Case 1 : For a given value of (T, σ), iffL1 ≤0andfL2 ≥0and dTdt >0, then the current point is in the reverse transformation zone. Now in case of posteriori estimate, ifPstatus ∼= 1, then first update the memory parameters and assignPstatus= 1. Then following the phase kinetics given by Eqns. (3.64) and (3.66),ψ1 ψ2,ψ3, andψ4, are determined.
Case 2 : For a given value of (T, σ), iffL3 ≤0andfL4 ≥0and dTdt <0, then the current point is in the forward transformation zone. Now, in case of posteriori estimate, ifPstatus ∼
= 2, then first update the memory parameters and assign Pstatus = 2. Then following the phase kinetics given by Eqns. (3.60) and (3.62),ψ1ψ2,ψ3, andψ4,are calculated.
Case 3 : In all the other cases, transformation is inactive. However, in the posteriori step, if Pstatus∼= 3, then update the memory parameters as the final value of stress, temperature and martensite volume fractions obtained in the last time step. Then assignPstatus = 3. Else, the martensite volume fractions are same as their corresponding values in memory parameters.
Finally, the functions,ψ1ψ2,ψ3, andψ4, are assigned as zero.
between the Extended Kalman Filter estimation and the actual stretch of the spring. To evaluate the effectiveness of the Extended Kalman Filter based Artificial Neural Network model, the responses of the same are compared with that of another Artificial Neural Network model, trained only using the experimental data. It has been observed that for the same number of neurons and same training data, Extended Kalman Filter based Artificial Neural Network model yields better results, even at higher frequencies.
6.3.1 Description of the System
The system as described in section 5.3 is used to explore the self-sensing capability of SMA using EKF and ANN. In what follows, the details of the ANN models are discussed.
6.3.2 EKF based Artificial Neural Network (ANN)
For the system of interest, an EKF has already been developed to estimate the change in length of the spring from the measured electrical resistance (ER) variation of the SMA wire.
For a given applied voltage, and the corresponding change in electrical resistance, as mea- sured using the set-up discussed in section 4.2, the estimated displacement of the spring and the same measured experimentally are compared, and is again presented in Fig. 6.3a. One can note the quantitative discrepancy between the two. To reduce the gap, two ANN models are explored. Each ANN model consists of one hidden layer having ten neurons, three in- puts, and one output. The models differ from the perspective of inputs only. The first model, referred as ANN-I, intakes the temperature of the SMA as estimated by EKF (TEKF), its rate of change (T˙EKF) and change in length of the spring estimated by the EKF (δEKF). On the other hand, ANN-II model concedes the applied voltage across SMA (V), its rate of change
(
dVdt)
and the change in electrical resistance of SMA as obtained from the experiment. Inboth the models, the output is the displacement of the spring. The whole process is explained using a flow diagram as shown in Fig. 6.3b. Thus the main difference between the two mod- els is that, ANN-I intakes the output of the estimation of the EKF; whereas ANN-II accepts the measured voltages and resistance. The ANN models are depicted in Figs. 6.3c and 6.3d, respectively. The rate of temperature and voltage are included as inputs to distinguish the heating and cooling part of the actuation cycles, taking care of hysteresis in SMA behavior.
Both ANN-I and ANN-II are trained using the data corresponding to the response shown in Fig. 6.3a. The neural network toolbox of MATLAB©MathWorks is used in this study.
0 20 40 60 80 100 120
0 2 4 6 8
0 30 60 90 2
4 6
Exp
EKF
Displacement(mm)
Time (sec)
Appliedvoltage(V)
T ime (sec)
(a) (b)
(c) (d)
Figure 6.3: (a) Comparison between EKF estimated and experimental response, (b) schematic of the ANN models, (c) details of ANN-I and (d) details of ANN-II.
6.3.3 Experimental Details
For measuring the electrical resistance of SMA and the actual displacement of the SMA actuated system, the experimental set-up presented in section 4.2 is used.
6.3.4 Results and Discussion
First, few continuous voltage signals, having different frequency and amplitude are applied across the SMA wire, and the corresponding change in electrical resistance of the SMA wire and the corresponding spring displacement are measured experimentally. Then EKF is used to estimate the temperature of the SMA wire (TEKF) and spring displacement (δEKF) from the experimentally measured electrical resistance of the SMA wire. All these are then used in ANN-I, which has been trained using the estimated data obtained for the input is shown in Fig. 6.3a, so as to obtain the actual displacement δ1. Besides, ANN-II directly uses the experimental data and yields actual displacement δ2. These are then compared and shown in Figs. 6.4 and 6.5. The responses, shown in Fig. 6.4a correspond to continuous periodic voltage signals having, frequency 0.1 Hz and amplitude 6.4 V. It can be observed that both the ANN models predict the displacement of the spring with reasonable accuracy. The outputs of the ANN models, for a similar voltage signal of frequency 0.2 Hz rad/sec, are presented in Fig. 6.4b. It can be observed that the output of ANN-II model is slightly erroneous, particularly in the last part of cooling. This may be because ANN-II has been trained for different input frequency as shown in Fig. 6.3a. Though the ANN-I model has also been trained for the same input as that of ANN-II, however, the governing equations, representing SMA behavior, inculcated in the EKF helps ANN-I model in avoiding the discrepancy. The responses for similar inputs with amplitude 4.5 V and frequency 0.1 Hz are presented in Fig. 6.4a. In this case, the SMA wire undergoes partial transformation and both models have
accurately captured the same.
However, with inputs having slightly higher frequency (0.2 Hz), the responses from these models are depicted in Fig. 6.4b. This result clearly reveals the limitation of ANN-II over ANN-I. It might be possible to improve the performance of ANN-II model by increasing more hidden layers and training data; however, that requires more computational resource and training time.
0 20 40 60 80 100
0 2 4 6 8
0 20 40
2 4 6
Displacement(mm)
Time (sec)
Exp
ANN-I
ANN-II
Appliedvoltage(V)
Time (sec)
(a)
0 10 20 30 40 50
0 2 4 6 8
0 10 20
2 4 6
Displacement(mm)
Time (sec)
Exp
ANN-I
ANN-II
Appliedvoltage(V)
Time (sec)
(b)
Figure 6.4: Comparison between EKF estimated and experimental responses for continuous periodic input voltage signal of (a) amplitude = 6.4 V and frequency = 0.1 Hz, and (b) amplitude = 6.4 V and frequency = 0.2 Hz.
0 20 40 60 80 100
0 1 2 3 4
0 20 40
2 3 4
Displacement(mm)
Time (sec)
Exp
ANN-I
ANN-II
Appliedvoltage(V)
Time (sec)
(a)
0 10 20 30 40 50
0 2 4 6 8
0 10 20
2 3 4 5
Displacement(mm)
Time (sec)
Exp
ANN-I
ANN-II
Appliedvoltage(V)
Time (sec)
(b)
Figure 6.5: Comparison between EKF estimated and experimental responses for continuous periodic input voltage signal of (a) amplitude = 4.5 V and frequency = 0.1 Hz, and (b) amplitude = 4.5 V and frequency = 0.2 Hz.
Thus with the EKF based ANN, one can estimate the response of an SMA actuated system, for both partial and complete transformation cases, with significant accuracy. More- over, it requires lesser training data and relatively small number of neuron and hidden layers.