Safe model-based design of experiments using Gaussian processes

https://doi.org/10.1016/j.compchemeng.2021.107339Get rights and content

Highlights

  • Safe application of experimental design using Gaussian processes.

  • Combination of trust-region ideas to balance exploration vs exploitation.

  • Surrogate based approximation for Fisher information matrix.

  • Construction of hybrid model for constraint satisfaction.

Abstract

The construction of kinetic models has become an indispensable step in developing and scale-up of processes in the industry. Model-based design of experiments (MBDoE) has been widely used to improve parameter precision in nonlinear dynamic systems. Such a framework needs to account for both parametric and structural uncertainty, as the physical or safety constraints imposed on the system may well turn out to be violated, leading to unsafe experimental conditions when an optimally designed experiment is performed. In this work, Gaussian processes are utilized in a two-fold manner: 1) to quantify the uncertainty realization of the physical system and calculate the plant-model mismatch, 2) to compute the optimal experimental design while accounting for the parametric uncertainty. The proposed method, Gaussian process-based MBDoE (GP-MBDoE), guarantees the probabilistic satisfaction of the constraints in the context of model-based design of experiments. GP-MBDoE is assisted with the use of adaptive trust regions to facilitate a satisfactory local approximation. The proposed method can allow the design of optimal experiments starting from limited preliminary knowledge of the parameter set, leading to a safe exploration of the parameter space. This method’s performance is demonstrated through illustrative case studies regarding the parameter identification of kinetic models in flow reactors.

Introduction

Biochemical, physicochemical or catalytic processes are often too complex to develop accurate physics-based only models, and plant-model mismatch is inevitable. It is common to develop approximate kinetic models under some assumptions regarding the system’s mechanism to represent the physical system accurately. The availability of a trustworthy kinetic model could be used to predict the system’s behaviour outside of the validated experimental conditions and then be employed for design, optimization and control in process systems engineering applications (Bonvin et al. (2016)). The model identification requires experimental data to evaluate the validity of a proposed model and estimate the model parameters to match the physical system’s behaviour over the selected range of operating conditions. Nevertheless, the experimental procedure may become a costly, very time consuming and infeasible task when poor experimental designs are implemented. Identifying an appropriate approximated model is highly dependent on limitations due to the certain observability of the physical phenomena. Additionally, the respected data may be difficult and expensive to obtain because of physical accessibility and/or limitations on the experimental budget.

Model-based design of experiments (MBDoE) has been widely utilized to improve parameter precision in highly nonlinear dynamic systems. The main goal of MBDoE is to design control inputs and sampling schedules (Franceschini and Macchietto (2008)) to achieve the most informative experiment possible by satisfying constraints on feasibility and safety of operation. The optimal MBDoE problem for improving parameter estimation involves the maximization of a measure of the expected information (or the minimization of a measure of the expected variance of model parameters) by acting on experiment decision variables to ensure the feasibility of the experimental trials. However, since the optimization problem is based on the available model, both plant-model mismatch and parametric uncertainty may significantly affect the result (Petsagkourakis et al. (2020b); Pan et al. (2020)).

The literature consists of various approaches to ensure optimality and feasibility, particularly under the presence of parametric uncertainty) Pronzato and Walter (1985)). Two distinctly different scenarios are considered regarding the information and the constraint satisfaction, robust and stochastic approaches. A robust approach for the information content was proposed in Asprey and Macchietto (2002), where the predicted information content of experiments needs to be optimal for the whole space of the uncertain parameters. The robust optimization problem is usually solved in terms of min-max optimization, in which uncertainties are typically assumed to be deterministic and bounded (Pronzato and Walter (1985); Körkel et al. (2004); Welsh and Kong (2011); Rojas et al. (2007)). This problem is often numerically intractable when candidate models are highly nonlinear. Various methods have been proposed to accommodate this limitation, including linearization of the inner optimization (Körkel et al. (2004)) and a sensitivity-based approximate robust approach assuming ellipsoidal joint confidence region model parameters (Nimmegeers et al. (2020)). The robust approaches are typically conservative as they require the optimality and satisfaction of constraints to hold for all the possible values within the predefined uncertainty region. On the contrary, a stochastic approach has been popularized, where the information content is maximized in expectation, and the constraints are satisfied with a given probability (Asprey and Macchietto (2002). The stochastic (also called probabilistic) experimental design framework avoids the typical strong conservatism of the worst-case MBDoE techniques, as the probability of occurrence of different realizations is accounted for differently. Various approaches (Telen et al. (2014); Nimmegeers et al. (2020); Galvanin et al. (2010); Streif et al. (2014); Mesbah and Streif (2015)) have been proposed in this family of problems usually inspired by the techniques originated from optimal control problem.

The satisfaction of chance constraints has been widely studied in the field of optimal control, especially from the stochastic model predictive control community (SMPC). Comprehensive reviews regarding SMPC can be found in (Mesbah (2016); Farina et al. (2016); Geletu et al. (2013). Additionally, both parametric and structural mismatches have been incorporated in (Hewing et al. (2020); Arellano-Garcia et al. (2020)). Probabilistic (chance) constraints are incorporated into the MBDoE problem to seek a trade-off between a designed experiment’s information content and allow for a user-defined level of risk during the experiment. Most approaches compute the expected information given the (previous) parameter identification procedure, where confidence regions are constructed using approximation techniques (Perić et al. (2018)) around the point estimates for the parameters.

Galvanin et al. (2010) directly utilized a Monte Carlo approach to compute the constraint tightening (backoff) given the approximated probability distribution of the parameters. Such a method is very accurate for estimating the constraint tightening for the user-defined constraint satisfaction but computationally costly. .Telen et al. (2014) proposed the use of the unscented transformation (Van Der Merwe and Wan (2001)) to propagate the parametric uncertainty in the objective function and the constraints. The unscented transformation has widely be utilized in the optimal control (Bradford and Imsland (2018a)) due to the low computational complexity. Nevertheless, unscented transformation approaches may fail in the presence of strong nonlinear relationships between the parameters and the states of the model. In this context, polynomial chaos expansion (PCE) has been used to compute the required moments of the objective and constraints in Mesbah and Streif (2015) and recently in Nimmegeers et al. (2020). Polynomial chaos expansions have been used in stochastic model predictive control for the purposes of uncertainty propagation through a nonlinear model Wiener (1938). Similarly, Gaussian process (GP) (Rasmussen and Williams (2006); Wang et al. (2020)) has been proposed as an alternative to PCE, as they are non-parametric models. Gaussian processes have been employed due to the estimation of uncertainty of prediction around points caused by an insufficient amount of data-points. The GP usually assumes an independent and identically distributed (iid) Gaussian noise added in the observed measurements, and the outputs obtained at different inputs are jointly Gaussian distributed, where a kernel function defines the covariance. Gaussian process has not found a significant application in MBDoE, with exception in Olofsson et al. (2018) in the context of model discrimination. In this case the GP is used to approximate the different models and marginalize the parameters of the models given the computed uncertainty. Nevertheless, GP has extensively be used in the area of optimal control, where the prior models and black-box techniques are combined to describe the system. GP may be coupled with approximate method for propagating the uncertainty as unscented Kalman Filter (Ko et al. (2007)), exact moment matching (Girard et al. (2003); Deisenroth et al. (2015)), linearization (Girard et al. (2003) and scenario-based approaches (Umlauft et al. (2018); Bradford et al. (2020)). Recently, GPs have been used in hybrid modeling rational, where they are coupled with the prior (nominal) model that is assumed for the system, and learning is conducted only for the GP (del Rio Chanona et al. (2021); Hewing et al. (2020)).

The methods of robust MBDoE, e.g. (Asprey and Macchietto (2002); Nimmegeers et al. (2020)) use the confidence regions computed in the parameter estimation step. Such an assumption may be sufficient to accommodate the uncertainty for the objective of the information content. However, constraint satisfaction should require different treatment. The satisfaction of the constraints is typically guaranteed given the uncertainty of the parameters and the required assumptions, including the availability of a large number of available prior experimental data points. In the typical scenario of paucity of data at the beginning and a presence of model mismatch, these assumptions do not hold. As the methodology is model-based, both model mismatch (i.e., a model structure inadequate to represent the physical systems) and parametric mismatch (i.e., incorrect values of the parameters) may affect the consistency of the whole design procedure (Perić et al. (2018); Petsagkourakis et al. (2020c)). These assumptions have been avoided in some works using a disturbance based estimation (Galvanin et al. (2012)) and the MBDoE in the context of guaranteed parameter estimation (Reddy et al. (2017)). Additionally, Quaglio et al. (2018a) developed a method to find the region where the approximated model is valid and avoid the intrinsic issues with the model-mismatch in MBDoE methods (Quaglio et al. (2018b)).

In this work, we propose a novel MBDoE framework suitable for guaranteeing operation feasibility when a plant-model mismatch is strongly present, i.e. possible disturbances/hardware malfunctions and the initial limited data. The plant-model mismatch is a critical issue in optimal control, where stability (Petsagkourakis et al. (2020a); Petsagkourakis et al. (2021)), offset-free tracking (Paulson et al. (2019)) and satisfaction of constraints has been extensively studied (Hewing et al. (2020)).

We treat the objective function of the MBDoE by utilizing a stochastic surrogate, i.e. GP. Additionally, we focus on the satisfaction of constraints that has not been addressed in the literature for the optimal experimental designs avoiding the use of covariance of the parameters that have been computed from the parameter estimation. Gaussian process is utilized to compute the model mismatch together with a trust region around operational point, that is updated iteratively. This is inspired by the derivative-free methods (Conn et al. (2005)) that were recently applied in del Rio Chanona et al. (2021). This way, restrictive assumptions of the nature of the ‘real’ model are avoided.

The structure of this paper is as follows. The problem definition is outlined in Section 2, the details of the proposed method for safe, optimal experimental design is presented in Section 3. A case study is presented in Section 4, where the framework is applied to two separate case studies, and in the last section, conclusions are outlined.

Section snippets

Problem definition

We assume that the process under consideration can be represented by a set of differential and algebraic equations (DAEs), where measured variables y can only be measured in finite sampling times {1,,N}f(x˙,x,u,w,θ*)=0y=h(x,u,ν),where xRnx, uRnu, yRny and θ*Rnθ* are the vector of state variables representing the true system, vector of control (manipulated) variables, vector of measured (output) variables and parameters of the process, correspondingly. Additionally, the physical system is

Methodology

This section presents the proposed methodology. Gaussian process is used in a two-fold manner: 1) to represent the model mismatch between the available model and the collected data is approximated using GP. To ensure the reliability of the predictions of the GP, trust regions will be applied to constrain the region of experimental conditions where predictions are reliable. 2) An additional GP is then employed to compute the expected value and variance of the objective (i.e. metric of expected

Case studies

The proposed methodology (GP-MBDoE) is illustrated using two case studies with different scenarios simulated in silico:

  • Case Study 1: a plug flow reactor where that the assumed mechanism is different than the ‘real’ one with an additional constant disturbance. The model contains 4 parameters and has 2 decision variables.

  • Case Study 2: a plug flow reactor that a Nucleophilic aromatic (SnAr) substitution Hone et al. (2017) reaction takes place, where the concentrations have a non-constant

Discussion

In this section, an additional discussion regarding the results of the case studies is added. In the case studies, theour proposed method (GP-MBDoE) was compared with two alternative methods, a method that deals with parametric only uncertainty (MC-MBDoE) and a disturbance based estimation method (DE-MBDoE). The GP-MBDoE has the main advantage: it can deal with potential model-mismatch and have as good performance as the standard MBDoE techniques in terms of information metrics. The

Conclusions

In this article a new methodology for the safe MBDoE based on GP and trust regions in the presence of structural uncertainty. GP-MBDoE was proposed and discussed; The optimal design of an experiment for improving parameter estimation is a particular form of optimization problem that can be very effective, but both optimality and feasibility of the designed experiment are important issues to consider. The proposed strategy allows to restrict the feasible space maintaining the optimal design in

CRediT authorship contribution statement

Panagiotis Petsagkourakis: Conceptualization, Software, Methodology, Formal analysis, Writing - original draft, Visualization. Federico Galvanin: Conceptualization, Methodology, Writing - review & editing, Supervision.

Declaration of Competing Interest

The authors declare no conflict of interest.

Acknowledgment

This project has received funding from EPSRC (EP/R032807/1). The support is gratefully acknowledged.

References (66)

  • J.A. Paulson et al.

    Mixed stochastic-deterministic tube MPC for offset-free tracking in the presence of plant-model mismatch

    J. Process Control

    (2019)
  • N.D. Perić et al.

    Set-membership nonlinear regression approach to parameter estimation

    J. Process Control

    (2018)
  • P. Petsagkourakis et al.

    Stability analysis of piecewise affine systems with multi-model predictive control

    Automatica

    (2020)
  • P. Petsagkourakis et al.

    Reinforcement learning for batch bioprocess optimization

    Computers & Chemical Engineering

    (2020)
  • L. Pronzato et al.

    Robust experiment design via stochastic approximation

    Math. Biosci.

    (1985)
  • M. Quaglio et al.

    A model-based data mining approach for determining the domain of validity of approximated models

    Chemometrics and Intelligent Laboratory Systems

    (2018)
  • M. Quaglio et al.

    Constrained model-based design of experiments for the identification of approximated models

    IFAC-PapersOnLine

    (2018)
  • M. Quaglio et al.

    An artificial neural network approach to recognise kinetic models from experimental data

    Computers & Chemical Engineering

    (2020)
  • M. Quaglio et al.

    An online reparametrisation approach for robust parameter estimation in automated model identification platforms

    Computers & Chemical Engineering

    (2019)
  • W.C. Rooney et al.

    Design for model parameter uncertainty using nonlinear confidence regions

    AlChE J.

    (2001)
  • D. Telen et al.

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

    Computers & Chemical Engineering

    (2014)
  • M.K. Titsias

    Variational learning of inducing variables in sparse Gaussian processes

  • J.S. Welsh et al.

    Robust experiment design through randomisation with chance constraints

    IFAC Proceedings Volumes

    (2011)
  • N. Wiener

    The homogeneous chaos

    American Journal of Mathematics

    (1938)
  • E. Arcari et al.

    Dual stochastic mpc for systems with parametric and structural uncertainty

    arXiv preprint arXiv:1912.10114

    (2019)
  • F. Berkenkamp et al.

    Safe model-based reinforcement learning with stability guarantees

    Adv. Neural. Inf. Process. Syst.

    (2017)
  • D. Bonvin et al.

    Linking models and experiments

    Ind. Eng. Chem. Res.

    (2016)
  • E. Bradford et al.

    Stochastic nonlinear model predictive control using gaussian processes

    2018 European Control Conference, ECC 2018

    (2018)
  • D.G. Brown et al.

    Analysis of past and present synthetic methodologies on medicinal chemistry: where have all the new reactions gone?

    J. Med. Chem.

    (2016)
  • C. Cartis et al.

    Adaptive cubic regularisation methods for unconstrained optimization. part i: motivation, convergence and numerical results

    Math. Program.

    (2011)
  • A.R. Conn et al.

    Introduction to derivative-free optimization

    (2005)
  • M.P. Deisenroth et al.

    Gaussian processes for data-Efficient learning in robotics and control

    IEEE Trans. Pattern Anal. Mach. Intell.

    (2015)
  • P.I. Frazier

    A tutorial on bayesian optimization, eprint: 1807.02811

    (2018)
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