State estimation incorporating infrequent, delayed and integral measurements

doi:10.1016/j.automatica.2015.05.001

Automatica

Volume 58, August 2015, Pages 32-38

https://doi.org/10.1016/j.automatica.2015.05.001 Get rights and content

Abstract

This paper investigates the problem of state estimation incorporating infrequent, delayed and integral measurements. In chemical process, there often exist two types of measurements. On one hand, the measurements for process variables such as flow rates and pressures are sampled frequently and are available nearly instantaneously. On the other hand, the measurements for quality variables such as concentration are sampled infrequently and are available with a delay due to long analysis time involved. Moreover, due to the interval time taken by chemical sample collection, the measurements for some quality variables have another important characteristic: it is a function of the states of the compositions over a period of time. This paper formulates the process with infrequent, delayed and integral measurements as an equivalent variable dimension system, whose measurements are both delay and integration free. Based on the new model, a variable dimension unscented Kalman filter (VD-UKF) is proposed to estimate the states. Furthermore, the stability of the proposed VD-UKF is analyzed. Compared with the existing results, the proposed stability condition is significantly relaxed and the invertibility condition of Jacobian matrices is no longer needed. Finally, a simulation example demonstrates the effectiveness of the proposed method.

Introduction

In chemical process systems, the measurements for process variables such as flow rates and pressures are usually sampled frequently and are available nearly instantaneously. However, the measurements for certain quality variables such as concentration are sampled infrequently and are only available with considerable time delays. Moreover, instead of depending on a state at certain past instant, the delayed measurements can also be a function of the integral of the states over certain past period of time. This type of measurements is called integral measurements in this paper. For example, in distillation columns, laboratory (lab) analysis is often required for measurement of the distillate and bottoms compositions as the use of online analyzers is often infeasible due to economic considerations or technological difficulty, though online analyzers can be used to measure other process variables such as tray temperatures frequently without time delay (Gopalakrishnan, Kaisare, & Narasimhan, 2011). In order to analyze the composition, a sufficient amount of samples needs to be collected. The time taken to collect the sample is not small compared to the sampling period of the estimator, such that it cannot be ignored. As will be seen in the next section, the compositions of the sample do not represent the compositions at a particular sampling instant but the integral of the compositions within some period of time. In addition, due to offline analysis of the collected samples in a lab the acquired measurement has an unavoidable delay. If the quality variable is inferred directly from the fast-sampled process variable through a model, it can be inaccurate due to model errors, sensor bias or unmodeled disturbances. In such cases the infrequent and delayed integral measurements need to be incorporated into the estimator as the lab analysis is usually more accurate.

By applying multi-rate state estimation techniques, Gudi, Shah, and Gray (1995) designed adaptive strategies to fuse the infrequent and delayed measurements for a fermentation process in a bioreactor. Zambare, Soroush, and Grady (2002) presented real-time implementation of a robust multirate state estimator on a continuous stirred-tank, free-radical, styrene polymerization reactor, where the estimator uses both frequent online measurements and infrequent offline and delayed measurements. Gopalakrishnan et al. (2011) analyzed several existing methods to incorporate delayed measurements and reinterpret their results under the extended Kalman filter framework. Bavdekar, Prakash, Patwardhan, and Shah (2014) proposed a recursive moving window Bayesian state estimator formulation to utilize delayed measurements to compute the state estimates. All the mentioned papers considered the fact that the measurement for quality variable is infrequent and delayed. However, another important characteristic about such a measurement, the integration nature of sample collection, has not been considered in the literature although the problem is commonly encountered in industries.

This paper investigates the problem of state estimation incorporating infrequent, delayed and integral measurements. First, the mathematical model of the considered process with integral measurements is formulated. Second, for the sake of filter design, true process is reformulated as an equivalent variable dimension system. Then a variable dimension unscented Kalman filter (VD-UKF) is proposed to estimate the states. Furthermore, the stability of the proposed VD-UKF is analyzed. Compared with the existing results (Li and Xia, 2012, Xiong et al., 2006), the proposed stability condition is significantly relaxed. The assumption that Jacobian matrices are required to be invertible is no longer needed. This is important since the Jacobian matrices of VD-UKF cannot be always a square matrix; let alone to be invertible. Finally, a simulation example demonstrates the effectiveness of the proposed method.

Notation

The notation used in this paper is fairly standard. The superscript “ $T$ ” stands for matrix transposition. $diag {\dots}$ stands for a block-diagonal matrix. $(\cdot)$ denotes the same content as that in the previous parenthesis. Given a real number $x$ , $⌈ x ⌉$ denotes the smallest integer greater than or equal to $x$ .

Section snippets

Problem formulation

Consider a continuous, nonlinear process described by the following stochastic differential equation: $d x (t) = f_{c} (x (t), u (t)) d t + d w_{c} (t),$ where $x \in R^{n_{x}}$ is the state of the process; $u \in R^{n_{u}}$ is the control input; $f_{c}$ is the drift function; $w_{c} (t) \in R^{n_{x}}$ is independent Brownian motions with diagonal diffusion matrix $Q_{c}$ .

We consider two types of measurements: fast-rate measurement such as online analyzer and slow-rate measurement such as lab analysis. Similar to Gopalakrishnan et al. (2011), we use the schematic

Reformulation of the filtering model

It is difficult to directly use the model (1), (2), (3) for filtering design, since the measurement $y^{l} (T k)$ in (3) not only is delayed but also contains an integral term. In order to design a filter, we need first to reformulate the model. Let us define ${\dot{q}}_{s} (t) = η_{s} (t) x (t) with q_{s} (t_{s}) = 0,$ where if $t \in [t_{s}, t_{s} + δ]$ , $η_{s} (t) = 1$ . Otherwise, $η_{s} (t) = 0$ .

From (5), we have $q_{s} (T k) = \int_{t_{s}}^{t_{s} + δ} x (t) d t$ , if $T k = t_{s} + δ + τ_{s}$ . Then (3) can be rewritten as $y^{l} (T k) = H^{l} q_{s} (T k) + v^{l} (t_{s}), if T k = t_{s} + δ + τ_{s} .$ One can see from (6) that the integral

VD-UKF algorithm

Different from EKF, UKF is a derivative-free estimation methods and it can reduce the linearization errors of the EKF (Julier and Uhlmann, 2004, Julier et al., 2000). In this paper we use UKF algorithm to estimate the states. Because the dimension of systems in (9) is variable, the number of sigma points in the proposed UKF is also variable, which is different from the traditional UKF algorithm. The procedure for implementing the VD-UKF can be summarized as follows:

Step 1: Select sigma points.

Simulation studies

In order to evaluate the performance of the proposed filter, we apply it to a highly nonlinear dynamic system describing the behavior of a bioreactor (Henson & Seborg, 1997 and Kandepu, Huang, Imsland, & Foss, 2007): ${\begin{matrix} d x_{1} = {- u x_{1} + μ (x_{2}, x_{3}) x_{1}} d t + d w_{c 1}, \\ d x_{2} = {u [S_{f} - x_{2}] - \frac{1}{Y} μ (x_{2}, x_{3}) x_{1}} d t + d w_{c 2}, \\ d x_{3} = {- u x_{3} (t) + [α μ (x_{2}, x_{3}) + β] x_{1}} d t + d w_{c 3} . \end{matrix}$

$x (t) = {[\begin{matrix} x_{1} (t) & x_{2} (t) & x_{3} (t) \end{matrix}]}^{T}$ is the system state vector, where $x_{1} (t)$ , $x_{2} (t)$ , and $x_{3} (t)$ are the biomass, substrate, and product concentrations, respectively. $w_{c} (t) = {[\begin{matrix} w_{c 1} (t) & w_{c 2} (t) & w_{c 3} (t) \end{matrix}]}^{T}$ is

Conclusions

This paper investigates the problem of state estimation incorporating infrequent, delayed and integral measurements. The challenge is how to deal with the integral characteristics contained in the measurements. This issue is solved by reformulating the true process as an equivalent variable dimension system, whose measurements are both delay and integration free. Based on the new model, a VD-UKF is proposed to estimate the states. Although the VD-UKF is a smooth extension of the traditional

Yafeng Guo received his B.Sc. degree in Automatic Control in 2001 and his M.Sc. degree in Control Theory and Control Engineering in 2004 both from Xiamen University, and his Ph.D. degree in Control Theory and Control Engineering in 2009 from Shanghai Jiao Tong University. Since then he joined the Department of Control Science and Engineering at Tongji University. He is currently a postdoctoral fellow in the Department of Chemical and Materials Engineering at the University of Alberta. His

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Biao Huang obtained his Ph.D. degree in Process Control from the University of Alberta, Canada, in 1997. He obtained M.Sc. degree (1986) and B.Sc. degree (1983) in Automatic Control from the Beijing University of Aeronautics and Astronautics. He joined the University of Alberta in 1997 as an Assistant Professor in the Department of Chemical and Materials Engineering, and is currently a Professor, NSERC Industrial Research Chair in Control of Oil Sands Processes and AITF Industry Chair in Process Control. He is a Fellow of the Canadian Academy of Engineering. His research interests include: process control, system identification, control performance assessment, Bayesian methods and state estimations. He has applied his expertise extensively in industrial practice.

^☆: This work was supported by NSERC (grant no. IRCPJ 417793) and AITF (grant no. PSI15502-2013). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Denis Dochain under the direction of Editor Ian R. Petersen.

¹: Tel.: +1 780 492 9016; fax: +1 780 492 2881.

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Brief paperState estimation incorporating infrequent, delayed and integral measurements☆

Abstract

Introduction

Section snippets

Problem formulation

Reformulation of the filtering model

VD-UKF algorithm

Simulation studies

Conclusions

Computers and Chemical Engineering

Information Fusion

Journal of Process Control

Automatica

Journal of Process Control

Automatica

Automatica

Computers and Chemical Engineering

A moving window formulation for recursive Bayesian state estimation of systems with irregularly sampled and variable delays in measurements

Industrial & Engineering Chemistry Research

A strong tracking extended Kalman observer for nonlinear discrete-time systems

IEEE Transactions on Automatic Control

Brief paper
State estimation incorporating infrequent, delayed and integral measurements☆