To accurately position an object with an actuator that exhibits load dependent hysteresis requires a hysteresis model that is capable of adjusting to a change in load. In this paper we investigate the specific problem of modeling the hysteresis of a simple shape memory alloy wire that is operated under changing tensile loads. A Preisach operator that incorporates load dependent parameters in the Preisach density function is proposed as the hysteresis model. In support of this selection, a relationship between the Preisach density function and the wire's thermal coefficient of expansion is established. It is then shown that the load dependent Austenite-Martensite transition temperatures of the wire can be used to estimate the parameters of the density function. Based on these findings a load dependent Preisach operator is defined. To test this approach, a bivariate density function that incorporates two load dependent parameters is substituted for the Preisach density function. Two load dependent linear estimators are developed from experimental data and used to estimate the parameters of the density function. These estimators and the load dependent Preisach operator are then used to estimate the length of a SMA wire that is operated under several tensile loads. The estimates are compared to experimental data and a discussion of the effectiveness of this approach is given.
KEYWORDS: Actuators, Data modeling, Control systems, Error analysis, Mathematical modeling, Systems modeling, Data analysis, Fiber optics sensors, Sensors, Complex systems
The accurate control of a system that exhibits hysteresis requires a control strategy that incorporates some form of compensation for the hysteresis. One approach is to develop an accurate model P of the hysteresis and then use the inverse P-1 as a compensator (inverse compensation). This paper focuses on the problem of determining P and P-1 for a piezoelectric stack actuator. The KP operator is used to formulate the hysteresis operator P and from this an approximation Pm is derived. Compensation is then defined in terms of the inverse of the approximation, Pm-1. Experimental data collected on an unloaded piezoelectric stack actuator is used to demonstrate the concept of inverse compensation. The parameters of Pm are identified using only major hysteresis loop data and results are given that show Pm provides an accurate model for both major and minor hysteresis loops. Results are then presented that demonstrate the capability of the inverse Pm-1 to compensate for the actuator hysteresis.
The accurate control of a system that exhibits hysteresis requires a control strategy that incorporates some form of compensation for the hysteresis. One approach is to develop a compensator based on an inverse hysteresis operator. If this can be accomplished, the composite operation will produce a linear relationship between the input and output. Thus, an open loop control can be developed in which the inverse operation adjusts the system input to compensate for the hysteresis in the physical system. One difficulty lies in developing a model of the hysteresis for which an inverse operator can be obtained. In this work, a system with hysteresis in modeled by a classical Preisach model. We show that in the case of certain bivariate distributions, a closed-form formula for the inverse operator can be obtained. The concept is illustrated by a computer simulation.
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