Perturbation Analysis with Qualitative Models

Perturbation analysis deals with the relation­ ships between small changes in a system's inputs or model and changes in its outputs. Reverse simulation is of particular interest, determining how to achieve desired outputs by perturbing inputs or model parameters. Some applications of this type of analysis are sug­ gested. Perturbation analysis is developed in the context of continuous systems whose dynamics, over small ranges of the system's behaviour, can be represented by linear models. A l l variables and signals are represented by intervals with qualitative end points. Qualita­ tive linear models are introduced to represent time-varying systems. These representations permit the use of network consistency algo­ rithms to solve perturbation analysis problems. This paper is dedicated to the memory of Dr . Murdoch McKinnon , late of C A R Electronics L td . and Concordia University, who faithfully supported this research since its beginning. 1. Introduct ion: Qualitative Perturbation Analysis 1.1 Reasoning about continuous systems Most work on qualitative physics [Bobr-84] has been device-centered (e.g. electric circuits, tanks and pipes) with models derived f rom component topol­ ogy [deKl-84]. Inferences about the behaviour of a device are made by constraint propagation. Qualita­ tive reasoning about processes [Forb-84], models the behaviour of a system as the combined effect of active processes which describe the relations and influences between objects. However, a system is still considered as a collection of objects and rela­ tions between them. In Q S I M [Kuip-86], continuous functions (over time) represent state variables and constraints model system structure. Components and interconnections are not the only models for dynamic systems. In some continuous systems, state variables depend on the aggregate behaviour of many elements. For example, the aerodynamic forces on an aircraft are the result of integrating the forces caused by airf low over the entire airframe. System models may be finiteelement approximations or differential equations; both types are useful for numerical simulations. Such models may be used in problem-solving, but are surely not the basis of human reasoning. When people design, control or diagnose such dynamic sys­ tems they use their understanding of physical pr inci­ ples and problem-solving skills. In particular, peo­ ple seem to reason about orders of magnitude of variables, and relations between variables and their rates of change. This paper considers how to make a computer program do the same. 1.2 Outl ine of the paper This paper describes QPA and the representations and algorithms which it requires. References to related research are included throughout the paper. The remainder of this section introduces the not ion of a perturbation to a system, discusses the types of models to which QPA is applicable, and summarizes the contributions of this research. Section two describes the qualitative representation of variables and signals, and the qualitative calculus. An exam­ ple Q L M is introduced in section two. Perturbations of Q L M s and a transformation to a CSPs are dis­ cussed in section three. Section four concludes wi th a summary and ideas for future work . 1.3 Perturbat ions and appl icat ions Engineers are frequently interested in how a system responds to perturbations. Consider a system A whose behaviour during a manoeuvre is described by a set M of init ial condit ions, inputs and outputs. Note that inputs and outputs are signals. One type of analysis is to change an input or init ial condit ion of a manoeuvre, or a parameter of the model , and perform a simulation to see the effects. A more dif­ f icult problem is to do the inverse. Given a desired perturbation on the outputs of a manoeuvre, how can this be achieved by perturbing inputs, ini t ial conditions or model parameters? The representa­ tions and algorithms used in answering these types of questions are called Qualitative Perturbation Analysis (QPA) and are the subject of this paper. 1180 Knowledge Representation QPA can be used to find causes of discrepancies between systems and models. If output discrepan­ cies can be expressed as perturbations, any input, init ial condit ion or parameter modif ied by QPA can be considered a cause of the original discrepancies. There are many potential applications of QPA: Design: A design model is being used to design a system A with desired behaviour M. If simulations do not match M, QPA can determine design changes so that A wi l l meet its specification. Diagnosis: Let A be a real, malfunctioning system, let M contain symptoms. If QPA discovers causes for the symptoms, any perturbed parameters are possible faults in A. Validation: When A is a real system and M contains real measurements, QPA can be applied to perturb simulation parameters to improve their accuracy. This research is part of a project studying AI techniques for validation of aerodynamic models (see [Prag-89] for an overview). A knowledge-based assistant system, called the Flite System, is being built for simulation engineers. QPA is designed for the key role of reasoning about discrepancies in simulations. 1.4 Linear models of a system Models for qualitative reasoning about continuous systems should have several properties: (a) related to human mental models (b) represent a wide variety of systems (c) represent relations between variables (d) represent time-varying signals (e) amenable to aggregation by subsystem (f) can be instantiated given recorded signals An appropriate class of models is first order linear differential equations (FOLDEs) , which have many applications in modern control theory [Frie-85] (e.g. to model spring-coupled masses, disti l lation columns etc.). For example, equations to model small motions in an aircraft's longitudinal axes are given in Figure 1. For some M a single set of FOLDEs may not be accurate, in which case M can be segmented and modeled by a sequence of F O L D E s , one per segment (see [Prag-89]). QPA is applicable to systems whose behaviour, after seg­ mentation, can be modeled by F O L D E s with con­ stant coefficients. Qualitative models can be derived f rom analytic models by representing all terms by qualitative values and interpreting equations as constraints [deKl-84], [Will-88]. Qualitative Linear Models (QLMs) are versions of F O L D E s , wi th a qualitative representation for signals and gains (coefficients of the FOLDEs are called gains). Q L M s clearly satisfy properties (b), (c) and (d) above. Property (e) is discussed in [lwas-88]. Given the model structure and signals, gains can be estimated by system iden­ ti f ication techniques [Eykh-74], thus (f) is satisfied. Whether QLM's satisfy (a) is more diff icult to argue. It does seem to be useful to reason about decoupled sub-systems, relative influences between variables, and relative magnitudes of signals. QLMs support these types of reasoning. The relation between linear models and complex simulation models is discussed in [Prag-89]. A map­ ping f rom QLMs to complex models wi l l in general be possible by exploiting the structure of the domain. Since this is a domain dependent problem, QPA is concerned only with linear models in their general form. 1.5 The QPA strategy Given A and M, the first step of QPA is to compute a Q L M L and the qualitative representation of sig­ nals in M. Knowledge of A is only used to deter­ mine the equations of L. Next, QPA uses L and a differentiation formula (see 2.4) to compute con­ straints on the derivatives of the Q L M . Derivative constraints are crit ical to Q P A since they constrain values of signals at successive t ime points. Th i rd , output perturbations are applied (usually all at the same time point) , making L inconsistent. The final step of QPA is to formulate a constraint satisfaction problem (CSP) and solve to f ind new values of sig­ nals, and possibly gains, consistent with the pertur­ bations. The transformation to a CSP is designed such that the general algorithms of [Mack-77] (see also [Mohr-86] and [Han-88]) can be applied.