The Performance of Event-Based Control for Scalar Systems with Packet Losses Rainer Blind and Frank Allg¨ ower Abstract— In this work, we analyze event-based control for scalar systems with an impulsive input that resets the state to the origin when applied. The event information is transmitted within packets, which are randomly dropped. We derive the cost as well as the expected times between events and discuss how to change the bounds, which generate the events, after a packet loss. Index Terms— Event-based control, Networked Control Sys- tem I. I NTRODUCTION Due to the steady increase in computer and communication technologies more and more control loops are closed by a digital communication network. Consequently, these systems are called Networked Control Systems (NCS). For such systems the number of packets should be as small as possible, and thus the usual choice of a constant sampling time is challenged. One alternative is event-based control, which is compared to time-triggered control in [1] for scalar systems. Under the assumption of ideal communications, event-based control needs less updates to get the same performance as time-triggered control. Consequently, [1] became one of the motivations for using event-based control for NCS, see, e.g., [2]–[8]. However, if the loop is closed over a packet based communication system, then packets might get lost or delayed. Since these effects are not taken into account in [1] it is important to extend this work to scenarios where packets might get lost or delayed. Event-based control of integrator systems with packet losses has been studied thoroughly, e.g., in [7]–[11]. In contrast, a similar analysis for scalar systems is still lacking. Thus, we analyze event-based control for scalar systems with packet losses in the present work. More precisely, we analytically derive its performance as well as the time between events and finally use an example to discuss the choice of the bounds. Thereby, we assume that the packet loss probability is constant and there is no delay. Studying the time between events is crucial when considering event- based control for NCS since the interevent times might affect loss and delay and thereby indirectly the performance, see [9]–[11]. Moreover, a proper choice of the bounds is very important because it directly affects the performance but also the interevent times. The remainder of this work is outlined as follows. First, we give the problem setup and introduce the event-based control scheme in Sec. II. Then, we derive analytic expressions for R. Blind and F. Allg¨ ower are with the Institute for Systems Theory and Automatic Control, University of Stuttgart, Stuttgart, Germany. {blind, allgower}@ist.uni-stuttgart.de. the time between two events in Sec. III and for the cost in Sec. IV. In Sec. V, we discuss how to increase the bounds after a loss with the help of an example. Finally, we conclude in Sec. VI. II. PROBLEM SETUP As in [1], we consider a scalar system dx = axdt + udt + dv (1) where x ∈ R is the state, u the input, and the disturbance v(t) is a Wiener process with unit incremental variance. This system is controlled by an impulsive input u, which resets the state to the origin if applied. Obviously, an impulsive input is not realistic for physical systems. However, when considering the error dynamics of a remotely observed sys- tem the error will be reset to zero whenever a measurement arrives. Moreover, we assume that each packet is lost with a probability p. Consequently, the probability that a packet arrives after i consecutive losses is p i = (1 - p)p i . (2) As in [1], [8], the cost is given by the variance of the state, i.e., the cost is J = lim T →∞ 1 T  T 0 E[x(t) 2 ]dt. (3) As already stated, we assume that an impulsive event- based control scheme is used, which is relatively simple when the communication is ideal: The state is reset to the origin whenever the normed state reaches the bound Δ 1 , i.e., whenever |x| =Δ 1 . This scheme becomes more complex, when packets might get lost. If packets can get lost, we need a sequence of bounds {Δ i } which is strictly increasing, i.e., Δ i < Δ i+1 . For consistency of notation, we define Δ 0 := 0. At each time, only one of these bounds is active. As in the lossless case, an event is generated, whenever the normed state reaches the active bound. In contrast to the lossless case, we have to change the active bound whenever it is reached as follows. If the packet is successfully transmitted, then the state will be reset to the origin. Consequently, the active bound is also reset to Δ 1 . In the other case, when the packet is lost and the state is not reset, the active bound is increased to the next larger one. Note that this is only possible when the sender knows whether or not its packets arrive. Using this scheme, after i consecutive losses the active bound is Δ i+1 . Finally, note that