parameter estimation in continuous-time dynamic models in the presence of unmeasured states and...

Parameter Estimation in Continuous-Time Dynamic Models in the Presence ofUnmeasured States and Nonstationary Disturbances

M. Saeed Varziri, Kim B. McAuley,* and P. James McLellanDepartment of Chemical Engineering, Queen’s UniVersity, Kingston, Ontario, Canada K7L 3N6

Mathematical models that describe chemical engineering processes are not exact. Therefore, it is importantto develop parameter-estimation algorithms that account for possible model uncertainties. In this article, asa follow-up to earlier work by Poyton et al. (Comput. Chem. Eng.2006, 30, 698) and Varziri et al. (Comput.Chem. Eng.2007), we investigate the performance of an approximate maximum likelihood estimation (AMLE)algorithm for parameter estimation in nonlinear dynamic models with model uncertainties and stochasticdisturbances. We examine the applicability of AMLE to cases in which some of the states are unmeasured,and we demonstrate that AMLE can be employed in models with nonstationary process disturbances. Theoreticalconfidence interval expressions are obtained and are compared to empirical box plots from Monte Carlosimulations. Use of the methodology is illustrated using a continuous stirred tank reactor model.

1. Introduction

Parameter estimation in nonlinear dynamic models is gener-ally treated as a traditional nonlinear least-squares (TNLS)minimization problem where a weighted sum of the squareddeviations of the model responses from the observed data isminimized, subject to the ordinary differential equations (ODEs)that describe the dynamic model.3-6 If the ODE model has ananalytical solution, then, by substituting this solution into theobjective function, the parameter-estimation problem can betransformed into an unconstrained nonlinear minimization prob-lem. Unfortunately, this is rarely the case, because it is usuallynot possible to find analytical solutions for nonlinear ODEmodels that represent dynamic chemical processes. In suchcases, TNLS methods require that the model ODEs, possiblyalong with sensitivity ODEs,7 be solved numerically and repeat-edly for each parameter perturbation. Repeated numerical sol-ution of the ODEs makes the traditional methods computation-ally expensive and also prone to numerical stability problems.8

The aforementioned problems have motivated new parameter-estimation algorithms that do not require repeated numericalsolution of the differential equation models. One method toavoid repeated integration is to fit an empirical curve (e.g.,smoothing splines) to the observed data.1,9-12 Once the empiricalcurve is obtained, it can be differentiated and substituted intothe model differential equations, thereby transforming the ODEsinto algebraic equations. However, these methods often resultin biased parameter estimates since there is no guarantee thatthe empirical curve is consistent with solution of the ODEs.13

Biegler and co-workers8,14 used collocation discretization toapproximate the solution of the ODEs, assuming that the solutionof the ODEs can be expressed as a linear combination of somebasis functions (e.g., polynomials). The constant coefficientsof the (polynomial) basis functions are obtained by requiringthe response trajectories to satisfy the ODEs. Basis-functionrepresentation of the response trajectories transforms the ODEsinto algebraic equations, and hence, the ODE constraints in thenonlinear parameter-estimation problem become algebraic con-straints. The main expense of collocation-based methods is thatthey usually require the solution of a large-scale nonlinearminimization problem to simultaneously select the basis-functioncoefficients and determine the optimal parameter estimates.

Fortunately, advanced optimization algorithms are capable ofcarrying out these kinds of problems quickly and efficiently.15

Traditional nonlinear least-squares and collocation-basedmethods enforce the model equations (in ODE and algebraicform, respectively) as hard constraints, implying that thestructure of the model is perfect. More often than not, however,mathematical models describing chemical processes are onlyapproximately true, and discrepancies between the model and

* To whom correspondence should be addressed. E-mail:[email protected]. Phone:+1 (613) 533 2768. Fax:+1

(613) 533 6637.

Figure 1. Input scheme for MIMO nonlinear CSTR.

Figure 2. Box plots fora using TNLS and AMLE.

380 Ind. Eng. Chem. Res.2008,47, 380-393

10.1021/ie070824q CCC: $40.75 © 2008 American Chemical SocietyPublished on Web 12/15/2007

the physical reality should not be neglected. When it is notappropriate to assume that the model structure is perfect,implementing the model equations as hard constraints in theparameter-estimation problem results in biased or inconsistentparameter estimates.

Poyton et al.1 showed that iterative principal differentialanalysis (iPDA) is an effective method for parameter estimation

in ODE models. In iPDA, basis functions (i.e., B-splines) areused to discretize the model ODEs, transforming them intoalgebraic equations. Unlike the previous collocation-based

Figure 3. Box plots forb using TNLS and AMLE.

Figure 4. Box plots forE/R using TNLS and AMLE.

Figure 5. Box plots forkref using TNLS and AMLE.

Figure 6. Observed, true, and predicted concentration response for AMLE.(b simulated data;- - response with true parameters and true stochasticnoise;s AMLE response).

Figure 7. Observed, true, and predicted concentration response for TNLS.(b simulated data;- - response with true parameters and true stochasticnoise,s TNLS response).

Figure 8. Observed, true, and predicted temperature response for AMLE.(b simulated data;- - response with true parameters and true stochasticnoise;s AMLE response).

Ind. Eng. Chem. Res., Vol. 47, No. 2, 2008381

methods, Poyton’s technique treats the model equations as softconstraints, allowing for possible model imperfections. TheiPDA objective function was originally minimized in an iterativeway, with each iteration consisting of two steps. OptimalB-spline coefficients are obtained in the first step, given themost recent parameter estimates; in the second step, fundamentalmodel parameters are estimated using the B-spline coefficientsobtained from the first step. The iterations continue until theparameter estimates converge.

Recently, Ramsay et al.16 proposed another spline-basedmethod that also accounts for possible model uncertainties. Inthis profile-based method, the dimensionality of the parameter-estimation problem is reduced because the spline coefficientsare treated as functions of the fundamental model parameters.Ramsay’s method uses a two-level optimization scheme whereinthe outer (primary) optimizer determines optimal values offundamental model parameters to minimize the sum of squaredprediction errors. The inner (secondary) optimizer selects splinecoefficients to minimize a weighted sum of the squaredprediction errors and a model-based penalty using the parameterestimates from the outer optimization loop. In the present article,we focus on Poyton’s iPDA objective function since it has adirect relationship to the likelihood function of stochasticdifferential equation models that are appropriate for describingchemical processes.

We have shown that minimizing the iPDA objective functionis equivalent to maximizing the log-likelihood of the conditionaljoint density function of the states and measurements, given

the fundamental model parameters, when model mismatchresults from additive stochastic white-noise disturbance inputs.2

We have also shown that fundamental model parameters andB-spline coefficients can be minimized simultaneously. Since

Figure 9. Observed, true, and predicted temperature response for TNLS.(b simulated data;- - response with true parameters and true stochasticnoise,s TNLA response).

Table 1. 95% Confidence Intervals for AMLE Parameter Estimates

parameterestimates lower bound upper bound

a 1.4501 2.6322b 0.3307 0.5400E/R 8.3152 8.7592kref 0.4575 0.4866

Table 2. 95% Confidence Intervals for AMLE Parameter Estimates(Unmeasured Concentration)


a 1.0750 2.4559b 0.3062 0.5765E/R 8.0477 8.9000kref 0.3932 0.4577

Figure 10. Box plots for a using TNLS and AMLE (unmeasuredconcentration).

Figure 11. Box plots for b using TNLS and AMLE (unmeasuredconcentration).

Figure 12. Box plots for E/R using TNLS and AMLE (unmeasuredconcentration).

382 Ind. Eng. Chem. Res., Vol. 47, No. 2, 2008

we no longer solve the optimization problem using the two-step iterative method, and because a likelihood criterion isapproximated using B-splines, we call the technique approximatemaximum likelihood estimation (AMLE) throughout the re-mainder of this article.

The contributions in the current article are as follows. Wedemonstrate that AMLE can be readily used in parameterestimation in cases in which some of the states are not observed.We also show that AMLE can estimate the unmeasured statesalong with fundamental model parameters. This capabilitynaturally leads to the application of AMLE to parameterestimation in dynamic models driven by nonstationary processdisturbances, because nonstationary disturbances can be con-sidered to be unmeasured states.17 Another important contribu-tion is the development of theoretical confidence-intervalexpressions for parameter estimates and B-spline coefficients.The application of the AMLE technique is examined using amulti-input multioutput (MIMO) nonlinear continuous stirredtank reactor (CSTR) model. In the CSTR case study, theoreticalconfidence intervals are compared to confidence intervals fromMonte Carlo simulations to confirm the theoretical results.

The manuscript is organized as follows. First, we brieflyreview the AMLE algorithm using a multivariate nonlineardynamic model. Detailed information about the algorithm and

its mathematical basis can be found in papers by Poyton et al.1

and Varziri et al.2 We then show how AMLE can be used forcases in which unmeasured states or nonstationary disturbancesare present, and we obtain theoretical confidence-intervalexpressions for the fundamental model parameters. Next, weuse a nonlinear CSTR example to study the effectiveness ofAMLE in several different scenarios: (i) when all states aremeasured, (ii) when temperature is measured, but the concentra-tion is not, and (iii) when a nonstationary disturbance entersthe material-balance differential equation and both states aremeasured. Finally, we highlight some of the remaining chal-lenges that need to be addressed to make AMLE more applicablefor parameter estimation in dynamic chemical process models.

1.1. Review of the AMLE Algorithm. Varziri et al.2 derivedthe AMLE objective function using a Bayesian argument for amulti-input multioutput first-order nonlinear dynamic system.Here, for simplicity, we use a first-order model with two inputsand two states to review the AMLE algorithm:

x1andx2 are state variables,u1 andu2 are input variables, andy1 andy2 are output variables.f1 andf2 are suitably well-behavednonlinear functions (Lipschitz continuous) andθ is the vectorof fundamental model parameters.η1(t) andη2(t) are indepen-dent continuous zero-mean stationary white-noise processes withintensitiesQp1 andQp2, respectively;E{η1(t)η1(t - τ)} ) Qp1δ-(τ), andE{η2(t)η2(t - τ)} ) Qp2δ(τ) whereδ(.) is the Diracdelta function.ε1 and ε2 are zero-mean independent normalrandom variables with variancesσm1

2 and σm22, respectively.

tm1j and tm2j are the time points at which outputsy1 andy2 aremeasured. We assume that there areN1 and N2 measurementtimes fory1 andy2, respectively. The model uncertainty, whichis represented byη1(t) and η2(t), could be due to physicalphenomena that have not properly been taken into account inthe model, simplifications and approximations during modeldevelopment, or uncertainties in the input variables. Note thatthis approach addresses some of the concerns that could alsobe addressed using the errors-in-variables methods.3 When themodel disturbance is propagated through the dynamic system,this noise influences the states and their future values.Qp1 andQp2 are intensities of continuous Gaussian noise processesη1-(t) andη2(t); in other words, they represent the energies of thesedisturbances. The intensity of a continuous Gaussian noiseprocess is closely related to the variance of the analogousdiscrete Gaussian noise sequence. If the continuous noisesequence is sampled at every∆t min, then the variance of thediscrete-time noise process becomes20

wherej1 and j2 are integers.In AMLE, the system states trajectories are approximated

using linear combinations of B-spline basis functions.18,19

Figure 13. Box plots for kref using TNLS and AMLE (unmeasuredconcentration).

Figure 14. Observed, true, and predicted concentration response for AMLE(unmeasured concentration). (- - response with true parameters and truestochastic noise;s AMLE response).

{x1(t) ) f1(x1(t),x2(t),u1(t),u2(t),θ) + η1(t)x2(t) ) f2(x1(t),x2(t),u1(t),u2(t),θ) + η2(t)x1(0) ) x10

x2(0) ) x20

y1(tm1j) ) x1(tm1j) + ε1(tm1j)y2(tm2j) ) x2(tm2j) + ε2(tm2j)

} (1)

E{η(j1∆t)η(j2∆t)} ) {σp2 ) Q

∆tj1 ) j2

0 j1 * j2}


whereâkι, i ) 1, ..., ck, andφkι(t), i ) 1, ..., ck, are B-splinecoefficients and B-spline basis functions, respectively, for thekth state. The knot sequences for the B-spline basis functionsshould be selected so that they provide enough flexibility forthe B-spline curves to follow the features of the state trajectories.However, caution should be exercised since using a knotsequence that is too fine will significantly slow down thealgorithm. We have found that, if the sampling interval is nottoo large, placing one knot at each observation point and oneor two knots in between every two observation points leads tosatisfactory parameter-estimation results, without excessivecomputation. If the knot sequence is rich enough to capture theimportant features of the state trajectories, not much is gainedby further refining the grid. Please refer to Poyton et al.1 forfurther discussions on knot-sequence strategies. Equation 2 canbe written in matrix form

whereæk(t) is a vector containing theck basis functions andâk

is vector ofck spline coefficients. Note that the “∼” subscriptis used to imply an empirical curve that can be easilydifferentiated:

In AMLE, the spline coefficientsâk and the vector offundamental model parametersθ are obtained so that thefollowing objective function is minimized:

We will refer to the first two terms∑(yk(tmj) - x∼k(tmj))2 in theobjective function as SSE (the sum of squared prediction errors),while PEN (the model-based penalty) will be used to refer tothe third and fourth terms∫(x∼k(t) - fk(x∼1(t),x∼2(t),u1(t),u2-(t),θ))2 dt. The SSE and PEN terms in eq 5 are weighted by thereciprocals of the measurement noise variances and processdisturbance intensities, respectively. In its original form, AMLE(which was called iPDA by Poyton et al.1) left the values ofthese weighting coefficients unspecified as tuning parametersthat could be adjusted by trial and error, and the tuningrequirement was, therefore, listed as a disadvantage of themethodology. Recently, we used a Bayesian argument to showthat, in the dynamic model described in eq 1, the optimalweighting coefficients are the reciprocals of the measurement

noise variances for the SSE terms and the reciprocals of theprocess disturbance intensities for the PEN terms, and hence,the objective function in eq 5 was derived.2 Note that selectingthe optimal weighting factors requires knowledge of measure-ment variances and process disturbance intensities. In real-worldapplications, the true values of these parameters are not knownand estimates must be obtained. Measurement noise variancescan be estimated from replicate measurements; however,estimating the process disturbance intensities is a difficult taskrequiring expert knowledge of the dynamic system and thecorresponding mathematical model. Therefore, efficient algo-rithms for estimating the unknown variances and intensitiesQp1

andQp2 need to be developed, which is a subject of our ongoingresearch.

Another shortcoming highlighted in our previous publicationson AMLE was the lack of a systematic method to assess thevariability of the parameter estimates (and the B-spline coef-ficients). We address this problem in Section 3 of the currentarticle, where we present the confidence interval results. Webriefly review the major advantages of AMLE:

• AMLE provides an easy-to-implement method for estimat-ing fundamental model parameters and system states in dynamicmodels in which stochastic process disturbances are present.

• The AMLE objective considers model uncertainties andmeasurement noise at the same time, and the trade-off betweenpoor measurements and model imperfections is addressed bythe optimal choice of weighting coefficients.

• AMLE uses B-spline basis functions to discretize thedifferential equations in the dynamic model, thereby transform-ing them into algebraic equations. Hence, AMLE circumventsthe repeated numerical integration used by traditional methods.

• The discretization approach used in AMLE removes therequirement of deriving and integrating the sensitivity dif-ferential equations used by conventional methods.

• AMLE inherits and combines the advantages of Kalmanfilters for state and parameter estimation in stochastic dynamicmodels and the benefits of the collocation-based methods forstate and parameter estimation in deterministic dynamic models.

In addition to these advantages, we will demonstrate in thenext section that AMLE can readily be applied to problems withunmeasured states and nonstationary disturbances, which areimportant in chemical processes.

2. Unmeasured States and Nonstationary Disturbances

Handling unmeasured states in AMLE is straightforward.There is no SSE term for the unmeasured state in the objectivefunction, because the number of measurements in the corre-sponding summation is zero. For instance, ifx1 is not measured,the new objective function is as follows:

When a state is measured, there are two sources of informa-tion (i.e., the measured data and the corresponding differentialequation) that can be used to estimate the states and eventuallythe model parameters. However, with an unmeasured state, weare only left with the differential equation, and hence, the

x∼k(t) ) ∑i)1

ck

âkιφkι k ) 1, 2 (2)

x∼k(t) ) ækT(t)âk k ) 1, 2 (3)

x∼k(t) )d

dt(∑

i)1

ck

âkιφkι(t)) )

∑i)1

ck

âkιφkι(t) ) ækTâk k ) 1, 2 (4)

1

σm12∑j)1

N1

(y1(tm1j) - x∼1(tm1j))2 +

1

σm22∑j)1

N2

(y2(tm2j) - x∼2(tm2j))2 +

1

Qp1

∫(x∼1(t) - f1(x∼1(t),x∼2(t),u1(t),u2(t),θ))2 dt +

1

Qp2

∫(x∼2(t) - f2(x∼1(t),x∼2(t),u1(t),u2(t),θ))2 dt (5)

1

σm22∑j)1

N2

(y2(tm2j) - x∼2(tm2j))2 +

1

Qp1

∫(x∼1(t) - f1(x∼1(t),x∼2(t),u1(t),u2(t),θ))2 dt +

1

Qp2

∫(x∼2(t) - f2(x∼1(t),x∼2(t),u1(t),u2(t),θ))2 dt (6)


B-spline curve should be flexible enough to follow the solutionof the differential equation quite closely. More flexibility canbe achieved by placing more knots within the time frame overwhich the integrals in eq 6 are evaluated. Parameter estimationin a nonlinear MIMO CSTR with an unmeasured state is studiedin Section 4.2 as an example.

So far, we have considered dynamic models with stationaryGaussian process disturbances. Dynamic models with nonsta-tionary disturbances can be handled by considering thesedisturbances as unmeasured states. To accommodate suchproblems, we consider the model in eq 7 below, which is aslightly modified version of the model in eq 1. We have addeda nonstationary disturbance (d1(t)) to the right-hand side of thefirst differential equation. Please note that it is possible to adda nonstationary disturbance to any of the differential equationsin the model.

In eq 7,d1(t) is a nonstationary disturbance driven byηd1(t),which is a continuous zero-mean stationary white-noise processwith intensityQpd1. fd1(.) is a function defining the nonstationarynoise model, andθd1 is the vector of noise model parameters.20

If θd1 is not known (which generally is the case), it can beestimated along with the vector of fundamental model param-etersθ.

By treatingd1(t) like any other unmeasured state, assumingthatx1 andx2 are measured, the AMLE objective function canbe written as

where d∼1(t) ) φd1

T (t)âd1. Note that eq 8 is minimized withrespect toθ, θd1, â, andâd1. As an example, parameter estimationin a nonlinear MIMO CSTR with a nonstationary disturbanceis studied in Section 4.3.

3. Theoretical Confidence IntervalsVarziri et al.2 considered the model in eq 1 and showed that

minimizing the AMLE objective function is equivalent tomaximizing the likelihood of the conditional joint densityfunction of states and measured output values given the model

parameter values,p(x,ym|θ) , wherex is the vector of statesandym is the vector of measured outputs (this likelihood functionis recommended by Maybeck21 as the most appropriate for

{x1(t) ) f1(x1(t),x2(t),u1(t),u2(t),θ) + d1(t) + η1(t)x2(t) ) f2(x1(t),x2(t),u1(t),u2(t),θ) + η2(t)d1(t) ) fd1(d1(t),θd1

) + ηd1(t)

x1(0) ) x10

x2(0) ) x20

d1(0) ) 0y1(tm1j) ) x1(tm1j) + ε1(tm1j)y2(tm2j) ) x2(tm2j) + ε2(tm2j)

} (7)

1

σm12∑j)1

N1

(y1(tm1j) - x∼1(tm1j))2 +

1

σm22∑j)1

N2

(y2(tm2j) - x∼2(tm2j))2 +

1

Qp1

∫(x∼1(t) - f1(x∼1(t),x∼2(t),d∼1(t),u1(t),u2(t),θ))2 dt +

1

Qp2

∫(x∼2(t) - f2(x∼1(t),x∼2(t),u1(t),u2(t),θ))2 dt +

1

Qd1

∫(d∼1(t) - fd1(d∼1(t),θd1

))2 dt (8)

Figure 15. Observed, true, and predicted concentration response for TNLS(unmeasured concentration). (- - response with true parameters and truestochastic noise;s AMLE response).

Figure 16. Observed, true, and predicted temperature response for AMLE(unmeasured concentration). (b simulated data;- - response with trueparameters and true stochastic noise;s AMLE response).

Figure 17. Observed, true, and predicted temperature response for TNLS(unmeasured concentration). (b simulated data;- - response with trueparameters and true stochastic noise;s AMLE response).


combined state and parameter estimation). Therefore, in AMLE,we maximize the log-likelihood ofp(x,ym|θ), denoted byL(x,θ,ym):

Analogous objective functions have been used in nonlinearfiltering problems.22-24 Under general regularity conditions,parameter estimates obtained from the minimization problemin eq 10 are consistent, asymptotically unbiased, asymptoticallynormally distributed, and asymptotically efficient.21,25,26 Toobtain confidence intervals for the model parameters (andB-spline coefficients), we exploit the properties of the maximumlikelihood estimator.

Assuming thatx can be represented using B-splines as in eq3, the parameter-estimation problem reduces to

By stacking the fundamental model parameters and splinecoefficients in a vectorτ ) [θT,âT]T the problem becomes

We assume thatL satisfies the required regularity conditions(e.g., Kay,26 Appendix 7B, p 212). Then, from the asymptoticproperties of maximum likelihood estimators (e.g., Kay,26

Theorem 7.3, p 183) we have

where I (τ) is the Fisher information matrix3 evaluated at thetrue values ofτ:

In the examples of Section 4, we have approximatedI (τ) by eqA5 in Appendix A evaluated atτ.

Approximate 100(1- R)% confidence intervals for the modelparameters and spline coefficients can be obtained as follows:

The confidence intervals obtained from eq 15 are approximatebecause

(1) The model is nonlinear with respect to the parametersand spline coefficients.

(2) The maximum-likelihood estimates are only asymptoti-cally normal.

(3) The state trajectories are approximated by B-splines.(4) The true parameter values are not known; therefore, the

approximate Fisher information matrix is evaluated at theestimated parameter values rather than true parameter valuesand at the observed measurements rather than taking expectedvalues over the joint distribution of the observations.

Note also that we have used the standard normal randomdeviate in eq 15, rather than a value from the student’stdistribution, because we are assuming that the noise varianceis known.

4. Case Study

In this section, we examine our results using a MIMOnonlinear CSTR model.27 In Section 4.1, we estimate fourparameters in the CSTR model, assuming that both temperatureand concentration can be measured. In Section 4.2, we assume

Figure 18. Box plots fora using TNLS and AMLE (with nonstationarydisturbance).

Figure 19. Box plots forb using TNLS and AMLE (with nonstationarydisturbance).

Figure 20. Box plots forE/Rusing TNLS and AMLE (with nonstationarydisturbance).

L(x,θ,ym) ) ln(p(x,ym|θ)) (9)

θ, x ) arg minθ,x

{-L(x,θ,ym)} (10)

minθ,â

{-L(â,θ,ym)} (11)

minτ

{-L(τ,ym)} (12)

τ ≈ N(τ,I -1(τ)) (13)

I (τ) ) -E{∂2L(τ,ym)

∂τ2 } ) E{∂L(τ,ym)T

∂τ∂L(τ,ym)

∂τ } (14)

τ ) τ ( zR/2 × xdiag(I-1(τ)) (15)


that only temperature can be measured and concentration isunmeasured. In Section 4.3, we consider the CSTR where bothstates are measured and nonstationary disturbances are presentin the system.

We use the simulations to compare parameter-estimationresults obtained using the proposed AMLE algorithm and TNLS.In all case studies, the Simulink toolbox of MATLAB was usedto solve the nonlinear dynamic models (using the ode45 solver)and to generate noisy measurements. TNLS parameter estimateswere obtained using the lsqnonlin optimizer (default solveroption), which employs a subspace trust region method and isbased on the interior-reflective Newton method. Sensitivityequations were solved along with the model differential equa-tions.

The AMLE objective function, in all examples, was optimizedsimultaneously over the fundamental model parameters andB-spline coefficients using the IP-OPT solver15 via AMPL.Initially, we tried to use lsqnonlin of MATLAB for minimizingthe AMLE objective function; although we had satisfactoryconvergence, the computation time was prohibitively highbecause of the large dimension of the combined vector of modelparameters and B-spline coefficients. Hence, we switched toIP-OPT, which converged very quickly so that the computationtime was not an issue. We note again that AMLE, andcollocation-based methods in general, circumvents potentialproblems associated with numerical integration at the expenseof solving a large nonlinear programming problem; hence, fastand efficient nonlinear programming solvers such as IP-OPT15

are essential for the successful implementation of these algo-rithms, especially for larger problems.

4.1. Nonlinear MIMO CSTR with Measured Temperatureand Concentration. We consider the following model thatrepresents a MIMO nonlinear CSTR. The model equationsconsist of material and energy balances27 with additionalstochastic disturbance terms:

whereε1(tmj), j ) 1, ...,N1, andε2(tmj), j ) 1, ...,N2, are white-noise sequences with variancesσm1

2 andσm22, respectively. We

also assume thatη1, η2, ε1, and ε2 are independent. Thisstochastic differential equation model is nonlinear in the states(CA, T) and parameters and does not have an analytical solution.

CA is the concentration of reactant A,T is the reactortemperature,V is the volume, andTref ) 350 K is a referencetemperature. The true values of the parameters to be estimatedare as follows:E/R ) 8330.1 K,kref ) 0.461 min-1, a ) 1.678

Figure 21. Box plots forkref using TNLS and AMLE (with nonstationarydisturbance).

Figure 22. Observed, true, and predicted concentration response for AMLE(with nonstationary disturbance). (b simulated data;- - response withtrue parameters and true stochastic noise;s AMLE response).

Figure 23. Observed, true, and predicted concentration response for TNLS(with nonstationary disturbance). (b simulated data;- - response withtrue parameters and true stochastic noise,s TNLS response).

dCA(t)

dt)

F(t)V

(CA0(t) - CA(t) - gCA(t) + η1(t))

dT(t)dt

)F(t)V

(T0(t) - T(t)) + â1(T(t) - Tcin(t)) -

â2gCA(t) + η2(t)

CA(0) ) 1.569 (kmol m-3)

T(0) ) 341.37 (K) (16)

y1(ti) ) CA(ti) + ε1(ti)

y2(tj) ) CA(tj) + ε2(tj)

g ) kref exp(- ER (1

T- 1

Tref)),

â1 ) -aFc

b+1(t)

VFCp(Fc(t) +aFc

b(t)

2FcCpc),

â2 )(-∆Hrxn)

FCp


× 106, andb ) 0.5. The initial parameter guesses were set at50% of the true parameter values. Parametersa andb accountfor the effect of the coolant flow rateFc on the heat-transfercoefficient. This nonlinear system has five inputs: the reactantflow rate F, the inlet reactant concentrationCA0, the inlettemperatureT0, the coolant inlet temperatureTcin, and the coolantflow rate Fc. Values for the various other known constants27

are as follows:V ) 1.0 m3, Cp ) 1 cal g-1 K-1, F ) 1 × 106

g m-3, Cpc ) 1 cal g-1 K-1, Fc ) 1 × 106 g m-3, and-∆Hrxn

) 130× 106 cal kmol-1. The initial steady-state operating pointis CAs ) 1.569 kmol m-3 andTs ) 341.37 K.

In this example, there is no temperature controller, andperturbations are introduced into each of the five inputs usingthe input scheme shown in Figure 1.28 Each input consists of astep up, followed by a step down, and then a step back to thesteady-state point.

We assume that concentration and temperature are measuredand initial values are known. Temperature is measured onceevery 0.3 min, while concentration is measured once per minute.The duration of the simulated experiment is 64 min, so thatthere are 213 temperature measurements and 64 concentrationmeasurements. The noise variance for the concentration and

temperature measurements areσm12 ) 4 × 10-4 (kmol/m3)2 and

σm22 ) 6.4× 10-1 K2, respectively. The corresponding process

noise intensities for the stochastic disturbances are assumed tobeQp1 ) 4 × 10-3 (kmol/m3)2/min andQp2 ) 4 K2/min. Fromeq 5, the AMLE objective function is

For the temperature trajectory, B-spline knots were placedat the observation times (one knot at every 0.3 min), and forthe concentration trajectory, we placed one knot at every 0.2min. For both the concentration and temperature trajectories,five Gaussian quadrature points were used between every twoknots to numerically calculate the integrals in eq 17.

We repeated this parameter-estimation problem using 100different sets of noisy observations to study the samplingproperties of the parameter estimates. The Monte Carlo box plotsfor AMLE and TNLS parameter estimates are shown in Figures2-5. The AMLE and TNLS predicted responses along withthe true responses are shown in Figures 6-9. On average, theAMLE parameter estimates are better than the TNLS parameterestimates (more precise and less biased), and the AMLEresponse trajectories are closer to the true trajectories than arethe TNLS trajectories.

Please note that maximum likelihood parameter estimates areonly asymptotically normally distributed and asymptoticallyunbiased, which suggests that, for small data sets, the parameterestimates may tend to be somewhat biased.

Table 1 presents 95% confidence intervals (derived in Section3) for this case study. Please note that the confidence intervalsshown correspond to one particular data set chosen randomly

Figure 24. Observed, true, and predicted temperature response for AMLE(with nonstationary disturbance). (b simulated data;- - response withtrue parameters and true stochastic noise;s AMLE response).

Figure 25. Observed, true, and predicted temperature response for TNLS(with nonstationary disturbance). (b simulated data;- - response withtrue parameters and true stochastic noise,s TNLS response).

Figure 26. True and estimated nonstationary disturbance using AMLE.(- - nonstationary stochastic disturbance;s AMLE estimate).

1

σm12∑j)1

64

(y1(tm1j) - CA∼(tm1j))2 +

1

σm22∑j)1

213

(y2(tm2j) -

T∼(tm2j))2 +

1

Qp1

∫t)0

64 (dCA∼(t)

dt-

F(t)

V(CA0(t) -

CA∼(t)) + gCA∼(t))2

dt +1

Qp2

∫t)0

64 (dT∼(t)

dt-

F(t)

V(T0(t) - T∼(t)) - â1(T∼(t) -

Tcin(t)) + â2gCA∼(t))2

dt (17)


from the 100 data sets used to generate the box plots, and theapproximate Fisher information matrix is evaluated using theseparameter estimates. By inspecting the Monte Carlo box plotsand the scatter plot matrix given in Appendix B and comparingthem to the theoretical results in Table 2, we see that theempirical and theoretical confidence intervals agree.

4.2. Nonlinear MIMO CSTR with Unmeasured Concen-tration. In this example, we consider the same CSTR modelas in the previous section. The only difference is that concentra-tion is unmeasured.

From eq 6, the appropriate AMLE objective function is

becauseN1 ) 0. We used 100 sets of noisy observations tostudy the sampling properties of the parameter estimates. TheMonte Carlo box plots are shown in Figures 10-13. The AMLEand TNLS predicted responses are compared against the trueresponses in Figures 14-17.

From Figures 10-17, we observe that, on average, AMLEparameter estimates are better than those of TNLS, and theAMLE response trajectories are closer to the true trajectoriesthan are the TNLS trajectories. We also observe that, sinceconcentration is not measured, both TNLS and AMLE parameterestimates are not as good as those in the previous section whenboth states were measured. Theoretical 95% confidence intervalsfor AMLE parameter estimates are presented in Table 2.

Note that removing the concentration measurements resultedin slightly wider confidence intervals. However, since thetemperature measurements are more frequent and more precisethan the concentration measurements, removing the concentra-tion measurements did not worsen the confidence intervals asdramatically as removing the temperature measurements does(not shown). A scatter plot matrix of the parameter estimatesfor this case study is presented in Appendix B.

4.3. Nonlinear MIMO CSTR with Nonstationary Distur-bance. In the third case study, we consider the same CSTRmodel as in previous sections but with an additional nonsta-tionary disturbance affecting the concentration differentialequation. This nonstationary disturbance could be used toaccount for a meandering input that is not included in thefundamental part of the material balance equation. We assumethat both concentration and temperature are measured, and weestimate the vector of fundamental model parameters along withthe nonstationary disturbance:

E{η3(t)η3(t - τ)} ) Qp3δ(τ) whereQp3 ) 6 × 10-2 (kmol/m3/min)2 and everything else remains the same as in Section 4.1.

From eq 8, the AMLE objective function is

The input scheme and the knot placements for the temperatureand concentration trajectories are the same as for the previousexamples. For the nonstationary disturbance trajectory,d1∼,knots were placed at 0.3 min intervals.

Figures 18-21 show the Monte Carlo box plots for theparameter estimates. Estimated concentration and temperaturetrajectories are presented (for one data set) in Figures 22-25.

As expected, the advantage of AMLE over TNLS is morepronounced when there are nonstationary disturbance inputs,which are not accounted for using TNLS. The AMLE parameterestimates are less biased and more precise than TNLS parameterestimates. AMLE does a significantly better job at estimating(smoothing) the concentration and temperature trajectories andcaptures the sharp and meandering features in the response thatappear due to the stationary and nonstationary disturbance terms

Table 3. 95% Confidence Intervals for AMLE Parameter Estimates(with Nonstationary Disturbance)


a 0.6019 2.0615b 0.3337 0.7259E/R 7.8400 8.8890kref 0.3924 0.4549

1

σm22∑j)1

213

(y2(tm2j) - T∼(tm2j))2 +

1

Qp1

∫t)0

64 (dCA∼(t)

dt-

F(t)

V(CA0(t) - CA∼(t)) + gCA∼(t))2

dt +

1

Qp2

∫t)0

64 (dT∼(t)

dt-

F(t)

V(T0(t) - T∼(t)) -

â1(T∼(t) - Tcin(t)) + â2gCA∼(t))2

dt (18)

dCA(t)

dt)

F(t)V

(CA0(t) - CA(t)) - gCA(t) + d1(t) + η1(t)

dT(t)dt

)F(t)V

(T0(t) - T(t)) + â1(T(t) - Tcin(t)) -

â2gCA(t) + η2(t)

dd1(t)

dt) η3(t)

CA(0) ) 1.569 (kmol m-3)

T(0) ) 341.37 (K) (19)

d1(0) ) 0 (kmol m-3 min-1)

y1(ti) ) CA(ti) + ε1(ti)

y2(tj) ) CA(tj) + ε2(tj)

g ) kref exp(- ER (1

T- 1

Tref)),

â1 ) -aFc

b+1(t)

VFCp(Fc(t) +aFc

b(t)

2FcCpc),

â2 )(-∆Hrxn)

FCp

1

σm12∑j)1

64

(y1(tm1j) - CA∼(tm1j))2 +

1

σm22∑j)1

213

(y2(tm2j) -

T∼(tm2j))2 +

1

Qp1

∫t)0

64 (dCA∼(t)

dt-

F(t)

V(CA0(t) -

CA∼(t)) + gCA∼(t) - d1∼(t))2

dt +1

Qp2

∫t)0

64 (dT∼(t)

dt-

F(t)

V(T0(t) - T∼(t)) - â1(T∼(t) - Tcin(t)) +

â2gCA∼(t))2

dt +1

Qp3

∫t)0

64(d1∼(t))2 dt (20)


in the model. The estimated disturbance in one of the data setsis illustrated in Figure 26.

Theoretical confidence intervals for the AMLE parameterestimates for the second example are presented in Table 3. Asexpected, these confidence intervals are wider than when therewas no meandering disturbance (Table 1). A scatter plot matrixfor the parameter estimates is presented in Appendix B.

To compare the quality of the state estimates in all of thethree examples in Sections 4.1, 4.2, and 4.3 using AMLE andTNLS, the root-mean-square errors (RMSEs) of the estimatedresponses are reported in Table 4.

Overall, the RMSE values of the AMLE state estimates aresmaller than those of the TNLS state estimates. This advantageis more pronounced in the third case, where the nonstationarydisturbance is present.

5. Summary and Conclusions

We show that AMLE can be readily used for parameter andstate estimation in nonlinear dynamic models in which stochasticprocess disturbances and measurement noise are present andsome states are not observed. We also show that the AMLEcan be modified to accommodate unmeasured states by simplyremoving the corresponding sum-of-squared-error terms fromthe objective function. AMLE can handle parameter estimationin nonlinear dynamic systems in which nonstationary distur-bances are present by treating these disturbances as unmeasuredstates. We develop theoretical confidence interval expressionsfor AMLE parameter estimates. The inference method is basedon the asymptotic properties of maximum-likelihood estimators.

We test our results using a MIMO nonlinear CSTR model.In the first scenario, both states are measured. In the second,the temperature is measured, but the concentration is unmea-sured. We estimate the concentration and temperature trajec-tories, along with four fundamental-model parameters, usingboth AMLE and TNLS. The parameter-estimation results fromAMLE are, on average, more precise and less biased than theTNLS results because AMLE is able to properly account forthe process disturbances and measurement noise.

We also consider an additive nonstationary input disturbance(Brownian motion) in the material balance equation. AgainAMLE and TNLS methods are compared and, in the case ofAMLE, the nonstationary disturbance is estimated. The AMLEparameter estimates are more precise and less biased than TNLSparameter estimates. AMLE obtains significantly better estimatesof the state trajectories. Confidence intervals for AMLEparameter estimates are obtained and compared with empiricalbox plots generated by Monte Carlo simulations. Confidenceintervals and box plots are consistent.

Application of AMLE relies on knowledge of measurementnoise variances and stochastic process intensities. These con-stants, however, are usually unknown in real-world applications.Before AMLE can enjoy widespread use, a means for estimatingthe noise and process disturbance constants, along with the

model parameters, needs to be developed. This is the subjectof our ongoing research.

Acknowledgment

The authors thank MITACS, Cybernetica, DuPont Engineer-ing Research and Development, and SAS for financialsupport and Drs. L. T. Biegler from Carnegie Mellon Universityand J. O. Ramsay from McGill University for technicaladvice.

Appendix A

Derivation of the Hessian for the AMLE ObjectiveFunction. In this appendix, we show how the exact orapproximate Hessian matrix for the AMLE objective functioncan be derived by exploiting the least-squares structure of theobjective function.

Rewriting eq 5 in the matrix-vector form where we replacethe integrals by Gaussian quadrature sums, we have

whereyi and xi, respectively, are vectors of outputsyi(t) andstatesx∼i(t) evaluated at the observation points.x∼qi, x3 ∼qi, f i,and ui are vectors containingx∼i(t), x∼i(t), fi(t), and ui(t),respectively, evaluated at Gaussian quadrature points.w1 andw2 are diagonal matrices containing the Gaussian quadratureweights. Equation A1 can further be simplified by introducingthe following vectors and matrices:

Note thatæ1 andæq1 are concatenated vectors ofæ1(t) evaluatedat observation times and Gaussian quadrature points, respec-tively. Analogous definitions apply toæ2 andæq2.

We letΣ1 andΣ2 beN1 × N1 andN2 × N2 diagonal matriceswith σm1

2 and σm22, respectively, on the diagonals. Then, the

AMLE objective function in eq A1 can be written in thefollowing form,

where

andW is a diagonal matrix with

Table 4. Root-Mean-Square Error of the State Estimates

AMLE TNLS

case 1: concentration andtemperature measured

RMSE_C (kmol/m3)RMSE_T (K)

0.02390.6601

0.03121.1506

case 2: only temperaturemeasured

RMSE_C (kmol/m3)RMSE_T (K)

0.04680.6714

0.04621.1232

case 3: concentration andtemperature measured;nonstationary disturbancepresent

RMSE_C (kmol/m3)RMSE_T (K)RMSE_D

(kmol/m3/min)

0.02580.63170.0297

0.17722.4671NA

1

σm12

(y1 - x∼1)T(y1 - x∼1) +

1

σm22

(y2 - x∼2)T(y2 - x∼2) +

1Qpl

(x3 ∼q1 - f1(x∼q1,x∼q2,u1,u2,θ))Tw1(x3 ∼q1 -

f1(x∼q1,x∼q2,u1,u2,θ)) + 1Qp2

(x3 ∼q2 -

f2(x∼q1,x∼q2,u1,u2,θ))Tw2(x3 ∼q2 -f2(x∼q1,x∼q2,u1,u2,θ)) (A1)

x∼ ) [x∼1

x∼2] ) [æ1â1

æ2â2], x∼q ) [x∼q1

x∼q2] ) [æq1â1

æq2â2],f ) [f1

f2], u ) [u1

u2], ym ) [y1

y2], â ) [â1

â2],æ ) [æ1

æ2], andæq ) [æq1

æq2]

L(τ,ym) ) -g(τ,ym)TWg(τ,ym) (A2)

g(τ,ym) ) [ym - Oâφqâ - f(θ,â) ] (A3)


Since, according to eq A2, the objective function is in a least-squares form, the Hessian matrix can be written as

If the second-order-derivative terms in the summation onthe right-hand side of the above expression can be neglected,then eq A4 can be further simplified by the following ap-proximation:

Note that∂g/∂τ ) [∂g/∂θ, ∂g/∂â] and from eq A3:

Hence,

where

Note that, to obtain the inverse of the matrix in eq A7, thefollowing property can be used.3 If all inverses exist, then

where B22 ) A22 - A21A11-1A12, B12 ) A11

-1A12, B21 )A21A11

-1, C11 ) A11 - A12A22-1A21, C12 ) A12A22

-1, andC22

) A22-1A21.

Appendix B

See the scatter plot matrices in Figures B1-B3.

Nomenclature

a ) CSTR model parameter relating heat-transfer coefficientto coolant flow rate

diag(W) )

[diag(Σ 1-1), diag(Σ2

-1), diag( 1Qp1

w1), diag( 1Qp2

w2)]

∂2L(τ,ym)

∂τ∂τT)

∂g(τ,ym)T

∂τT ( ∂2L

∂g∂gT) ∂g(τ,ym)

∂τ+

∑i

∂L

∂gi

∂2gi

∂τ∂τT(A4)

∂2L(τ,ym)

∂τ∂τT≈ -2

∂g(τ,ym)T

∂τW

∂g(τ,ym)

∂τ(A5)

∂g∂θ

) [0- ∂f∂θ ], ∂g

∂â) [-æ

æq - ∂f∂â ] (A6)

(∂g∂τ)T

W(∂g∂τ) )

[(∂f∂θ)T

w12(∂f∂θ) - (∂f

∂θ)Tw12(æ3 q - ∂f

∂â)- (æ3 q - ∂f

∂â)Tw12(∂f

∂θ) æTΣ-1æ + (æ3 q - ∂f∂â)T

w12(æ3 q - ∂f∂â) ](A7)

w12 ) [w1

w2]

[A11 A12

A21 A22]-1

) [A11-1 + B12B22

-1B21 -B12B22-1

-B22-1B21 B22

-1 ]) C11

-1-C11-1C12-C21C11

-1A22-1 + C21C11

-1C12 (A8)

Figure B1. Scatter plot matrix for 100 sets of parameter estimates obtainedusing AMLE with both states measured.

Figure B2. Scatter plot matrix for 100 sets of parameter estimates obtainedusing AMLE without concentration measurements.

Figure B3. Scatter plot matrix for 100 sets of parameter estimates obtainedusing AMLE with both temperature and concentration measured and anonstationary disturbance in the material balance.


b ) CSTR model exponent relating heat-transfer coefficient tocoolant flow rate

ci ) number of spline coefficients for stateiCA ) concentration of reactant A, kmol m-3

CA0 ) feed concentration of reactant A, kmol m-3

CAs ) concentration of reactant A at steady state, kmol m-3

Cp ) reactant heat capacity, cal g-1 K-1

Cpc ) coolant heat capacity, cal g-1 K-1

d1 ) nonstationary disturbance term, kmol m-3 min-1

d1∼ ) B-spline approximation of the nonstationary disturbance,kmol m-3 min-1

E{.} ) expected valueE/R ) activation energy over the ideal gas constant, KF ) reactant volumetric flow rate, m3 min-1

Fc ) coolant volumetric flow rate, m3 min-1

fi, f i ) nonlinear function on the right-hand side of thedifferential equation for statei

fd ) nonlinear function on the right-hand side of the nonsta-tionary disturbance differential equation

g ) vector of combined sum-of-squared errors and model-basedpenalties

I ) Fisher information matrixkref ) kinetic rate constant at temperatureTref, min-1

L ) log-likelihood functionNi ) number of observations of stateip(.) ) probability density functionQpi ) process noise intensity of stochastic differential equation

for stateiQpd ) process noise intensity of stochastic differential equation

for nonstationary disturbancetmij ) jth measurement time for theith state, minT ) temperature of reactor contents, KT0 ) reactant feed temperature, KTcin ) inlet temperature of coolant, KTs ) temperature of reactant at steady-state value, KTref ) reference temperature, Kui, ui ) input to the differential equation for stateiV ) volume of the reactor, m3

wi ) matrix of Gaussian quadrature weights for calculating themodel-based penalty integrals of differential equation for statei

W ) overall weighting matrix to define the log-likelihoodfunction in a least-squares form

x, X ) state variablesxi∼ ) B-spline approximation of theith statey, Y ) noisy output measurementsym ) stacked vector of measured outputszR/2 ) normal random deviate corresponding to an upper tail

area ofR/2R ) significance level for confidence intervalsâij ) jth B-spline coefficient of theith stateâi ) vector of B-spline coefficients of theith stateâd ) vector of B-spline coefficients of the disturbance termδ(.) ) Dirac delta function∆Hrxn ) enthalpy of reaction, cal g-1 K-1

εI ) normally distributed measurement noise for stateIηi ) white Gaussian process disturbance for differential equation

of the ith stateηd ) white Gaussian process disturbance for the nonstationary

disturbance differential equationθ ) vector of model parametersθd ) vector of disturbance model parametersF ) density of reactor contents, g m-3

Fc ) coolant density, g m-3

σmi2 ) measurement noise variance for theith state

Σi ) measurement noise variance for theith stateΣi ) measurement noise covariance matrix of theith state vectorΣ ) measurement noise covariance matrixτ ) combined vector of model parameters and spline coefficientsφij ) jth B-spline basis function of theith stateæi ) vector of B-spline basis functions for theith stateæd1 ) vector of B-spline basis functions for disturbanced1

æi ) matrix of all æi’s evaluated at theith state observationtimes

æ ) matrix containing allæi’sæqi ) matrix of all æis evaluated at the quadrature points of the

differential equation corresponding to theith stateæq ) matrix containing allæqi’sAMLE ) approximate maximum likelihood estimationCSTR) continuous stirred tank reactordiag ) diagonal elements of a matrixiPDA ) iteratively refined principal differential analysisMIMO ) multi-input multioutputODE ) ordinary differential equationPEN ) model-based penaltySSE) sum of squared errorsTNLS ) traditional nonlinear least-squares

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ReceiVed for reView June 15, 2007ReVised manuscript receiVed September 28, 2007

AcceptedSeptember 28, 2007

IE070824Q


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