## 3 Model Identification  
One of the main shortcomings of the SMA actuators is the difficulties in motion control due to hysteresis and other nonlinearities. The reason that gives rise to hysteresis is that the material's crystalline structure shifts between martensite and austenite phases depending on the applied temperature and stress. Martensite is the rather soft and easily deformed phase of shape memory alloys. This phase exists at lower temperatures, whereas at higher temperatures, the austenite phase is relatively hard. The hysteresis of the shape memory actuators of the experimental platform for four consecutive heating up and cooling down cycles is shown in Figure 3.  
Figure 3: Four consecutive heating up and cooling down cycles of shape memory alloy with hysteresis.  
Figure 2: Deflection of the textile-reinforced composite with side view of the test bench unit.  
The system responds with a time lag to the applied voltage. In addition, there is almost no overshoot. For these reasons, we want to describe the system's essential behavior by a first order linear time-invariant system. In the Laplace frequency domain, the system is modeled by the transfer function  
\[G(s)=\frac{K}{1+sT} \tag{1}\]  
with the complex variable \(s\) representing the excitation frequency. In addition, the parameter \(K>0\) describes the gain and the parameter \(T>0\) is the system's time constant. Roughly speaking, the transfer function in Equation (1) can be understood as frequency-dependent gain factor [27].  
The actuator specimen has been tested with different step inputs and different operating points. Due to these different basic conditions, each experiment delivers more or less differing parameter values of \(K\) and \(T\). The set of parameters obtained can be described by intervals  
\[\underline{K}\leq K\leq\overline{K}\quad\text{and}\quad\underline{T}\leq T \leq\overline{T}. \tag{2}\]  
This interval model is later extended to a dynamic uncertainty model in order to capture the hysteresis and other unmodeled dynamics.  
For the identification, the System Identification ToolboxTM of MATLAB(r) is used [28]. This toolbox offers an easy and efficient way to generate various mathematical models of systems such as continuous-time and discrete-time transfer functions as well as state space models. To describe the characteristics of the SMA actuator over a wide range of operation points, open-loop tests of the actuator using step inputs are conducted. The input and output values of these experiments are used to identify the mathematical model (Equation (1)), of the system. Each step response yields different values of the parameters \(K\) and \(T\). Over all experiments we obtained the following upper and lower bounds of these parameters [21]:  
\[\underline{K}=1.969,\quad\overline{K}=2.6328,\quad\underline{T}=112.676,\quad \overline{T}=382.995. \tag{3}\]  
Thereafter, the plant's nominal transfer function  
\[\widetilde{G}(s)=\frac{\widetilde{K}}{1+s\widetilde{T}} \tag{4}\]  
can be derived, where the (nominal) parameter values \(\widetilde{K}\approx 2.3009\) and \(\widetilde{T}\approx 247.8355\) are obtained by the arithmetic mean values of the bounds seen in Equation (3).