Observer and controller design for a methane bioconversion process

Highlights

• A methane bioconversion model is constructed as an unstructured mechanistic model.

• An observability and controllability analysis are executed with empirical Gramians.

• The control scheme estimates non-measured states by an extended Kalman filter.

• An LQR controller with integral feedback is designed to enable reference tracking.

Abstract

Methane is currently an emerging alternative feedstock for biological processes. In this paper, we study a particular methane to lactate bioconversion process based on the bacterium Methylomicrobium buryatense 5GB1, with the aim of designing suitable state estimators and controllers for the process. First, a nonlinear unsegregated, unstructured model, consisting of six dynamic mass flow balance equations and six state variables, is constructed and implemented as a dynamic simulator in MATLAB and Simulink. Simulation results match qualitatively with experimental data from literature. Second, an observability analysis is performed with the aim of finding suitable sensor configurations for state estimation. For each of the resulting observable configurations a linear Kalman filter is constructed, of which the performance is evaluated. This performance is not satisfactory and, therefore, extended and unscented Kalman filters are designed to overcome the nonlinear nature of the system. Third, a controllability analysis is executed to establish controllability with respect to the process inputs. Both the observability and controllability give an assessment of practical observability and controllability by using empirical Gramians. A linear quadratic regulator (LQR) with integral feedback is constructed to control the biomass and lactate concentrations. The results for a combined observer and controller scheme are promising and give reliable simulation results.

Introduction

Methane emissions account for 16% of the total greenhouse gas emissions worldwide [17]. Together with an estimated global warming potential between 25 and 47 for an atmospheric residence time of 100 years, methane can be considered as a major contributor to global warming [10], [52]. 50–65% of these emissions originate from anthropogenic sources, including venting and flaring off of natural gas [19], [54]. Alternatively, using methane as feedstock for the production of various products has become an interesting option due to lower methane costs as a consequence of increased shale gas extraction [19]. The biological conversion of methane to a potentially broad range of valuable chemicals and fuels with the aid of methanotrophs, i.e. prokaryotes that use methane as their main source of carbon and energy as defined by Hanson and Hanson [25], is considered as a promising route to obtain such sustainable (bio)processes [13], [19], [35]. However, the market integration of these methane bioconversion processes is currently hampered by different hurdles, related to strain development and process efficiency [8], [55]. As mentioned by Fei et al. [19], one of these hurdles to take is optimising the culture conditions. Specifically for methane bioconversion processes, the methane-oxygen balance should be carefully addressed [19].

To increase process efficiency and to maintain optimal culture conditions, a well designed control strategy is indispensable. The goal of this work is to design a well performing control strategy for a methane bioconversion process by the bacterium Methylomicrobium buryatense with a focus on lactate and biomass production. In order to design such a control strategy, an unstructured unsegregated ODE model for the respective bioprocess is constructed. To the authors’ best knowledge, this paper is the first paper describing a control strategy for methane bioconversion processes in conventional bioreactors. Petersen et al. [42] present a modelling approach for a U-shaped bioreactor for a methane bioconversion process to produce single cell proteins (SCP), coupled to an observability and controllabilty analysis, without discussing actual observer and controller designs.

As mentioned by Alford [1], and Caramihai and Severin [6], the most common control approaches for general bioprocesses on industrial scale are on/off-control and proportional-integral-derivative (PID) controllers [2], mostly applied for temperature, pH and dissolved oxygen (DO) control. Since PID controllers are linear, the operating range in which reliable control is possible, is limited around a predefined operating point due to the highly nonlinear nature of bioprocesses. An alternative to the classic PID controller is gain scheduling during which the controller gain of a classic PID controller is regularly tuned to better fit the process dynamics [49].

Alternatively, Bastin and Dochain [3] explain the design of output feedback controllers for nonlinear bioprocesses by feedback linearization. Unfortunately, this technique suffers from two principal problems [9]: (1) it requires exact parameter values, and (2) the measurement of key biological variables is needed.

More advanced controllers apply off-line calculated optimal feeding profiles based on an optimal analytic solution from the Pontryagin maximum principle [26]. Typical objective functions involve productivity and yield maximization. Other optimal control approaches involve genetic algorithms and differential evolution algorithms [20]. Both, however, suffer from extensive calculation efforts [33]. A recent trend for bioprocesses is the use of model integrated control strategies such as model predictive control (MPC) and sliding-mode control [6], [18].

Controllers often work with a state space representation of the model. Such state feedback controllers rely on frequent state measurements. As not all states are measurable due to cost-efficiency or since sensors for online application are not available, reliable estimates are required. Komives and Parker [33] distinguish different observer types. Black box observers estimate the process states, based on the available measurements without using first principles knowledge. Examples are artificial neural networks (ANN) [11], [32], [40] and other data driven methods such as principal component analysis (PCA) or partial least squares (PLS) [30]. Model driven methods, often based on mass balance equations, involve Luenberger type observers [4], [12], [41], asymptotic observers [3], [29], [48], (extended or unscented) Kalman filters [3], [5], [7], [21], [59] and (super-twisting) sliding mode observers [39], [56].

Before observer and controller development, the observability and controllability of the system for the specific methane bioconversion process of interest will be assessed. The purpose of these analyses is to provide insight in possible measurement and input configurations. Traditional controllability and observability approaches for linear time-invariant systems involve calculating the observability and controllability Gramians [31], or using the Kalman or Hautus rank criteria [53]. For nonlinear systems, the above mentioned methods require a linearization around a predefined operating point, resulting in notions of local controllability and observability. Hermann and Krener [27] define notions of local weak controllability and observability for nonlinear systems based on Lie algebra. Zeitz [58] investigates controllability and observability of nonlinear systems by converting these nonlinear systems to canonical (normal) forms, under certain assumptions generally applicable to bioprocess systems. Gauthier and Kupka [22] come to similar conclusions for observability of nonlinear systems. A drawback of the previously mentioned approaches for nonlinear systems is that they only provide binary results, i.e. the model is either fully observable/controllable or not at all. In this work, we take another novel approach to assess practical controllability and observability of this specific nonlinear bioprocess model. This method relies on calculating the condition number of the empirical controllability and observability Gramians as defined by Lall et al. [34], which allow for a more quantitative evaluation of observability and controllability.

This paper is structured as follows. Section 2 explains the construction of the mechanistic methane bioprocess model. Sections 4 describes the observability and controllability analysis, based on the above mentioned methods for a linearized model around an operating point, and based on empirical Gramians. In Section 5, more detail is devoted to the observer design in which extended and unscented Kalman filters will be tested under different measurement combinations. The design of an LQR controller and its evaluation are described in Sections 6. Novel to the control approach is that both the methane and the air incoming flow rates are adjustable inputs. This fact results in a more flexible control compared to the traditional approach in which only one gas flow rate is controlled. The model development, observability analysis and observer design are extensions from our earlier conference paper [38].