Emulation-based stabilization of networked control systems implemented on FlexRay

doi:10.1016/j.automatica.2015.06.010

Automatica

Volume 59, September 2015, Pages 73-83

https://doi.org/10.1016/j.automatica.2015.06.010 Get rights and content

Abstract

We investigate the emulation controller design approach for nonlinear networked control systems (NCS) with FlexRay. FlexRay is a deterministic communication protocol which is increasingly used in the automotive industry as it provides a high bandwidth and allows for safety critical applications. It is characterized by pre-set communication cycles that are subdivided into static and dynamic segments; the data transmissions are scheduled by different rules depending on the segment. We propose for the first time a hybrid model of NCS with FlexRay for this purpose. We show, under reasonable assumptions, that the asymptotic stability property ensured by the controller in the absence of communication constraints is preserved when the latter is implemented over FlexRay with sufficiently frequent data transmission. In particular, we assume that on each communication segment, the data transmissions are governed by uniformly globally exponentially stable protocols. This covers the case when the round-robin protocol is implemented on the static segment and the try-once-discard protocol is implemented on the dynamic segment. We provide explicit maximum allowable transmission interval bounds that guarantee stability.

Introduction

Networked control systems (NCS) is a new generation of control systems in which the controller and the plant communicate via a digital network. The network is typically modeled as a serial communication channel with a finite bandwidth. NCS attract a lot of attention due to the benefits they offer in terms of reduced cost, weight and volume and ease of maintenance and installation. On the other hand, the communication channel induces undesirable constraints like time-varying sampling, time delays, scheduling, quantization and packet dropouts, which complicate the analysis and the design and in general adversely affect the behavior of NCS. In the present paper, we concentrate on the effect of time-varying data sampling and scheduling. Many solutions have been proposed to handle these constraints for NCS, see Heemels and van de Wouw (2010) for a good summary. However, it is not clear whether these theoretical results can be directly applied to NCS with some specific communication networks. This paper aims to bridge this gap between theory and practice by considering NCS with Flexray. There are works that deal with feedback control systems implemented on FlexRay, see Goswami, Schneider, and Chakraborty (2011) and Naghshtabrizi and Hespanha (2009) for instance. We generalize these results by considering nonlinear systems (as opposed to linear) and we propose a novel model of NCS with FlexRay, which we believe would be very useful in future studies of this important class of systems.

FlexRay was developed by BMW, Daimler-Chrysler, Philips and Freescale in 2000 (Flexray Consortium, 2005, Schmidt and Schmidt, 2009) to provide appropriate communications for implementing x-by-wire technology in automotive control. FlexRay network sends data packets in pre-set communication cycles that consist of a static segment and a dynamic segment that are periodically repeated (Flexray Consortium, 2005). During the static segment, the network capacity is assigned to nodes in a prefixed manner: we say that the scheduling rule is static. The dynamic segment enables messages to be sent whenever it is required which helps to meet varying bandwidth requirements that can emerge at system run time. A dynamic policy schedules transmissions of the nodes in this case is based on the online information.

Our first contribution is a novel hybrid model for NCS with FlexRay based on the formalism of Goebel, Sanfelice, and Teel (2012) that we derive under some simplifying but reasonable assumptions. Our goal is to provide a high fidelity model that is amenable to controller design and stability analysis; the main challenge is to model the data transmissions on the one hand and the switches between the static and the dynamic segments on the other hand. To achieve this, we first describe in detail the Flexray communication cycles and then introduce two clock variables in our model which respectively represent the time elapsed since the last data transmission and the time elapsed since the last segment switch.

Our second contribution is an emulation result for NCS with FlexRay. The main idea in emulation is to first design a controller that stabilizes the plant in the absence of the network; at this step, one can use any nonlinear continuous-time design technique to construct the controller. Then, in the second step, the controller is implemented over the network and it is shown that the stability of the system is preserved for sufficiently high communication bandwidth, which is measured in terms of the so-called maximal allowable transmission interval (MATI). This approach is applicable to a wide class of nonlinear NCS and network protocols, see Carnevale, Teel, and Nešić (2007), Heemels and van de Wouw (2010), Nešić and Teel (2004a), Tabbara, Nešić, and Teel (2008) and Walsh and Ye (2001) and references therein. However, none of the available results in the literature are directly applicable to the model of NCS with FlexRay that we propose; this is mainly due to the complexity of the FlexRay communication cycle that requires more complicated models not considered in prior works. Hence, we develop novel emulation results that are applicable in this case. In particular, we assume that, for instance, Round-Robin (RR) protocol is implemented during the static segment and Try-Once-Discard (TOD) protocol is implemented during the dynamic segment of the FlexRay communication cycle. We show that asymptotic stability of the system in the absence of network is maintained when the controller is implemented over FlexRay provided that the MATI of each segment satisfies a given bound.

The analysis relies on an original hybrid Lyapunov function construction which generalizes the one proposed in Carnevale et al. (2007). Moreover, we derive segment-dependent MATI bounds which seems to be more adequate in practice. We believe that the results of this paper, in particular the hybrid model and the constructed Lyapunov function, could be used as a starting point to investigate other problems such as estimation or tracking control for NCS over FlexRay. A preliminary version of this work is presented in Wang, Nešić, and Postoyan (2014b).

The paper is organized as follows. Preliminaries and definitions related to hybrid systems are given in Section 2. We explain the emulation approach for controller design in Section 3 and describe FlexRay in more detail in Section 4. We present the hybrid model in Section 5. The stability results are stated in Section 6 and an example is proposed in Section 7. Conclusions are given in Section 8. Most proofs are postponed to the Appendix A Technical lemmas, Appendix B Proof of the main results.

Section snippets

Preliminaries

Let $Z_{> 0} ≔ {1, 2, \dots}$ , $Z_{\geq 0} ≔ {0, 1, 2, \dots}$ and $R_{\geq 0} ≔ [0, \infty)$ . Let $| x |$ denote the Euclidean norm of the vector $x \in R^{n}$ and $I_{n}$ be the identity matrix of dimension $n$ . For $(x, y) \in R^{n + m}$ , $(x, y)$ stands for ${[x^{T}, y^{T}]}^{T}$ . Given a closed set $A \subset R^{n}$ and $x \in R^{n}$ , we define the distance of a vector $x$ to $A$ as ${| x |}_{A} ≔ {inf}_{y \in A} | x - y |$ . A set-valued mapping $M : R^{m} ⇉ R^{n}$ is locally bounded at $x \in R^{m}$ if there exists a neighborhood $U_{x}$ of $x$ such that $M (U_{x}) \subset R^{n}$ is bounded. The mapping $M$ is locally bounded if it is locally bounded at each $x \in R^{m}$ . A set-valued

Emulation for NCS

The purpose of this section is twofold. First, we explain the emulation method for controller design for NCS as proposed in Carnevale et al. (2007) and Nešić and Teel, 2004a, Nešić and Teel, 2004b that we follow in this paper. Second, we present an impulsive model and a hybrid model of NCS that was considered in prior work (Carnevale et al., 2007, Nešić and Teel, 2004a, Nešić and Teel, 2004b) and explain why that the modeling approach is also pertinent to NCS with FlexRay. Our main observation

FlexRay network

In this section, we provide a detailed description of the FlexRay communication cycle and introduce the modeling assumptions that allow us to represent NCS with FlexRay as a hybrid system of the form (1).

Impulsive model

In view of Section 4, NCS with FlexRay can be modeled by the impulsive system below under Assumption 2, Assumption 3, Assumption 4, $\begin{matrix} \dot{x} = f (x, e) t \in [t_{i}, t_{i + 1}] \\ \dot{e} = g (x, e) t \in [t_{i}, t_{i + 1}] \\ \dot{q} = 0 t \in [t_{j}^{m}, t_{j + 1}^{m}] \end{matrix}$ $\begin{matrix} x (t_{i}^{+}) = x (t_{i}) \\ e (t_{i}^{+}) = h (i, e (t_{i}), q (t_{i})) \\ q (t_{i}^{+}) = q (t_{i}) \end{matrix}$ $\begin{matrix} x ({t_{j}^{m}}^{+}) = x (t_{j}^{m}) \\ e ({t_{j}^{m}}^{+}) = e (t_{j}^{m}) \\ q ({t_{j}^{m}}^{+}) = 3 - q (t_{j}^{m}) \end{matrix}$ where $x ≔ (x_{p}, x_{c}) \in R^{n_{x}}$ , $i, j, m \in Z_{> 0}$ , the vector fields $f$ and $g$ are obtained by direct calculations from (3), (4) and are assumed to be continuous. We also assume that $h$ is continuous, which is the case for RR and TOD

Asymptotic stability properties

In this section, we first state the assumptions we make on system (23). We then construct a novel Lyapunov function for the hybrid model (23) which is shown to decrease on flows and not to increase at jumps, from which we derive that a global asymptotic stability property holds for system (23).

Illustrative example

In this section, we apply our results to stabilize the origin of a batch reactor in Nešić and Teel (2004a) over FlexRay. Consider the batch reactor plant of the form $x_{p} = A_{p} x_{p} + B_{p} u y = C_{p} x_{p},$ and the PI controller given by ${\dot{x}}_{c} = A_{c} x_{c} + B_{c} y u = C_{c} x_{c} + D_{c} y,$ where $x_{p} \in R^{4}$ , $x_{c} \in R^{2}$ , $y \in R^{2}$ and $u \in R^{2}$ , $A_{p}$ , $B_{p}$ , $C_{p}$ , $A_{c}$ , $B_{c}$ , $C_{c}$ and $D_{c}$ are constant matrices that can be found in Section 4 of Nešić and Teel (2004a). We consider the scenario where the controller is directly connected to the actuator and the sensor measurements

Conclusion

We have shown how the emulation approach can be used to design controllers implemented over FlexRay for nonlinear systems. In contrast with the previous related works (Carnevale et al., 2007, Nešić and Teel, 2004a, Tabbara et al., 2008, Walsh and Ye, 2001), the protocol which governs transmissions in FlexRay switches in the sense that a different policy is used during the static and the dynamic segments. As a consequence, we need to develop a novel hybrid model which describes NCS implemented

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 from 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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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 is a Fellow of IEEE and a Fellow of IEAust. He is currently a Distinguished Lecturer of CSS, IEEE (since 2008). He served as an Associate Editor for the journals Automatica, IEEE Transactions on Automatic Control, Systems and Control Letters and European Journal of Control.

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).

^☆: Supported by the Australian Research Council under the Discovery Project and Future Fellow program, the ANR under the grant COMPACS (ANR-13- BS03-0004-02) and the European 7th Framework Network of Excellence “Highly-complex and networked control systems” (HYCON2 No. 257462). The material in this paper was partially presented at the 53rd IEEE Conference on Decision and Control, December 15–17, 2014, Los Angeles, CA, 2014. This paper was recommended for publication in revised form by Associate Editor Maurice Heemels under the direction of Editor Andrew R. Teel.

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Emulation-based stabilization of networked control systems implemented on FlexRay☆

Abstract

Introduction

Section snippets

Preliminaries

Emulation for NCS

FlexRay network

Impulsive model

Asymptotic stability properties

Illustrative example

Conclusion

Automatica

A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems

IEEE Transactions on Automatic Control

On the properties of the flexible time division multiple access technique

IEEE Transactions on Industrial Informatics

Hybrid dynamical systems: modeling, stability, and robustness

Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance

IEEE Transactions on Automatic Control

Stability and stabilization of networked control systems

Nonlinear control systems

Nonlinear control systems: part II