Event-triggered transmission for linear control over communication channels

Abstract

We consider an exponentially stable closed loop interconnection between a continuous-time linear plant and a continuous-time linear controller, and we study the problem of interconnecting the plant output to the controller input through a digital channel. We propose an event-triggered transmission policy whose goal is to transmit the measured plant output information as little as possible while preserving closed-loop stability. Global asymptotic stability is guaranteed when the plant state is available or when an estimate of the state is available (provided by a classical continuous-time linear observer). Under further assumptions, the transmission policy guarantees global exponential stability of the origin.

Introduction

The study of event-triggered and self-triggered systems (Anta and Tabuada, 2010, Carnevale et al., 2007, Cervin and Henningsson, 2008, Mazo and Tabuada, 2008, Nešić and Teel, 2004, Postoyan et al., 2011, Tabuada, 2007, Wang and Lemmon, 2008a, Wang and Lemmon, 2008b) led to a significant amount of research results where the core problem under consideration is that of two nodes (the sensing node and the actuating one) communicating through a (low capacity) digital channel where the transmission policy is determined based on suitable Lyapunov-like conditions involving some (more or less coarse) measurement of the plant state. A natural way to represent and suitably write the dynamics of this specific two-nodes configuration is to use the hybrid systems notation, namely a state-space description wherein the state flows according to some continuous-time rules and, at some specific times, called jump times, it jumps following some discrete-time jump rule. A framework for the representation of hybrid systems that has been recently proposed and is surveyed in Goebel et al., 2009, Goebel et al., 2012 allows for a quite natural description of these phenomena with useful Lyapunov like results that have been proven to apply to large classes of systems described using this framework. This framework was used in connection with event-triggered control in Forni, Galeani, Nešić, and Zaccarian (2010), which is a preliminary version of the results of this paper. Then, in Postoyan et al. (2011), Lyapunov tools are used to model ISS properties of networked control systems and the MATI (maximum allowable transfer interval), to preserve asymptotic stability. Later, in Seuret, Prieur, and Marchand (2013) a similar approach to the one of this paper (and Forni et al., 2010) has been developed for the nonlinear case.

Here, we consider a closed-loop system that consists of a linear controller driving a linear plant to guarantee closed-loop asymptotic stability, as shown in Fig. 1, and we break the continuity of the transmission of the measured plant output $y$ to the controller input $u$ by introducing transmission devices/sensors which measure the output $y$ and decide whether or not sending this measurement to the controller input $u$ through a transmission channel, based on non-periodic Lyapunov-based policies. The ideal scenario where this approach is relevant corresponds to cases where due to some technological constraint, there is a transmission line between a location where all the sensors are installed and a second location where the actuators are placed with a transmission channel in between (see Fig. 1). An example of an industrial control problem where this type of constraints appear is described in Jijón, Canudas de Wit, Niculescu, and Dumon (2010) and corresponds to oil well drilling operations where the stick–slip self excited oscillations happening at the bit need to be measured locally and then suppressed by the action of an actuator which is located at the top of the drill string.

Our paper is a constructive solution along the general lines of Carnevale et al. (2007), Nešić and Teel (2004) and Postoyan et al. (2011), where Lyapunov tools and the hybrid framework are used in similar contexts. Our approach pairs with many interesting results on event-triggered control (Mazo & Tabuada, 2008, Postoyan et al., 2011, Tabuada, 2007, Wang & Lemmon, 2008a and references therein). Additional work sharing the scenario of Fig. 1 is that of Nešić and Liberzon (2009), Sharon and Liberzon (2012) and references therein, where the feedback signal is affected by an undesired quantization effect, rather than the presence of the communication channel. We show in the paper how restricting the attention to linear systems (whereas Tabuada, 2007 and Wang & Lemmon, 2008a considers nonlinear systems) allows us to design transmission policies which lead to improved results, both in terms of architectures and of achievable performance, as compared to those in Postoyan et al. (2011), Tabuada (2007) and Wang and Lemmon (2008a) where, since a much more general nonlinear scenario is considered, the results obtained are more conservative. Finally, this work extends the results proposed in Forni et al. (2010) by enforcing a dwell-time between transmissions and by introducing exponential bounds on the asymptotic stability guaranteed by the transmission policies. Then, the practical stability results of the output feedback case in Forni et al. (2010) are here strengthened to asymptotic (and exponential) stability results.

The paper is organized as follows. We introduce nominal and event-triggered closed-loop systems in Section  2. The state-feedback transmission policy is illustrated in Section  3. The output-feedback case and robustness of the policy are considered in Sections  4 Output-feedback transmission policy, 5 Robustness of the transmission policies. Simulation examples are given in Section  6.

Notation: Given a vector $v,v^T$ denotes the transpose vector of $v$. Given two vectors $w$ and $v,〈v,w〉=w^{T}v$. Given a set $a=\{a_1,\dots ,a_n\}$ where $a_i\in \mathbb{R}$ for each $i=1\dots ,n,\mathrm{diag}\left(a\right)$ denotes a diagonal matrix having the entries of $a$ on the main diagonal. Both the Euclidean norm of a vector and the corresponding induced matrix norm are denoted by $|\cdot |$. For a vector $v\in {\mathbb{R}}^n$ and a set $\mathcal{A}\subset {\mathbb{R}}^n$${\left|v\right|}_{\mathcal{A}}≔{inf}_{y\in \mathcal{A}}|y-v|$. Given a set $\mathcal{A}\in {\mathbb{R}}^n$, the set $\mathcal{A}+\epsilon \mathbb{B},\epsilon \ge 0$, is the set of vectors $v$ such that ${\left|v\right|}_{\mathcal{A}}\le \epsilon$. A continuous function $\alpha :{\mathbb{R}}_{\ge 0}\to {\mathbb{R}}_{\ge 0}$ is said to belong to class $\mathcal{K}$ if it is strictly increasing and $\alpha \left(0\right)=0$; it belongs to class ${\mathcal{K}}_{\infty }$ if, moreover, ${lim}_{r\to +\infty }\alpha \left(r\right)=+\infty$. For any $s\in \mathbb{R}$, consider the function $f:\mathbb{R}\to \mathbb{R}$ defined by $f\left(s\right)=0$ if $\left|s\right|\le 1$, and $f\left(s\right)=\mathrm{sgn}\left(s\right)(\left|s\right|-1)$ if $\left|s\right|\ge 1$. Then, for any $s={\left[\begin{array}{ccc}\hfill s_1\hfill & \hfill \cdots \hfill & \hfill s_n\hfill \end{array}\right]}^T\in {\mathbb{R}}^n$, the deadzone function $\mathrm{dz}:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is given by $\mathrm{dz}\left(s\right)=\mathrm{diag}(f\left(s_1\right),\dots ,f\left(s_n\right))$.