Linear Parameter Varying Modeling and Estimation of a SMA Wire Actuator

Shape Memory Alloys (SMAs) actuators operate via a nonlinear and hysteretic relationship between input power and mechanical motion. This nonlinearity presents a serious challenge when developing methods for controlling these actuators. Because this hysteresis and nonlinearity is caused by the crystal phase transformation however, the SMA constitutive and kinetic models can be written in Linear Parameter Varying (LPV) form, with the partial derivative of crystal phase fraction with respect to temperature as the varying parameter. This allows a SMA system to be written in a state-space format where the coefficients in the state matrices vary as a function of the state variables, allowing for the application of powerful linear system analysis tools to this model without simplifying assumptions. This LPV model can then be used to create an estimator for the system, allowing for real-time approximations of the system states, including temperature and phase fraction. This paper presents the derivation of one such LPV model and explores its ability to accurately represent a physical SMA actuator system by comparison with an instrumented SMA muscle system.