1999 Third International Conference on Knowledge-Based Intelligent Information Engineeing Systems, 3 I It Aug- Id Sept 1999, Adelaide, Australia zyxw . , Design and Analysis of Fuzzy Schedulers using Fuzzy Lyapunov Synthesis . z Machine , Michael Margaliot Gideon Langholz Department of Electrical Engineering - Systems Tel Aviv University, Israel 69978 zyxw michaelm,langholz@eng.tau.ac.il Keywords: fuzzy scheduling, fuzzy Lyapunov synthesis, computing with words. Abstract zyxwvutsrqpo Recently, the authors suggested a new approach to the design offuzzy control rules. The method, re- ferred to as fuzzy Lyapunov synthesis, extends clas- sical Lyapunov synthesis to the domain of "comput- ing with words", and allows the systematic, instead of heuristic, design and analysis of fuzzy controllers given linguistic information about the plant. In this papes we use the fuzzy Lyapunov synthe- sis method to design and analyze the rule-base of a zyxwvutsr fuzzy scheduler. We show thatfuzzy rules, previously suggested based on heuristics, can be derived system- atically and, therefore, that the entire process can be automated. This may lead to a novel "computing with words '' algorithm: the input is linguistic information concerning the "plant '' and the "control"objective, and the output is a suitable fuzzy rule-base. 1 Introduction Consider a manufacturing system consisting of several machines that process parts of different types. Parts of type i zyxwvut (i = 1,2, ..., P) must be processed in order by a series of machines, say machine Mi, for Ti,il seconds, then machine Mi, for ~i,i, seconds, and so on until machine Mi, where the production of the parts is completed. Because each of the ma- chines can only process one part at a time, each ma- chine must also contain a buffer, for each part-type it inputs, which stores the parts that entered the machine and are waiting to be processed. A scheduler is an algorithm that specifies the or- der in which the machines should process the waiting parts. To assure adequate performance, schedulers must also guarantee that the buffer levels in the ma- chines remain bounded. A scheduler that guarantees this property is called stable. Various performance measures (e.g.. the average load of the buffers) can be used to rate different sched- ulers, where scheduler 1 is considered better than scheduler 2 if it yields a smaller value of the perfor- mance measure. Hence, the scheduling problem is to find a scheduler that stabilizes the system while min- imizing the performance measure. Scheduling a fixed number of parts with known arrival and processing times is referred to as static scheduling. On the other hand, if at time t the sched- uler determines which part-type to process next based on quantities that are known at time t (e.g., the current state of the machines' buffers), then the scheduling is referred to as dynamic scheduling. Perkins and Ku- mar zyxwvu [5] showed that the problem of dynamic schedul- ing can be handled in a distributed manner, that is, the scheduling decisions of machine Mi are based only on the properties and the current state of machine Mi regardless of the other machines in the system. Thus, the problem can be decomposed into a set of indepen- dent single-machine scheduling problems. zyxw Figure 1. Single-machine dynamic scheduler Fig. 1 depicts the organization of a single-machine dynamic scheduler. Here, x(t) = (zl(t) ... zp(t)), where zi(t) is the level of buffer zi at time t, and i* is the index of the part-type to process next. Note 207