A sampling-based stochastic optimal experiment ...

A sampling-based stochastic optimal experiment design formulation with application to the Williams-Otto reactor Philippe Nimmegeers ∗ Dries Telen ∗ Jan Van Impe ∗ ∗

KU Leuven, Chemical Engineering Department, BioTeC+ & OPTEC, Gebroeders De Smetstraat 1, 9000 Gent, Belgium (e-mail: [email protected]) Abstract: Governmental pressure and an industrially competitive climate force chemical companies to strive for a sustainable design and operation. Model-based optimization has been proven to be an indespensable tool to achieve these goals. For the development and maintenance of process models, experimental data with a high information content are required. Model-based optimal experiment design techniques have been developed to obtain experiments that yield a high information content for estimating the model parameters. Since the true parameter values are unknown, these experiment designs start from the current best guesses. In practice, this can result in a lower information content than originally expected. In addition, the experiment design can become practically infeasible (e.g., in terms of safety), due to the violation of operational constraints. In this work a sampling-based stochastic optimal experiment design formulation is employed to address these problems. If the uncertainty on the model parameters can be considered to be a priori known, the parametric uncertainty can be propagated towards the states, which makes the experiment design more robust with respect to information content. The presented approach is based on polynomial chaos expansion and is compared with the sigma points approach. As a practical case study of this formulation, the optimal experiment design of a Williams-Otto fed-batch reactor is made robust with respect to information content and reactor temperature state constraint violations. Keywords: stochastic optimal control problems, experiment design, robust control, process modeling and identification, control problems under uncertainty 1. INTRODUCTION Sustainability is a key factor for chemical companies to survive in the competitive worldwide market and to comply with legislation. Therefore continuous improvements need to be realized to increase economic margins and to reduce the burden on the environment and society. Modelbased optimization techniques have been proven to be successful in many applications (Cappuyns et al. (2007); Schenkendorf et al. (2009); Logist et al. (2012); Telen et al. (2012)). To construct process models, experiments need to be performed which yield a maximum information content. Typically in a (bio)chemical setting, such experiments are expensive and the experimental burden has to be kept low. Model-based optimal experiment design (OED) techniques can be applied to reduce the experimental burden and yield informative experiments (Franceschini and Macchietto (2008)). In this contribution optimal experiment design for parameter estimation is studied, which aims at designing a high informative experiment to estimate the model parameters. Traditional optimal experiment design techniques start from the current best guesses for the parameter values, since the true parameter values are unknown. This forms a

challenge, since as a result of these current best guesses for the parameter values, (i) the designed experiment can yield a lower information content than originally predicted and (ii) critical operational constraints are potentially violated making the experiment design infeasible in practice (e.g., due to safety considerations) (Telen et al. (2014)). Hence, the designed experiment should be robust with respect to information content and robust with respect to critical constraint satisfaction (Asprey and Macchietto (2002)). Different techniques have been proposed to design robust experiments with respect to information content: Pronzato and Walter (1985) and Asprey and Macchietto (2002) proposed to calculate the expected value and statistical moments of the objective function by an integration over the parameter space, K¨orkel et al. (2004) formulated the robust OED problem as a max-min problem, in which the inner loop is solved using a linear approximation and a Bayesian robust optimal experiment design approach is presented in Liepe et al. (2013). Robustness with respect to state constraints has also been studied in Srinivasan et al. (2003), Galvanin et al. (2010) and Telen et al. (2015). In these references and also in this work, it is assumed that a previous parameter identification procedure has been performed to characterize the parametric uncertainty distribution. Alternative ap-

proaches guarantee robustness when the uncertainty is known to lie in a bounded set (Houska et al. (2012)).

V ∈ Rnh ×nh the measurement variance-covariance matrix and c(x, u, θ, t) ∈ Rnc the constraints.

Alternatively, guaranteed parameter estimation approaches can be used to account for the parametric uncertainty. Such computationally challenging approaches determine all parameter values, consistent with the measurements within prespecified error bounds. In Paulen et al. (2016) an improved set-inversion algorithm is presented, exploiting (reduced order) Taylor models and optimization-based domain reduction techniques to increase convergence speed.

Considering nx states and nθ parameters in the dynamic model, nx nθ states related to sensitivity equations and nθ 2 (nθ + 1) Fisher elements (due to symmetry), the total number of states for the optimal experiment design problem equals: nθ nx + nx nθ + (nθ + 1) (2) 2

In this paper a sampling-based stochastic optimal experiment design formulation is presented based on polynomial chaos expansion. Assuming that the uncertainty on the model parameters is a priori known, this parametric uncertainty is propagated towards the states and constraints and the expected value and variance-covariance matrix are computed. This allows to make the experiment design more robust with respect to (i) the information content and (ii) critical constraint violation. Outline: In the next section the mathematical OED problem formulation is presented, together with the concepts of the unscented transformation/sigma points and the polynomial chaos expansion. In Section 3 the samplingbased stochastic OED formulation is discussed. In Section 4, the optimal experiment design of a Williams-Otto fedbatch reactor with two uncertain parameters (Hannemann and Marquardt (2010); Logist et al. (2012)) is made robust with the sampling-based stochastic OED formulation based on polynomial chaos expansion (Wiener (1938); Xiu and Karniadakis (2002); Mesbah and Streif (2015); Nimmegeers et al. (2016)). The results are compared with the results obtained with the sigma points approach (Julier and Uhlmann (1996); Telen et al. (2014)). Section 5 summarizes the main conclusions and the future work. 2. MATHEMATICAL PROBLEM FORMULATION 2.1 OED problem formulation In this paper, the optimal experimental design problem is formulated as a dynamic optimization problem. The Fisher information matrix F is used as a measure for the information content (Telen et al. (2014)). min J(F(tf )) u,x, ∂x ,F  ∂θ x˙ = f (x, u, θ, t)     x(0) = x0      ∂x ∂f ∂x ∂f d    = +   dt ∂θ ∂x ∂θ ∂θ   ∂x ∂x0 (0) = s.t. (1)  ∂θ ∂θ >  >    ∂x dh dh ∂x   dF = V−1   dt ∂θ dx dx ∂θ     F(0) = 0   0 ≥ c(x, u, θ, t) with x ∈ Rnx the state vector, u ∈ Rnu the control vector, x0 ∈ Rnx the vector of initial conditions, ∂x ∂θ ∈ Rnx ×nθ , the sensitivity of the states with respect to the parameter vector θ ∈ Rnθ , h ∈ Rnh a measurement function which can depend on the states, F ∈ Rnθ ×nθ ,

Since the aim of optimal experiment design is to design maximum informative experiments with respect to estimating the model parameters, the objective function has to be a measure for this information content. A scalar function of the Fisher information matrix is typically used as objective function for OED. In this work the D-criterion (i.e., J(F(tf ) = −det(F(tf )) is used as objective function. 2.2 The unscented tranformation/sigma points The first sampling-based technique considered for propagating parametric uncertainty towards the states, is the sigma points approach (Julier and Uhlmann (1996); Telen et al. (2014)). In this approach the distribution after a nonlinear transformation is approximated by evaluating the dynamic system at the sigma points πi,SP . Note that this method can only be applied to symmetric, unimodal distributions (e.g., normal distribution). In Equations (3)(4) the approximations of the expected value of the response function q(t) are presented. ! 2n 1 1 Xθ q¯(t) = κq(x0 (t)) + q(xi (t)) (3) nθ + κ 2 i=1 Pqq (t) =

1 κ(q(x0 (t)) − q¯(t))(q(x0 (t)) − q¯(t))> nθ + κ 2n 1 Xθ + (q(xi (t)) − q¯(t))(q(xi (t)) − q¯(t))> ) (4) 2 i=1

with:   π0,SP = θ¯ , p πi,SP = θ¯ + p(nθ + κ)Σi with i = 1, . . . , nθ ,  πi,SP = θ¯ − (nθ + κ)Σi−nθ with i = nθ + 1, . . . , 2nθ . In these formulations, the term κ accounts for knowledge on higher moments of the given probability distribution. In this work it is chosen to set κ = 3 − nθ . The mean parameter vector and parameter variance-covariance matrix, denoted by θ¯ and Σ respectively, are known. This approach requires ns,SP = 2nθ +1 model evaluations, which only differ in the parameter values, i.e., the sigma points. 2.3 Polynomial chaos expansion The non-intrusive implementation of the polynomial chaos expansion (PCE) method (Wiener (1938), Xiu and Karniadakis (2002)) can be viewed as another sampling-based approach, which is applicable to any type of probability distribution. Based on the probability density functions of the parametric uncertainties, orthogonal polynomials are derived. The sum of these orthogonal polynomials is subsequently used to approximate the model states. Note

that these polynomials are a function of the uncertain parameters. For simplicity, the polynomial chaos expansion (p) qPCE of order p for the response function q(t) is formulated as (Xiu and Karniadakis (2002); Mesbah and Streif (2015); Nimmegeers et al. (2016)): (p)

q(t) ≈ qPCE (θ) =

L−1 X

(p)

γq,j Γj (θ),

(5)

j=0

expansion (Mesbah and Streif (2015); Nimmegeers et al. (2016)).

3. A SAMPLING-BASED STOCHASTIC OED FORMULATION

(p)

with γq,j the unknown PCE coefficients and Γj (θ) the multivariate orthogonal polynomials and j a term based index (j = 0, . . . , ns,PCE ). (p)

The coefficients γq,j in Equation (5) are obtained by evaluating the model response in the ns,PCE polynomial chaos expansion sampling points πi,PCE (resulting in the (p) vector containing these model evaluations qs ∈ Rns,PCE ) and setting the model response equal to its polynomial chaos expansion of order p. Hence the coefficients can explicitly be written as the solution of the system in Equation (6). >  (p) q(p) = Λ γq(p) , (6) s h i> (p) (p) (p) with γq = γq,0 . . . γq,L−1 , πi,PCE the parameter vector corresponding to the ith PCE sampling point and Λ(p) ∈ Rns,PCE ×ns,PCE given in Equation (7).  ... Γ0 (πns,PCE −1,PCE )   .. .. =  . . Γns,PCE −1 (π0,PCE ) . . . Γns,PCE −1 (πns,PCE −1,PCE ) (7) 

Λ(p)

Γ0 (π0,PCE ) .. .

In order to robustify the experiment design with respect to information content the expected value of the objective function can be computed via a sampling-based approach as the sigma points or the polynomial chaos expansion approach (i.e., setting the response function q(t) = J(F(tf )). The expected value and variance-covariance of the state constraint cr,j1 can also be computed via sampling-based approach (by setting the response function q(t) = cr,j1 (t). The sampling-based stochastic OED formulation is presented in Equation (11):

min

u,x, ∂x ∂θ ,F

 x˙ i (t)    xi (0)     d ∂xi     dt ∂θ    ∂xi     ∂θ (0) s.t. dFi    dt    Fi (0)    0      0   

¯ J(F(t f )) = f (xi (t), u(t), πi ) with i = 0, . . . , ns − 1 = x0 ∂f ∂xi ∂f = + ∂xi ∂θ ∂θ ∂x0 = ∂θ ∂xi > dh > −1 dh ∂xi = V ∂θ dxi dxi ∂θ =0 ≥ c(xi , u, πi , t) q ≥ c¯r,j1 (t) + αcr,j1 Pcr cr ,j1 j1 (t) with j1 = 1, . . . , ncr

From Equation (7) it follows that Λ(p) only depends on the evaluation of the orthogonal polynomials in the polynomial chaos expansion sampling points, thus this (p) matrix is known upfront. Hence, the coefficients γq,j are computed during the optimization as a weighting of the response function evaluated at the sampling points, which can be expressed by the least squares approximation in Equation (8) (Nimmegeers et al. (2016)).   > −1 (p) (p) (p) γq = Λ Λ Λ(p) q(p) (8) s

with xi ∈ Rnx the state vector at the ith sampling point, ¯ u ∈ Rnu the control vector, J(F(t f )) the approximated expected value of the objective function, πi the parameter vector corresponding to the i + 1th sampling point, 0 ≥ cr,j1 (t) a constraint that is robustified and ncr the number of robustified constraints.

The sampling points can be determined via the roots of the higher order orthogonal polynomials. A large number of combinations exists for the selection of the sampling points, since many parameter sets span equally high probability regions. A necessary condition for using one of
> these parameter sets is that the matrix Λ(p) Λ(p) is nonsingular. The expected value and variance of q are computed in the polynomial chaos expansion approach as:

In the sampling-based OED formulation (Equation (11)), the sigma points or polynomial chaos expansion approach can be used to propagate the parametric uncertainty towards the states and model responses. Hence the expected ¯ value of the objective function J(F(t f )), the expected value of the constraint function to be robustified c¯r,j1 (t) and the variance-covariance matrix Pcr cr ,j1 j1 (t) can be approximated with the sigma points and polynomial chaos expansion approach.

(p)

(p)

(9)

q¯P CE = γ0 (p)

Pqq,PCE =

L−1 X

(p)

γq,j

2

  E Γ2j (θ)

(10)

j=1

The number of model evaluations for the polynomial chaos expansion approach equals ns,PCE = (nnθθ+p)! !p! with nθ the number of uncertain parameters and p the order of the

(11)

In case that αcr,j1 = 0, the constraint 0 ≥ c¯r,j1 (t) is not robustified and only its expected value is considered. Therefore the case in which αcr,j1 = 0 is referred to as expected value experiment design. If α 6= 0, the variance on the constraint function is taken into account and the constraint 0 ≥ cr,j1 (t) is robustified. Hence, the approach in which αcr,j1 6= 0 is referred to as robustified expected value experiment design.

4. RESULTS AND DISCUSSION

the states and controls are discretized with an orthogonal collocation scheme (Biegler (2007)). For the numerical calculation of derivatives, casADi (Andersson (2013)) is used. The number of control intervals is set to 50.

4.1 Case study As a case study, the sampling-based OED formulation has been implemented for the Williams-Otto fed-batch reactor from (Logist et al., 2012; Hannemann and Marquardt, 2010). Three exothermic reactions take place in this reactor, producing products P , E, and G, from reactant A which is initially present in the reactor and reactant B which is fed to the reactor during the process with a feed rate u1 which can be manipulated throughout the process: A + B −→ C, C + B −→ P + E and P + C −→ G. The temperature of the liquid in the surrounding cooling jacket can also be manipulated throughout the process to remove the generated heat. Hence, the second control variable for this case study is the dimensionless jacket fluid temperature u2 . The model equations and paramaters for the Williams-Otto reactor model can be consulted in Logist et al. (2012) and Hannemann and Marquardt (2010). Following constraints are considered:

The number of states to be calculated, is 25 for the nominal D-design, 121 for the sigma points D-designs, 73 for the first order polynomial chaos expansion D-designs and 145 for the second order polynomial chaos expansion (PCE2) D-designs. Since the reactor volume does not depend on the uncertain parameters, its corresponding state is not considered in the parametric uncertainty propagation.

60 ≤ T (t) ≤ 90 0 ≤ u1 (t) ≤ 5.784 0.02 ≤ u2 (t) ≤ 0.2 V (tf ) ≤ 5

In Table 1, the D-criterion objective function values are shown that are obtained from evaluating the model with the current best guess of the parameter values and the controls from the nominal, expected and robustified expected D-designs for the sigma points, PCE1 and PCE2 approaches. From this table it is clear that the price for the robustification to be paid is the loss in information content compared with the nominal OED, measured by the determinant of the Fisher information matrix. Table 1. D-criterion values resulting from simulation with current best guess of the parameters

(12) (13) (14) (15)

The model contains 7 states: xA , xB , xC , xP , xE , xG representing dimensionless concentrations, T the reactor temperature and V the liquid volume. The initial conditions are denoted by x0 = [1, 0, 0, 0, 0, 0, 65, 2]> and the final time tf is set to 1000h. Two uncertain parameters are considered for the OED of the Williams-Otto reactor: k1 , kinetic reaction coefficient related to A −→ B and l1 , the heat transfer coefficient. The current best guesses of the parameter values are 1.6599e6 and 2.435e-4, respectively. Unless stated differently, both parameters are assumed to be normally distributed with as mean values 1.6599e6 and 2.435e-4 and assumed variances (1.6599e5)2 and (2.435e-5)2 for k1 and l1 , respectively. 4.2 Numerical results The nominal D-design, the expected D-design (ED-design) and the robustified expected D-design (robust ED-design) are computed for the Williams-Otto reactor. The state constraint with respect to reactor temperature in Equation (12), is robustified. More specifically, to ensure that the expected temperature evolution satisfies the state constraint with approximately 95 % confidence, it is chosen to set αcr,j1 = 2. This holds if a normally distributed state evolution can be considered. Otherwise, CantelliChebychev’s inequality can be used (Mesbah and Streif (2015)). p E[T (t)] + 2 PT T ≤ 90 (16) p E[T (t)] − 2 PT T ≥ 60 (17) The dynamic optimization problems are solved with an inhouse developed software, called Pomodoro (Bhonsale et al. (2016a), Bhonsale et al. (2016b)). In this software,

Note that the sigma points and PCE2 yield the same approximation for the expected value computation (i.e., five sampling points in common and the same weights for the sampling points are used). As a consequence, the minor differences between the sigma points and PCE2 approach are due to the approximation of the constraint function’s (i.e. the constraint on the reactor temperature) variance-covariance matrix. Therefore only the graphs corresponding with PCE2 are shown in the remainder of this work and are referred to as SP/PCE2.

D(×1010 ) D/Dnom

Nominal 2.46 1

D(×1010 ) D/Dnom

Nominal 2.46 1

Expected D-designs SP PCE1 PCE2 2.031 2.24 2.031 0.8257 0.9097 0.8257 Robustified Expected D-designs SP PCE1 PCE2 2.00 2.015 2.00 0.8132 0.8190 0.8134

In Figure 1 the control profiles are shown for the different D designs. The initial feeding and heating of the system stop earlier in the stochastic designs (i.e.,EDdesign/robustified ED-design) compared with the Ddesign. By stopping earlier the uncertainty on the reactor temperature is taken into account to reduce the probability of violating the constraints on this temperature. To investigate the number of practically feasible experiments, i.e. experiments for which the state constraint on the temperature is not violated, Monte Carlo simulations have been performed. From a normal parametric uncertainty distribution for k1 and l1 , 500 samples have been taken. As illustration of the robustification of the state constraint on the reactor temperature, the empirical mean and 95 % confidence bounds are shown in Figure 2. The empirical 95% confidence bound of the nominal D design clearly violates the states constraint, leading to a lower number of feasible experiments. The ED and the robustified ED designs clearly show a substantial improvement

6

6 D-design ED-design Robust ED-design

5

5

4

4

u 1 : Feed rate B [-]

u1 : Feed rate B [-]

D-design ED-design Robust ED-design

3

2

3

2

1

1

0

0

0

200

400

600

800

1000

0

200

400 600 Time [min]

Time [min]

(a) Feed rate u1 SP/PCE2. 0.2

1000

0.14 0.12 0.1 0.08

D-design ED-design Robust ED-design

0.18

u 2 : Jacket temperature [-]

0.16 u2 : Jacket temperature [-]

0.2

D-design ED-design Robust ED-design

0.18

800

(b) Feed rate u1 PCE1.

0.06

0.16 0.14 0.12 0.1 0.08 0.06 0.04

0.04

0.02

0.02 0

200

400

600

800

0

1000

200

Time [min]

(c) Jacket temperature u2 SP/PCE2.

400 600 Time [min]

800

1000

105

105

100

100

95

95

90

90

Reactor temperature [-]

Reactor temperature [-]

with respect to the reactor temperature state constraint satisfaction.

85 80 75 Mean D-design Mean ED-design Mean robust ED-design Empirical 95% CB D-design Empirical 95% CB ED-design Empirical 95% CB robust ED-design

65 60

85 80 75 Mean D-design Mean ED-design Mean robust ED-design Empirical 95% CB D-design Empirical 95% CB ED-design Empirical 95% CB robust ED-design

70 65 60

0

200

400

600

800

Time [h]

(a) Empirical Temperature profile SP/PCE2.

The Monte Carlo simulations for the Beta distribution, results in 182 nominally designed feasible experiments. The experiments designed, based on the Beta distribution resulted in a lower number of feasible experiments then expected, i.e., for PCE1 324 (64.8%) and 399 (79.8%) and for PCE2 419 (83.8%) and 436 (87.2%) for the expected and robustified D-designs. A possible explanation is the fact that the state constraints due to the parametric uncertainty cannot be expected to be normally distributed. Performing a Lilliefors test with a 5 % significance level has shown that the reactor temperature is not normally distributed at the discretization points present.

(d) Jacket temperature u2 PCE1.

Fig. 1. Control profiles for the nominal D, expected D and robustified D designs using the (a) sigma points (and PCE2) and (b) PCE1 approaches.

70

observed for this case study that for the second order polynomial chaos expansion there are 426 (85.2%) feasible experiments for the expected D-design, while there are 474 (94.8%) feasible experiments for the robustified expected D-design. This is just below the target of 95% set in the experiment design phase.

1000

0

200

400

600

800

1000

Time [h]

(b) Empirical Temperature profile PCE1.

Fig. 2. Empirical temperature profile obtained from 500 Monte Carlo simulations (normally distributed parameters) for (a) SP/PCE2 and (b) PCE1. The nominal experiment design, results in 212 (42.2%) feasible experiments. For the sigma points and PCE2 approach these simulations result for the ED-design in 471 (94.2%) feasible designs and for the robust ED-design in 479 (95.8%) feasible designs . However, the polynomial chaos expansion of first order results into 429 (85.8%) feasible designs, while the robust ED-design results in 477 (95.4%) feasible designs. Hence, for this case study, a first order technique (PCE1) sufficiently robustifies the experiment design. In addition, two other distributions have been studied for both k1 and l1 , i.e. a uniform parametric uncertainty and an asymmetric Beta distribution with shape parameters a = 2, b = 3. Polynomial chaos expansions of first and second order have been derived with respect to these parametric uncertainty distributions. Note that in theory the sigma points approach cannot be applied to the second distribution. Similarly, 500 samples have been taken from the parametric uncertainty distribution. From the Monte Carlo simulations with uniformly distributed parameters it is

5. CONCLUSIONS AND FUTURE WORK In this paper, the problem of planning optimal experiments for parameter estimation in the presence of parametric uncertainty is addressed. Not taking this uncertainty into account, potentially results in practically infeasible experiments or a loss in predicted information content of the designed experiment. Therefore, a sampling-based formulation has been presented in this work to approximate the expected value of the considered OED criterion with a sampling-based uncertainty propagation technique. In addition, the expected value and variance-covariance can also be approximated with these sampling-based techniques to avoid (or limit) state constraint violations and as such robustify the designed experiments. More specifically, this formulation has been illustrated on the case study of optimal experimental design of a Williams-Otto fedbatch reactor for the sigma points and polynomial chaos expansion approach. It has been proven that the number of practically feasible experiments increases by applying this formulation, compared with the nominal experimental design. Furthermore it is concluded for this case study that a first order polynomial chaos expansion (PCE1) yields sufficient robustification, which is comparable to the second order techniques (sigma points and PCE2). One of the typical drawbacks of sampling-based stochastic optimal experiment design approaches is the increasing size of the optimization problem, with the number of uncertain parameters and the order of the orthogonal polynomials. In future work, the aim is to reformulate the stochastic OED as a distributed optimization problem, consisting of ns decoupled subsystems, that are coupled by the same controls and constraint and objective functions as a weighted sum. A novel distributed optimization algorithm, ALADIN (Houska et al. (2016)), will be used to decouple the large optimization problem and solve the stochastic optimal control problem in a computationally more efficient way (Jiang et al. (2017)). This should allow to apply sampling-based approaches of higher order and to cases with more uncertain parameters.

ACKNOWLEDGEMENTS

Group, Department of Engineering Science, University of Oxford. This work was supported by KU Leuven [PFV/10/002 K¨orkel, S., Kostina, E., Bock, H., and Schl¨oder, J. (2004). Center-of-Excellence Optimization in Engineering (OPTEC), Numerical methods for optimal control problems in design of robust optimal experiments for nonlinear dyFonds Wetenschappelijk Onderzoek Vlaanderen [G.0930.13] namic processes. Optimization Methods and Software and the Belgian Science Policy Office (DYSCO) [IAP Journal, 19(3-4), 327–338. VII/19]. Liepe, J., Filippi, S., Komorowski, M., and Stumpf, M.P.H. (2013). Maximizing the information content of experiments in systems biology. PLOS Computational Biology, REFERENCES 9(1), 1–13. Andersson, J. (2013). A General-Purpose Software Frame- Logist, F., Vallerio, M., Houska, B., Diehl, M., and work for Dynamic Optimization. PhD thesis, KU LeuVan Impe, J. (2012). Multi-objective optimal control of ven, Department of Electrical Engineering. chemical processes using ACADO toolkit. Computers Asprey, S. and Macchietto, S. (2002). Designing robust opand Chemical Engineering, 37, 191–199. timal dynamic experiments. Journal of Process Control, Mesbah, A. and Streif, S. (2015). A probabilistic ap12(4), 545 – 556. proach to robust optimal experiment design with chance Bhonsale, S., Vallerio, M., Telen, D., Vercammen, D., constraints. In International Symposium on Advanced Logist, F., and Van Impe, J. (2016a). Solace: An open Control of Chemical Processes (ADCHEM), 2015, 100– source package for nolinear model predictive control and 105. state estimation for (bio)chemical processes. In Pro- Nimmegeers, P., Telen, D., Logist, F., and Van Impe, J. ceedings of the 26th European Symposium on Computer (2016). Dynamic optimization of biological networks Aided Process Engineering. Portoroz, Slovenia., June under parametric uncertainty. BMC Systems Biology, 12th - 15th. 10(1), 1–20. Bhonsale, S.S., Vercammen, D., Vallerio, M., Hufkens, J., Paulen, R., Villanueva, M.E., and Chachuat, B. (2016). Cabianca, L., Nimmegeers, P., Telen, D., Logist, F., Guaranteed parameter estimation of non-linear dynamic and Van Impe, J. (2016b). Pomodoro - an open source systems using high-order bounding techniques with dotoolkit for multiobject optimal control, and model based main and CPU-time reduction strategies. IMA Journal control and estimation. Preprint submitted to Expert of Mathematical Control and Information, 33, 563–587. Systems with Applications. Pronzato, L. and Walter, E. (1985). Robust experiment Biegler, L. (2007). An overview of simultaneous strategies design via stochastic approximation. Mathematical Biofor dynamic optimization. Chemical Engineering and sciences, 75, 103–120. Processing: Process Intensification, 46, 1043–1053. Schenkendorf, R., Kremling, A., and Mangold, M. (2009). Cappuyns, A., Bernaerts, K., Smets, I., Ona, O., Prinsen, Optimal experimental design with the sigma point E., Vanderleyden, J., and Van Impe, J. (2007). Optimal method. IET Systems Biology, 3, 10–23. fed batch experiment design for estimation of monod Srinivasan, B., Bonvin, D., Visser, E., and Palanki, S. kinetics of Azospirillum brasilense: From theory to (2003). Dynamic optimization of batch processes II. practice. Biotechnology Progress, 23(5), 1074–1081. Role of measurements in handling uncertainty. ComFranceschini, G. and Macchietto, S. (2008). Model-based puters and Chemical Engineering, 27, 27–44. design of experiments for parameter precision: State of Telen, D., Logist, F., Vanderlinden, E., and Van Impe, the art. Chemical Engineering Science, 63, 4846–4872. J. (2012). Optimal experiment design for dynamic Galvanin, F., Barolo, M., Bezzo, F., and Macchietto, S. bioprocesses: a multi-objective approach. Chemical (2010). A backoff strategy for model-based experiment Engineering Science, 78, 82–97. design under parametric uncertainty. AIChE Journal, Telen, D., Vallerio, M., Cabianca, L., Houska, B., 56, 2088–2102. Van Impe, J., and Logist, F. (2015). Approximate robust Hannemann, R. and Marquardt, W. (2010). Continuous optimization of nonlinear systems under parametric unand discrete adjoints for the Hessian of the Lagrangian certainty and process noise. Journal of Process Control, in shooting algorithms for dynamic optimization. SIAM 33, 140 – 154. Journal on Scientific Computing, 31, 4675–4695. Telen, D., Vercammen, D., Logist, F., and Van Impe, Houska, B., Frasch, J., and Diehl, M. (2016). An AugJ. (2014). Robustifying optimal experiment design for mented Lagrangian Based Algorithm for Distributed nonlinear, dynamic (bio)chemical systems. Computers Non-Convex Optimization. SIAM Journal on Optimizaand Chemical Engineering, 71, 415–425. tion, 26(2), 1101–1127. Wiener, N. (1938). The homogeneous chaos. American Houska, B., Logist, F., Van Impe, J., and Diehl, M. (2012). Journal of Mathematics, 60, 897–936. Robust optimization of nonlinear dynamic systems with Xiu, D. and Karniadakis, G. (2002). The Wiener-Askey application to a jacketed tubular reactor. Journal of polynomial chaos for stochastic differential equations. Process Control, 22, 1152–1160. SIAM Journal of Scientific Computation, 24, 619–644. Jiang, Y., Nimmegeers, P., Telen, D., Van Impe, J., and Houska, B. (2017). A distributed optimization algorithm for sampling-based stochastic optimal control. In Submitted to the 20th IFAC World Congress. Julier, S. and Uhlmann, J. (1996). A general method for approximating nonlinear transformations of probability distributions. Technical report, Robotics Research