## 5 Stability and Robustness  
In general, the stability of the closed-loop system can be calculated from the roots of the closed-loop transfer function denominator (\(den_{CL}\)):  
\[G_{CL}=\frac{G_{P}G_{CR}}{1+G_{P}G_{CY}}=\frac{num_{CL}}{den_{CL}}, \tag{48}\]  
where  
\[den_{CL}=s\Big{(}1+a_{1}s+a_{2}s^{2}+\dots\Big{)}(1+sT_{F})e^{sT_{delay}}+ \Big{(}1+b_{1}s+b_{2}s^{2}+\dots\Big{)}\Big{(}K_{D}s^{2}+K_{PS}+K_{I}\Big{)}. \tag{49}\]  
Clearly, the denominator consists of a pure time-delay, which complicates the stability analysis. The time-delay in \(den_{CL}\) (49) can be developed into an infinite-order Taylor series. However, the analysis of infinite-order series using, e.g., the Routh-Hurwitz stability criterion, is far from straightforward and outside the scope of our research. However, the stability and robustness of closed-loop systems can be tested on process models that cover most of the integrating processes in various industries. One such process model is the second-order integrating process with a time-delay:  
\[G_{P}(s)=\frac{K_{PR}e^{-T_{delay}s}}{s\Big{(}1+a_{1}s+\alpha a_{1}^{2}s^{2} \Big{)}}. \tag{50}\]  
The process \(G_{P}\) (s) (50) also covers pure time-delay and the first-order integrating processes. The stability and robustness of the closed-loop system will be tested with different ratios of \(T_{delay}/a_{1}\) and factor \(\alpha\). Note that this process has real poles only when \(\alpha\leq 0.25\). When choosing the PID\({}_{b}\) controller reference weighting factor \(b=0\), the closed-loop system is stable for any positive values of \(\alpha\) and \(T_{delay}\). The system is also stable for all the processes with real process poles for \(b=0\), \(b=0.5\), and \(b=0.9\). As shown by Figure 20, increasing factor \(b\) has a negative impact on system stability. This system is stable for any time-delay with \(b=0.5\) if \(\alpha\leq 1.5\), and \(\alpha\leq 1.05\) with \(b=0.9\). The overall stability improves by increasing the time-delay or decreasing \(b\) and the factor \(\alpha\). Figure 20: Stability region for process \(G_{P}\) (s) at different values of \(T_{delay}/a_{1}\), \(\alpha\), and the reference weighting factor \(b\). Note that the system is stable when \(b=0\). The robustness of the proposed tuning method (PID\({}_{\text{b}}\) and PID\({}_{\text{fh}}\) controller) is next studied for the following process model:  
\[G_{\text{P1}}(s)=\frac{e^{-s}}{s(1+s)^{2}}. \tag{51}\]  
First, the PID\({}_{\text{b}}\) controller filter time constant \(T_{F}\) = 0.01 s was added to Process (51). Using Expression (8), the following characteristic areas were calculated from Process (51)'s transfer function:  
\[A_{0}=1,\ A_{1}=3.01,\ A_{2}=5.5301,\ A_{3}=8.22197,\ A_{4}=10.9572. \tag{52}\]  
In the next step, the PID\({}_{\text{b}}\) controller parameters for reference weighting factors \(b\) = 0.5 and \(b\) = 0 were calculated using Expressions (21) and (24):  
\[ \begin{array}{l}b=0.5,\ K_{P}=0.46336,\ K_{I}=0.062992,\ K_{D}=0.5563,\\ b=0,\ K_{P}=0.51394,\ K_{I}=0.083446,\ K_{D}=0.58265.\end{array}  \tag{53}\]  
Additionally, the parameters of the PID\({}_{\text{fh}}\) controller reference filter were calculated using Expression (43) (for chosen \(K_{CL}\) = 3):  
\[ \begin{array}{l}b_{\text{F5}}=8.2389,b_{\text{F4}}=25.8477,b_{F3}=27.1014,b_ {F2}=15.97,b_{\text{F1}}=5.6589,\\ T_{dm}=1,T_{\text{CL}}=1.0155,\ n=2.\end{array}  \tag{54}\]  
The calculated parameters were applied to the PID\({}_{\text{b}}\) and PID\({}_{\text{fh}}\) controllers. The robustness was studied by assuming a \(\pm\)10% change in the Process (51) steady-state gain and time-delay. Figures 24 and 25 show the corresponding responses for the input disturbance step change (\(d\) = 1) and the reference step change (\(r\) = 1) for the PID\({}_{\text{b}}\) controller with \(b\) = 0.5 and for the PID\({}_{\text{fh}}\) controller, respectively. Figure 23: The values of M\({}_{\text{S}}\) for \(G_{P}\) (s) and controller parameter \(b\) = 0.9 at different values of \(T_{delay}\) and \(\alpha\).