IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 54, NO.1, FEBRUARY 2005 391 Iterative-Type Evaluation of PSGS Fuzzy Systems for Anytime Use Orsolya Takács and Annamária R. Várkonyi-Kóczy, Senior Member, IEEE Abstract—While fuzzy systems can advantageously be used in system modeling and control, their use in time-critical applications is limited because of complexity problems, especially in cases when not only low, but also flexibly changeable complexity is needed. Previously, a method has been proposed to use fuzzy and other soft-computational tools in the frame of modular anytime archi- tectures; however, the applicability needs the a priori knowledge of the temporarily available time and resources. This paper pro- poses a new transformation method, which makes possible the it- erative-type evaluation of product-sum-gravity-singleton (PSGS) fuzzy systems, with a really flexibly changeable complexity, and with an easily estimable error at any step of the evaluation. More- over, the transformation also ensures the fastest possible decrease of the error. Index Terms—Anytime systems, fuzzy systems, iterative evalua- tion. I. INTRODUCTION T ODAY, MOST of the industrial applications can be charac- terized by two important features: their time-critical prop- erty and the increasing complexity of the problems to be solved. The time-critical property means that most of the tasks such as measurement, signal processing, fault detection and diag- nosis, system control, etc.—should be carried out “online,” with a guaranteed response time. Because usually not only the time, but other resources are also limited, computing methods with low complexity are highly desirable. Furthermore, the amount of the available time and resources may depend on the actual circumstances as well: Temporal overloading of the system caused by tasks or an alarm signal, needing immediate attention, can cause some of the modules of the system to not obtain as many resources, as would be op- timal. These temporal shortages of processing time/resources, along with the possible fallouts of input data, may cause a critical breakdown in the performance of the system or cause the data processing to not be carried out at all. On the other hand, especially in measurement, if the design of the model of the measurement is based upon some a priori knowledge, we can build a better measurement scheme, being able to collect more precise information about the system or pa- rameters to be measured, and these can be used again for im- proving the measurement system. Manuscript received June 15, 2002; revised July 26, 2004. This work was supported by the Hungarian Fund for Scientific Research under Grant OTKA T 035190 and the I2S Integrated Intelligent Systems Japanese–Hungarian Laboratory. The authors are with the Department of Measurement and Information Sys- tems, Budapest University of Technology and Economics, Budapest H-1521, Hungary (e-mail: takacs@mit.bme.hu; koczy@mit.bme.hu). Digital Object Identifier 10.1109/TIM.2004.838118 To improve the model design and to avoid the mentioned breakdowns, the so-called “anytime” techniques can be ap- plied advantageously. These systems are able to provide short response time (possibly with less accuracy) and are able to maintain the information processing even in cases of temporary shortage of time or computational power [1], [2]. The aim of the use of these techniques is to help early decision making and to ensure the continuous operation of the system in case of changing circumstances, thus, to provide optimal overall performance for the whole system. Another important tendency is the increasing complexity of the problems to be solved. While for the simpler, basically linear problems, theoretically well-established methods and tools are available, and they are successfully combined with adaptive techniques to provide optimum performance [3], it is problematic to find any systematic method which is suitable to solve larger classes of nonlinear modeling and control problems. In practical industrial applications the typically nonlinear processes are often too complex to be modeled by classical tools: The exact physical/mathematical description of the system is unknown, too complex to be handled, or could be obtained only with two much difficulties. On the contrary, fuzzy systems can be well used even in cases when the problem is too complicated to be solved by classical tools. Knowledge originating from human experts can be built into fuzzy systems easily, and there are also known training methods for the improvement of fuzzy systems by the use of sample data. Thus, fuzzy systems can be applied when exact mathematical description is not, but expert knowledge and/or sample data is available. [4], [5] Another advantage of fuzzy-based systems is that they are able to handle not only imprecise data, but also inexactly for- mulated concepts which can be expressed by classical tools only with difficulties. It results in that fuzzy systems can be well used in cases when the available data is imprecise or the interpreta- tion may depend on the context [6], [7]. Although, the usability of fuzzy tools in time-critical systems is limited by the lack of any systematic method for determining the needed complexity of the system. In case of fuzzy systems, higher granularity (higher number of antecedent fuzzy sets) usu- ally results in better approximation of the original system. To achieve the needed accuracy, one can be tempted to overesti- mate the needed granularity, which results in huge and redun- dant systems with too many rules and too high complexity. Another problem is the flexibility. In case of temporal shortage of time resources, a solution could be evaluating only a subset of the rules. To this, we have to give an estimation for the accuracy of the approximation, which may be problematic 0018-9456/$20.00 © 2005 IEEE