Modeling Approaches and Robust Stability Conditions for Networked Controlled Systems with Uncertain Delays ⋆ Leonidas Dritsas and Anthony Tzes ∗ ∗ University of Patras Electrical & Computer Engineering Department Rio 26500, Greece Abstract: In this article two modeling approaches for Networked Controlled Systems (NCS) with different types of uncertainly varying bounded transmission delays and static discrete– time control laws are presented. Different models are offered for each case, all linked to the objective of designing robust discrete-time controllers. It is analytically shown how the careful mixing of asynchronous (event–driven) and synchronized (clocked) signals can lead to discrete time uncertain (possibly switched) systems, where results form robust control analysis and synthesis can be applied. After showing the implications of these modelling results for control synthesis purposes, sufficient conditions for the robust stability are given for each approach and a comparison of the conservatism of results is discussed. Keywords: Networked Control Systems, Uncertain dynamic systems, Robust Stability, Control over networks. 1. INTRODUCTION It is well known that one of the primary effects and major control challenges in Networked Controlled System (NCS) is the presence of uncertain network-induced de- lays stemming from the very fact of utilizing a common communication channel for closing the loop Baillieul and Antsaklis [2007], Hespanha et al. [2007]. Network-induced delays in NCS commonly appear in the information flow between the sensor and the controller (delay τ sc (k)), as also between the controller and the actuator (delay τ ca (k)), where ‘k’ denotes the dependence on the k th sampling period. As has been shown, when a static feedback law is employed, it is allowed to lump τ sc (k) and τ ca (k) into one delay τ k △ = τ sc (k)+ τ ca (k) (Nilsson et al. [1998], Zhang et al. [2001], Tzes et al. [2005]). The type and the characteristics of the underlying delays are varying and in most of the cases depend on the utilized network protocol, the scheduling methods, the communication overhead (collisions/retrasmissions), the packet losses, and in general to uncertain factors that can deteriorate the stability and performance of the controlled system, sometimes even driving it to instability Zhang et al. [2001]. Significant effort has hence been invested in developing control methodologies to handle the network delay effect in NCSs. A survey of control methodologies for a closed–loop control system over a data network has been presented in Tipsuwan and Chow [2003], Hespanha et al. [2007]. ⋆ This work was partially funded by EU’s FP6 Network of Excellence HYCON, contract number FP6-IST-511368 For NCS using random access MAC protocols (Ether- net, DeviceNet) the assumption of equidistant sampling and constant network delay may no longer be valid (see Naghshtabrizi and Hespanha [2006], Hespanha et al. [2007] for the variable sampling case). Hence a more cau- tious treatment of the modeling and discretization pro- cedure is necessary, and even more so for the control synthesis. The remainder of this work starts with the general setup regarding the modeling of NCS in Section 2. The two proposed modeling schemes (presented in sections 3 and 4), allow the control designer to embed in a combined NCS dynamic model (plant,controller,network, sample and hold devices), network–induced delays smaller that one sampling period,with known bounds, uncertainly varying, or constant and unknown. Moreover the ensuing robust stability conditions do not need the a-priori knowledge of the probability distribution functions of the uncertain delays. Comparison of the two approaches is presented in 5, while the conclusions are drawn in Section 6. Inhere, the case of SISO systems with “less than one sampling period delay”, (τ k <h) is examined. Recent works concerning Maximum Allowable Transfer Interval (“M.A.T.I.”) computations, have revived the interest for this case of systems( Kim et al. [2003]). 2. MODELING ISSUES FOR NCS The dynamics of the NCS under investigation is described by the combination of a continuous–time linear time– invariant plant with a discrete–time controller Zhang et al. [2001]. The sampling period h is assumed to be constant and known, whereas both controller and actuator (includ- ing the zero-order-hold ZOH) are event-driven devices in 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 6353 10.3182/20080706-5-KR-1001.1322