Dries Telen

Robustifying optimal experiment design for nonlinear, dynamic (bio)chemical systems

Abstract Dynamic experiments that yield as much information as possible are highly valuable for estimating parameters in nonlinear dynamic processes. Techniques for model-based optimal experiment design enable to systematically design such experiments. However, these experiments depend on the current best estimate of the parameters, which are not necessarily the true values. Consequently, in real experiments (i) the information content can be lower than predicted and (ii) state constraints can be violated. This paper presents a novel, computationally tractable formulation that enables the robustification of optimally designed experiments with respect to (i) information content and (ii) constraint satisfaction. To this end, the objective function is the expected value of a scalar function of the Fisher information matrix, which is efficiently computed using the sigma point method. This approach already has a robustifying effect. The sigma point method also enables the efficient computation of constraints’ variance–covariance matrix, this can be exploited for further robustification.

Multi-objective optimal control of dynamic bioprocesses using ACADO Toolkit

The optimal design and operation of dynamic bioprocesses gives in practice often rise to optimisation problems with multiple and conflicting objectives. As a result typically not a single optimal solution but a set of Pareto optimal solutions exist. From this set of Pareto optimal solutions, one has to be chosen by the decision maker. Hence, efficient approaches are required for a fast and accurate generation of the Pareto set such that the decision maker can easily and systematically evaluate optimal alternatives. In the current paper the multi-objective optimisation of several dynamic bioprocess examples is performed using the freely available ACADO Multi-Objective Toolkit (http://www.acadotoolkit.org). This toolkit integrates efficient multiple objective scalarisation strategies (e.g., Normal Boundary Intersection and (Enhanced) Normalised Normal Constraint) with fast deterministic approaches for dynamic optimisation (e.g., single and multiple shooting). It has been found that the toolkit is able to efficiently and accurately produce the Pareto sets for all bioprocess examples. The resulting Pareto sets are added as supplementary material to this paper.

Dynamic optimization of biological networks under parametric uncertainty

BackgroundMicro-organisms play an important role in various industrial sectors (including biochemical, food and pharmaceutical industries). A profound insight in the biochemical reactions inside micro-organisms enables an improved biochemical process control. Biological networks are an important tool in systems biology for incorporating microscopic level knowledge. Biochemical processes are typically dynamic and the cells have often more than one objective which are typically conflicting, e.g., minimizing the energy consumption while maximizing the production of a specific metabolite. Therefore multi-objective optimization is needed to compute trade-offs between those conflicting objectives. In model-based optimization, one of the inherent problems is the presence of uncertainty. In biological processes, this uncertainty can be present due to, e.g., inherent biological variability. Not taking this uncertainty into account, possibly leads to the violation of constraints and erroneous estimates of the actual objective function(s). To account for the variance in model predictions and compute a prediction interval, this uncertainty should be taken into account during process optimization. This leads to a challenging optimization problem under uncertainty, which requires a robustified solution.ResultsThree techniques for uncertainty propagation: linearization, sigma points and polynomial chaos expansion, are compared for the dynamic optimization of biological networks under parametric uncertainty. These approaches are compared in two case studies: (i) a three-step linear pathway model in which the accumulation of intermediate metabolites has to be minimized and (ii) a glycolysis inspired network model in which a multi-objective optimization problem is considered, being the minimization of the enzymatic cost and the minimization of the end time before reaching a minimum extracellular metabolite concentration. A Monte Carlo simulation procedure has been applied for the assessment of the constraint violations. For the multi-objective case study one Pareto point has been considered for the assessment of the constraint violations. However, this analysis can be performed for any Pareto point.ConclusionsThe different uncertainty propagation strategies each offer a robustified solution under parametric uncertainty. When making the trade-off between computation time and the robustness of the obtained profiles, the sigma points and polynomial chaos expansion strategies score better in reducing the percentage of constraint violations. This has been investigated for a normal and a uniform parametric uncertainty distribution. The polynomial chaos expansion approach allows to directly take prior knowledge of the parametric uncertainty distribution into account.

Symmetric algorithmic differentiation based exact Hessian SQP method and software for Economic MPC

Economic Model Predictive Control (EMPC) is an advanced receding horizon based control technique which optimizes an economic objective subject to potentially nonlinear dynamic equations as well as control and state constraints. The main contribution of this paper is an algorithmic differentiation (AD) based real-time EMPC algorithm including a software implementation in ACADO Code Generation. The scheme is based on a novel memory efficient, symmetric AD approach for real-time propagation of second order derivatives. This is used inside a tailored multiple-shooting based SQP method, which employs a mirrored version of the exact Hessian. The performance of the proposed auto-generated EMPC algorithm is demonstrated for the optimal control of a nonlinear biochemical reactor benchmark case-study. A speedup of a factor more than 2 can be shown in the CPU time for integration and Hessian computation of this example.

An economic objective for the optimal experiment design of nonlinear dynamic processes

State-of-the-art formulations of optimal experiment design problems are typically based on a design criterion which allows us to optimize a scalar map of the predicted variance-covariance matrix of the parameter estimate. Famous examples for such scalar objectives are the A-criterion, the E-criterion, or the D-criterion, which aim at minimizing the trace, maximum eigenvalue, or determinant of the variance-covariance matrix. In this paper, we propose a different way of deriving an economic design criterion for the optimal experiment design. Here, the corresponding analysis is based on the assumption that our ultimate goal is to solve an optimization problem with a given economic objective that depends on uncertain parameters, which have to be estimated by the experiment. We illustrate the approach by studying a fedbatch bioreactor.

Gasds: a Kinetic-based Package for Biomass and Coal Gasification

GASDS: a Kinetic-Based Package for Biomass and Coal Gasification Lorenzo Cabianca, Andrea Bassani, André F. Amaral, Francesco Rossi, Giulia Bozzano, Eliseo Ranzi, Dries Telen, Filip Logist, Jan Van Impe, Flavio Manenti a Politecnico di Milano, Dipartimento di Chimica, Materiali ed Ingegneria Chimica “Giulio Natta”, Piazza Leonardo da Vinci 32, 20133 Milano, Italy b KULeuven, University of Leuven, Department of Chemical Engineering, BioTeC+, Gebroeders De Smetstraat 1, B-9000 GENT, BELGIUM flavio.manenti@polimi.it

A differentiable reformulation for E-optimal design of experiments in nonlinear dynamic biosystems

Informative experiments are highly valuable for estimating parameters in nonlinear dynamic bioprocesses. Techniques for optimal experiment design ensure the systematic design of such informative experiments. The E-criterion which can be used as objective function in optimal experiment design requires the maximization of the smallest eigenvalue of the Fisher information matrix. However, one problem with the minimal eigenvalue function is that it can be nondifferentiable. In addition, no closed form expression exists for the computation of eigenvalues of a matrix larger than a 4 by 4 one. As eigenvalues are normally computed with iterative methods, state-of-the-art optimal control solvers are not able to exploit automatic differentiation to compute the derivatives with respect to the decision variables. In the current paper a reformulation strategy from the field of convex optimization is suggested to circumvent these difficulties. This reformulation requires the inclusion of a matrix inequality constraint involving positive semidefiniteness. In this paper, this positive semidefiniteness constraint is imposed via Sylversters criterion. As a result the maximization of the minimum eigenvalue function can be formulated in standard optimal control solvers through the addition of nonlinear constraints. The presented methodology is successfully illustrated with a case study from the field of predictive microbiology.

Robust Optimal Experiment Design: A Multi-Objective Approach

Abstract Optimal Experiment Design (OED) is an indispensable tool in order to reduce the amount of labour and cost intensive experiments in the modelling phase. The unknown parameters are often non-linearly present in the dynamic process models. This means that the Fisher Information Matrix also depends on the current guess for the parameters. In the early stage of the modelling phase these estimates are often highly uncertain. So designing an optimal experiment without taking this uncertainty into account is troublesome. In order to obtain an informative experiment, a robust optimisation approach is necessary. In recent work a formulation using an implicit weighted sum approach is proposed where the objective function is split in a nominal optimal experiment design part and a robust counterpart. This weighted sum has well known drawbacks in a Multi-Objective Optimisation approach. In this work these objectives are studied using advanced methods like the Normal Boundary Intersection and the Normalised Normal Constraint. In this way, the experimenter gets an overview of the different experiments possible. Furthermore, in past work the necessary third order derivatives are approximated using a finite different approach. The results in this work are obtained using exact third order and fourth order derivatives by exploiting the symbolic and automatic derivation methods implemented in the ACADO-toolkit.

Self-reflective model predictive control

This paper proposes a novel control scheme, named self-reflective model predictive control (MPC), which takes its own limitations in the presence of process noise and measurement errors into account. In contrast to existing output-feedback MPC and persistently exciting MPC controllers, the proposed self-reflective MPC controller not only propagates a matrix-valued state forward in time in order to predict the variance of future state estimates, but it also propagates a matrix-valued adjoint state backward in time. This adjoint state is used by the controller to compute and minimize a second order approximation of its own expected loss of control performance in the presence of random process noise and inexact state estimates. The properties of the proposed controller are illustrated with a small but nontrivial case study.

SolACE: An Open Source Package for Nolinear Model Predictive Control and State Estimation for (Bio)Chemical Processes

Abstract In spite of the wide spread use of Nonlinear Model Predictive Control (NMPC) in large chemical companies, the small and medium enterprises (SMEs) remain oblivious of its potential mostly due to the large investment costs and in-house expertise required. This paper presents an open source python based simulation environment - SolACE, which can aid SMEs in realizing the full potential of the advanced control techniques. The syntax to introduce the controller and plant models is straightforward, enabling even non-experts to easily formulate the control (and estimation) problems. With the developed package, SMEs can consider the implementation of NMPC on their processes without any overlaying costs or large technical know-how. From a research perspective, the current package can be used as a building block to develop toolkits for advanced control strategies like robust or economic NMPC. It also provides researchers a way to test various models in an NMPC framework without the hustle of having to write the discretization and optimization routines themselves.