Journal of Process Control 23 (2013) 826–838 Contents lists available at SciVerse ScienceDirect Journal of Process Control jou r n al hom ep age: www.elsevier.com/locate/jprocont Characterization of limit cycles for self-regulating and integral processes with PI control and send-on-delta sampling Jesús Chacón a,∗ , José Sánchez a , Antonio Visioli c , Luis Yebra b , Sebastián Dormido a a Universidad Nacional de Educación a Distancia, Spain b CIEMAT - Plataforma Solar de Almería, Spain c Università degli Studi di Brescia, Italy a r t i c l e i n f o Article history: Received 6 November 2012 Received in revised form 7 February 2013 Accepted 2 April 2013 Available online 31 May 2013 Keywords: PID control Limit cycles Time delayed processes a b s t r a c t This work is focused on the study of limit cycles that appear in a control scheme which is based on the use of a PI controller with an event-based send-on-delta sampling (SOD). The processes investigated are integrator processes plus time delay (IPTD) and first and second order processes plus time delay, which are of interest because they are frequently used to model many industrial processes. The SOD sampling is characterized as a non-linearity of n levels with hysteresis. An algorithm to calculate the limit cycles properties is proposed, and then the results obtained in simulations are compared with experiments performed on a real plant, a distributed solar collector field at the Solar Platform of Almería (PSA, Spain). © 2013 Elsevier Ltd. All rights reserved. 1. Introduction Traditionally, the control systems used in the different engineer- ing areas have been based on periodic sampling, for which there is a wide range of mathematical tools and well established theoretical results (for example, [1]). On the contrary, event-based sampling schemes appear in a natural way in wireless sensor networks (WSN) where the controller or the sensors send their outputs according to the violation of some condition. Also, event-based strategies have been used for long time in areas such as control of industrial pro- cesses [2], robot path planning [3] and engine control [4]. However, event-based control has been used mainly in an ad-hoc way, due to the lack of theoretical results, which have begun to be available only in the last years (for example [5–7]). Recently, event-based control is also being used in multi-agent [8] and distributed systems [9]. Event-based systems are frequently classified into two types, event-triggered systems [10], which usually do not have a model of the plant, where the changes measured in the state provoke spo- radic updates of the controller, and self-triggered systems [11,12], where the controller calculates the time when the next event is triggered, based on an estimation of the state of the plant obtained from a model. Some works related to event-triggered systems are [9], where authors develop a distributed event-triggered estimation algorithm for networked systems, or [13,14], where it is used a periodic event- ∗ Corresponding author. Tel.: +34 913987147. E-mail address: jchacon@bec.uned.es (J. Chacón). triggered strategy, which is a variant of event-triggered where the event condition is verified with a constant sampling period. In [15], authors implement a distributed event-triggered control sys- tem based on local event generators and prove practical stability. Finally, in [16], authors analyze the relationship between sampled- data control and two event-triggered strategies, namely, deadband control and model-based event-triggered control. With respect to self-triggered systems, a simple self-triggered sampler for nonlinear systems is presented in [17]. Another self-triggered sampler for nonlinear systems is developed in [18,19], under the assumption that the closed-loop system with continuous-time control is input-to-state stable (ISS) with respect to measurement errors. In [20], authors present a scheme that ensures finite-gain L2 stability of the resulting self-triggered feed- back systems. Proportional-Integral-Derivative (PID) controllers are widespread in industry, mainly because of their satisfactory performance for many processes and because they are relatively easy to design and tune. Thus, in recent times many authors have addressed the design and implementation of event-based PID controllers ([21,22]). A comprehensive survey of the different methods proposed in the literature for event based PID control can be found on [23]. This work is focused on event-triggered sampling, and in par- ticular on the level crossing or send-on-delta sampling [21], which consists in taking a new sample when a change greater than a pre- defined threshold ı is detected in the signal. In practice, this scheme is equivalent to introduce a non-linearity that can be characterized as a quantization of n levels with hysteresis. 0959-1524/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jprocont.2013.04.001