# PID Control Assessment Using L-Moment Ratio Diagrams
## 3 Simulation Research
Following the above presentation scheme, Figure 4 shows the analogous relationship related to the settling time \(T_{\text{set}}\). We see quite a contrary behavior, as the loops that exhibited a long settling time are represented by points with a lower L-kurtosis value. Generally, these two plots confirm the contradictory meaning of the overshoot and the settling time. Higher accuracy was achieved at the cost of the settling time, and conversely speeding up the control led to a lower accuracy and introduced oscillations represented by the overshoot. Figure 3: The LMRD(\(\tau_{3}\), \(\tau_{4}\)) for \(G_{1}(s)\) related to overshoot \(\kappa\). Figure 2: Time series for the loop \(G_{1}(s)\), which was run using the so-called well-tuned PID algorithm (blue line—process variable, green—setpoint, magenta—manipulated variable). The next two diagrams extend the analysis, as they introduce the integral indexes. Figure 5 shows how the varying ISE index (norm \(\ell_{2}\)) is represented in the LMRD(\(\tau_{3}\), \(\tau_{4}\)), while Figure 6 presents the effect of the norm \(\ell_{1}\). We can observe that both integrals ISE and IAE reacted similarly to the settling time, while the direct difference between ISE and IAE was less significant. We can only observe that IAE made a bit more of a mutual distinction between control loops, while ISE was, one might say, more binary in making a sharper distinction. This separation between the norms \(\ell_{2}\) and \(\ell_{1}\) is visible. This observation confirms that the meaning of the ISE and IAE is significant and should be considered in the selection of a performance index. Additionally, the LMRD(\(\tau_{3}\), \(\tau_{4}\)) diagram is characterized by one shortcoming, as it has no possibility of reflecting loop fluctuations, which is naturally addressed by scale factors, like L-scale \(l_{2}\) or L-Cv \(\tau_{2}\). One may assume that similar observations of the overshoot and three other indexes might also appear for the other plots. As the same effect was observed, the following analysis is presented in the form of two types of diagrams  
Figure 5: The LMRD\((\tau_{3},\tau_{4})\) for \(G_{1}(s)\) related to ISE. Figure 6: The LMRD\((\tau_{3},\tau_{4})\) for \(G_{1}(s)\) related to IAE. Unfortunately, for the LMRDs(\(\tau_{2},\tau_{3}\)), we observed that the L-Cv \(\tau_{2}\) value was not limited, and the OX axis scaling depended on data values. The use of the scaled \(l_{2}\) moment should have addressed this issue. Figures 9 and 10 show LMRDs(\(l_{2},\tau_{3}\)) related to the overshoot and settling time. Unfortunately, we did not obtain a full improvement. The plots were scaled according to the known limits, but the separation related to the overshoot was lost. In such a case, there is no way to determine between high and low \(\kappa\) values using the LMRD(\(l_{2}\), \(\tau_{3}\)). This meant that the control performance could not be fully assessed using the LMRD(\(l_{2}\), \(\tau_{3}\)). Finally, we introduced a new type of LMRD that compares L-kurtosis with L-moment \(l_{2}\). Figures 11 and 12 show the LMRD(\(l_{2}\), \(\tau_{4}\)) plots related to the overshoot and settling time. We still observed a good separation for the settling time, while high overshoot performance detection was hard to accomplish. We may conclude that any single LMRD does not allow solving the assessment issue. There are two reasons for this: the overshoot detectability, and there being three statistical factors related to the control performance: scale, skewness, and kurtosis.Therefore, it is suggested to use a combination of two plots: the LMRD(\(\tau_{2}\), \(\tau_{3}\)) and the LMRD(\(l_{2}\), \(\tau_{4}\)) or LMRD(\(\tau_{3}\), \(\tau_{4}\)). Further simulation analysis was conducted for two other transfer functions: \(G_{4}(s)\) and \(G_{5}(s)\).