Process Systems Engineering and Process Safety
Nonlinear state estimation for fermentation process using cubature Kalman filter to incorporate delayed measurements

https://doi.org/10.1016/j.cjche.2015.09.005Get rights and content

Abstract

State estimation of biological process variables directly influences the performance of on-line monitoring and optimal control for fermentation process. A novel nonlinear state estimation method for fermentation process is proposed using cubature Kalman filter (CKF) to incorporate delayed measurements. The square-root version of CKF (SCKF) algorithm is given and the system with delayed measurements is described. On this basis, the sample-state augmentation method for the SCKF algorithm is provided and the implementation of the proposed algorithm is constructed. Then a nonlinear state space model for fermentation process is established and the SCKF algorithm incorporating delayed measurements based on fermentation process model is presented to implement the nonlinear state estimation. Finally, the proposed nonlinear state estimation methodology is applied to the state estimation for penicillin and industrial yeast fermentation processes. The simulation results show that the on-line state estimation for fermentation process can be achieved by the proposed method with higher estimation accuracy and better stability.

Graphical abstract

This figure shows the estimation results of biomass concentration for penicillin fermentation process using the SCKF algorithm to incorporate delayed measurements and the SCKF algorithm with secondary measurements only. The estimates with secondary measurements only cannot track the true state values lacking of sufficient measurement information under incorrect initial conditions, while the SCKF algorithm incorporating delayed measurements converges to the true process state much faster after the first time arrival of the primary measurements. The estimation accuracy of the biomass concentration is improved greatly using the SCKF algorithm with sample-state augmentation, which incorporates delayed measurements efficiently.

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Introduction

Nonlinear state estimation is a highly active research area and plays a major role in many applications including the optimal control for fermentation process. Fermentation process is one of typical nonlinear processes, involving cell growth and reproduction. Various process parameters in a fermentation process can be divided into two categories: on-line measurements and off-line measurements. On-line measurements can be taken directly by hardware sensors without any time delay, including pH, temperature, culture volume, concentrations of carbon dioxide and dissolved oxygen, etc. Off-line measurements are mainly for biological variables, such as concentrations of biomass, substrate and product, generally obtained through laboratory analytical instruments and usually with uncertain time delays because of the limitation of operating conditions. The real-time monitoring of biological process variables is the key factor of the optimal control for fermentation process [1]. However, the measurement of biological process variables is difficult due to the lack of hardware sensors. In the last two decades, soft sensors have been applied to fermentation process to solve this problem based on some easy-to-measure variables [2], [3]. Generally, soft sensors can be classified into two groups: model-driven and data-driven [4]. The focus of this work is the state estimation method by means of measurement data and suitable process model based on nonlinear filtering, which belongs to model-driven soft sensor.

As the nonlinear characteristic of fermentation model, nonlinear filters are the most popular state estimation technique used for bioprocess monitoring. The extended Kalman filter (EKF), based on a first-order linearization of process model, has been widely applied to on-line estimation of biological process states [5], [6]. It is generally known that the EKF has two main drawbacks: the linearized approximation makes the filter divergence if the assumption of local linearity is violated and calculation of Jacobian matrices may be very difficult or even infeasible for complex nonlinear systems. In order to overcome these drawbacks, the unscented Kalman filter (UKF) [7] is developed to solve nonlinear state estimation problems, which is better than the EKF in terms of robustness and speed of convergence [8]. Wang et al. [9] has focused on the on-line estimation in fed-batch fermentation process using the UKF and their proposed method to improve estimation accuracy of biological process variables. Fermentation process model is highly nonlinear and usually considered as a high-dimensional nonlinear system, which is a challenge to nonlinear filters. Therefore, the performance of nonlinear filter is important to the accuracy and stability of state estimation for fermentation process.

The cubature Kalman filter (CKF) [10] is an emerging nonlinear state estimation method, especially for complex high-dimensional nonlinear system. The CKF algorithm can achieve at least second-order accuracy and does not need to calculate Jacobian matrices. Furthermore, the performance of the CKF is more stable and more accurate in high-dimensional state estimation comparing with the UKF [11]. As the CKF algorithm requires the square root operation of the state covariance matrix in the calculation, if the calculation conditions are not met, which will lead to the interruption of filter, it will affect the stability of the algorithm. Arasaratnam and Haykin [10] have also presented a square-root version of cubature Kalman (SCKF) algorithm. The SCKF algorithm solves the problem of numerical stability and reduces the amount of computation, so it has better filtering performance. Consequently, the SCKF algorithm is introduced to solve the nonlinear state estimation for complex fermentation process in this paper.

Generally, off-line measurements in fermentation process belong to delayed measurements. They are usually considered as unsuitable for control and estimation purposes, because they are available with a time delay and usually infrequent and irregular [12]. However, delayed measurements contain valuable information about biological process variables and if they can be utilized in appropriate ways, the accuracy of the estimation will be improved. The state estimation with delayed measurements for fermentation process has been studied using the multi-rate state estimation. Gudi et al. [13], [14] and Soons et al. [15] implemented a multi-rate estimator using the EKF to estimate the key state variables based on delayed measurements for a bioreactor. Myers et al. [16] used a two-stage EKF to amalgamate on-line and time-delayed off-line measurements into a common estimate for state variables of a fed-batch biochemical reactor. To overcome the inadequacy of the EKF for severely nonlinear processes, Tatiraju et al. [17] presented a multi-rate state estimation method using nonlinear state observers to estimate state variables of a pilot-scale biochemical reactor. Cao and Soh [18] designed a multi-rate nonlinear estimator based on Taylor series expansion and applied it to a class of biological systems to estimate unavailable state variables. The processing methods of delayed measurements in the multi-rate estimation are important to computational efficiency, accuracy and stability of the on-line estimation. It is therefore essential to further explore more effective methods to handle measurement delays in the multi-rate estimation to improve the performance of state estimation for fermentation process.

It is straightforward to deal with delayed measurements utilizing the filter recalculation method [19], which will recompute the estimates when delayed measurements are available from corresponding sampling time instant. The main drawback of this method is the huge storage cost and computational burden so it is not suitable for on-line nonlinear state estimation. Several state augmentation approaches have been explored in chemical and biochemical processes and discussed in detail by Gopalakrishnan et al. [20]. There are mainly three schemes for assimilating the delayed measurements in recursive estimators: fixed-lag smoothing [21], measurement augmentation [22] and sample-state augmentation [23]. The first two approaches require all the state information between sampled time instant and arrival time instant of the delayed measurements to obtain estimates when the state is updated to incorporate delayed measurements, leading to a heavy computational load. The sample-state augmentation method is a more efficient approach, only augmenting the state with the sample-state, which is the state at sampling time of the delayed measurements, and is applicable to the case with uncertain and time-varying delays [24]. The sample-state augmentation based on EKF algorithm has been used to solve nonlinear state estimation problem of chemical process [20]. In order to overcome the limitation of the EKF, it is promising to study the sample-state augmentation method in the SCKF algorithm frame to improve the state estimation considerably.

Hence, we focus on the nonlinear state estimation utilizing the SCKF algorithm and delayed measurements for fermentation process in this paper. The nonlinear state estimation method using square-root version of SCKF algorithm to incorporate delayed measurements for fermentation process is explored. Two simulation examples are given for the state estimation of penicillin and industrial yeast fermentation processes, to demonstrate the validity and effectiveness of the proposed method.

Section snippets

Square-root version of CKF algorithm

Consider the following nonlinear discrete-time systemxk=fk1xk1uk1+wk1zk=hkxk+vkwhere xk and zk are the n-dimensional state vector and l-dimensional measurement vector of the system, respectively; wk  1 and vk are the independent process and measurement Gaussian noise with zero means and covariances Qk  1 and Rk, respectively; uk  1 is the control input vector; fk  1(⋅) and hk(⋅) are the state function and measurement function of the nonlinear system.

The specific SCKF algorithm process based on

Simulation Results

Two case studies for fermentation process are used to validate the efficacy of the nonlinear state estimation method presented in this paper. The first case is the state estimation of a penicillin fed-batch fermentation process, which is a benchmark simulation model for monitoring and fault diagnosis of fermentation process. The second case focuses on the state estimation of an industrial yeast fermentation process, which is often used for the test of optimization control methods for

Conclusions

The key motivation for this paper is to introduce a novel nonlinear state estimation methodology for fermentation processes, which can make full use of the benefits of the SCKF algorithm and delayed measurements of biological process variables. The SCKF is a promising nonlinear filter, especially for high-dimensional nonlinear systems, which has potential to improve the performance of state estimation for complex fermentation process. We design a nonlinear state estimation methodology using the

Nomenclature

    A

    unitary matrix obtained through QR decomposition

    B

    upper triangular matrix obtained through QR decomposition

    C

    penalty parameter for error term in SVM model

    Ci

    concentration of component i, g·L 1

    ei

    ith column of a n-dimensional unit matrix

    F

    feed flow rate, L·h 1

    F0

    initial value of feed flow rate (= 500 L·h 1)

    f(⋅)

    state function of nonlinear system

    fa(⋅)

    augmented state function of nonlinear system

    g

    major instance at which the primary measurements arrive

    h(⋅)

    measurement function of nonlinear system

    hk1(⋅)

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Supported by the National Natural Science Foundation of China (61503019), the Beijing Natural Science Foundation (4152041) and Beijing Higher Education Young Elite Teacher Project (YETP0504).

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