Toyoaki Nishida and Shuji Departs of Pnformation Science Oto University an I . Introdmctisn

Intuitively, discontinuous changes can be seen as very rapid continuous changes. A couple of alternative methods based on this ontology are presented and compared. One, called the approximation method, approximates discontinuous change by continuous function and then calculates a limit. The other, called the direct method, directly creates a chain of hypothetical intermediate states (mythical instants) which a given circuit is supposed to go through during a discontinuous change. Although the direct method may fail to predict certain properties of discontinuity and its applicability is limited, it is more efficient than the approximation method. The direct method has been fully implemented and incorporated into an existing qualitative reasoning program. I . Introdmctisn Continuous change is a notion in which quantities are assumed to take a certain amount of time to change value. Discontinuous changes are those to which this assumption does not apply; quantities can change value in a moment. Notion of discontinuous change plays a crucial role in characterizing the behavior of dynamic systems, such as nonlinear oscillators or flip-flops, without worrying about unmotivated details. At the commonsense level, the notion of discontinuous change seems to be natural; things appear to suddenly stop moving, collide, disappear and so on. Unfortunately, analysis of discontinuous changes is not easy. This is mainly because ordinary models for physical systems (e.g., circuit equations) do not always specify the system’s behavior under discontinuous change in full detail. In textbooks, this problem is often solved by using an ontology in which discontinuous change is very rapid continuous change. A couple of alternative methods are possible to implement this view. One, called the approximation method, approximates discontinuous change by a continuous function and then calculates a limit. The other, called the direct method, uses a notion of mythical instants to describe hypothetical intermediate states which a given circuit is supposed to go through during a discontinuous change. In this paper, we present and compare these two algorithms. We base our theory on qualitative reasoning, a formal theory for causal understanding, and we choose electronic circuits as a subject domain. In the next section, we study properties of discontinuous changes. In section ill, we will briefly overview previous work in qualitative reasoning and see how discontinuity has been handled. In sections IV and V, we will describe the two algorithms separately, and in section Vl, we will compare the two and summarize the discussion. The varieties of discontinuous changes depend on the physical model employed. In this paper, we study discontinuous changes arising in piecewise linear equation models for electronic circuits, since the use of piecewise linear equations is one of the most popular techniques in the electronic circuit domain. In this modeling, nonlinear circuit elements, such as diodes or transistors, are described with multiple operating regions. Circuit devices modeled with multiple operating regions will be called multiple-mode devices. Figure 1 shows the models for diodes and transistors we employ for explanation in this paper. Although they might appear too simple, they suffice for the discussion below, since the same kind of phenomena arise even when more complex models are used, as will be seen below. Possible causes of a single occurrence of discontinuous change arising in piecewise linear circuit models can be classified into three categories: (Al) discontinuous input (A2) mode transition of a multiple-mode device (A3) positive feedback without time delay. Nishida and Doshita 643 From: AAAI-87 Proceedings. Copyright ©1987, AAAI (www.aaai.org). All rights reserved.