This section illustrates the use of extent-based incremental model identification solved to global optimality on a problem that exhibits more than one local minimum. For this, let us consider a batch reactor of constant volume, in which the enzymatic decomposition S → 2 I and the product formation I → P take place. The objective is to identify the maximal rate
Vma x and the inhibition constant KDof the enzymatic decomposition. The concentrations of
S, I and P are denoted as cS, cI and cP, and c =
cS cI cP
T
. The stoichiometry is given by N = −1 2 0
0 −1 1. The kinetics of the decomposition reaction expresses the behavior of
an enzyme with two binding sites of equal binding affinity, no cooperativity, and previously known substrate inhibition [107]:
r(c, α, θ) = Vma x cS KD+0.1 c2S K2D 1 + 2cS KD + c2S K2D , (4.82)
with the parameter α = Vma x = 3 mol L−1 min−1 appearing linearly and the parameter
θ = KD=0.32 mol L−1appearing nonlinearly in the rate law, that is, L = 1 and N = 1. The
dynamics of cS is described by
˙cS= −r(c, α, θ ), cS(0) = cS,0=2 mol L−1. (4.83) An experiment is run for 3 min and noise-free measurements of the concentration cS are collected every 5 s. If the measurements ˜cS were corrupted by noise, it would be possible
to compute unbiased estimates of r(c, α, θ) as shown in Section 4.3.2. In addition, if the measurements ˜cP were also available, one would be able to compute uncorrelated extents 102
4.6. Conclusion 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 t[min] cS [m o l L − 1]
Figure 4.5 – Measured concentrations (circles) and fitted profiles that result from the so- lution to the identification problem using standard (dashed line) and convex (solid line) optimization algorithms.
˜xr,1and ˜xr,2according to Section 4.3.1, using
˜xr,1= V
cS,0− ˜cS , (4.84a)
˜xr,2= V˜cP− cP,0 . (4.84b)
The noise-free case is considered here. Even in this case, the contour plot of the identifi- cation cost function J(Vma x, KD)in Figure 4.4 shows the presence of two local minima. De- pending on the initial guess that is used, a standard gradient-based optimization algorithm may not converge to the correct values of the parameters Vma x and KD. For example, a stan-
dard optimization algorithm with user-supplied gradients and using the initial guess KD= 0.04 mol L−1 yields the solution V∗
ma x = 6.63 mol L−1 min−1, KD∗ = 0.001 mol L−1, with
J (Vma x∗ , K∗D) = 0.0702. However, the convex optimization algorithm presented in Section 4.4, using n = 20 and ¯θ =1 mol L−1, yields the global solution V∗
ma x = 3 mol L−1min
−1, KD∗ =0.32 mol L−1, with J(Vma x∗ , KD∗) =4.4 × 10−9. Figure 4.5 shows that the fitted curves that result from these two solutions are different, and only the convex algorithm predicts a concentration profile that matches the measurements.
In addition, one can simulate the substrate concentration using a different initial condi- tion, that is, cS,0=3 instead of 2 mol L−1. Figure 4.6 shows that the two models are indeed
significantly different.
4.6
Conclusion
This chapter has shown that extent-based incremental model identification can be used to converge efficiently to global optimality. Several features of extent-based incremental model identification contribute to this result. Firstly, the cost function that results from this
Chapter 4. Estimation of Kinetic Parameters via the Incremental Approach 0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 3 t[min] cS [m o l L − 1]
Figure 4.6 – Simulated concentrations with a different initial condition corresponding to the two models that were obtained using standard (dashed line) and convex (solid line) optimization algorithms for model identification.
approach involves only the parameters of a single rate candidate and is a quadratic func- tion of the parameters in which the rate expression is linear. Since these parameters can be determined via matrix inversion, the number of parameters that need to be determined via optimization is much smaller than in identification problems that result from the simultane- ous approach. Secondly, this cost function can be approximated via a Taylor series expansion as a rational function of the parameters that appear nonlinearly in the rate expression. This rational function is used in the formulation of the identification problem as a polynomial optimization problem with constant coefficients computed only once prior to optimization. Finally, this polynomial optimization problem can be converted to an SDP, which can be handled by SDP solvers that efficiently attain the global solution upon convergence.
As a consequence, guaranteed convergence to global optimality via the extent-based incremental approach exists for virtually all identification problems in reaction systems, provided that some mild technical conditions are satisfied. For many of these problems, it would be practically infeasible to obtain global optimality via the standard simultaneous approach, due to the large number of model parameters and the many possible combina- tions of rate candidates. As shown by the simulated example in this chapter, identification problems with more than one local minimum exist, and standard optimization algorithms may converge to a local minimum that is not the global one.
This chapter has also shown that extent-based incremental model identification not only converges to global optimality, but can also be used to provide maximum-likelihood parameter estimates, with quality similar to simultaneous model identification. Maximum- likelihood parameter estimation relies on (i) a method to obtain uncorrelated experimental extents from a set of uncorrelated measurements that depend linearly on these extents, and (ii) a method to obtain unbiased rate estimates computed from measurements cor- rupted by zero-mean noise. In other words, the experimental extents are computed from 104
4.6. Conclusion
measured concentrations such that they are uncorrelated, whereas the modeled extents correspond to the integral of unbiased rate estimates computed from measured concentra- tions. Both the computation of uncorrelated experimental extents and the use of unbiased rate estimation contribute decisively to being able to estimate optimal parameters in the maximum-likelihood sense. Future work shall focus on ensuring that uncorrelated exper- imental extents can be obtained even when the measurements are correlated or depend nonlinearly on the extents.
Another relevant extension of this work is the application of the extent-based incre- mental approach to reaction systems other than the lumped homogeneous reaction systems considered in this chapter. Since the concept of extents has been developed for other re- action systems, as mentioned in Section 4.2.1, such an extension seems to be relatively straightforward. In general, any reaction system can be described by a model in terms of (ordinary or partial) differential and algebraic equations that express the material and heat balances in the system. These models include reaction rates that can always be written as analytical expressions that relate concentrations and temperature to reaction rates. Hence, the existence of such analytical expressions assumed by the incremental approach does not represent a limitation for the extension of the approach to other systems. The only aspect that might hinder the use of the incremental approach in certain situations is the fact that it requires that the reaction rates be expressed as functions of measured quantities. This is realistic in the case of lumped homogeneous reaction systems but may be difficult in other reaction systems due to experimental constraints. In summary, one can foresee that it will be possible to design an extent-based incremental approach that guarantees globally opti- mal, maximum-likelihood estimates of kinetic parameters for any reaction system where the reaction rates can be expressed as functions of measured quantities.
5
Estimation of Rate Signals without
Kinetic Models
Part of this chapter is adapted from the postprint of the following article [108]:
D. Rodrigues, M. Amrhein, J. Billeter, and D. Bonvin. Fast estimation of plant steady state for imperfectly known dynamic systems, with application to real-time optimization.
Ind. Eng. Chem. Res., 57(10):3699–3716, 2018.
Link: http://doi.org/10.1021/acs.iecr.7b04631. Copyright © 2018 American Chemical Society
The author of this thesis contributed to that article by developing the main novel ideas, implementing the simulations, and writing a significant part of the text. Hence, the author retains the right to include the article in this thesis since it is not published commercially and the journal is referenced as the original source.
5.1
Introduction
Model identification, controller design, and process optimization are often regarded as closely related tasks since the control laws and the optimal decision policies are typically calculated using the plant model. For example, efficient control of reaction systems typically requires good kinetic models to predict the dynamic effects, namely the reaction rates.
Since the identification of reaction systems can be rather difficult and time consuming, one would ideally like to avoid it as much as possible. Hence, one could try to infer the reaction rates directly from measurements, that is, without the help of a kinetic model, which can be done if the various rates can be decoupled [99]. This decoupling would be an alternative to the use of observers for measurement-based rate estimation without kinetic models [2], the latter being difficult to design due to the coupling between the estimated states and the estimated rates.
The concept of variants and invariants has been proposed to decouple the dynamic ef- fects in reaction systems, thereby facilitating their analysis, control and monitoring [12, 11]. A finer separation of the various dynamic effects in both homogeneous and heterogeneous open reaction systems has been proposed along with a linear transformation of the numbers
Chapter 5. Estimation of Rate Signals without Kinetic Models
of moles and heat to a particular type of variants in chemical reaction systems, the so-called vessel extents [18, 19].
Since the concept of variants allows isolating the different rates in reaction systems, it can also be used to estimate unknown rates without the need for identifying the correspond- ing (kinetic) models. Subsequently, it is legitimate to ask whether applications of variants to control and optimization can be found.
Regarding the case of reaction systems, this chapter introduces methods that will al- low estimating reaction rates from concentration and temperature measurements via the concept of variants, which will be applied in subsequent chapters.
This chapter is structured as follows. The systems considered in this chapter are de- scribed in Section 5.2, a numerical differentiation filter that is relevant for the purpose of rate estimation, the Savitzky-Golay filter, is presented in Section 5.3, and the rate estima- tion method and its properties are shown in Section 5.4, while Section 5.5 concludes this chapter.