Brief paperObserver design for networked control systems with FlexRay☆
Introduction
Networked control systems (NCS) attract great attention due to the features they offer in terms of ease of installation and flexibility. NCS are characterized by the use of a shared serial communication channel to connect spatially distributed sensors and actuators with the control unit. While the usage of a network offers great advantages over traditional point-to-point set-ups, it also introduces communication constraints in terms of limited sampling, scheduling, delay, packet loss, quantization, which may have a severe impact on the desired requirements. In this paper, we focus on FlexRay networks and we concentrate on the effect of sampling and scheduling. FlexRay is a protocol developed by BMW, Daimler–Chrysler, Philips and Freescale in 2000 to provide appropriate communications for implementing X-by-wire technology in automotive control (Consortium, 2005, Schmidt and Schmidt, 2009). It works with communication cycles, that alternate between a static and a dynamic segment during which a specific scheduling rule is used (Consortium, 2005); we thus have to deal with a switched protocol. Our objective is to design observers for uncertain nonlinear NCS with FlexRay. This study is motivated by the fact that this communication protocol is increasingly used in the automotive industry and estimation methods for such systems are currently missing in the literature; only stabilization results are available to the best of our knowledge, see e.g. Naghshtabrizi and Hespanha (2009) and Wang, Nešić, and Postoyan (2015).
We consider the scenario where the plant input and the measurements are sent to the observer via the network. Actuators and sensors are grouped into nodes, and only one of these transmits its data to the observer at each transmission instant. The latter therefore only has access to partial and sampled information of the plant input and output. The solution we propose is based on emulation. The idea is to first synthesize the observer while ignoring the communication constraints. At this stage, any of the continuous-time observer design techniques available in the literature can be applied. Then, the observer is implemented over the network and conditions on the latter are derived to preserve the desired error convergence properties. Similar results exist for NCS with non-switched protocols see Postoyan and Nešić (2012) and Postoyan, van de Wouw, Nešić, and Heemels (2014), however none of these apply to our problem because of the switches between the two scheduling rules exhibited by FlexRay. Indeed, these switches require a new model, appropriate assumptions on the observer and the network, and a new stability analysis. In addition, we consider plants with input, perturbed dynamics and noisy measurements, which are more general and induce additional technical difficulties compared with (Postoyan & Nešić, 2012) and Postoyan et al. (2014).
We model the overall system as a hybrid system in the formalism of Goebel, Sanfelice, and Teel (2012), for which a jump either describes a segment switch or a transmission. We assume that the transmissions during the static and dynamic segments are governed by possibly (distinct) input-to-state stable (ISS) protocols, which include round-robin (RR) and maximum-error-first try-once-discard (TOD) (Walsh & Ye, 2001), as particular cases. The concept of ISS protocol was introduced in Tabbara and Nešić (2008) and appears to be very useful for NCS subject to measurement noise. We provide explicit bounds on the maximum allowable transmission intervals (MATI) for each segment and we guarantee an input-to-state stability property for the estimation error system. The analysis relies on a novel hybrid Lyapunov function.
Our contributions are threefold. First, we present an observer design strategy for NCS with FlexRay for the first time to the best of our knowledge. Second, the MATI bounds we propose are much simpler to compute than those derived in Wang et al. (2015), where the corresponding stabilization problem is addressed. This is due to the novel hybrid Lyapunov function we construct. Third, our results extend the works in Postoyan and Nešić (2012) and Postoyan et al. (2014) to perturbed systems with control inputs and noisy measurements in the particular case where there is a single segment.
Section snippets
Problem statement
Consider the nonlinear plant where is the state, is the control input, is the external disturbance, is the plant output affected by the noise . The functions and are assumed to be Lebesgue measurable and differentiable. Moreover, these functions and their time-derivatives are assumed to have a finite norm. We assume that we know an observer of the form where is the estimate of the state
FlexRay
In this section, we briefly present FlexRay and the assumptions we make on the network; for more details, see Wang et al. (2015). FlexRay works with pre-set communication cycles of length . Each cycle contains a static segment of length , a dynamic segment of length and two protocol segments called symbol window and network idle time, see Consortium (2005). The lengths of the protocol segments can be considered as negligible compared to and , and hence we ignore them in the
Observer emulation
We emulate observer (2) as follows As already mentioned in Section 2, the emulated observer (4) no longer depends on , but on because of the network. These variables are generated by the observer based on the received data. Furthermore, observer (4) does not depend on its own output , as in (2), but on . The variable is an artificially introduced networked version of . The idea to use instead of was suggested in Postoyan et al. (2014) and
Switched protocols
In this section, we model the transmission mechanisms of FlexRay under the assumptions made in Section 3 and we present the class of scheduling rules we consider.
NCS model
We now write the overall system. We introduce for this purpose the variable to count the number of transmissions, which is useful to model scheduling policies such as RR. We also introduce the estimation error with , and , where and respectively denote the time-derivative of the input signal and of the noise . Let be the full state vector. In view of Sections 2 Problem statement, 3 FlexRay, 4 Observer emulation, 5 Switched
Stability analysis
We first assume an exponential growth condition on the -subsystem during two consecutive transmissions, like in Postoyan and Nešić (2012) and Postoyan et al. (2014). Assumption 3 For each , there exist a continuous function , and such that for all , , , , and almost all : , where the function comes from Assumption 2. □
The above condition is satisfied when is globally Lipschitz in
Conclusions
We proposed an observer design approach for nonlinear NCS with FlexRay. FlexRay is composed of communications cycles and each cycle consists of a static and a dynamic segment for which different scheduling policies apply. For an observer designed in the absence of communication constraints, we investigated the conditions on the network to preserve its convergence property. In particular, we provided explicit segment-dependent maximal allowable transmission interval (MATI) bounds and showed that
Wei Wang received her B.E. and M.E. degrees in Mechanical Engineering from Qingdao University of Science and Technology in 1994 and 2002, respectively. She obtained her Ph.D.degree in Electrical Engineering at the University of Melbourne, Australia, in 2011. Since July 2011, she is a post-doctoral researcher in the Department of Electrical and Electronic Engineering at the University of Melbourne. Her research interests include nonlinear control systems, hybrid systems and networked control
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Wei Wang received her B.E. and M.E. degrees in Mechanical Engineering from Qingdao University of Science and Technology in 1994 and 2002, respectively. She obtained her Ph.D.degree in Electrical Engineering at the University of Melbourne, Australia, in 2011. Since July 2011, she is a post-doctoral researcher in the Department of Electrical and Electronic Engineering at the University of Melbourne. Her research interests include nonlinear control systems, hybrid systems and networked control systems.
Dragan Nešić is a Professor in the Department of Electrical and Electronic Engineering (DEEE) at The University of Melbourne, Australia. He received his B.E. degree in Mechanical Engineering from The University of Belgrade, Yugoslavia in 1990, and his Ph.D. degree from Systems Engineering, RSISE, Australian National University, Canberra, Australia in 1997. Since February 1999 he has been with The University of Melbourne. His research interests include networked control systems, discrete-time, sampled-data and continuous-time nonlinear control systems, input-to-state stability, extremum seeking control, applications of symbolic computation in control theory, hybrid control systems, and so on. He was awarded a Humboldt Research Fellowship (2003) by the Alexander von Humboldt Foundation, an Australian Professorial Fellowship (2004–2009) and Future Fellowship (2010–2014) by the Australian Research Council. He was a Distinguished Lecturer of CSS, IEEE. He is a Fellow of IEEE and a Fellow of IEAust. He served as an Associate Editor for the journals Automatica, IEEE Transactions on Automatic Control, IEEE Transactions on Control of Networked Systems, Systems and Control Letters and European Journal of Control.
Romain Postoyan received the master degree (diplôme díngénieur) 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). He serves as an Associate Editor at the Conference Editorial Board of the IEEE Control Systems Society and for the journals: Automatica, IEEE Control Systems Letters, and IMA Journal of Mathematical Control and Information.
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Supported by the Australian Research Council under the Discovery Project (DP170104099), and the ANR under the grant COMPACS (ANR-13- BS03-0004-02). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Hideaki Ishii under the direction of Editor Christos G. Cassandras.