ISA Transactions 51 (2012) 351–361 Contents lists available at SciVerse ScienceDirect ISA Transactions journal homepage: www.elsevier.com/locate/isatrans On-line delay estimation for stable, unstable and integrating systems under step response J. Herrera ∗ , A. Ibeas Departament de Telecomunicació i d’Enginyeria de Sistemes, Escola d’Enginyeria, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain article info Article history: Received 24 May 2011 Received in revised form 3 November 2011 Accepted 18 November 2011 Available online 10 December 2011 Keywords: Delay estimation Modified Smith Predictor Unstable systems abstract A simple but effective on-line method to estimate the delay from step response, which can be used for stable, unstable and integrating systems, is proposed in this paper. The estimation and control are made simultaneously since the nominal delay is updated in closed-loop based on certain calculus on the output signal. Moreover, the approach is based on a Modified Smith Predictor and the delay estimation is implemented using a multi-model scheme with fixed models. Additionally, the convergence properties of the estimation algorithm and the stability analysis of the closed-loop are well-defined. Simulation examples show the effectiveness of the proposed method, where the delay estimation leads to an optimal and robust controller, tackling the uncertainty in the delay. © 2011 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction Delay is present in many system models due to different reasons, especially: (i) the network delays, generally small, which are due to the transport of the information (control law, measures, etc.) between the process and the controller [1]; (ii) the mass transport delay, whose size is related with the physical properties of the systems [2]. The effect of the delay over the system is related principally with the ratio between the system time constant and the delay size. Different approaches can be used to deal with the delay effect. On the one hand, if the delay is small in relation with the system time constant, it is possible to make a model approximation (for instance, a Padé approximation) to design the controller. Tuning of PID controllers where the delay is approximated can be found in [1,3]. On the other hand, when the delay is large in comparison with the system time constant, approximation is not recom- mended. Generally, this problem is addressed using a Delay Compensation Scheme (DCS), where a system model is necessary beforehand in order to perform the compensation of the delay. In an ideal case, the Smith Predictor (SP) is undoubtedly the best well-known DCS for delay-system control since it allows the controller to be optimally designed without taking into account the delay. However, the basic shortcomings of the SP are also well- known: its reduced robustness with respect to delay uncertainty and its application to stable systems only [4]. ∗ Corresponding author. Tel.: +34 935 813027; fax: +34 935 814031. E-mail addresses: jorgeaurelio.herrera@uab.es (J. Herrera), asier.ibeas@uab.es (A. Ibeas). The SP has been extended to stable, unstable and integrating systems in [5], where a perfectly known delay is also necessary. Nevertheless, this restriction reduces the commercial use of these controllers. For this reason, a great deal of research has focused their efforts to counteract the delay uncertainty making the SP and its variants robust. Different approaches such as the robust PID tuning [6,7], iterative learning control [8,9], two-degree- of-freedom (2DOF) controllers [10] and Lyapunov–Krasovskii functional methods [11,12], have been proposed for this purpose. Generally, the obtained robustness is only in the stability property while the performance is strongly limited in some cases. Thus, its scope is reduced since good performance of the controller is often a prerequisite in practice. Recently, a framework focused on the identification and control of systems with delay uncertainty, has been proposed for both stable [13–15] and unstable cases [16]. The approach presented in [13–15] is based on the classical SP and a multi-model scheme. The multi-model scheme contains a battery of time-varying models which are updated using a modification rule. Each model possesses the same rational component but a different delay value. The algorithm compares the mismatch between the actual system and each model and selects, at each time interval, the one that best describes the behaviour of the actual system, providing online identification of the delay while simultaneously ensuring the closed-loop stability. The way in which the delay varies is determined by a heuristic optimization; this allows the delay identification and the system control simultaneously. Additionally, this approach leads to a robustly stable closed-loop system while achieving a great performance for systems with unknown long delays. This work was extended to stable, unstable and integrating systems in [16]. The approach has the same framework but, in this 0019-0578/$ – see front matter © 2011 ISA. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.isatra.2011.11.005