Model-based periodic event-triggered control for linear systems☆
Introduction
In most digital control applications, sampling of the outputs of the plant, and computing and implementing new actuator signals are executed periodically. Although periodic sampling might be preferable from an analysis and design point of view, it is sometimes less preferable from a resource utilization point of view. Executing the control task at times when no disturbances are acting on the system and the system is operating desirably is clearly a waste of communication resources. To mitigate the unnecessary waste of communication resources, it is of interest to consider an alternative control paradigm, namely event-triggered control (ETC), which started to attract quite some interest in the late 1990s (Arzén, 1999, A˚ström and Bernhardsson, 1999, Heemels et al., 1999, Hendricks et al., 1994), although mainly for different reasons including the reduction of computational resources and dealing with the event-based nature of the plants’ sensors. Also some earlier attempts towards event-triggered sampling and ETC, going back to the 1960s (Draper, Wrigley, & Hovorka, 1960), can be found, see, e.g., the overview paper (A˚ström, 2008). ETC is a control strategy in which the control task is executed after the occurrence of an event, generated by some well-designed event-triggering condition, rather than the elapse of a certain fixed period of time, as in conventional periodic sampled-data control. In this way, ETC is capable of significantly reducing the number of control task executions, while retaining a satisfactory closed-loop performance, as simulation and experimental results show in, e.g., Arzén (1999), A˚ström and Bernhardsson (1999), Heemels et al. (1999), Hendricks et al. (1994), Henningsson and Cervin (2009), Kwon, Kim, Lee, and Paek (1999), Lehmann and Lunze (2010) and Yook, Tilbury, and Soparkar (2002). For these appealing reasons, ETC received ample attention in the last decade (A˚ström and Bernhardsson, 2002, Donkers and Heemels, 2012, Gawthrop and Wang, 2009, Heemels et al., 2008, Henningsson et al., 2008, Kofman and Braslavsky, 2006, Lunze and Lehmann, 2010, Miskowicz, 2006, Otanez et al., 2002, Tabuada, 2007, Wang and Lemmon, 2009). In most of these ETC schemes the event-triggering condition has to be verified continuously, and therefore they are sometimes referred to as “continuous event-triggered control” (CETC) schemes.
Recently, interest was shown in a class of event-triggered control algorithms that aim at integrating ideas from conventional periodic time-triggered control and ETC paradigms. This results in so-called periodic event-triggered control (PETC) (Arzén, 1999, Heemels et al., 2011, Heemels et al., 2013, Heemels et al., 2008, Henningsson et al., 2008, Yook et al., 2002), in which the event-triggering condition is verified periodically, and every sampling time it is decided whether or not to transmit new measurement and control values. The network resources are used only when needed to guarantee stability or certain performance properties. The resulting PETC laws can be considered in a continuous-time framework, see, e.g., Heemels et al., 2011, Heemels et al., 2013 or can be studied as event-triggered controllers for discrete-time systems (disregarding the intersample behavior) as in, e.g., Cogill (2009), Eqtami, Dimarogonas, and Kyriakopoulos (2010), Lehmann (2011, Section 4.5), Li and Lemmon (2011) and Yook et al. (2002). In this paper we study them from the latter perspective.
The objective of this paper is to develop new and improved classes of event-triggering mechanisms (ETMs). Besides some notable exceptions (Garcia and Antsaklis, 2011, Lehmann and Lunze, 2011, Li and Lemmon, 2011, Lunze and Lehmann, 2010, Yook et al., 2002), which will be discussed in more detail below, the ETMs currently available in the literature are rather basic for both CETC and PETC. Indeed, often sensors transmit their information to the controller when the difference between the current measured output and the most recently transmitted output value exceeds a certain absolute threshold value or a relative bound with respect to the measured output value, see, for instance, A˚ström and Bernhardsson (2002), Gawthrop and Wang (2009), Heemels et al. (2008), Henningsson et al. (2008), Miskowicz (2006), Otanez et al. (2002), Tabuada (2007) and Wang and Lemmon (2009) for the case of state-feedback and Donkers and Heemels (2012) and Kofman and Braslavsky (2006) for the case of output-feedback controllers. This is a rather basic strategy for the ETMs, which we will refer to as the baseline strategy. In this paper, we will propose model-based ETMs for both the sensor-to-controller () and the controller-to-actuator () channels that result in significant reductions of the communication traffic if compared to both time-triggered periodic implementations as well as ETMs using this baseline strategy, while preserving desirable stability and performance properties. In this paper, the focus will be on PETC strategies, although we believe that the main ideas also apply in the context of CETC.
The main idea of the proposed ETMs is the use of a Luenberger observer at the sensor system, and the use of a model-based predictor that runs both at the sensor and the controller system. This observer and this predictor both produce an estimate of the state of the plant. The observer will generally produce better estimates than the predictor, as it has access to all the measured outputs, while the predictor has not. Due to the fact that the sensor system runs a copy of the predictor, it is aware of the estimate the controller system has. The ETM triggers a transmission of the estimated state of the observer to the controller system, only if the estimate of the controller system deviates too much from the (better) estimate the observer has.
The main principle behind the ETMs proposed in this paper, is to utilize predictive techniques as used for networked control systems (NCSs) with unreliable channels exhibiting delays and packet loss, see, e.g., Bemporad (1998), Chaillet and Bicchi (2008) and Hu, Liu, and Rees (2007) and the references therein. In the mentioned references, the predictive control ideas are used for counteracting packet loss and transmission delays. In this paper, we will exploit the idea of predictive control and buffering future control values for a different purpose. Namely, we will exploit it for reducing the number of transmissions. In fact, both the and the ETMs can result in significant reductions of the number of transmissions, by introducing more computations. In this sense “computations are traded for bandwidth” (Yook et al., 2002). In addition, if computation is cheap compared to communication in terms of power usage (which is often the case), the proposed PETC strategies can also result in considerable reductions in the overall energy usage of battery-powered wireless sensors, controllers and actuators, thereby having positive effects on their battery lives.
In the literature, related model-based ETMs were presented for the channels, see Garcia and Antsaklis (2011), Lehmann and Lunze (2011), Li and Lemmon (2011), Lunze and Lehmann (2010) and Yook et al. (2002). There are, however, some essential differences. First of all, (Lehmann and Lunze, 2011, Lunze and Lehmann, 2010, Yook et al., 2002) use ETMs based on absolute thresholds resulting in ultimate boundedness and BIBO stability types of conditions, while in this paper we are interested in developing ETMs that achieve exponential stability and guaranteed (finite) -gains. This does not only require a different type of ETMs, but also different mathematical analysis frameworks. Secondly, the works (Garcia and Antsaklis, 2011, Lehmann and Lunze, 2011, Lunze and Lehmann, 2010) are formulated in the context of CETC, while in this paper we focus on PETC implementations. Thirdly, in this paper we will also consider communication saving measures in the channels by using advanced model-based ETMs, as opposed to Garcia and Antsaklis (2011), Lehmann and Lunze (2011), Lunze and Lehmann (2010), and Yook et al. (2002), which assume controllers and actuators to be collocated or hard-wired, and to Li and Lemmon (2011), which does not use predictive techniques in the channel. Fourthly, in Garcia and Antsaklis (2011) and Lunze and Lehmann (2010) state-feedback control laws are considered, while here the focus is on output-based dynamic controllers. Fifthly, in Garcia and Antsaklis (2011), Lehmann and Lunze (2011), Li and Lemmon (2011) and Lunze and Lehmann (2010) the ETMs and controllers are implemented in a centralized setting, while in the present paper we will also provide decentralized implementations of both the controller and the ETMs based on simple low-order local models. Our decentralized strategy does not require running a global high-order estimator at each local node as in Yook et al. (2002).
As such, the main contributions of this paper are the proposition and the formalization of the mentioned and ETMs leading to model-based PETC strategies for discrete-time linear plants subject to disturbances, not requiring the full state to be available for feedback. We provide two general modeling and analysis frameworks based on perturbed linear (PL) systems and piecewise linear (PWL) systems, and derive linear matrix inequality (LMI)-based conditions for global exponential stability and guaranteed -gains of the closed-loop systems. In fact, these frameworks for modeling and analysis form another distinctive difference with the works (Garcia and Antsaklis, 2011, Lehmann and Lunze, 2011, Li and Lemmon, 2011, Lunze and Lehmann, 2010, Yook et al., 2002). In these works analysis methods are adopted that are in nature closer to the PL system approach, and they do not consider the PWL system approach. This latter approach can be shown to be never outperformed by the PL system approach and in most cases actually provides (much) better guarantees (see Section 4.3 below). The PL system approach, on the other hand, will be used to show that the performance of any observer-based controller, implemented in a conventional time-triggered fashion, can be recovered by a PETC implementation. This result forms a strong justification for so-called emulation-based design, which is a two-stage design procedure. In the first design stage the controller parameters are chosen using standard discrete-time () synthesis tools (assuming a periodic time-triggered implementation and thus ignoring the event-triggered implementation). In the second stage, the parameters of the ETMs are chosen by making appropriate tradeoffs between the network utilization on the one hand and the performance in terms of -gains on the other hand. This justification for the emulation-based design forms an important motivation for presenting the PL system approach, next to the facts that it is closer to the existing literature, see, e.g., Garcia and Antsaklis (2011), Lehmann and Lunze (2011), Li and Lemmon (2011), Lunze and Lehmann (2010), Tabuada (2007) and Yook et al. (2002), and that it results in simpler -norm conditions that guarantee closed-loop stability when compared to the PWL system approach. The PWL system approach is of importance as it generically results in less conservative results, as already mentioned above.
The results mentioned above will be presented for a model-based PETC setup with a centralized controller and centralized ETMs in the first part of the paper. In the second part, we will indicate how these ideas and methodologies can be applied in the context of decentralized observer-based control laws using small local observers and predictors. The latter setup is highly relevant for large-scale plants in which sensors, actuators and controllers are physically distributed over a wide area. In fact, centralized ETMs can be prohibitive in this case, as the conditions that generate events would need access to all the plant or controller outputs at every sampling time, which can be an unrealistic assumption for large-scale systems. In addition, the size of the complete plant model also might be too large for usage in the ETMs for computational reasons. The solution proposed in the second part will overcome both issues.
Notations For a vector , we denote by its 2-norm. For a symmetric matrix , and denote the maximum and minimum eigenvalue of , respectively. For a matrix , we denote by the transposed of , and by its induced 2-norm. For vectors , , the vector , where , is given by . By , we denote a block-diagonal matrix with the matrices on the diagonal, and for the sake of brevity we sometimes write symmetric matrices of the form as . We call a matrix positive definite (positive semidefinite) and write (), if is symmetric and () for all . A matrix is called Schur, if all its eigenvalues are within the open unit circle of the complex plane. The set consists of all infinite sequences with , , and . For each , we write .
Section snippets
Problem formulation
In this paper, we will study the networked control configuration as shown in Fig. 1, in which the plant is given by a discrete-time linear time-invariant (LTI) model of the form where , , and denote the state, control input, disturbance and measured output, respectively, at discrete time instant . The sensors of the plant transmit their measurement information to the controller, and the controller transmits the control data to the
Model-based PETC strategy for channels
In this section, we propose a solution for the problem formulated in Section 2 in the context of Fig. 2.
Stability and -gain analysis
In this section, we define the notions of performance and stability used in this paper and we will analyze stability and performance of the closed-loop ETC system based on both the PWL model (10) and the PL model (12). To do so, let us introduce the performance output given for by
Definition 4.1 The system (10) with is said to be globally exponentially stable (GES), if there exist constants and such that for all initial states and , , the corresponding
Including controller-to-actuator ETMs
In this section, we will consider the network configuration as shown in Fig. 1, in which there is also a shared communication network between the controller and actuator system for which the communication and/or energy resources are limited. For this case, we propose to combine the ideas of Section 3 for the ETM with a new class of ETMs for the channel. The ETMs will be based on the transmission of model-based predictions of future control values to a buffer located in the actuator
Analysis of PETC with and ETMs
In this section, we carry out the stability and performance analysis of the PETC strategies with both and ETMs using the PWL and the PL system approaches.
Decentralized PETC and ETMs
In this section, we will show how the ideas presented in the previous sections can be implemented in a decentralized fashion. As already mentioned in the introduction, this setup is particularly relevant for large-scale plants in which sensors, actuators and controllers are physically distributed over a wide area. In fact, centralized ETMs and controllers can be prohibitive in this case, as the conditions that generate events and the controllers would need access to all the plant or controller
Illustrative examples
In this section, we illustrate the newly proposed model-based PETC strategies for both the case with only a ETM as well as the case with both and ETMs. For the former case, we will compare the newly proposed model-based PETC scheme with a periodic time-triggered controller and with the baseline PETC strategy as discussed in Section 3.3. For the latter case, we will discuss the influence of the parameters , and on the upper bound on the -gain and the number of transmissions.
Conclusions
In this paper, we proposed and formalized new model-based PETC strategies for both the sensor-to-controller () and the controller-to-actuator () channels. The results apply to discrete-time linear plants subject to disturbances, and do not require the full state to be available for feedback. The event-triggering mechanisms (ETMs) are based on exploiting a Luenberger observer in the sensor system and two identical predictors in both the controller and sensor system. The sensor system
W.P.M.H. Heemels received the M.Sc. degree in mathematics and the Ph.D. degree in control theory (both summa cum laude) from the Eindhoven University of Technology (TU/e), Eindhoven, The Netherlands, in 1995 and 1999, respectively. From 2000 to 2004, he was with the Electrical Engineering Department, TU/e, as an Assistant Professor and from 2004 to 2006 with the Embedded Systems Institute (ESI) as a Research Fellow. Since 2006, he has been with the Department of Mechanical Engineering, TU/e,
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W.P.M.H. Heemels received the M.Sc. degree in mathematics and the Ph.D. degree in control theory (both summa cum laude) from the Eindhoven University of Technology (TU/e), Eindhoven, The Netherlands, in 1995 and 1999, respectively. From 2000 to 2004, he was with the Electrical Engineering Department, TU/e, as an Assistant Professor and from 2004 to 2006 with the Embedded Systems Institute (ESI) as a Research Fellow. Since 2006, he has been with the Department of Mechanical Engineering, TU/e, where he is currently a Full Professor in the Hybrid and Networked Systems Group. He held visiting research positions at the Swiss Federal Institute of Technology (ETH), Zurich, Switzerland (2001) and at the University of California at Santa Barbara (2008). In 2004, he was also at the Research and Development Laboratory, Océ, Venlo, The Netherlands. His current research interests include hybrid and cyber-physical systems, networked and event-triggered control systems and constrained systems including model predictive control. Dr. Heemels was an Associate Editor for the journal Nonlinear Analysis: Hybrid Systems, and currently is an Associate Editor for the journal Automatica. In addition, he served as the general chair of the 4th IFAC Conference on Analysis and Design of Hybrid Systems (ADHS) 2012 in Eindhoven, The Netherlands, and will be the IPC chair for the 4th IFAC Workshop on Distributed Estimation and Control in Networked Systems (NECSYS) 2013 in Koblenz, Germany.
M.C.F Donkers received the M.Sc. degree and the Ph.D. degree (both summa cum laude) in systems and controls in 2008 and 2011, respectively, from Eindhoven University of Technology, The Netherlands. In 2010, he was a visiting researcher at the Cyber-Physical Systems Laboratory of the University of California at Los Angeles, CA, USA. His current research interests include networked and event-driven control, distributed control, and switched systems.
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This work is supported by the Dutch Science Foundation (STW) and the Dutch Organization for Scientific Research (NWO) under the VICI grant “Wireless controls systems: A new frontier in automation” (No. 11382), and the European 7th Framework Network of Excellence under grant HYCON2-257462. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Dragan Nešić under the direction of Editor Andrew R. Teel.
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