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

doi:10.1016/j.automatica.2014.10.100

Automatica

Volume 51, January 2015, Pages 98-103

https://doi.org/10.1016/j.automatica.2014.10.100 Get rights and content

Abstract

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.

Introduction

Nonlinear differential equation models are nowadays indispensable tools for the analysis, design, operation and optimization of dynamic processes. For an accurate modeling, it is necessary to collect experimental data by performing experiments. To limit this experimental burden optimal experiment design (OED) methods have been developed. The idea is to design experiments which reveal the highest amount of information. The field of OED (for parameter estimation) has been founded by Fisher (1935) and has been extended to static linear and nonlinear models in Box and Lucas (1959) and Kiefer and Wolfowitz (1959). The transition to dynamic systems has been accomplished in Gevers and Ljung (1986) and Mehra (1974) for the linear and in Espie and Machietto (1989) for the nonlinear case. For a more detailed overview, the reader is referred to Franceschini and Macchietto (2008) and Pukelsheim (1993). With respect to numerical implementations, state-of-the-art methods are described in Balsa-Canto, Alonso, and Banga (2010), Hoang, Barz, Merchan, Biegler, and Arellano-Garcia (2013), Körkel, Kostina, Bock, and Schlöder (2004), Schenkendorf, Kremling, and Mangold (2009) and Telen et al. (2013). In practice, model-based process optimization is meant to improve the performance of the process without spending (too) much effort on performing experiments.

This paper follows the philosophy to design experiments with respect to the intended model application, a well-established concept for linear systems (Gevers & Ljung, 1986), in particular, in the context of joint design for control and identification (Gevers, 1993, Hjalmarsson, 2005). However, in OED for nonlinear dynamic processes these concepts are less established, and thus we propose in this paper a way to formulate a design criterion that leads to a new concept named the economic optimal experiment design for nonlinear dynamic systems. We assume that our ultimate goal is to solve an optimal control problem with economic objective that depends on an unknown parameter vector $p$ . If we solve this optimal control problem based on an estimate of the parameters $p$ in place of their unknown exact values, we will find a sub-optimal control input. Now, the aim of economic OED is to reduce the expected optimality gap that is associated with solving the optimal control problem based on an estimate for $p$ . The contribution is that we formulate and approximately solve such economic OED problems, yielding an optimally weighted A-criterion that is invariant under affine parameter transformations.

We start in Section 2 with a motivating example and briefly review the idea of OED in Section 3. Our contribution is presented in Sections 4 Second order expansion of optimality loss, 5 Economic optimal experiment design, where we discuss how to formulate economic OED problems. Section 6 presents a case study and Section 7 the conclusions.

Section snippets

A motivating example

We consider a dynamic model for a continuously stirred tank reactor (CSTR) in which a Van de Vusse reaction takes place (Bonilla, Diehl, Logist, De Moor, & Van Impe, 2010): $A \overset{k_{1}}{\to} B \overset{k_{2}}{\to} C and 2 A \overset{k_{3}}{\to} D$ . Since the substances C and D are unwanted and do not react further, our dynamic model is given by ${\dot{c}}_{A} = \frac{\dot{V}}{V_{R}} (c_{A 0} - c_{A}) - k_{1} c_{A} - k_{3} c_{A}^{2}$ ${\dot{c}}_{B} = - \frac{\dot{V}}{V_{R}} c_{B} + k_{1} c_{A} - k_{2} c_{B},$ where $c_{A}$ and $c_{B}$ are the concentrations of the substances A and B. The feed inflow has a known concentration $c_{A 0} = 5.1 \frac{mol}{L}$ and its flow rate $\dot{V}$ can be

Optimal experiment design

We are interested in a maximum likelihood parameter estimation problem of the form $min_{x, p} \frac{1}{2} {‖ M (x, p) - η ‖}_{Σ^{- 1}}^{2} + \frac{1}{2} {‖ p - \hat{p} ‖}_{Σ_{0}^{- 1}}^{2}$ $s.t. G (x, u, p) = 0 .$ Here, $p \in R^{n_{p}}$ is the parameter which we want to estimate, $u \in R^{n_{z}}$ is a given control input which can be adjusted for taking the measurements, and $η \in R^{n_{M}}$ is the measurement value. The measurement function $M$ and the right-hand side function $G$ are assumed to be continuously differentiable. The function $G$ can for example denote a steady state equation, where $x$ would

Second order expansion of optimality loss

Our ultimate goal is to solve the “economic” optimization problem $min_{x, u} F (x, u, p_{real}) s.t. {\begin{matrix} G (x, u, p_{real}) = 0 \\ H (u) \leq 0, \end{matrix}$ which depends on an unknown parameter $p_{real} \in R^{n_{p}}$ . The functions $F$ , $G$ , and $H$ are assumed to be twice continuously differentiable and the state $x_{s} (u, p_{real})$ of the real dynamic process is for any given $u$ assumed to be determined uniquely by the equation $G (x_{s} (u, p_{real}), u, p_{real}) = 0 .$ Similar to the non-degeneracy condition from the previous section, the matrix $\frac{\partial G (x_{s} (u, p_{real}), u, p_{real})}{\partial x}$ is

Economic optimal experiment design

Let $p$ be a random variable with expectation value $E {p} = p_{real}$ and let the variance–covariance matrix $E_{p} (p - p_{real}) {(p - p_{real})}^{T} = Σ_{p} \in S_{+}^{n_{p}}$ be given. Now, if either

A1
the function $Δ$ is smooth and has bounded moments, or
A2
the function $Δ$ is twice continuously differentiable and the diameter of the support of the probability distribution of $p$ is of order $O (‖ Σ_{p} ‖)$ ,

then we have

E_{p} {Δ (p)} = \frac{1}{2} Tr (W (p_{real}) Σ_{p}) + O ({‖ Σ_{p} ‖}^{\frac{3}{2}}) .

This statement follows by using Theorem 1 in combination with well-known moment expansion techniques

Economic optimal experiment design for a dynamic fedbatch bioreactor

The dynamic model equations of a well mixed fedbatch bioreactor benchmark are given by Telen et al. (2013): $\frac{d C_{s}}{d t} = - σ (C_{s}) C_{x} + \frac{u}{V_{R}} C_{s, in} - \frac{u}{V_{R}} C_{s}$ $\frac{d C_{x}}{d t} = μ (C_{s}) C_{x} - \frac{u}{V_{R}} C_{x}$ $\frac{d V_{R}}{d t} = u .$ Here, $C_{s} (g / L)$ is the concentration limiting substrate, $C_{x} (g / L)$ the biomass concentration and $V_{R} (L)$ the volume of liquid in the reactor. The control input $u (L / h)$ is the volumetric feed rate, containing a substrate concentration $C_{s, in}$ . The specific growth rate $μ (C_{s})$ is of the Monod type: $μ (C_{s}) = μ_{\max} \frac{C_{s}}{K_{s} + C_{s}}$ . The substrate consumption

Conclusions

In this paper we have discussed a new economic design objective for optimal experiment design problems for nonlinear dynamic systems. This design objective aims at minimizing the expected optimality gap that is associated with solving a given optimal control problem based on the parameter estimate in place of the real but unknown parameter values. One of the main results of this paper has been presented in Section 5, where we have proven that economic design objectives can be interpreted as

Acknowledgments

D. Telen has a Ph.D. grant of the Agency for Innovation through Science and Technology in Flanders (IWT), Grant-Nr.: 101643. J. Van Impe holds the chair Safety Engineering sponsored by the Belgian Chemistry and Life Sciences Federation essenscia.

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Boris Houska received a diploma in mathematics and physics from the University of Heidelberg in 2007, and a Ph.D. degree in Electrical Engineering from KU Leuven in 2011. From 2012 to 2013 he was a postdoctoral researcher at the Centre for Process Systems Engineering at Imperial College London. From 2013 to 2014 Boris Houska worked as a faculty member at the Department of Automation at Shanghai Jiao Tong University. In August and September 2014 he was a guest professor at the Institute for Microsystems Engineering at the University of Freiburg. Since October 2014 Boris Houska is an assistant professor at the School of Information Science and Technology at ShanghaiTech university. His research interests include numerical optimization and optimal control, robust and global optimization, as well as fast model predictive control algorithms.

Dries Telen received the M.Sc. degree in Mathematical Engineering from KU Leuven in 2010 and is pursuing a Ph.D. degree at the same university. His research focuses on optimal experiment design for nonlinear dynamic systems, robust dynamic optimization, multiobjective optimization and sequential convex optimization.

Filip Logist is an assistant professor at the chemical engineering department of KU Leuven. He received his M.Sc. and Ph.D. degrees in 2003 and 2008, respectively, at the same university. His main research interests include model-based optimization and control of dynamic processes with a focus on (bio)chemical applications. Methodologies related to the modeling research line are model structure selection, model parameter estimation, and optimal experiment design. Approaches connected to the optimization and control research line relate to: dynamic optimization and optimal control, optimization-based control, multiple-objective optimization, and robust optimization. He is one of the two Belgian representatives in the Computer-Aided Process Engineering (CAPE) Working Party of the European Federation of Chemical Engineering (EFCE).

Moritz Diehl studied Mathematics and Physics at the universities of Heidelberg and Cambridge in 1993–1999 and received his Ph.D. degree in scientific computing in 2001 from Heidelberg University. From 2006 to 2013 he was an associate professor at the electrical engineering department of KU Leuven and the Principal Investigator of KU Leuven’s Optimization in Engineering Center (OPTEC). Since 2013 he is a full professor at the Institute for Microsystems Engineering at the university of Freiburg. His research interests are in numerical optimization and control, in particular algorithms for embedded, nonlinear, convex, as well as robust optimization, and their applications in engineering, mostly in mechatronics and robotics, signal processing, process control, and renewable energy systems. Recent research focuses on the automatic control of tethered air-planes for high altitude wind power generation, for which he obtained a grant of the European Research Council (ERC) from 2011 to 2016.

Jan F.M. Van Impe is a full professor at the chemical engineering department of KU Leuven. He received his M.Sc. degree from the University of Gent in 1988, and his Ph.D. degree from the KU Leuven in 1993. Immediately thereafter he founded the BioTeC (Chemical and Biochemical Process Technology and Control) research group. In the period 2005–2011 he has served as the Departmental Head. Since academic year 2006–2007 he is also a visiting professor at the UA (University Antwerpen). He is a founding partner of the KU Leuven Center-of-Excellence OPTEC (Optimization in Engineering) in 2005, and at present he is coordinating OPTECs continuation (Phase II: 2010–2017). In 2008 he started the Flemish Cluster for Predictive Modeling in Foods—CPMF2 [KU Leuven/BioTeC & UGent/LFMFP], to facilitate the transfer of the broad expertise in the area of predictive microbiology to industry/government. From 2009 on, he holds the essenscia-chair, funded by the Belgian Chemical and Life Sciences Industry platform.

^☆: The research was supported by the KU Leuven Research Fund: PFV/10/002 (OPTEC), OT/10/035, GOA/10/09, GOA/10/11, the KU Leuven Industrial Research Fund: KP SCORES4CHEM, FWO: projects: FWO KAN2013 1.5.189.13, FWO-G.0930.13 and BelSPO: IAP VII/19 (DYSCO), SBO LeCoPro, FP7-SADCO (MC ITN-264735), Eurostars SMART, ERC ST HIGHWIND (259 166). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor David Angeli under the direction of Editor Andrew R. Teel.

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Brief paperAn economic objective for the optimal experiment design of nonlinear dynamic processes☆

Abstract

Introduction

Section snippets

A motivating example

Optimal experiment design

Second order expansion of optimality loss

Economic optimal experiment design

Economic optimal experiment design for a dynamic fedbatch bioreactor

Conclusions

Acknowledgments

Chemical Engineering Science

Automatica

Automatica

ESCAPE

Journal of Process Control

Chemical Engineering Science

An iterative identification procedure for dynamic modeling of biochemical networks

BMC Systems Biology

A multiple shooting algorithm for direct solution of optimal control problems

A convexity-based homotopy method for nonlinear optimization in model predictive control

Optimal Control Applications & Methods

Design of experiments in non-linear situations

Biometrika

On the role of constraints in optimization under uncertainty

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