Number 36 March 93 The Huntington Technical Brief By David Brubaker Ph.D. A FUZZY STATE MACHINE MODEL --------------------------- TERMINOLOGY A state machine consists of a collection of states interconnected by transitions that are activated by events. A system moves from its current state to its next state via a transition. Transitions are triggered by events, which are typically received asynchronously by the state machine as inputs. The next system state is a function of the current system state and the triggering event. Traditional state machines require that the system reside fully in only one of its states at any given time. The occurrence of an appropriate event will force a transition from the current state to the next state. Two types of events are common, absolute events with no associated data, called trigger events, and those associated with data, called conditional events. A fuzzy state machine is an extension of a traditional state machine, and consists of states in which the system can partially reside. States themselves are crisp, more analogous to points than sets, but the system can exist in a given state to varying degrees. The degree to which a system resides in a state shall be called the degree of being of that state, represented by the symbol beta. Conditional events can be fuzzy. An event's degree of fuzziness is the degree to which the associated condition is true, and is represented by the truth value, mu. DESCRIPTION The primary differences between the fuzzy state machine model being described and traditional state machines are that in the fuzzy state machine: 1) a system can reside in a state with less than unity degree of being; and 2) a system can simultaneously reside in more than one state so long as system "being" is conserved. For each executed transition, the degree of being of the source state is reduced by the product of the source state's degree of being and the truth value of the event. Because the overall "being" of the system is conserved, the degree of being of the destination state is the sum of its current value with this same product. Fuzzy events can occur over several to many data sample intervals, resulting in smooth transitions. Stronger events cause faster inter-state transitions, and weaker events cause slower inter-state transitions. EVENTS Trigger events carry an implicit truth value of 1. They are crisp, are instantaneous, and a "partial occurrence" is not allowed. When used in a fuzzy state machine, trigger events cause complete transitions: the entire degree of being currently in the source state is transferred to the destination state. Conditional events can be either crisp or fuzzy. In a crisp state machine, the condition necessary for a transition is stated in a true/false relational statement, as in "x = 3" or "x * 2 AND y * ", where x is a numeric variable. When the condition is satisfied, the transition is executed. In a fuzzy state machine, the condition of a conditional event is expressed as a fuzzy relational statement, for example "DISTANCE is NEAR" or "DISTANCE is NEAR and SPEED is SLOW". NEAR and SLOW are fuzzy values of the fuzzy variables DISTANCE and SPEED, respectively. The degree to which the expression is true is the truth value of the event, and when the condition is evaluated with non-zero truth value, transition processing is performed as described in the previous section. CONCLUSION We have discussed a new model for a fuzzy state machine, with which the system can reside in system states with partial strength, or degree of being. The model also allows multiple states to be simultaneously active (to have non-zero degrees of being) so long as the sum of all degrees of being remains unity. A state machine is used to introduce sequence into a system. A fuzzy state machine further allows smooth transitions between states, thereby allowing smooth transitions in the system parameters (outputs, for example) controlled by the associated states. The model presented is not intended to be a totally general structure, although it also is not restricted to any given application. Rather, it is a practical, working state machine with definite, clearly evident attributes. ---------------------------------------------------------------- The Huntington Group provides technical consulting services in complex, real-time, embedded processor, and fuzzy system design. The Huntington Technical Brief, of which this is a much edited version, is a marketing tool, published once a month on fuzzy topics of interest - a yearly subscription costs $24 and back issues are available at $2.00 per issue. A report, "Introduction to Fuzzy Logic Systems" is also available for $50. For more information, call David Brubaker at 415-325-7554. ---------------------------------------------------------------- Copyright 1992 by The Huntington Group 883 Santa Cruz Avenue, Suite 31 Menlo Park, CA 94025-4608 This information is provided by Aptronix FuzzyNet 408-428-1883 Data USR V.32bis