# Performance Optimal PI controller Tuning Based on Integrating Plus Time Delay Models
## 4 Optimal Performance Settings  
Consider the following Pareto performance objective defined as  
\[J(p)=s_{r}\frac{\text{IAE}_{x}(p)}{\text{IAE}_{x}^{o}}+(1-s_{r})\frac{\text{IAE}_{ cuu}(p)}{\text{IAE}_{pu}^{o}}, \tag{41}\]  
where \(s_{r}\) is the servo-regulator parameter originally introduced in [2], and is chosen in the interval \(0\leq s_{r}\leq 1.0\) for the weighting between output disturbance (servo) weighting \(s_{r}=1.0\) and input disturbance (regulator) weighting \(s_{r}=0\). In Equation (41), the function argument is \(p=[K_{p},T_{i}]^{T}\). In this work, we set \(s_{r}=0.5\) ([16]). Furthermore, we set \(x=vy\), which was argued in [17] to be the equivalent of setting \(x=r\), which was used in the original work of [16] and also [2]. The reference/weight values are calculated as following, \(\text{IAE}_{vy}^{o}=\min_{p}\text{IAE}_{vy}(p)\), and, \(\text{IAE}_{pu}^{o}=\min_{p}\text{IAE}_{pu}(p)\), for a prescribed robustness \(M_{s}^{pre}\). We set \(M_{s}^{pre}=1.59\) which is the robustness value corresponding to a SIMC-tuned PI controller with \(T_{c}=\tau\) for a FOPTD process where \(K=T=\tau=1\) ([16]). We consider the reference example where we are given an IPTD process with \(k=\tau=1\). We find the same reference values as in [18], viz. \(\text{IAE}_{vy}^{o}=2.17\) where \(K_{p}=0.5\) and \(T_{i}=\infty\), and \(\text{IAE}_{pu}^{o}=15.10\) where \(T_{i}=5.8\) and \(K_{p}=0.4\). The following main performance objective is defined in a mean square error sense,  
\[V_{M}(x,\,y)=\frac{1}{M}\sum_{i=1}^{M}(J_{x}(i)-J_{y}(i))^{2}, \tag{42}\]  
where \(x\) is a tuning method and, \(y=\text{PO}\) (default) and \(\text{M}=\text{length($M_{s}$)}\). A couple of optimal suggestions for the choice of the MP parameter are worked out in the following. The first MP parameter setting may be found by solving the following optimization problem,  
\[\bar{c} = \arg\min_{\bar{c}}V_{M}(\text{Alg. 1}(\bar{c}),\,\text{Alg. 1}^{o})=2.7, \tag{43}\]  
where Alg. 1 (\(\bar{c},\delta_{i}\)) and Alg. 1\({}^{o}\) (\(\delta_{i}\)) is pre-calculated as follows  
\[J_{\text{Alg. 1}_{i}}=\min_{\bar{c}}J_{\text{Alg. 1}}(\bar{c},\delta_{i})\, \,\forall\,\,1.1\leq\delta_{i}\leq 3.4. \tag{44}\]  
Interestingly, the MP parameter setting in Equation (43) is approximately equal to the setting which is deduced in Section 3.3. Additional MP parameter settings are given in Table 1 based on solving Equation (43) for different servo-regulator parameters \(0\leq s_{r}\leq 1.0\) in the Pareto performance objective \(J\) (Equation (41)). The second MP parameter is found by  
\[\bar{c}=\arg\min_{\bar{c}}V_{M}(\text{Alg. 1}(\bar{c}),\,\text{PO})=2.5, \tag{45}\]  
where Alg. 1 (\(\bar{c},\delta(M_{s}^{i})\)) and PO (\(M_{s}^{i}\)) are pre-calculated as follows  
\[J_{\text{PO}_{i}}=\min_{p}J(p,M_{s}^{i})\,\,\forall\,\,1.1\leq M_{s}^{i}\leq 3.4. \tag{46}\]  
Notice that \(\bar{c}=2.5\) is equal to the recommended MP parameter in [2]. However, the MP parameter in this paper results from an optimization problem, while the one proposed in [2] originated from an ad hoc approach. Figure 3 illustrates the two MP parameters described above. In terms of the main performance objective \(V_{M}\) (Equation (42)), Table 2 shows that \(\bar{c}=2.5\) is \(\frac{V_{c-4}}{V_{c-2.5}}=3\text{e}+4\) times better than SIMC (arguably \(\bar{c}=4\)), and \(\frac{V_{c-2.7}}{V_{c-2.5}}=78\) times better than \(\bar{c}=2.7\). Based on numerical simulations, we present the recommended settings for choosing the MP parameter \(\tilde{c}\) as proposed in Table 3.  
Consider PI controller settings for an IPTD system, \(H_{p}(s)=k\frac{c^{-s}}{s}\), with varying gain velocity, \(k\), and time delay \(\tau\geq 0\). Tables 4 and 5 illustrate the \(\tilde{c}_{\mbox{min}}=\mbox{arg min}_{\tilde{c}}M_{s}\), i.e., the minimum of \(M_{s}\), \(\mbox{IAE}_{eu}\), \(\tilde{\mbox{ITAE}}_{eu}\) and \(\mbox{IAE}_{r}\), TV, ISE, \(\tilde{\mbox{ITAE}}_{r}\), respectively, as a function of \(\tilde{c}\).