# Causality in Control Systems Based on Data-Driven Oscillation Identification
## 1 Introduction  
Causality analysis requires a specific approach and designated methods that allow one to determine the internal relations between different variables [1]. One may find various approaches, which have been developed in different domains. Actually. there are four leading approaches to detecting causality from historical process data: Granger Causality, Transfer Entropy, Cross-correlation, and Partial Directed Coherence [2]. Cross-correlation is simple, but it assumes linearity [3]. Some of them are based on the process model, like the coherence method [4], the Granger causality [5] and its extension Partial Directed Coherence. Spectral Granger causality uses Fourier Transform, which limits the method to the linear and stationary cases. Others belong to the group of data-driven and model-free methods, like the Transfer Entropy approach [6]. The analysis of complex and varying control systems, for which the exact model is unknown or its estimation is highly uncertain, encourages the use of the data-driven method. It is a great simplification in the field of causality determination as well as fault propagation research. Faults can be observed as system oscillations, among others. The industrial demands for the causality analysis have been defined quite early [7], but the he research on the causality root-cause analysis in the area of process control is relatively new and only few works are published. The introductory paper [2] has been followed by some examples of the Gragner causality and Transfer Entropy comparison in [8; 9]. Transfer Entropy research has been followed in [10] with the hybrid combination of TE and the modified conditional mutual information (CMI) approach, which uses not data time series but multi-valued alarm series. Authors of [11] used an opposite approach. They are assuming that periodical elements are not bringing any value so they use denoising and periodicity-removing time-delayed convergent cross mapping (TD-CCM), which is an extension of the original algorithm [12] and its time-delayed version [13]. The Transfer Entropy approach has been addressed by a few works. The Transfer Entropy systematic workflow for oscillation diagnosis is proposed in [14], while the outliers impact is addressed in [15]. Finally, the TE approach is improved by information granulation in [16]. AI-driven methods, like neural or Bayesian networks, are uncommon in the process industry applications. They are many times harder to train because of the increasing size and complexity of the used data sets [11]. Moreover, we would need to know the causality examples to train it. The authors decided not to use AI and develop the TE approach. Our approach assumes that periodicity contains important information and propagates through the system. Therefore, the Transfer Entropy approach uses decomposed oscillation elements. An example of the oscillatory behavior propagation through the control system is discussed in [17]. Oscillation diagnosis may allow one to trace the root cause of these faults. It allows answering the question of which control loop of a given system introduces disturbances into the system causing its improper performance. As the result, too aggressive or too sluggish operation [18] is observed. Thus, assuming that the oscillations in a multi-loop control system carry out valuable information, it is imperative to conduct the research in such a direction. In the considered analysis, the innovative application of the Transfer Entropy approach is combined with the oscillation decomposition leading toward time-frequency causality analysis. The algorithm should work not only in the simulation or laboratory environment but also in industrial reality, which introduces various disturbances and limitations. Non-stationarity and non-Gaussian properties are common. They often turn out in the form of the variables distribution tails and outliers. One has to be aware that outlying observations are very frequent in industrial practice [19].