(588g) Online Steady and Transient State Detection Using the Dickey-Fuller Test

Identification of when a process is at steady state or not, despite noise in the signal is an important task in the process industry. For example, adapting steady state models with online data, e.g. real time optimization, requires the online identification of when the process is at steady state. Using a steady state model on transient data could result in inaccurate parameter estimation, leading to inferior performance of the process, and potentially unstable or hazardous operation. Automatic detection of when process variables are at steady state can also be useful as inputs for an abnormal event detection algorithm or similar. In these use cases, steady state detection must take place online, hence an automatic process to detect steady state is desirable. In this work we propose to use the Dickey-Fuller test [2] to detect when a process is at steady or transient state.

Steady state detection is not a trivial task as a real process variable will never be completely unchanging because of system noise and stochasticity, i.e. if there was no noise one could simply check for any change in the data to identify steady state. Instead, the closest a process will come to complete steady state is to operate in a region near some set point, where it is continuously disturbed by measurement noise, stochastic shocks or similar. This means that any steady state detection algorithm must estimate a measure of the local trend to identify whether the process is at steady or not [7]. Examples of proposed steady state detection algorithms include computing an F-test on two estimates of the process variance computed by applying different filters to the process data [1,7], and detecting whether in a time window the process signal minus its mean is within some band defined by the standard deviation of the signal times a critical value from a Student’s t-test [4].

The novel idea in this work is that we propose is to use the Dickey-Fuller test [2] in a sliding window fashion to identify if the process is in a transient or steady state. We compare the tuning and performance of this method to other steady state detection algorithms [1,4]. The Dickey-Fuller test is a statistical test used in econometrics [3] for detecting if a time series is trend-stationary, however, to our best knowledge it has not been used for identifying steady state in the context of process systems engineering. The test requires fitting the auto-regressive model of order 1 (AR(1)):

y_t+1 = a₀ + a₁t + py_t + u_t

where a₀, a₁ and p are parameters, ut is white noise (mean zero, finite variance), and y_t a measurement value at time t. For steady state detection, we wish to detect if the process is stationary, so we set a₀ and a₁ to zero, i.e. we use the AR(1) model:

y_t+1 = py_t + u_t

Note that as the AR(1) models are linear, we have an analytic solution for p, when solving the least squares problem. The null hypothesis is p=1 (transient state), with the alternative hypothesis |p|<1 (steady state). To understand why this test is useful for steady state detection, consider a controlled process operating around some set point, disturbed by noise that enters the system. Any large deviation from the set point would likely be followed by smaller deviations, which requires |p|<1. When p=1 it means that the process is entirely driven by the system noise, i.e. there is not a deterministic part of the process that returns the process to the set point (steady state). Lastly, when |p|>= 1 means that system does not revert to its set point, i.e. it is not at steady state. We test the null hypothesis (transient state) using a response surface for the Dickey-Fuller distribution [5].

We propose to use this test in a sliding window fashion to decide if the data within that window is from a process at steady state. We can either test whether the data reverts to the set point, or alternatively the mean of the data within the time window. Our steady state detection method requires two parameters from the user: the window size and the significance level of the hypothesis test.

The comparison of the proposed algorithm to other steady state detection algorithms in the literature [1, 4] is performed using real and synthetic data. For the synthetic data, normal and student-t distributed noise is added to the data at different noise levels. The real data consists of a time series from an experimental rig (described in [6]), and industrial data. To compare the performance we use confusion matrices, the F1 score and the phi coefficient (also known as the mean square contingency coefficient). When tuning the method the parameters are relatively easy to choose as the significance level is readily interpretable while a reasonable window size can be estimated from process knowledge, e.g. the time constant and sampling interval of the system [4].The Dickey-Fuller test performs well compared to the other algorithms, especially when examining the phi coefficient, which we argue is an intuitive and natural metric for comparing steady state detection algorithms.

References

[1] Cao, S. and Rhinehart, R.R., 1995. An efficient method for on-line identification of steady state. Journal of Process Control, 5(6), pp.363-374.

[2] Dickey, D.A. and Fuller, W.A., 1979. Distribution of the estimators for autoregressive time series with a unit root. Journal of the American statistical association, 74(366a), pp.427-431.

[3] Elder, J. and Kennedy, P.E., 2001. Testing for unit roots: what should students be taught?. The Journal of Economic Education, 32(2), pp.137-146.

[4] Kelly, J.D. and Hedengren, J.D., 2013. A steady-state detection (SSD) algorithm to detect non-stationary drifts in processes. Journal of Process Control, 23(3), pp.326-331.

[5] MacKinnon, J.G., 2010. Critical values for cointegration tests (No. 1227). Queen's Economics Department Working Paper.

[6] Matias, J., de Castro Oliveira, J.P., Le Roux, G.A. and Jäschke, J., 2021. Real-time optimization with persistent parameter adaptation applied to experimental rig. Ifac-papersonline, 54(3), pp.475-480.

[7] Rhinehart, R.R., 2013. Automated steady and transient state identification in noisy processes. In 2013 American Control Conference (pp. 4477-4493). IEEE.