Brief paperEvent-triggered control of nonlinear singularly perturbed systems based only on the slow dynamics☆
Introduction
The increasing popularity of embedded systems and networked control systems has motivated the development of new implementation paradigms in order to handle the resources limitations of these systems. Indeed, although periodic sampling is appealing from the analysis and implementation point of view, it may yield a conservative solution as it may unnecessarily use the network. Event-triggered control has been proposed as an alternative where it is the occurrence of an event, typically a variation of the plant state and not a clock, which closes the feedback loop (Årzén, 1999, Åström and Bernhardsson, 1999). This may allow to significantly reduce the utilization of the resources compared to the periodic implementation, see e.g. Donkers and Heemels (2012), Heemels, Johansson, and Tabuada (2012), Postoyan, Anta, Nešić, and Tabuada (2011), Tabuada (2007) and Wang and Lemmon (2011). Available techniques rely on the knowledge of an accurate model of the plant (which may be affected by uncertainties or external disturbances). However, the controller is often designed based on a reduced or simplified model of the plant dynamics. For two time-scale systems for instance, singular perturbation theory can be used to approximate the slow and the fast dynamics, see Khalil (2002) and Kokotović, Khalil, and O’Reilly (1986). In this context, it is possible to design the controller based only on the slow model, when the origin of the fast model is asymptotically stable, for stabilizable linear time-invariant (LTI) systems (Kokotović et al., 1986), classes of nonlinear systems (see Section 5.4 in Khalil, 2002) and linear time-varying sampled data systems with periodic sampling (Pan & Başar, 1994). In this paper, we investigate whether this approach is applicable for event-triggered control.
We consider the scenario where the controller communicates with a two-time scale nonlinear system via a digital communication channel. Our objective is to design a stabilizing event-triggered feedback law based only on an approximate model of the slow dynamics. This problem is motivated by the fact that engineers often neglect the fast stable dynamics in practice when these are stable and design the feedback law based only on the slow model. To the best of our knowledge, this is the first paper in that direction. It is important to stress that the stability of the approximate slow and fast models is not sufficient to conclude the overall stability of the original system even for continuous-time systems, see Lobry and Sari (2005) for a counter example. Hence, although the approximate fast subsystem is assumed to be stable and the approximate slow dynamics is stabilized by the event-triggered controller, a thorough analysis has to be made to ensure the stability of the original system.
We cast the overall problem as a hybrid singularly perturbed system with the formalism of Goebel, Sanfelice, and Teel (2012). The stability of this type of systems is analyzed in Sanfelice and Teel (2011), Wang et al., 2012a, Wang et al., 2012b. In this study, we address a design problem as we construct the flow and jump sets (i.e. the triggering condition) and we propose different stability analyses under a different set of assumptions. We highlight a specific challenge which arises with the event-triggered implementation: the state of the fast model experiences a jump at each transmission due to the change of variables we introduce to separate the slow and the fast dynamics using singular perturbation theory. These jumps induce non-trivial difficulties in the stability analysis. That is a feature of the problem which is not present in available results on event-triggered control where only the sampling-induced error is reset to zero at each transmission, see e.g. Donkers and Heemels (2012), Heemels et al. (2012), Postoyan et al. (2011), Tabuada (2007) and Wang and Lemmon (2011).
We follow an emulation-like approach to design the event-triggered controllers (see Postoyan et al., 2011, Tabuada, 2007). We first synthesize a stabilizing controller for the approximate slow model obtained by singular perturbation theory, in the absence of communication constraints. Afterwards, we take into account the effect of the network and we design the event-triggering condition. The first observation we make is that, even if the triggering law guarantees the asymptotic stability of the origin of the slow model and the existence of a strictly positive lower bound on the inter-transmission times, such a time is no longer guaranteed to exist for the overall system. As a consequence, the controller is not implementable in practice. We then propose two classes of event-triggered controllers which overcome this issue. The first policy relies on the event-triggering conditions (Donkers and Heemels, 2012, Mazo and Cao, 2012) but it requires to fully modify the stability analysis to handle the features of the problem due to the two-time scale nature of the system. We show that a semiglobal practical stability property holds where the adjustable parameter appears in the event-triggering condition. The second technique combines the event-triggered implementation of Tabuada (2007) with the time-triggered results in Nešić, Teel, and Carnevale (2009), like in Mazo and Tabuada (2011), Tallapragada and Chopra, 2012, Tallapragada and Chopra, 2013 and Wang, Sun, and Hovakimyan (2012), in the sense that transmissions are only allowed after a predefined amount of time has elapsed since the last transmission. This allows us to directly tune the minimum transmission interval. We show that a global asymptotic stability property is satisfied in this case, under an additional assumption. The results are shown to be applicable to a class of globally Lipschitz systems, which include stabilizable LTI systems as a particular case.
The remainder of the paper is organized as follows. The problem is stated in Section 3. The main assumptions are presented in Section 4. In Section 5, we state the main results. In Section 6, we show that the proposed event-triggered control strategies are applicable to a class of globally Lipschitz systems. The proofs are given in the Appendix.
Section snippets
Preliminaries
We denote . The Euclidean norm is denoted as . We use the notation to represent the vector for and . A continuous function is of class if it is zero at zero, strictly increasing, and it is of class if in addition as . A continuous function is of class if for each is of class , and, for each is decreasing to zero. We denote the minimum and maximum eigenvalues of the
Problem statement
Consider the following nonlinear time-invariant singularly perturbed system where and are the states, is the control input and is a small parameter. We use singular perturbation theory to approximate the slow and the fast dynamics. We rely on the following standard assumption (see (11.3)–(11.4) in Khalil, 2002). Assumption 1 The equation has isolated real roots where is continuously differentiable.
Assumptions
We present the assumptions made on system (12). We will show in Section 6 that all the conditions are satisfied by a class of globally Lipschitz systems. The approximate slow and fast models (3), (5) are now in view of (8), (9) First, we assume that the slow system (14) is input-to-state stable (ISS) with respect to .
Assumption 2 There exist a continuously differentiable function and class functions with
Main results
First, we show that the design of triggering conditions of the same form as in Tabuada (2007) for the slow model may not ensure the existence of a strictly positive minimum amount of time between two jumps for the overall system. We then present our main results.
The case of globally Lipschitz systems
In this section, we show that all the conditions of Section 4 are verified by a class of globally Lipschitz systems, which includes LTI systems as a particular case. We assume that the vector fields and are globally Lipschitz and that the stability of the slow and the fast model can be verified using quadratic functions and . Under these conditions, the proposition below states that Assumption 1, Assumption 2, Assumption 3, Assumption 4, Assumption 5, Assumption 6 hold. Hence, the
Conclusion
We have investigated the event-triggered stabilization of nonlinear singularly perturbed systems based only on the slow dynamics. Two classes of controllers have been developed which ensure different asymptotic stability properties. We believe that this work can be extended along two important directions. First, the design of event-triggered controllers for singularly perturbed systems with potentially unstable fast dynamics can be pursued based on the model (12). Second, it would be
Mahmoud Abdelrahim received the M.Sc. degree in Mechatronics Engineering from Assiut university (Egypt) in 2010. He obtained the Ph.D. in Control Theory from the Centre de Recherche en Automatique de Nancy, Université de Lorraine (France) in 2014. His current research interests include event-triggered and self-triggered control, networked control systems, hybrid dynamical systems, singularly perturbed systems, robust stabilization and optimization.
References (23)
- et al.
On singular perturbations due to fast actuators in hybrid control systems
Automatica
(2011) - et al.
Event-triggered decentralized dynamic output feedback control for LTI systems
Estimation and Control of Networked Systems
(2012) - et al.
Asynchronous task execution in networked control systems using decentralized event-triggering
Systems & Control Letters
(2012) - et al.
Analysis for a class of singularly perturbed hybrid systems via averaging
Automatica
(2012) - Abdelrahim, M., Postoyan, R., & Daafouz, J. (2013). Event-triggered control of nonlinear singularly perturbed systems...
- Årzén, K.E. (1999). A simple event-based PID controller. In 14th IFAC world congress, Beijing, China, 18 (pp....
- Åström, K.J., & Bernhardsson, B. (1999). Comparison of periodic and event based sampling for first-order stochastic...
- et al.
Output-based event-triggered control with guaranteed -gain and improved and decentralised event-triggering
IEEE Transactions on Automatic Control
(2012) - et al.
Hybrid dynamical systems: modeling, stability, and robustness
(2012) - Heemels, W.P.M.H., Johansson, K.H., & Tabuada, P. (2012). An introduction to event-triggered and self-triggered...
Nonlinear systems
Cited by (0)
Mahmoud Abdelrahim received the M.Sc. degree in Mechatronics Engineering from Assiut university (Egypt) in 2010. He obtained the Ph.D. in Control Theory from the Centre de Recherche en Automatique de Nancy, Université de Lorraine (France) in 2014. His current research interests include event-triggered and self-triggered control, networked control systems, hybrid dynamical systems, singularly perturbed systems, robust stabilization and optimization.
Romain Postoyan received the M.Sc. degree in Electrical and Control Engineering from ENSEEIHT (France) in 2005. He obtained the M.Sc. by Research in Control Theory & Application from Coventry University (United Kingdom) in 2006 and the Ph.D. in Control Theory from Université Paris-Sud (France) in 2009. In 2010, he was a research assistant at the University of Melbourne (Australia). Since 2011, he is a CNRS researcher at the Centre de Recherche en Automatique de Nancy (France).
Jamal Daafouz received the M.S. degree in automatic control from the INSA Toulouse, France, in 1994 and the Ph.D. degree in automatic control from the INSA Toulouse, in 1997. In 1998, he joined the Institut National Polytechnique de Lorraine (INPL) as an assistant professor and the Research Centre of Automatic Control (CRAN UMR 7039 CNRS). In 2005, he got the French Habilitation degree from the Institut National Polytechnique de Lorraine (INPL). In 2005, he was engaged as a professor of automatic control at “Université de Lorraine” in Nancy, France. Prof. Daafouz is an IUF (Institut Universitaire de France) junior member. He is an Associate Editor at the Conference Editorial Board of the IEEE Control Systems Society and serves also as an associate editor for the journals: IEEE Transactions on Automatic Control and European Journal of Control. His research interests include hybrid and switched systems, networked control systems, robust control and applications in secure communications and metallurgy.
- ☆
This work has been supported by the European 7th Framework Network of Excellence ‘Highly-complex and networked control systems’ (HYCON2) (grant agreement no. 257462) and also by the ANR under the grant COMPACS (ANR-13-BS03-0004-02). The material in this paper was partially presented at the 9th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2013), September 4–6, 2013, Toulouse, France. This paper was recommended for publication in revised form by Associate Editor Maurice Heemels under the direction of Editor Andrew R. Teel.