Optimization of the recursive least squares algorithm for capacitive strain sensing

Dielectric elastomer actuators (DEAs) consist of an elastomer membrane sandwiched between two compliant electrodes. Applying a voltage across the electrodes compresses the membrane which then expands in the in-plane direction due to the incompressible nature of the material. This mechanism provide a cheap, simple, and robust means to achieve high-strain actuation for a number of applications, including soft robotics [1], tunable lenses [2], and reconfigurable keyboards [3]. In addition to their use as actuators, DEAs can also be used as an integrated strain sensor and thereby provide the opportunity for simultaneous strain sensing and actuation. The use of dielectric elastomers as integrated self-sensors was first proposed by Sommer-Larson et al [4], who noted that changing the geometry of a DEA would result in changing its electrical properties, which could be exploited to measure the strain of the elastomer itself. This self-sensing capability is applicable both to strains induced by high voltage actuation as well as mechanically induced strains. Additionally, because the method relies on geometric changes of the DEA membrane, self-sensing is independent of material characterization, and is generally robust even for highly hysteretic DEA materials.

A DEA may be modeled using an equivalent electrical circuit consisting of three components: the membrane and electrodes are treated as a variable capacitor, which lies in series with a variable resistor (due to the resistive nature of the compliant electrodes), and in parallel with an additional resistor to account for any leakage current across the dielectric material [5]. Most models neglect the leakage current, allowing the DEA to be treated as a simple series RC circuit (figure 1(a)), in which the capacitance varies as a function of the in-plane strain [6].

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A number of methods have been proposed to track the changing capacitance as the membrane thickness changes. For example, the use of an external resistor in series with the membrane creates a first order high-pass filter, which may be used to measure the capacitance [7]. Gisby et al [5], developed a low bandwidth digital control method in which the amplitude and duty cycle of a pulse width modulation signal are used to approach a stable constant charge system. Finally, Gisby et al [8] demonstrated an approach in which, by tracking voltage, current, and charge as a function of time, the slope of the resulting hyperplane can be used to track changes in the capacitance, resistance and leakage current.

For applications requiring fast response times, real-time recursive algorithms have proven to be both computationally inexpensive and experimentally robust. These algorithms monitor voltage and current to estimate capacitance and resistance. Two types of recursive algorithms—the least mean squares (LMS) and the recursive least squares (RLS) - were investigated by Rizzello et al [6, 9] for application in DE self-sensing and the RLS approach was found to be slightly more accurate.

In the RLS algorithm the DEA is treated as a simple series RC circuit (figure 1(a)), where both the resistance and capacitance change with time at a typical frequency, f_c . The dielectric is 'probed' by injecting a low-amplitude sinusoidal voltage, V_p , at a frequency, f_p  ≫ f_c , and measuring the current. The discretized constitutive equations of a series RC circuit can be manipulated into linear-in-parameters form:

$$\begin{eqnarray}&&{V}_{k}-{V}_{k-1}={R}_{k}{i}_{k}+\left(\displaystyle \frac{1}{{f}_{s}{C}_{k}}-{R}_{k}\right){i}_{{k}_{1}},\end{eqnarray} \tag{ 1 }$$

where f_s is the sampling frequency, V is the applied voltage, i is the measured current, and C and R are the capacitance and resistance, respectively, at the k-th time-step. The capacitance can be predicted at each time-step using the RLS algorithm:

$$\begin{eqnarray}&&\left\{\begin{array}{l}{\hat{\theta }}_{k+1}={\hat{\theta }}_{k}+{K}_{k+1}{\varepsilon }_{k+1}\\ {K}_{k+1}=\tfrac{{P}_{k}{\varphi }_{k+1}}{1+{r}_{k+1}}\\ {\varepsilon }_{k+1}={y}_{k+1}-{\varphi }_{k+1}^{T}{\hat{\theta }}_{k}\\ {P}_{k+1}=\tfrac{1}{\mu }\left({P}_{k}-\tfrac{{P}_{k}{\varphi }_{k+1}{\varphi }_{k+1}^{T}{P}_{k}}{1+{r}_{k+1}}\right)\\ {r}_{k+1}={\varphi }_{k+1}^{T}{P}_{k}{\phi }_{k+1},\end{array}\right.\end{eqnarray} \tag{ 2 }$$

where

$$\begin{eqnarray}&&{y}_{k}={V}_{k}-{V}_{k+1}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{\varphi }_{k}={[{i}_{k}{i}_{k-1}]}^{T}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{\theta }_{k}={\left[{R}_{k}\displaystyle \frac{1}{{f}_{s}{C}_{k}-{R}_{k}}\right]}^{T},\end{eqnarray} \tag{ 5 }$$

and μ is the 'forgetting factor'. A forgetting factor of unity weights the entire time series equally, while a greater bias for recent samples is introduced as the forgetting factor is decreased. Typical values range from 0.9 to 1.

The effect of the forgetting factor can be represented by μ = 1 − 1/W, where W is the number of samples that will be weighted more heavily by the algorithm [6]. However, as the sampling frequency changes, a constant forgetting factor will result in weighting different durations of a capacitance cycle. We can account for this with the use of a normalized variable, $\bar{W}$:

$$\begin{eqnarray}&&\bar{W}=\displaystyle \frac{{f}_{c}}{{f}_{s}(1-\mu )}.\end{eqnarray} \tag{ 6 }$$

Here, $\bar{W}$ reflects the fraction of a capacitance cycle that is more heavily weighted by the RLS algorithm, regardless of the frequencies. When capturing dynamic changes in the capacitance, this variable will be shown to be a primary driver of the accuracy of the algorithm output, as will be discussed in section 3.1.

For the linear-in-parameters assumption to be valid, it is necessary to probe the equivalent circuit at a frequency significantly higher than the expected changes in capacitance and resistance, f_p  ≫ f_c . Additionally, because the voltage and current are discretized in the analysis, the sampling frequency, f_s , must be high relative to the probe frequency, f_s  ≫ f_p . However, a practical goal is to keep the sampling frequency as low as possible, which leads to the natural question of how much higher the sampling frequency, f_s , must be relative to the probe frequency, f_p , and similarly how much higher the probe frequency must be relative to the capacitance frequency, f_c . In addition, we would like to have guidance regarding the optimal choice of the forgetting factor, μ.

To address these questions, we use the RLS algorithm to estimate the state of a DEA membrane and study its performance as a function of the RLS parameters. Analytic sinusoidal strain profiles over a range of frequencies were used to derive synthetic voltage and current signals. These signals were fed into the RLS algorithm to generate a predicted capacitance which was then compared with the known capacitance input. The optimized algorithm was then validated experimentally using a dielectric elastomer membrane that was strained over various timescales. The capacitance predicted by the RLS algorithm was then compared to the capacitance derived from a visual analysis of the membrane strain.

2.1. Algorithm optimization

Sinusoidal signals for capacitance and resistance, with white noise superimposed, were generated using MATLAB. Using the equivalent RC circuit and superimposing an applied probe voltage, the expected current output was calculated and fed into the RLS algorithm. The RLS output was low-pass filtered, with a cut-off frequency of 0.9f_p , and the estimated capacitance was compared with the analytical input in terms of amplitude, offset, and noise. For the purposes of this analysis, amplitude was defined as the peak-to-peak amplitude of the low-pass filtered output, and offset was defined as the mean of that signal. Noise was defined as the root-mean-squared (RMS) amplitude of the estimated signal after high-pass filtering with a cut-off frequency of 1.1f_c . The performance was assessed for a range of frequencies, f_c , f_p , and f_s , as well as values of the forgetting factor, μ, ranging from 0.5 to 1 (table 1).

Table 1. The range of variables used in the RLS algorithm optimization.

	fc (Hz)	fp /fc	fs /fp	μ	$\bar{W}$
Minimum	10	3	20	0.5	0.0013
Maximum	200	80	100	0.99	0.333

2.2. Benchtop validation

A silicone membrane of 1 mm nominal thickness (TC-5005 A/BC, 35% C, BJB Enterprises, Tustin, CA, USA) was prepared, and the thickness was measured to be within 15 μm of the nominal value across the surface area. Carbon grease electrodes were applied to the top and bottom surfaces. The membrane was adhered to the top of an open cylindrical dish of diameter D, creating a pressure vessel (figure 1(b)). The strain in the membrane was varied by changing the pressure inside the dish over the course of 3600 seconds while the electrodes were probed with a sinusoidal voltage (V_p  = 10V_pp , f_p  = 1 kHz). The applied voltage and resulting current were recorded at f_s  = 60 kHz and fed into the RLS algorithm for capacitance estimation. The shape of the membrane was recorded from the side with a digital camera at 60 Hz, and the deflection, h, was determined using the MATLAB Image Processing Toolbox.

To a very good approximation, the pressurized membrane takes the shape of a spherical cap, in which case the membrane stretch, λ, is equal to 4h/D + 1 [10]. Treating the DEA as a flat plate capacitor and assuming that the membrane material exhibits incompressibility and stretch-independent permittivity, stretch can be converted to an estimated capacitance, C(t):

$$\begin{eqnarray}&&C(t)=\displaystyle \frac{{\varepsilon }_{0}{\varepsilon }_{r}A}{d}={C}_{0}{\lambda }^{2}={C}_{0}{\left(4\displaystyle \frac{h(t)}{D}+1\right)}^{2},\end{eqnarray} \tag{ 7 }$$

where C₀ is the capacitance of the unstrained membrane [11].

3.1. Algorithm optimization

The normalized forgetting factor, $\bar{W}$, was seen to be the primary driver of amplitude accuracy. As shown in figure 2, as $\bar{W}$ increases the RLS algorithm increasingly underestimates the amplitude of capacitance variation. This is consistent with the physical interpretation of $\bar{W}$ - if $\bar{W}=0.25$, for example, the algorithm must estimate the capacitance at the next time-step using 25% of the previous capacitance cycle, thus smoothing the variation in capacitance over an unacceptably long period of time. Though this analysis uses pure sinusoidal signals, it is equally applicable to more complex variations, with $\bar{W}$ acting similarly to the normalized cut-off frequency of a low-pass filter. Recognizing that high accuracy requires that only a small portion of the capacitance period should be heavily weighted, we see that in order to maintain a low value of $\bar{W}$, the sampling frequency must increase proportionally with the maximum capacitance frequency that must be resolved (equation (6)).

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Additionally, it can be seen from equation (6) that, if the goal is to minimize sampling rate, the lowest feasible forgetting factor should be chosen. However, figure 3 indicates the limitation inherent to setting the forgetting factor. As the forgetting factor is lowered, noise becomes more prominent in the output. This effect is likely caused by amplification of noise in the input signal, due to the numerical differentiation upon which the algorithm relies. As the forgetting factor is lowered, fewer and fewer samples are weighted by the RLS algorithm, increasing the impact of the relatively high-frequency noise. Figure 3 shows that setting the forgetting factor below 0.9 results in a drastic increase in noise in the algorithm output—such values should be avoided. However, the minimum feasible forgetting factor is likely to be strongly influenced by the amplitude and type of noise in the experimental set-up, and should be characterized for each application. For example, a set-up with minimal noise in the voltage and current signals would result in less numerical differentiation error, and could potentially tolerate a lower forgetting factor than a set-up with higher noise levels.

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The above analysis provides guidance on setting f_s and μ for a desired level of accuracy at a particular deformation frequency. However, the probe frequency was not seen to play a significant role in either amplitude or noise in the RLS output, with variations of less than 0.1% in both amplitude and noise seen for the range of f_p tested. Probe frequency, f_p , does, however, have a significant effect on the mean value of the estimated capacitance when considered relative to the sampling frequency, f_s , as shown in figure 4. The capacitance is seen to be increasingly underestimated as f_s /f_p decreases, with errors in excess of 1% when the sampling frequency is less than 35 times the probe frequency. For applications in which the mean capacitance value is critical, as opposed to the frequency content, this frequency ratio should be taken into consideration.

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The optimization demonstrates that the selection of probe frequency, sampling frequency, and forgetting factor are strongly dependent on the level of accuracy desired, as well as the characteristic frequency of deformation that is being sensed. Applications that require only quasi-steady capacitance measurement will require a much lower sampling rate for high accuracy. However, unsteady phenomenon will require increasingly high probe and sampling frequencies for accurate results. The optimization shown here allows these variables to be selected based on the needs of the application at hand.

3.2. Benchtop validation

The RLS-estimated capacitance is compared to the video-derived capacitance in figure 5. In a representative time series including both fast and slow changes in strain, it is seen that the RLS algorithm closely tracks the video-derived capacitance. Indeed, when comparing the entire time-series in figure 5(b), we see a strong correlation, with minimal scatter, which suggests that the capacitance measurements do not suffer from drift over the duration of the experiment, and that good agreement is seen over a wide range of capacitance values measured. Additionally, as seen in figure 5(c), the frequency content of the two measurement methods (offset for clarity) compares very well. This is particularly important in the context of unsteady applications, in which determining the frequency spectra of deformation is the primary goal.

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Additional conclusions can be drawn on the limitations of this technique by plotting the error as a function of normalized capacitance, as shown in figure 5(d), where the output capacitance is normalized by the unstrained value. The error is seen to be maximum at low strains, where the change in capacitance is small, deceasing to less than 1% at higher strains where C/C_o is larger than approximately 2.

An RLS adaptive algorithm allows capacitance to be estimated with good accuracy and at low computational cost. However, its accuracy is strongly dependent on the ratio of the frequencies used for probing and sampling the data relative to the characteristic frequency of deformation (f_p /f_c and f_s /f_c ), as well as the amplitude of the forgetting factor applied during processing. Though the frequencies of deformation considered in this work are higher than those in the original demonstration of this self-sensing algorithm[6], some of the same conclusions are drawn. Higher probe frequencies are seen to improve performance, and similar errors between measured and calculated deformation are observed. This work seeks to provide further guidance on the degree to which each parameters effects algorithm performance.

If deformation is expected to occur over relatively long timescales, the sampling frequency can be lowered without significantly decreasing accuracy, as captured using the variable $\bar{W}$. The presence or relative absence of noise in the experiment may allow for the tuning of the forgetting factor without loss of accuracy, which also affects the value of $\bar{W}$. Similarly, if estimation of unsteady deformation is less important than the long-time-averaged values of the capacitance, then the ratio of sampling to probe frequency should be treated as the primary driver of accuracy.