Discrete-event modelling and diagnosis of quantised dynamical systems

The paper deals with the diagnosis of quantised continuous-variable systems whose state can be measured only by means of a quantiser . Hence, the on-line information used in the diagnosis is given by the sequences of input and output events . The diagnostic algorithm uses a representation of the quantised system by means of discrete-event models . Four different forms of such models will be explained and their usefulness for the solution of diagnostic tasks discussed . The paper shows that a timed discrete-event representation is necessary if the diagnostic task should be solved as quickly as possible under real-time constraints . The results are illustrated by diagnosing a batch process. Introduction Diagnosis of quantised systems . This paper is concerned with the diagnosis of dynamical systems with discrete inputs and outputs . As shown in Figure 1, the system under consideration is a continuous-variable continuous-time system, which can be described by some analytical model (set of differential equations) . However, the system state x is accessible only through a quantiser, which generates an event whenever the state changes its qualitative value . The input assumes a sequence of discrete values v, which is transformed into a continuous input function u(t) by the injector . Since the observations are based on the quantised signals, a qualitative model has to be used for the diagnosis . The system consisting of the continuous--variable system, the quantiser and the injector is called the quantised system. Aim of the paper. This workshop paper should show how diagnostic methods can be elaborated for quantised continuous-variable dynamical systems . The development consists of two major steps . " First, four different discrete-event representations of the quantised system are described . " Second, diagnostic algorithms that use these models and the observed input and output sequences are given . QR99 Loch Awe, Scotland Jan Lunze Technische Universitat Hamburg-Hamburg Arbeitsbereich R.egelungstechnik Eissendorfer Str . 40, D-21071 Hamburg email : LunzeU_tu-harburg.de Discrete control actions I I I I F___ l I Event sequence Fig . 1 : Diagnosis of quantised systems As the models distinguish concerning the information about the dynamical properties of the quantised systems, the diagnostic results differ concerning their precision . The severeness of these differences are shown bya numerical example . Relevant literature . Results along this line of research have been obtained in two fields . The modelling problem for quantised systems has been investigated, for example, in (Lunze 1992), (Lunze 1994), (Lunze 1999), (Raisch, O'Young 1997) or (Stursberg, Kowalewski, Engell 1997) . On the other hand, diagnosing quantised systems by means of a discreteevent representation has been investigated in (Lichtenberg, Steele 1996), (Lunze 1998), (Lunze, Schiller 1997), (Lunze, Schiller, 1999), (Sampath, Sengupta, Lafurtune, Sinnamohideen, Teneketzis 1995) or (Srinivasan, Jafari 1993) . This paper uses the principle of consistency-based diagnosis (Hamscher, Console, and de Kleer 1992) which will be applied here to four different discrete-event representations . Example: Diagnosis of a batch process The class of diagnostic problems considered in this paper is illustrated by the batch process depicted in Figure 2 . The dashed lines mark liquid levels, which are measured by sensors that indicate only if the level is