Elsevier

Automatica

Volume 89, March 2018, Pages 376-381
Automatica

Brief paper
Stochastic thresholds in event-triggered control: A consistent policy for quadratic control

https://doi.org/10.1016/j.automatica.2017.12.043Get rights and content

Abstract

We propose an event-triggered control scheme for discrete-time linear systems subject to Gaussian white noise disturbances. The event-conditions are given in terms of the deviation between the actual system state and the state of a nominal undisturbed system whose state is identical to the real system state at the event times. In order to ensure that the conditional distribution of the deviation between the two systems, under the condition that no event occurs, remains a normal distribution, we employ thresholds that are themselves random variables. This allows us to: (i) provide expressions for the probability mass function of the times between events and, in turn, arbitrarily select this function; (ii) synthesize controllers associated with the proposed transmissions scheduler that are optimal in terms of an average quadratic cost. In particular, these two properties allow us to show that our event-triggered scheme is consistent in the sense that it outperforms (in a quadratic cost sense) traditional periodic control for the same average transmission rate and does not generate transmissions in the absence of disturbances. We demonstrate the effectiveness of our scheme in a numerical example and describe a way to solve the non-convex optimization problem arising in the approach.

Introduction

Event-triggered control has been proposed in recent years to reduce the communication burden of traditional periodic control in networked control systems. A standard networked control system for a single control loop is depicted in Fig. 1. An event-triggered control scheme is composed of: (i) a scheduler which based on sensor measurements of the process determines when transmissions to a remote controller occur; (ii) a controller which based on the received sensor measurements decides the control input. In most of the previous works (e.g. Anderson et al. (2015), Årzén (1999), Åström & Bernhardsson (2002), Cassandras (2013), Demirel et al. (2013), Garcia & Antsaklis (2013), Heemels et al. (2012), Lunze & Lehmann (2010), Rabi & Johansson (2009), Tabuada (2007), Tallapragada et al. (2016), Yook et al. (2002)), the scheduler decides when transmissions should occur based on deterministic criteria, such as the deviation between process state and a state estimation on the controller side (e.g., the last transmitted state value or a model-based estimation) exceeding a given value or, more generally, a function of the information known to the trigger mechanism at a given time crossing a given threshold. While designing the controller and the scheduler in an optimal way, according to given performance specifications, is typically hard, it is still possible to design these to meet desired specifications. In particular a desired property, one of the consistency properties defined in Antunes and Khashooei (2016), is that the performance of such event-triggered control is better than that of periodic control for the same average transmission rate.

In this paper, we propose consistent event-triggered controllers that rely on stochastic thresholds. We consider a process disturbed by Gaussian white noise. As has been recognized recently, if deterministic threshold policies are employed in event-triggered control, the state does not preserve the Gaussian nature of the probability distribution of the disturbances (Wu, Ren, Han, Shi, & Shi, 2016). This severely complicates, or even totally prevents, the exact stochastic characterization of the state and, therefore, the closed-loop design and analysis. In the context of event-triggered estimation, the authors of Han et al. (2015), Shi, Chen, and Darouach (2016) and Wu et al. (2016) have therefore proposed stochastic threshold policies for which the state preserves the Gaussian nature of the probability distribution of the disturbances. The main idea is to define the trigger condition in terms of a threshold on the value of a quadratic form of the deviation between the actual system state and a nominally predicted system state. The value of the threshold is itself a random variable, drawn (independently from all other variables) from a certain exponential1 distribution. In Han et al. (2015) and Shi et al. (2016), the shaping matrix of the quadratic form is a fixed tuning parameter in the estimation scheme. Here, we choose this matrix equal to the covariance matrix of the current distribution of the deviation between the actual system state and the nominally predicted state, similarly to Wu et al. (2016). This allows us to obtain explicit expressions of the trigger probabilities in terms of the intensity of the exponential distributions defining the thresholds, and, in turn, enables us to assign these probabilities arbitrarily. The use of stochastic thresholds is then exploited in the context of closed-loop control to design an event-triggered controller that minimizes a quadratic cost, as in standard discrete-time (periodic) linear quadratic control, with communication constraints captured by constraints on the (average) transmission rate. This is in general a hard problem, see for example Antunes and Heemels (2014), Molin and Hirche (2013) and Ramesh, Sandberg, Bao, and Johansson (2011), which consider similar problems where the transmission rate is also penalized in the cost function. While several results have been established on the structural properties of optimal designs, the optimal – and combined – design of the state estimator, scheduler, and controller remains challenging. We propose to fix the scheduler to be a stochastic scheduler as discussed above, and show that in the state-feedback case the optimal controller can then be found and is, in fact, given by certainty equivalence feedback, consistent with the results in Molin and Hirche (2013). By optimizing the parameters of the proposed scheduler subject to a desired transmission rate constraint, we show that the proposed event-triggered controller is never outperformed by an optimal periodic control with the same average transmission rate, which is the first of two properties of a consistent periodic controller as defined in Antunes and Khashooei (2016). We also show that the second property is satisfied, that is, that no transmissions are generated in the absence of disturbances. While the resulting optimization problem needed to obtain the scheduling law is non-convex, we show how it can be solved such that the performance of the resulting closed loop is not worse than if periodic control is employed.

The remainder of the paper is organized as follows. This introductory section concludes with some remarks on notation. The stochastic scheduler and its main properties are presented in Section 2. In Section 3, consistency and optimality of the control scheme are discussed. A numerical example illustrating the results is presented in Section 4, certain aspects of the proposed scheme are discussed in Section 5, and Section 6 concludes the paper.

Notation: N denotes the set of natural numbers, N0 denotes the set of non-negative integers. For xRn and for a positive definite QRn×n we define xQxQx. The rank of a matrix MRn×n is denoted by rk(M), its Moore–Penrose pseudo-inverse by M, and the product of all of its non-zero eigenvalues (each according to its algebraic multiplicity) by det(M). Moreover, zN(z̄,Ξ) (z|IN(z̄,Ξ)) indicates that the (conditioned) probability distribution of zRn (on the information I) is Gaussian with mean z̄ and covariance Ξ0. Finally, rexp(λ) indicates that rR is an exponentially distributed random variable with intensity λ0.

Section snippets

Schedulers with stochastic thresholds

We detail the event-triggered setting in Section 2.1 and define the proposed class of stochastic schedulers in Section 2.2. We then state a first key result regarding the Gaussian nature of the state distribution in Section 2.3. In Section 2.4 we consider the special case of identically distributed disturbances. In Section 2.5, we investigate the stability of the closed-loop system.

Consistent ETC for linear quadratic control

In this section, we assume that the disturbances are identically distributed, that is, Wt=W for all tN0. Furthermore, we assume that the threshold parameters λ̄t are defined in terms of the probability of a transmission occurring steps after the last transmission, as described in Section 2.4.

Given the scheduler (3), suppose that we wish to design a controller policy γt, to minimize the average quadratic cost JetclimsupN1NEt=0N1g(xt,ut)where g(x,u)xQx+uRu, Q is positive semi-definite,

Example

As a proof of concept, we demonstrate the proposed method on an academic example. Consider the following system, which is obtained by an exact discretization of a continuous-time double-integrator, xt+1=1101xt+0.51ut+wt,where wtN0,[1001]. With the weighting matrices Q=[1001] and R=10, we obtain P=[3.26643.20163.20169.3569]and K=[0.20680.6756]. We used YALMIP (Löfberg, 2004) and Matlab’s general purpose solver fmincon 3 to obtain a

Discussion

While the numerical example demonstrates – for one particular system and objective function – that the proposed event-triggered scheme outperforms periodic control there are two open issues that need to be addressed.

The first issue is the complexity of the optimization problem in (14), which, even if the sums are restricted to be finite, precludes us from finding a globally optimal solution in general. However, considering the discussion leading up to Theorem 5, it is always possible to

Conclusion

We have presented an event-triggered control scheme based on stochastic trigger rules, which were shown, both theoretically and numerically, to outperform control schemes based on periodic sampling. Future research topics are optimal output-feedback control within this setting and the efficient solution of the optimal scheduling problem. Moreover, we want to investigate the setup where a bound on the covariances Θ is given and H is to be maximized. Further, imperfections of the network, such

Florian David Brunner received his Diploma in Engineering Cybernetics from the University of Stuttgart, Germany in 2012. He conducted his doctoral studies at the Institute for Systems Theory and Automatic Control at the University of Stuttgart. His research interests include robust predictive control and networked control systems with aperiodic communication.

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    Florian David Brunner received his Diploma in Engineering Cybernetics from the University of Stuttgart, Germany in 2012. He conducted his doctoral studies at the Institute for Systems Theory and Automatic Control at the University of Stuttgart. His research interests include robust predictive control and networked control systems with aperiodic communication.

    Duarte Antunes was born in Viseu, Portugal, in 1982. He received the Licenciatura in Electrical and Computer Engineering from the Instituto Superior Técnico (IST), Lisbon, in 2005. He did his Ph.D. from 2006 to 2011 in the research field of Automatic Control at the Institute for Systems and Robotics, IST, Lisbon. From 2011 to 2013 he held a postdoctoral position at the Eindhoven University of Technology (TU/e). He is currently an Assistant Professor at the Department of Mechanical Engineering of TU/e. His research interests include Networked Control Systems, Stochastic Control, Dynamic Programming, and Systems Biology.

    Frank Allgöwer studied Engineering Cybernetics and Applied Mathematics in Stuttgart and at the University of California, Los Angeles (UCLA), respectively, and received his Ph.D. degree from the University of Stuttgart in Germany. He is the Director of the Institute for Systems Theory and Automatic Control and Executive Director of the Stuttgart Research Centre Systems Biology at the University of Stuttgart. His research interests include cooperative control, predictive control, and nonlinear control with application to a wide range of fields including systems biology. For the years 2017–2020 Frank serves as President of the International Federation of Automatic Control (IFAC) and since 2012 as Vice President of the German Research Foundation DFG.

    The authors would like to thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/2) at the University of Stuttgart. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Dimos V. Dimarogonas under the direction of Editor Christos G. Cassandras.

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