# PID Control Assessment Using L-Moment Ratio Diagrams
## 1 Introduction  
The task of a control engineer is to design, tune, and maintain control systems, which allow a given installation to work according to the assumed requirements. To fulfill these activities, one must have enough skills, knowledge about the process, and updated information about the actual performance of the control system. Control performance assessment tasks support an engineer with information about how the control system operates [1]. Thus, CPA plays a very important role in maintaining the control system and the desired performance of the operation of the actual plant. When the control system does not fulfill its requirements, the whole installation also misses its targets and performs inconsistently with expectations [2]. Despite the fact that this knowledge has been well known for decades, it is still the case that industrial systems often do not operate as demanded [3; 4; 5], which is caused by process difficulties (non-stationarity, disturbances and noises), hardware problems (actuators, sensors, IT infrastructure), human errors or intention (inadequate maintenance, design or tuning), or just a general lack of time or proper knowledge [6; 7]. The key performance indicators (KPIs) depend on the type of plant, engineering culture, and tradition; however, no matter what these are, CPA activities and measures have to be applied using a reliable and transparent approach. In process industry, the proportional-integral-derivative (PID) algorithm constitutes the backbone of control strategies. It is used in an overwhelming majority, i.e., more than 90% or even 95% of industrial control loops [8; 9; 10]. The story started in 1967 with a pulp and paper plant and its performance assessment using variable standard deviation [11]. Since then, the method has evolved dramatically, exploring and exploiting various research areas and approaches [12]. Nowadays, investigations address numerous methods [13], like the ones that require plant parametric experiments [14], which use process model identification [15; 16], or that model-free techniques that need plant data only [17; 18]. Apart from homogeneous approaches,there are proposals to use hybrid techniques that use the fusion of various measures and approaches [19; 20]. As a result, we do not end up with a single and universal approach. There is a large spectrum of methods, and an engineer is generally lost, as he has to select an appropriate method. As a result, we demand hybrid and multi-criteria approaches that allow supporting control loop performance decision-making. In this task, statistical methods, which have been a bit forgotten but are still characterized by enduring advantages, can come to our rescue.Moment ratio diagrams (MRD) deliver such a tool [21], which have actually been known in statistics for decades [22] as MRDs originated from [PERSON]'s research in the early 19 th century. Moment ratio diagrams graphically compare different statistics of a given time series to visualize properties of the stochastic process, which is hidden behind data. A MRD can be used in various tasks, such as fitting of empirical data to theoretical probabilistic density functions (PDFs), comparison of the PDF shapes, and the classification of data [23]. A MRD is a two-dimensional graphical diagram in a Cartesian coordinate system for a pair of selected statistical moments. As L-moments constitute a robust version of the common definition of statistical moments [24], an L-moment ratio diagram (LRMD) exploits and extends the idea of the MRD. LMRD plots are extensively used in life sciences, like seismology [25], hydrology [26], astronomy [27], meteorology [28], medicine [29], and many others [30]. Their utilization in engineering sciences and specifically in control technology is extremely rare. This paper fills this observed research gap. Any research that addresses aspects of control performance validation, their design, or tuning remains unsatisfactory if it is limited to theoretical or simulation considerations only.