.032 kmol‚m-3 4.580 kmol‚m-3 2.9615 × 102 K 3.2315 × 102 K

The process has eight state variables: monomer concentration (Cm), initiator concentration (Ci), solvent concentration (Cs), reactor temperature (T), jacket temperature (Tj), and the first three leading moments of MWD (λ0, λ1, and λ2). The parameter values of the reactor model are the same as those given in Schmidt and Ray (1981) except that the heat transfer coefficient, U ) 6.27 × 10-6, kJ‚m-2‚s-1‚K-1, and the flow rate of cooling water, Fcw ) 3.45 × 10-8, m3‚s-1. The operating conditions of the reactor, given in Table 1, are chosen such that the gel and glass effects are not present (ensured by the sufficiently high fraction of solvent in the feed and a low steady-state monomer conversion, xm ) 0.29). Measurements in a solution polymerization reactor include those of density, viscosity, and MWD of the reacting mixture, temperature, and pressure. Density and viscosity of the reacting mixture, temperature, and pressure can be measured on-line at very high sampling frequencies and almost with no delays. Because these measurements are available at sufficiently high frequencies, these can be considered as continuous functions of time (as “fast” measurements). The monomer conversion, xm, can be inferred from the density measurements and, thus, can be calculated on-line. An average molecular weight of a polymer solution is known to be correlated to the reacting mixture viscosity that can be measured on-line or be inferred from fast and delay-free measurements of reactor stirrer torque (Chien and Penlidis, 1990; Ray, 1986). On the other hand, MWD measurements that are obtained by gel permeation chromatography (GPC) are available with significant delays and usually at low sampling rates. The first-principle mathematical model of the process described by (12) is used to represent the actual process. The following three measurement scenarios will be considered: • Case I, in which fast on-line measurements of the monomer concentration (Cm), reactor temperature (T), and jacket temperature (Tj) are available. • Case II, in which fast on-line measurements of the monomer concentration (Cm), reactor temperature (T), jacket temperature (Tj), and weight-average molecular weight (Mw) are available. • Case III, in which fast on-line measurements of the monomer concentration (Cm), reactor temperature (T), jacket temperature (Tj), and number-average molecular weight (Mw) are available. The weight or number average molecular weight is inferred from on-line viscosity measurements and the conversion (hence Cm) from on-line density measurements.


The issue of nonlinear estimation in the presence of multirate/delayed measurements are not considered in this paper. A multirate version of the nonlinear observer has been successfully implemented in real time on a bioreactor and the same polymerization reactor example via numerical simulations (Tatiraju et al., 1996, 1997).

governed by

5. State Estimation via the Nonlinear Observer

1 e˘ 3 ) - e3 + ∆f3(e1, e2, Ci, Cs, Cm, T) τ L31∆f6(e1, Ci, Cm, T) - L32∆f7(e1, Ci, Cm, T, Tj)

5.1. Case I. In this case, the measurable output vector y ) [Cm T Tj]T. The system (12) is already in the form of (3), and thus, the first five differential equations form the η subsystem and the last three differential equations represent the y subsystem. The initiator concentration Ci is observable almost everywhere, whereas the state variables Cs, λ0, λ1, and λ2 are not. Loosely speaking, according to the model, the available measurements do not include any information on these unobservable state variables. However, Cs, λ0, λ1, and λ2 are detectable everywhere [see Soroush (1997) for the definitions of the observability and detectability of a state variable]. The η subsystem satisfies the condition that all the eigenvalues of the Jacobian matrix of (10) lie in the left half of the complex plane. Thus, in this case, all the state variables can be reconstructed by using a reduced-order observer whose rate of convergence is not completely adjustable (rate of convergence of the unobservable state variables are set by the process itself).

[ ]

For this case, the nonlinear observer of (5) takes the form

[]

[

- ki +

]

[

][

1 e - L11∆f6(e1, Ci, Cm, T) τ 1 L12∆f7(e1, Ci, Cm, T, Tj)

]

1 e˘ 2 ) - e2 - L21∆f6(e1, Ci, Cm, T) τ L22∆f7(e1, Ci, Cm, T, Tj)

1 e˘ 4 ) - e4 + ∆f4(e1, e2, Ci, Cs, Cm, T) τ L41∆f6(e1, Ci, Cm, T) - L42∆f7(e1, Ci, Cm, T, Tj) 1 e˘ 5 ) - e5 + ∆f5(e1, e2, Ci, Cs, Cm, T) τ L51∆f6(e1, Ci, Cm, T) - L52∆f7(e1, Ci, Cm, T, Tj) (14) where

∆f3(e1, e2, Ci, Cs, Cm, T) ) f3(Ci + e1, Cs + e2, Cm, T) - f3(Ci, Cs, Cm, T) ∆f4(e1, e2, Ci, Cs, Cm, T) ) f4(Ci + e1, Cs + e2, Cm, T) - f4(Ci, Cs, Cm, T) ∆f5(e1, e2, Ci, Cs, Cm, T) ) f5(Ci + e1, Cs + e2, Cm, T) - f5(Ci, Cs, Cm, T) ∆f6(e1, Ci, Cm, T) ) f6(Ci + e1, Cm, T) - f6(Ci, Cm, T)

1 C ˆ +f τ i 1

C ˆs z˘ 1 + f2 τ z˘ 2 λˆ z˘ 3 ) - 0 + f3(C ˆ i, C ˆ s, Cm, T) τ z˘ 4 λˆ 1 - + f4(C ˆ i, C ˆ s, Cm, T) z˘ 5 τ λˆ 2 ˆ i, C ˆ s, Cm, T) - + f5(C τ L11 L12 L13 l l l L51 L52 L53

[

e˘ 1 ) - ki +

∆f7(e1, Ci, Cm, T, Tj) ) f7(Ci + e1, Cm, T, Tj) f7(Ci, Cm, T, Tj) As the first ordinary differential equation (ODE) of the preceding error dynamics shows, e1 is independent of e2, ..., e5, and is governed by

-

f6(C ˆ i, Cm, T) ˆ i, Cm, T, Tj) f7(C f8(T, Tj)

]

[

e˘ 1 ) - ki +

x

1 e - Cm τ 1

]

2f*ki [ C + e1 kt x i

(13)

Thus, choosing L11 and L12 such that

L11 e γL12

where

C ˆ i ) z1 + L11Cm + L12T + L13Tj C ˆ s ) z2 + L21Cm + L22T + L23Tj λˆ 0 ) z3 + L31Cm + L32T + L33Tj λˆ 1 ) z4 + L41Cm + L42T + L43Tj λˆ 2 ) z5 + L51Cm + L52T + L53Tj

5.1.1. Stability of the Observer Error Dynamics: Tuning the Observer. The observer error is

xCi]kp{L12γ - L11} (15)

ensures that globally e1(t) f 0 as t f ∞. Maple V software was used to obtain the elements of the Jacobian of the system (14) symbolically to study the effect of the observer gain, L, on the stability of the error dynamics around the process steady state and e ) 0. The Jacobian is a lower triangular matrix, and thus its diagonal entries are the eigenvalues of the linear approximation of the error dynamics. While the eigenvalue corresponding to e1 depends on the observer gain, the other four eigenvalues are in the left half of the plane and equal to -1/τ (independent of the observer gain). Recall that the four state variables Cs, λ0, λ1, and λ2 are not observable. Thus, the error components e2, ..., e5 all decay with time irrespective of the values of


[ ]

[ ]

60 0 L1 ) 0 0 0

2 0 0 0 0

0 0 0 0 0

0 0 L2 ) 0 0 0

0 0 0 0 0

0 0 0 0 0

L13 and Lij, i ) 2, ..., 5, j ) 1, ..., 3 (their decay times are independent of L13 and Lij, i ) 2, ..., 5, j ) 1, ..., 3 and are set by the process itself). Because of this feature of the observer dynamics, we simply set

L13 ) Li1 ) Li2 ) Li3 ) 0,

i ) 2, ..., 5

(16)

Therefore, if the observer gain, L, is chosen according to (15) and (16), then globally e(t) f 0 as t f ∞ for every profile of (Ci(t), Cs(t), λ0(t), λ1(t), λ2(t), Cm(t), T(t), Tj(t)) and for every e(0). Use of non-zero values for L13 and Lij, i ) 2, ..., 5, j ) 1, ..., 3, can affect the shape of the profiles of the estimated states but not the time needed for the convergence of each estimated state. The two observer gain values given in Table 2, which satisfy the conditions of (15) and (16), will be used in the numerical simulations. 5.2. Case II. In this case, because the weightaverage molecular weight, Mw ) λ2/λ1, is inferred from on-line viscosity measurements, in addition to the monomer and initiator concentrations, the solvent concentration, Cs, and the first and second moments of the MWD, λ1 and λ2, are observable. Thereby, all the state variables except the zeroth moment of MWD, λ0, are observable. The unobservable state variable is, however, detectable everywhere. Here an η subsystem is

{ {

dCi 1 ) - ki + Ci + f1 dt τ dCs Cs )+ f2 dt τ dλ1 λ1 ) - + f4(Ci, Cs, Cm, T) dt τ λ0 dλ0 ) - + f3(Ci, Cs, Cm, T) dt τ

[

]

(17)

[

[]

- ki +

1 C ˆ + f1 τ i

]

C ˆs z˘ 1 + f2 z˘ 2 τ ) λˆ 1 z˘ 3 - + f4(C ˆ i, Cs, Cm, T) z˘ 4 τ λˆ 0 ˆ i, C ˆ s, Cm, T) - + f3(C τ f*(C ˆ ,C ˆ , λˆ , M , C , T) L11 L12 L13 L14 5 i s 1 w m ˆ i, Cm, T) f6(C l l l l ˆ , C , T, Tj) f (C L41 L42 L43 L44 7 i m f8(T, Tj)

[

][

]

(19)

where

C ˆ i ) z1 + L11Mw + L12Cm + L13T + L14Tj C ˆ s ) z2 + L21Mw + L22Cm + L23T + L24Tj λ1 ) z3 + L31Mw + L32Cm + L33T + L34Tj λ0 ) z4 + L41Mw + L42Cm + L43T + L44Tj Once λˆ 1 is known, λˆ 2 is calculated from λˆ 2 ) λˆ 1Mw. 5.2.1. Stability of the Observer Error Dynamics: Tuning the Observer. The corresponding observer error is governed by

1 e τ 1 L11∆f*5(e1, e2, e3, Ci, Cs,λ1, Mw, Cm, T) L12∆f6(e1, Ci, Cm, T) - L13∆f7(e1, Ci, Cm, T, Tj)

[

e˘ 1 ) - ki +

]

1 e˘ 2 ) - e2 - L21∆f*5(e1, e2, e3, Ci, Cs,λ1, Mw, Cm, T) τ L22∆f6(e1, Ci, Cm, T) - L23∆f7(e1, Ci, Cm, T, Tj) 1 e˘ 3 ) - e3 + ∆f4(e1, e2, Ci, Cs, Cm, T) τ L31∆f* 5(e1, e2, e3, Ci, Cs, λ1, Mw, Cm, T) L32∆f6(e1, Ci, Cm, T) - L33∆f7(e1, Ci, Cm, T, Tj)

and the y subsystem is

dMw ) f*5(Ci, Cs,λ1, Mw, Cm, T) dt dCm ) f6(Ci, Cm, T) dt dT ) f7(Ci, Cm, T, Tj) dt dTj ) f8(T, Tj) dt

[ ]


Table 2. Observer Gain Values for Case I

1 e˘ 4 ) - e4 + ∆f3(e1, e2, Ci, Cs, Cm, T) τ L41∆f* 5(e1, e2, e3, Ci, Cs, λ1, Mw, Cm, T) L42∆f6(e1, Ci, Cm, T) - L43∆f7(e1, Ci, Cm, T, Tj) (20) (18)

where

1 f*5 (Ci, Cs, λ1, Mw, Cm, T) ) [f5(Ci, Cs, Cm, T) λ1 Mwf4(Ci, Cs, Cm, T)]

where

∆f* 5(e1, e2, e3, Ci, Cs, λ1, Mw, Cm, T) ) f* 5(Ci + e1, Cs + e2, λ1 + e3, Mw, Cm, T) f* 5(Ci, Cs, λ1, Mw, Cm, T) In this case, because of the complex dependence of ∆f*5 on e2 and e3, the proof of the global asymptotic stability of the error dynamics is not as simple as in case I. Thus, in this case, the value of the observer gain, L, is chosen to ensure that locally e(t) f 0 as t f ∞. If

2684 Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997 Table 3. Observer Gain Values for Case II

[

0 0.001 L1 ) -0.006 0

50 0 25 0

1.75 0 7 0

0 0 0 0

] [

0 0.001 L2 ) -0.01 0

60 0 25 0

2 0 10 0

0 0 0 0

]

{ {

state variable is, however, detectable everywhere. Here an η subsystem is

dCi 1 ) - ki + Ci + f1 dt τ dCs Cs )+ f2 dt τ dλ0 λ0 ) - + f3(Ci, Cs, Cm, T) dt τ λ2 dλ2 ) - + f5(Ci, Cs, Cm, T) dt τ

we set L11 ) 0, the Jacobian of the system (20) will be lower block triangular, i.e., of the form

[

J′11 J′ J′ ) J′21 31 J′41

0 J′22 J′32 J′41

0 J′23 J′33 J′43

0 0 0 J′44

]

[

The real eigenvalue J′11 depends on L12 and L13 only; it will be negative, if L12 and L13 are chosen according to where

[

Cm

1 e τ 1 2f*ki [ C + e1 - xCi]kp{L13γ - L12} (22) kt x i

]

x

Then, choosing L12 and L13 according to (21) ensures that globally e1(t) f 0 as t f ∞. The real eigenvalue J′44 does not depend on L at all and is equal to -1/τ (recall that λ0 is detectable but not observable). The 2 × 2 matrix δ′ and its eigenvalues depend on L21 and L31 only. Thus, the values of L21 and L31 should be chosen such that both eigenvalues of the 2 × 2 matrix δ′ evaluated at e ) 0 and at the process steady state lie in the left half plane. Since the eigenvalues of the Jacobian J′ do not depend on L41, L42, L43, L44, L14, L24, and L34, we simply set

L41 ) L42 ) L43 ) L44 ) L14 ) L24 ) L34 ) 0

(25)

(21)

The reason is that the first ODE of the error dynamics of (20), e˘ 1, is independent of e2, ..., e4, and e1 is governed by

e˘ 1 ) - ki +

(24)

dMn ) f*4(Ci, Cs, λ0, Mn, Cm, T) dt dCm ) f6(Ci, Cm, T) dt dT ) f7(Ci, Cm, T, Tj) dt dTj ) f8(T, Tj) dt

]

L12 e γL13

]

and the y subsystem is

Thus, the eigenvalues of the Jacobian will be J′11, J′44, and the eigenvalues of the 2 × 2 matrix

J′ J′ δ′ ) J′22 J′23 32 33

[

(23)

By this selection of the gain matrix, L, local stability of the error dynamics is ensured. The components of the observer gain already set to zero do not affect the decay time of the error dynamics locally but affect the shape of the profiles of the estimated states. Two observer gain values are given in Table 3; they will be used in the numerical simulations. 5.3. Case III. In this case, because the numberaverage molecular weight, Mn ) λ1/λ0, is inferred from on-line viscosity measurements, in addition to the monomer and initiator concentrations, the solvent concentration, Cs, and the moments λ0 and λ1, are observable. Thereby, all the state variables except the second moment of MWD, λ2, are observable. The unobservable

f*4(Ci, Cs, λ0, Mn, Cm, T) )

1 [f (C , C , C , T) λ0 4 i s m Mnf3(Ci, Cs, Cm, T)]

[ ]


[

[]

- ki +

1 C ˆ + f1 τ i

]

C ˆs z˘ 1 + f2 z˘ 2 τ ) λˆ 0 z˘ 3 - + f3(C ˆ i, Cs, Cm, T) z˘ 4 τ λˆ 2 ˆ i, Cs, Cm, T) - + f5(C τ f*(C ˆ ,C ˆ , λˆ , M , C , T) L11 L12 L13 L14 4 i s 0 n m ˆ i, Cm, T) f6(C l l l l ˆ , C , T, Tj) f (C L41 L42 L43 L44 7 i m f8(T, Tj)

[

][

]

(26)

where

C ˆ i ) z1 + L11Mn + L12Cm + L13T + L14Tj C ˆ s ) z2 + L21Mn + L22Cm + L23T + L24Tj λˆ 0 ) z3 + L31Mn + L32Cm + L33T + L34Tj λˆ 2 ) z4 + L41Mn + L42Cm + L43T + L44Tj Once λˆ 0 is known, λˆ 1 is calculated from λˆ 1 ) λˆ 0Mn. 5.3.1. Stability of the Observer Error Dynamics: Tuning the Observer. The corresponding ob-


server error is governed by

Table 4. Observer Gain Values for Case III

1 e τ 1 L11∆f*4(e1, e2, e3, Ci, Cs, λ0, Mn, Cm, T) L12∆f6(e1, Ci, Cm, T) - L13∆f7(e1, Ci, Cm, T, Tj)

[

e˘ 1 ) - ki +

]

1 e˘ 2 ) - e2 - L21∆f*4(e1, e2, e3, Ci, Cs, λ0, Mn, Cm, T) τ L22∆f6(e1, Ci, Cm, T) - L23∆f7(e1, Ci, Cm, T, Tj) 1 e˘ 3 ) - e3 + ∆f3(e1, e2, Ci, Cs, Cm T) τ L31∆f*4(e1, e2, e3, Ci, Cs, λ0, Mn, Cm, T) L32∆f6(e1, Ci, Cm, T) - L33∆f7(e1, Ci, Cm, T, Tj) 1 e˘ 4 ) - e4 + ∆f5(e1, e2, Ci, Cs, Cm T) τ L41∆f*4(e1, e2, e3, Ci, Cs, λ0, Mn, Cm, T) L42∆f6(e1, Ci, Cm, T) - L43∆f7(e1, Ci, Cm, T, Tj) (27) where

∆f* 4(e1, e2, e3, Ci, Cs, λ0, Mn, Cm, T) ) f* 4(Ci + e1, Cs+ (C , C e2, λ0 + e3, Mn, Cm, T) - f* 4 i s, λ0, Mn, Cm, T) In this case, because of the complex dependence of ∆f* 4 on e2 and e3, the proof of the global asymptotic stability of the error dynamics is not as simple as in case I. Thus, the value of the observer gain, L, is chosen to ensure that locally e(t) f 0 as t f ∞. The stability analysis of this case is almost the same as that of case II. If we set L11 ) 0, the Jacobian of the system (27) will be lower block triangular, i.e., of the form

[

J′′11 J′′ J′′ ) J′′21 31 J′′41

0 J′′22 J′′32 J′′41

0 J′′23 J′′33 J′′43

0 0 0 J′′44

]

Thus, the eigenvalues of the Jacobian will be J′′11, J′′44 and the eigenvalues of the 2 × 2 matrix

[

J′′ J′′ δ′′ ) J′′22 J′′23 32 33

]

The real eigenvalue J′′11 depends on L12 and L13 only; it will be negative, if L12 and L13 are chosen such that L12 e γL13. The real eigenvalue J′′44 does not depend on L at all and is equal to -1/τ (recall that λ2 is detectable but not observable). The 2 × 2 matrix δ′′ and its eigenvalues depend on L21 and L31 only. Thus, the values of L21 and L31 should be chosen such that both eigenvalues of the matrix δ′′ evaluated at e ) 0 and at the process steady state lie in the left half plane. Since the eigenvalues of the Jacobian J′′ do not depend on L41, L42, L43, L44, L14, L24, and L34, these are set according to (23). By this selection of the gain matrix, L, local stability of the error dynamics is ensured. Two observer

[

0 0.001 L ) -0.0007 0 1

60 0 0.15 0

2 0 0.15 0

0 0 0 0

] [

0 0.001 L ) -0.00018 0 2

60 0 0.14 0

2 0 0.14 0

0 0 0 0

]

Table 5. Initial Estimated Values for the Nonlinear Observer and the EKF case

C ˆ i(0)

C ˆ s(0)

λˆ 0(0)

λˆ 1(0)

λˆ 2(0)

C ˆ m(0)

T ˆ (0)

T ˆ j(0)

IC1 IC2

0.016 0.048

4.58 4.58

0.01 0.03

0.5 1.5

250.0 750.0

5.162 5.152

346.0 346.0

326.0 326.0

gain values are given in Table 4; they will be used in the numerical simulations. 6. State Estimation via the Extended Kalman Filter For this process, the EKF of (11) consists of 72 firstorder ODEs, and the Jacobian matrix F(x) is calculated symbolically and has a complex form. Because of the elaborate nature of the EKF equations, these equations are not presented in this paper. The off-diagonal elements of the initial error covariance matrix, P0, are set to zero. The EKF will be used to estimate the state variables from each of the same three sets of measurements considered in the previous section. 7. Results and Discussion: Nonlinear Observer vs Extended Kalman Filter In this section, the performance of the reduced-order nonlinear state-observer in estimating the inaccessible state variables of the polymerization reactor is presented and compared to that of the deterministic EKF, which is full order. For each case, the convergence of the observer for various initial estimated values (see Table 5) and for various observer gain values is presented. The convergence of the EKF is also tested for various initial estimates of the state variables and for different values of the filter gain (see Table 6). A step size of 10 s is used in all the numerical integrations. 7.1. Case I. The reduced-order observer of this case is given by (13). As discussed in section 5, in this case, the process is detectable everywhere, and the observer error dynamics are globally asymptotically stable (global asymptotic stability of the error dynamics was proved in section 5). However, the solvent concentration and the three leading moments are not observable. As a result, the rate of convergence of the initiator concentration estimate to its actual value can be adjusted by the observer gain, but the rates of the convergence of the unobservable state variables (Cs, λ0, λ1, and λ2) are not adjustable and are dictated by the process itself. Figure 1 compares the actual and estimated values of the process states. The estimated values are calculated by the nonlinear observer (NLO) and EKF for the two sets of initial estimated values IC1 and IC2 given in Table 5. These two sets of initial estimated values correspond to 50% and 150% of a subset of the initial conditions of the actual process, given in Table 1. The observer gain L ) L1 (given in Table 2), and the EKF gain c ) c1 (given in Table 6). In the case of the NLO, rapid convergence of the estimated value of the initiator concentration to its actual value can be seen in Figure 1, irrespective of the observer initial values (IC1 and IC2), whereas the rate of convergence of the EKFestimated values is much slower than that of the nonlinear observer for the same set of initial estimated


Figure 1. Actual and estimated values of the state variables for various estimator initial conditions (case I).

Figure 2. Actual and estimated values of the state variables for various gain values (case I).

Table 6. Deterministic EKF Gain Values

5). When L ) L2, the reduced-order observer is simply an open-loop reduced-order observer whose convergence is, of course, slower than when L ) L1. However, no significant change is seen in the rate of convergence of the EKF-estimated values of the observable states for different values of the filter gain. In fact, the EKF exhibits oscillatory behavior even in estimating the initiator concentration. As discussed in the previous paragraph, the rate of convergence of the unobservable state variables cannot be adjusted by changing the observer and EKF gain values. In the case of changing the EKF gain value, the estimated profile of the second moment is seen to be more oscillatory than in Figure 1. The performance of the EKF is thereby seen to be even worse than that of the open-loop nonlinear observer. Figure 3 compares the nonlinear-observer and EKF estimated values of the initiator concentration for the two sets of initial estimated values IC1 and IC2 when the inlet initiator concentration varies according to Ciin ) 0.032[1 + 0.5sin(0.01t)]. The observer gain L ) L1, and the EKF gain c ) c1. The convergence of the EKF estimated value of initiator concentration to its actual value is seen to be significantly slower than that of the nonlinear observer for the same initial estimated values. 7.2. Case II. In this case, in addition to the measurements of case I, measurement of the weight-

gain

case I

case II

case III

c1 c2

1.000 0.001

0.0001 0.001

0.0002 0.005

values. The convergence of the observer-estimated values of the leading moments to their actual values are very slow. This is expected, since the leading moments of the MWD are unobservable from the given measurements; with these measurements, there exist no means for improving the convergence of the estimated values of the leading moments by a choice of a proper observer gain value. A similar trend is seen in the case of EKF (even though the gain is calculated by the algorithm itself). Furthermore, the estimates of the second moment are highly oscillatory for IC1, and the estimates show initial convergence followed by a divergence. This effect can be attributed to the local linearization of the nonlinear process model in the EKF algorithm. Figure 2 compares the NLO and EKF estimated values of the initiator concentration and the three leading moments, for two different observer gain values (given in Table 2) and for two different EKF gain values (given in Table 6). The EKF and observer initial estimated values are set to those of IC1 (given in Table


Figure 3. Actual and estimated values of the initiator concentration for various estimator initial conditions (case I), in the presence of varying initiator feed concentration.

Figure 5. Actual and estimated values of the state variables for various gain values (case II).

Figure 4. Actual and estimated values of the state variables for various estimator initial conditions (case II).

average molecular weight, Mw, is available. From this additional measurement, the first and second moments of MWD as well as the solvent concentration are observable. As a result, an observer with adjustable rate of convergence for these two moments and the solvent concentration can be designed. Figure 4 compares the nonlinear-observer and EKF estimated values of the initiator concentration and the three leading moments for the two sets of initial estimated values, IC1 and IC2, given in Table 5. The observer gain L ) L1 (given in Table 3), and the EKF

gain c ) c1 (given in Table 6). In this case, the observer estimated values of the initiator concentration (and also solvent concentration, not shown in Figure 4), and the first and second moments converge to their actual values very rapidly. The convergence of the estimated value of the zeroth moment to its actual value is, however, very slow; the rate of the convergence is set by the process itself. Recall that the zeroth moment of the MWD is unobservable from the Mw measurement. The convergence of the EKF for this case was found to be somewhat better than that of the EKF of case I. But, there is a small mismatch between the EKF-estimated and the actual values of the observable process state variables, which is not seen in the case of the nonlinear observer. Note that there is no mismatch between actual and estimated initiator concentration in this case for IC1; this further substantiates the unpredictable nature of the EKF convergence, since the EKF now seems to exhibit a dependence on the initial estimated values. Figure 5 compares the nonlinear-observer and EKF estimated values of the initiator concentration and the three leading moments, for two different observer gain values (given in Table 3) and for two different EKF gain values (given in Table 6). The EKF and observer initial estimated values are set to those of IC1 (given in Table 5). It can be seen that in the case of the closed-loop


observer (with L ) L1 or L ) L2) the convergence of the initiator concentration and the first and second moments is much faster than those seen in Figure 1. This is because these states are observable from the measurements, thereby making it possible to adjust their rates of convergence by varying the gain values. As discussed in the previous paragraph, the rate of convergence of the unobservable zeroth moment cannot be adjusted by changing the gain value. Again, no significant change is seen in the rate of convergence of the EKF-estimated values for different values of the filter gain. In comparison to case I, in case II the additional weight-average molecular weight measurement made the first and second moments observable, leading to significant improvements in the rate of convergence of the first and second moment estimates. The observer of Case II is, of course, more robust to model errors and unmeasurable disturbances than the observer of Case I. 7.3. Case III. In this case, in addition to the measurements of case I, measurement of the numberaverage molecular weight, Mw, is available. From this measurement, the zeroth and first moments of MWD as well as the solvent concentration are observable. As a result, an observer with adjustable rate of convergence for these two moments can be designed. Figure 6 compares the nonlinear-observer and EKF estimated values of the initiator concentration and the three leading moments for the two sets of initial estimated values, IC1 and IC2, given in Table 5. The observer gain L ) L1 (given in Table 4), and the EKF gain c ) c1 (given in Table 6). In this case, the observer estimated values of the initiator concentration (and also solvent concentration, not shown in Figure 6) and the zeroth and first moments converge to their actual values very rapidly. The convergence of the estimated value of the second moment to its actual value is, however, very slow; the rate of the convergence is set by the process itself. Recall that the second moment of the MWD is unobservable from the Mn measurement. As in Figure 5, in this case, there is a small mismatch between the EKF-estimated and the actual values of the observable process state variables (which is clearly evident between t ) 20 min and t ) 30 min, which is not seen in the case of the nonlinear observer. Note that again the mismatch between actual and estimated initiator concentration in this case is present only for IC2 (not for IC1); this further substantiates the argument brought out in the analysis of Figure 4. Figure 7 compares the nonlinear-observer and EKF estimated values of the initiator concentration and the three leading moments, for two different observer gain values (given in Table 4) and for two different EKF gain values (given in Table 6). The EKF and observer initial estimated values are set to those of IC1 (given in Table 5). It can be seen that, in the case of the closed-loop observer (with L ) L1 and L ) L2), the convergence of the initiator concentration and the zeroth and first moments is much faster than those seen in Figure 1. This is because these states are now observable from the measurements, thereby making it possible to adjust their rates of convergence by varying the gain values. Note that in this case the convergence of the zeroth moment is much faster than that observed in Figure 5. As discussed in the previous paragraph, the rate of convergence of the unobservable second moment cannot be adjusted by changing the gain value. In comparison to case II, in case III, the numberaverage molecular weight measurement made the ze-

Figure 6. Actual and estimated values of the state variables calculated for various estimator initial conditions (case III).

roth moment observable, leading to significant improvement in the convergence of the zeroth moment estimate but at the expense of the unobservability of the second moment, resulting in the slower convergence of the estimate of the second moment. 8. Conclusions The problem of nonlinear state estimation in a continuous polymerization reactor was studied for three measurement cases. A deterministic EKF was applied to the same polymerization reactor to compare the performance of the nonlinear observer to that of the EKF. For case I, the global asymptotic stability of the observer-error dynamics was proven; this is the first time that the global convergence of the estimated states including the MWD moments has been proven for a class of free-radical polymerization reactors. The reduced-order nature of the nonlinear observer made this possible. For the observers of cases II and III, it was not possible to prove the global asymptotic stability of the error dynamics due to the complex form of the error dynamics expressions. In these two cases, first the local stability of the error dynamics was ensured and then the global asymptotic stability of the error dynamics was shown through numerical simulations.

Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997 2689 f* ) initiator efficiency kfm ) rate constant for chain-transfer-to-monomer reactions, m3‚kmol-1‚s-1 kfs ) rate constant for chain-transfer-to-solvent reactions, m3‚kmol-1‚s-1 ki ) dissociation rate constant for initiator, s-1 kp ) rate constant for propagation reactions, m3‚kmol-1‚s-1 kt ) rate constant for termination reactions (kt ) ktc + ktd), m3‚kmol-1‚s-1 ktd ) rate constant for termination by disproportionation reactions, m3‚kmol-1‚s-1 ktc ) rate constant for termination by combination reactions, m3‚kmol-1‚s-1 L ) observer gain matrix Mm ) molecular weight of monomer, kg‚kmol-1 Mn ) number-average molecular weight, kg‚kmol-1 Mw ) weight-average molecular weight, kg‚kmol-1 P ) live polymer molar concentration, kmol‚m-3 R ) universal gas constant, kJ‚kmol-1‚K-1 S ) jacket-reactor heat-transfer surface area, m2 T ) reactor temperature, K Tin ) temperature of the reactor inlet stream, K Tj ) jacket temperature, K Twin ) inlet coolant temperature, K t ) time, s U ) overall jacket-reactor heat-transfer coefficient, kJ‚m-2‚s-1‚K-1 V ) reactor volume, m3 Vj ) volume of jacket holdup, m3 Greek Letters -∆HP ) heat of propagation reactions, kJ‚kmol-1 γ ) (-∆HP)/(Fcp) λ0 ) 0th moment of the MWD, kmol‚m-3 λ1 ) 1st moment of the MWD, kg‚m-3 λ2 ) 2nd moment of the MWD, kg2‚kmol-1‚m-3 τ ) reactor residence time (τ ) V/F), s τj ) jacket residence time (τj ) Vj/Fcw), s

Literature Cited Figure 7. Actual and estimated values of the state variables for various gain values (case III).

Unlike the EKF, the nonlinear state observer is designed directly by using the original nonlinear process model without any linear approximation. The design and implementation of the nonlinear observer are easier that those of the EKF. The simulation results showed that in some cases the nonlinear state observer outperforms the EKF for the observable states, and exhibits a mixed performance with respect to the EKF for the cases when the states are unobservable from the available measurements. Acknowledgment Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of this research. Nomenclature Ci ) outlet initiator concentration, kmol‚m-3 Ciin ) inlet initiator concentration, kmol‚m-3 Cm ) outlet monomer concentration, kmol‚m-3 Cmin ) inlet monomer concentration, kmol‚m-3 Cs ) outlet solvent concentration kmol‚m-3 Csin ) inlet solvent concentration, kmol‚m-3 cp ) heat capacity of the reacting mixture, kJ‚kg-1‚K-1 cpj ) heat capacity of the jacket fluid, kJ‚kg-1‚K-1 e ) observer error

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Received for review December 26, 1996 Revised manuscript received April 7, 1997 Accepted April 9, 1997X IE960905E

X Abstract published in Advance ACS Abstracts, June 1, 1997.

Nonlinear State Estimation in a Polymerization Reactor - American

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