# Performance Optimal PI controller Tuning Based on Integrating Plus Time Delay Models  
[PERSON]  
1 Meiersvingen 38, Skien 3744, Norway; [EMAIL_ADDRESS]  
[PERSON]  
2 Department of Electrical Engineering, IT and Cybernetics, University of South-Eastern Norway,  
Porsgrgnum 3918, Norway2 [PERSON]; Tel. : +47-3557-516821377218  
where \(K_{p}\) is the PI controller proportional gain, \(T_{i}\) is the integral time constant, \(k\) is the gain velocity (slope) and \(\tau\geq 0\) is the time delay. \(\alpha\) and \(\beta\) in Equation (2) are dimensionless parameters. For instance, using the classical Ziegler-Nichols (ZN) PI controller tuning rules, proposed in the works [9; 10; 11], gives \(\alpha=\frac{\pi}{4.4}\), \(\beta=\frac{4}{1.2}\) (i.e., the ZN closed loop method). Using the Internal Model Control (IMC) PI controller tuning rules in Table 1 of [7] with closed loop time constant \(T_{c}=\sqrt{10}\tau\), as proposed in [6], gives parameters \(\alpha=0.42\) and \(\beta=7.32\). Using the Simple/Skogestad IMC (SIMC) PI controller tuning rules, presented in the works of [8; 12; 13], with closed loop time constant \(T_{c}=\tau\) (i.e., is the only tuning parameter in SIMC) gives \(\alpha=0.5\) and \(\beta=8\). To find PI controller settings with good robustness properties (i.e., one could have uncertainties in the gain velocity and time delay) and simultaneously obtain reasonable fast reference and disturbance properties, for IPTD processes, the size and balanced relation between the parameters \(\alpha\) and \(\beta\) are of importance. Using the PI controller setting in Equation (2), we may define a Method Product (MP) parameter \(\bar{c}\) as,  
\[\bar{c}=\alpha\beta=K_{p}T_{i}\,k. \tag{3}\]  
The defined MP parameter \(\bar{c}\) in Equation (3) is constant for numerous PI controller tuning methods. The SIMC PI controller settings yield an MP parameter \(\bar{c}=4\). The original ZN method gives an MP parameter \(\bar{c}=2.38\) (i.e., the ZN closed loop method). In this paper, we search for optimal MP parameters, i.e., choosing \(\bar{c}\) which ensures the closed loop system some optimal robustness or performance setting, e.g., minimisation of the Integrated Absolute Error (IAE) or sensitivity index \(M_{s}\) given a prescribed robustness. Figure 1 shows that \(M_{s}\) is approximately minimised for \(\bar{c}=2.0\). However, it might be argued that the changes in \(M_{s}\) is negligible, and that \(M_{s}\) is optimal over the MP parameter interval \(1.5\leq\bar{c}\leq 4.0\). Figure 1: Consider PI control of the FOPTD process model, \(H_{p}(s)=\frac{c^{-s}}{s}\). The figure shows the robustness \(M_{s}\) as a function of the MP parameter \(\bar{c}\), given constant robustness, for the interval \(1.5\leq\bar{c}\leq 4.0\). It has been pointed out that there is usually a high degree of trial-and-error in choosing the closed loop time constant tuning parameter \(T_{c}\) in SIMC and IMC (e.g., [1] for SIMC and [6] for IMC). Note that one also may focus on the maximum sensitivity peak \(M_{s}\) of the sensitivity function as described in [14], where some inequalities relating to the Gain Margin (GM) and the Phase Margin (PM) to the robustness \(M_{s}\) are proposed on p. 126. Consider that the values of the minimum robustness \(M_{s}\) are in the interval \(1.3\leq M_{s}\leq 2\)[14]. The contributions of this work are itemised in the incoming:  
* The PI controller tuning method in the work of [1, 2] is further developed with more optimal settings for the MP parameter as well as tuning for some special instance integrating systems. * In the instance of a small or zero time delay \(\tau=0\), we propose a variant in which the Maximum Time Delay Error (MTDE) \(d\tau_{\text{max}}>0\) is the tuning parameter (see Section 3.2). * Two optimal settings for the MP parameter are presented in Section 4. These are optimal in the sense that they minimise a Pareto performance objective (i.e., integrated absolute error for combined step changes in output and input disturbances) on two different aspects.