# PID Control Assessment Using L-Moment Ratio Diagrams
## 3 Simulation Research  
A dedicated Matlab simulation layout was designed to analyze the single element PID control loop. It is sketched in Figure 1. We used a parallel form of the PID algorithm defined as  
\[G_{PID}(s)=k_{\mathrm{P}}(1+\frac{1}{T_{\mathrm{i}S}}+T_{\mathrm{d}}s). \tag{8}\]  
To assess the effectiveness of the LMRD diagrams, we simulated three univariate PID controls with three plants, which were proposed by [PERSON] [42] as control benchmarks:  
* multiple equal pole transfer function, \[G_{1}(s)=\frac{1}{(s+1)^{4}},\] (9)
* first-order transfer function with a dead time, \[G_{4}(s)=\frac{1}{(0.2s+1)^{2}}e^{-s},\] (10)  
Figure 1: The simulation close-loop system diagram. * fast and slow mode transfer function, \[G_{5}(s)=\frac{1}{(s+1)(0.04s^{2}+0.04s+1)}.\] (11)  
Such a selection allowed running the analysis taking into account a wide spectrum of possible processes, which can be met in process industry. The simulation loops were sampled with a time period of \(\Delta T=0.1\) s. The setpoint signal was set to a constant zero value. Moreover, two disturbances were introduced into the loops. They simulated potential industry-like effects:  
* the Gaussian \(N(0,\sigma^{2})\) measurement noise with standard deviation \(\sigma=0.1\cdot\sqrt{2}\), which was added at the plant output to the process variable;
* filtered heavy-tailed disturbance, which was added before the plant transfer function, and which was simulated as a S\(\alpha\)S stochastic process based on the \(\alpha\)-stable distribution random number generator with the stability exponent \(\alpha=1.95\) and scaling factor \(\gamma=2.0\)[43]. All PID controllers were tuned using performance index optimization, according to three weighted criteria: the integral of time-weighted absolute error (ITAE) criterion (weight \(w_{1}=1\)), maximum overshoot (\(w_{2}=10\)), and sensitivity (\(w_{3}=20\)), as proposed in [44]. These settings are referred to hereafter as well-tuned and are shown in Table 1.  
A detailed analysis with the whole set of considered diagrams is presented for the first transfer function \(G_{1}(s)\), while the other two processes are assessed by selecting the most promising diagram, i.e., the LMRD(\(l_{2},\tau_{4}\)). Figure 2 shows the sample control error time series for the \(G_{1}(s)\) transfer function controlled by the so called well-tuned controller, with its parameters as in Table 1.  
The analysis assessed and compared various settings of the PID algorithm. The gain \(k_{p}\in[0.05;2.05]\) was changed every 0.25 step, the integration time \(T_{i}\in[0.2;10.2]\) was changed every 1.0, while the derivative time constant was set to a constant \(T_{d}=0.9293\). Altogether, we used further comparison results for 99 various control loops. To exclude statistical effects, each set of parameters was run 50 times and the evaluated L-moment values were averaged. We related the obtained controller performance to common and well-understood loop quality indexes. We used overshoot, settling time, ISE, and IAE for that purpose. This relationship was included in all LMRDs in the form of the respective shading of the circles representing each tuning. Figure 3 shows the respective LMRDs(\(\tau_{3},\tau_{4}\)) related to the overshoot. We can observe that, generally, the majority of loops exhibited a low overshoot. Those few that had significant value of overshoot were characterized by an increased L-kurtosis \(\tau_{4}\) compared to the normal distribution and \(\tau_{4}=0.1226\). We may also observe that the scatter of points with respect to the OX axis denoting skewness \(\tau_{3}\) was small and insignificant within a narrow set of \(\tau_{3}\in(-0.01,0.01)\). \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(k_{p}\) & \(T_{i}\) & \(T_{d}\) \\ \hline G\({}_{1}(s)\) & 1.0503 & 2.9977 & 0.9293 \\ G\({}_{4}(s)\) & 0.2653 & 0.6066 & 0.2121 \\ G\({}_{5}(s)\) & 0.1330 & 0.2585 & 0.0808 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters for the well-tuned PID controllers.