Research article A PI tuning rule for integrating plus dead time processes with parametric uncertainty Pedro Mercader n , Alfonso Baños Dpt. Informática y Sistemas, University of Murcia, 30071 Murcia, Spain article info Article history: Received 7 September 2015 Received in revised form 19 September 2016 Accepted 23 January 2017 Keywords: PI control Robust control Integrating plus dead time process abstract A novel method to tune a Proportional–Integral (PI) compensator for an integrating plus dead time (IPDT) process, in presence of interval parametric uncertainty, is presented. The design is based on optimization of load disturbance rejection with constraints on the magnitude of the sensitivity and complementary sensitivity functions, that must be satisfied for any element belonging to a set of plants. Instead of solving this problem with a brute force approach (grid the uncertainty set), we prove that this problem can be solved by considering only two plants. That lets us to obtain a tuning rule, after using some approx- imations. To conclude, some examples will be given in order to elucidate the usefulness of the proposed tuning rule. & 2017 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction Proportional–Integral (PI) and Proportional–Integral–Derivative (PID) compensation are a classical and still active topic in control engineering research (see for example [1–3]). Although several advanced control strategies have been proposed, PI (D) compensators are widely used in industrial control practice. This fact is mainly due to its noticeable effectiveness and simple structure, which is conceptually easy to understand. This work is primarily focused on the problem of tuning PI compensators for integrating plus dead time (IPDT) processes having interval parametric uncertainty. IPDT processes are a common and important part in the process industry; they appear, for example, in level systems, pulp and paper plants, oil-water-gas separators in oil industry, etc. [2,4]. They also appear on simplified Saint-Venant equations to model irrigation canals for control de- sign purpose [5–7], where this model is referred to as integrator delay (ID) model. Recent works on IPDT controlled by PID-type controllers have been reported in [8–14]. The method proposed here is based on the parameter space approach, without using dead time approximations or ignoring parameter uncertainties. These are two important features: on the one hand, it is well known that some dead time approximations can lead to instability [15]; on the other hand, most of tuning rules are usually based on a nominal process model, whereas in practice the model usually varies with the operation point, or simply it is not known precisely. Specifically in PI(D) compensation, the formulation of the control design problem with parametric uncertainty has previously been approached in a number of works, see for example [16–19]. However, in all these previous works, the design method comes up with an algorithmic or graphical procedure to obtain the optimum value of the compensator parameters. This problem is usually solved with a brute force approach (by gridding the uncertainty set). This work will show that this problem can be solved by considering only two IPDT plants, and that a solution in form of a tuning rule can be obtained with a minimum degree of conservatism. The outline of this paper is as follows. Section 2 describes the problem at hand. The main results of this work are presented in Sections 3 and 4. Section 3 provides an equivalent control design problem that only involves two plants, making a key simplification of the original problem. Section 4 presents a tuning rule for an uncertain IPDT process, which is based on the solution of the simplified control problem. Some design examples that illustrate the proposed tuning rule are given in Section 5. In a final section, general remarks and conclusions are presented. 2. Problem formulation The considered control system setup is the single–degree–of– freedom feedback control system shown in Fig. 1, where r is the reference input, y is the plant output, = − e r y is the error, d is the disturbance input, and u is the compensator output. Here, C(s) is the compensator and P(s) is the plant. Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/isatrans ISA Transactions http://dx.doi.org/10.1016/j.isatra.2017.01.025 0019-0578/& 2017 ISA. Published by Elsevier Ltd. All rights reserved. n Corresponding author. Please cite this article as: Mercader P, Baños A. A PI tuning rule for integrating plus dead time processes with parametric uncertainty. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.01.025i ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎