Constrained Control of Event-Driven Networked Systems ⋆ Leonidas Dritsas and Anthony Tzes ∗ ∗ University of Patras Electrical & Computer Engineering Department Rio 26500, Greece (e-mail: tzes@ece.upatras.gr) Abstract: This article focuses on the control of Networked Systems in which the packets are time-stamped and suffer from long (more than one sampling period) transmission delays. The inner sample arrival of the packet, coupled with other constraints posed by the system’s characteristics such as control and/or state saturation impedes the system’s performance. A constrained finite time optimal controller is designed for this system that is robust against the inner sample delays. The presented simulation studies investigate on the performance of the suggested controller. Keywords: Networked Control Systems, Uncertain dynamic systems, Robust Stability, Control over networks. 1. INTRODUCTION It is well known that in a Networked Control System (NCS) the inclusion of the delays between the sensor and the controller (delay τ sc (k)) and between the controller and the actuator (delay τ ca (k)) can destabilize the closed– loop system. In a typical closed loop NCS along with network–induced delays, the state vector x is sampled, transmitted through the network, fed to the controller which computes the con- trol effort and transmits it to the plant. The plant receives this command after a certain delay. The network–induced delays have in general different characteristics depending on the utilized network protocol and scheduling methods used in the NCS, while their presence can deteriorate the performance of the controlled system, sometimes even driving it to instability (Zhang et al. [2001], Walsh et al. [2001], Lian et al. [2001], F.L. Lian [2001]). The network–induced delays in general can be catego- rized based on their characteristics as: a) Constant and Exactly known (e.g deterministic scheduling, intentional delay buffers), b) Constant and Unknown, with known bounds (τ min <τ<τ max ), c) Time–Varying and Exactly known (“Time–stamps” included in the data), d) Time– Varying (Uncertain) with known bounds (τ min <τ (t) < τ max ), and e) Time–Varying (stochastic process). The usual approach for NCS design consists of the fol- lowing steps: (i) design a controller ignoring the network, and (ii) analyze stability, performance and robustness with respect to the effects and the presented characteristics of network–delays and scheduling policies. The second step usually results in the selection of an appropriate scheduling protocol as well as setting bounds, so–called Maximum– Allowable–Transfer–Interval (or “M.A.T.I”) on the trans- ⋆ This work was partially funded by EU’s FP6 Network of Excellence HYCON, contract number FP6-IST-511368 mission rate so that the desired properties of the network– free control system are preserved (G. Walsh and H. Ye and L. Bushnell [2002]). If the delay τ k is measured (assuming “time–stamps” in the packets arriving from the sensor to the controller) then appropriate compensation techniques can be applied (Nilsson et al. [1998]). This research effort focuses on the case–(d), of bounded, varying but unknown time controller to actuator delays ((d − 1)h ≤ τ k < dh, where h is the sampling period, and d fixed and known integer) and instead of following the usual approach (design a controller ignoring the network and then analyze stability and performance), it takes into account the network delays in the controller design process. In this work the case of SISO NCS is covered and the results of a “delay less than one sampling period delay” (τ k <h), are extended. This effort has been sparked by the recent work in the area of NCS concerning the M.A.T.I (Kim et al. [2003]), which has revived the interest even for this seemingly limiting case. This is due to the fact that for NCSs using Random (Ethernet– Type) Access Networks, M.A.T.I is actually the effective sampling–period and hence large values on M.A.T.I allow the employment of a slower (larger) sampling period which has also the beneficial effect of reducing network traffic. In the best of authors’ knowledge the issue of input con- straints is not covered adequately in the NCS literature, although the incorporation of constraints would make the system modelling more realistic since it takes into account the inevitable limitations present in any actu- ator. Recently there are some relevant results from the research into constrained Time Delayed Systems (TDS). In reference (Fridman et al. [2003]) the case of time– delayed systems with saturating actuators is examined us- ing the Lyapunov-Krasovskii functional and the descriptor Proceedings of the 17th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-11, 2008 978-1-1234-7890-2/08/$20.00 © 2008 IFAC 11600 10.3182/20080706-5-KR-1001.3609