# Evaluation of PID-Based Algorithms for UGVs
## 3 Methodology
### Evaluation Criteria for PID Control Algorithms  
In the methodology for designing a PID controller, one of the most important performance criteria is the difference (error) between the system's output and the reference value. Generally, optimization functions are formulated based on error equations, with the three most common being [45]:  
* Integral of the Absolute Error Magnitude (IAE);
* Integral of the Squared Error (ISE);
* Integral of the Time multiplied by the Absolute Error Magnitude (ITAE).  
These are given as follows [45]:  
\[\text{IAE}=\int_{0}^{\text{T}}|\text{e}(\text{t})|\text{dt} \tag{6}\]  
\[\text{ISE}=\int_{0}^{\text{T}}\Big{|}\text{e}(\text{t})^{2}\Big{|}\text{dt} \tag{7}\]  
\[\text{ITAE}=\int_{0}^{\text{T}}t|\text{e}(\text{t})|\text{dt} \tag{8}\]  
To complement the values provided by the metrics given in formulas 6-8, multi-objective optimization (MOO) techniques were used with the selection of Pareto-optimal solutions. This method, as mentioned in [46], explains that optimization aims to maximize performance with minimal resources and is divided into exact methods, which guarantee optimal solutions but have limited applicability, and meta-heuristic methods, which effectively handle complex, real-world problems. In MOO, multiple conflicting objectives are addressed simultaneously, resulting in a set of Pareto-optimal solutions where no single solution is universally better, requiring trade-offs.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Cohen–Coon** & \(\mathbf{K_{C}}\) & \(\mathbf{T_{I}}\) & \(\mathbf{T_{D}}\) & \(\boldsymbol{\tau}\) & \(\mathbf{a}\) \\ \hline PID & \(\frac{1.35}{a}\Big{(}1+\frac{0.18\text{T}}{1-\tau}\Big{)}\) & \(\frac{2.5-2.07}{1-0.99\tau}*L\) & \(\frac{0.37-0.37\tau}{1-0.81\text{T}}*L\) & \(\frac{L}{L+T}\) & \(\frac{K_{strong}*L}{T}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Gain calculation formulas [43].  
Figure 15: Open-loop reaction curve [44].  
A decision vector \(x^{*}\in S\) is Pareto-optimal if no other vector \(x\in S\) can improve all components without worsening at least one [47]. Mathematically, point \(x^{*}\in S\) is Pareto-optimal only if there is no point \(x\in S\) such that \(f(x)\leq f(x^{*})\) with at least one \(fi(x)>fi(x^{*})\)[46].  
As noted by [48], all Pareto-optimal solutions in decision variable space form the Pareto-optimal set. Furthermore the representation of the Pareto optimal set in the objective function space is called the Pareto-optimal front [49], expressed as:  
\[PF^{*}=\{f(x^{*})|x^{*}\in PS^{*}\} \tag{9}\]  
This indicates that the Pareto-optimal front (PF) is the set of objective function values for all Pareto-optimal solutions (PSs) in the decision space, showcasing the trade-offs between conflicting objectives.  
In addition to the criteria mentioned, further evaluation metrics were added to complement the above criteria. These metrics consist of the mean, standard deviation, and maximum, with the scale being in meters. With these three metrics, the aim was to analyze the position error of the robot relative to the trajectory that the UGV should follow. Finally, the navigation time was also considered to determine which of the control algorithms could execute a given trajectory in the shortest possible time.