3.1 Proposed algorithm

In this section, we explain the realization of our proposed algorithm and show how it works in stabilizing the outputs without any filtering. Setting thresholds is very crucial in any control systems. In embedded control systems, microcontrollers are programmed to set the threshold for the operation of actuators⁽²³⁾. We especially take the example in Bang-bang or Hysteresis control type (see Fig. 1) systems.

In general cases, irrespective of the level of noise, a single threshold will ensure smooth operation of the actuators. Suppose in a normal environment, readings of a temperature sensor triggers a motor when a specified threshold is reached. It turns on if the signal crosses above the threshold and turns off if the signal goes below the threshold. However, if the signal stays around the threshold, it will bring instability in the operation of the motor. The frequent change in the state eventually will bring mechanical failure in the motor. Many practical methods do exist to prevent such events from occurring. Some of them include reading the sensor data only at certain time intervals or introducing delays in reading the signals. Other methods include tracking the threshold where signals will be ignored for a specified time once the threshold is reached. These type of methods are useful especially in environmental monitoring and control where continuous data is not of much importance. However, in other applications where the instantaneous response is required per observation, such methods may not be helpful.

Algorithm 1 Threshold Tuning

1: Begin: Determine threshold;

2: Read n samples from the sensor at t=0 to t=n;

3: Compute σ from n samples;

4: Assign a Boolean variable Tracker to FALSE

5: Start Loop

#Condition 1

if (current sensor reading $>$ threshold)

do Actuator → ON

do Tracker → TRUE

#Condition 2

else if ((current sensor reading $>$ (threshold –3σ)) AND (Tracker → TRUE))

do Actuator → ON

else do Actuator OFF

do Tracker FALSE

6: End

Here, Algorithm 1 shows the working procedure of the Threshold tuning algorithm. The working concept of the algorithm is also shown in flowchart in the Fig. 2. In step 1, a threshold is determined and assigned to the variable threshold. In step 2 and 3, n number of samples are read from the sensor to calculate the standard deviation σ. In step 4, a Boolean variable Tracker is assigned a FALSE or 0 value initially. This Boolean variable will be assigned TRUE value only if the signal reaches the threshold. This will help in determining whether the signal has crossed the threshold or not. In step 5, the system loop begins, and Condition 1 is checked first. If the sensor reading reaches the threshold, the actuator will be triggered ON, and the Tracker variable will be assigned TRUE simultaneously. Activation of Condition 1 leads to the activation of Condition 2 immediately where the previous status of the actuator will be maintained. Once the signal is below 3σ range, the actuator is triggered OFF, and the Tracker variable will be assigned FALSE. This creates a stable zone for the signal from the original threshold to the 3σ range.

3.2 Dynamic Threshold tuning

Readings of sensor data in single sensing and multi- sensing environment often different. The measurement data of the multi-sensing environment has more noise and disturbances than the single sensing environment⁽¹⁷⁾. Even the sensors of the same type manufactured by same vendor company drift slightly when actual measurements are done⁽⁹⁾. To address these problems we added a feature to the Algorithm 1, i.e. dynamically tuning the threshold every loop.

Algorithm 2 represents dynamic threshold tuning algorithm. The only difference between this algorithm and Algorithm 1 is, it computes the standard deviation every time a new sample is taken. A new 3σ range is set every time the loop runs.

Algorithm 2 Dynamic threshold Tuning

1: Begin: Determine threshold;

2: Assign a Boolean variable Tracker to False

3: Start Loop

Read current sample from the sensor and Store;

Compute and update σ in every loop;

if (current sensor reading $>$ threshold)

do Actuator → ON

do Tracker → TRUE

else if (current sensor reading $>$ (threshold – 3σ)) AND (Tracker → TRUE)

do Actuator → ON

else do Actuator → OFF

do Tracker → FALSE

4: End

4.1 Actuator output with known methods

In this section, we compare our method with other known methods: Exponential smoothing and Kalman filter, in the actuator stability criteria. Exponential smoothing method smooths the noisy readings from the sensor and acts as a low-pass filter. Equation (6) and (7) shows the mathematical expression of the Exponential smoothing method:

where represents sensor data at time t=0, represents the best estimate of the data and α is the smoothing factor. The value of α close to zero will have a greater smoothing effect but less responsive to the recent changes, whereas value closer to 1 will have lower smoothing effect but more responsive to the recent changes. The value of the estimator s will be updated every time the sensor makes a new reading.

Figure. 3 shows the application of the Exponential smoothing algorithm at α = 0. For the experimental purpose, we used a microcontroller board based on ATmega32u4 providing 16 MHz clock speed and 10 bit ADC. The algorithm successfully managed to smooth the signal and filter the high frequencies. However, it was not able to reduce the frequent changes in the state of the output. Another common yet complex algorithm is Kalman filter which has a wide range of application including sensor signal processing. It is based on a state-space approach where it optimally estimates the states of the model containing statistical noise with known parameters⁽²⁴⁾.

The state dynamics and output equations are described by equation (8) and (9) respectively. Here, x, y, u, w, v, F, G, H represents state vector, output vector, input vector, process noise vector, measurement noise vector, system matrix-state, system matrix-input and observation matrix respectively. These equations can further be divided into two groups:

a. Time update equations

b. Measurement update equations

Here, equation (10) and (11) represent the time update equations and (12), (13) and (14) represent the measurement update equations. Q, R, K and Z represents process noise covariance, measurement noise covariance, Kalman gain and measured value respectively. Figure. 4 shows the application of Kalman filter algorithm. For our experiment in sensor signal processing the parameters of Kalman filter are assumed as follows:

i. x is the sensor output voltage,

ii.F=1; meaning the sensor reading does not vary instantaneously,

iii. G = 0; meaning there is no control input and

iv. H=1; meaning the output voltage is only observable.

v. R is the variance of the sensor reading

Estimating Q and R is very challenging and often require complex calculations⁽¹⁷⁾. We arbitrarily assigned a very small value of 1×10^-4 to Q, whereas R was the variance of the sensor reading calculated after taking 50 samples. It successfully managed to get rid of the noise and estimate the signal. However, it could not reduce the frequent changes in the state of the output satisfactorily. It is because filtered signals are not 100% smooth and they also hover around the threshold just like the original signal.

4.2 Actuator output with the proposed method

The proposed method controlled the threshold based on the measurement of the probability of the random Gaussian variable. We empirically found that the observed series of sensor measurements indeed converged to Gaussian distribution when the sufficiently large number of samples were taken. We took 2000 sample readings of a light sensor (CdS photo sensor) in an illuminated room of 500 lux. With the total power supply of 5 V, the µ of sensor readings was 1.33 V with the σ being 0.12. The maximum and minimum value observed was 1.71V and 0.96V respectively. Operating frequency of the sensor was observed to be approximately 120 Hz.

Figure. 5 shows the extraction of the sensor data using an oscilloscope. Figure. 6 shows sensor readings in using an embedded computer (ATmega32u4) where 1σ, 2σ and 3σ range cover approximately 68%, 95% and 100% of the data. Figure. 7 shows PDF of the collected sensor data fitted to Gaussian distribution.

As shown in Figure. 7 we manipulated the environment for the experimental purpose to demonstrate the usefulness of the algorithm. In a normal environment, the output of the system is completely stable irrespective of any level of noise as shown in Figure. 8. Once the signal is at the threshold the actuator status is ON, and it keeps on maintaining the same status till the signal is at the 3σ range. Once it goes below the 3σ range, the actuator status is OFF. Now after this, even if the signal crosses above 3σ range its status would not change to ON. This brings great stability to 99.7% to the output of a control system. In addition to that, unexpected events can be avoided, life of the actuators can be prolonged, and maintenance cost can be reduced as well.

4.3 Discussions

The average stability using Exponential smoothing algorithm, Kalman filter and the proposed algorithm was approximately 50%, 70% and 99.7% respectively. Our initial hypothesis before beginning the experiment was all the data of the readings of sensors in an uncontrolled environment would converge to a Gaussian distribution. However, not all sensors exactly converge to the Gaussian distribution and the shape of the distribution of those sensors are somehow complex. A Gaussian distribution is merely an approximation of such complex probability distributions. Also, the variance of a filtered signal is much less than a raw sensor signal. The algorithm proposed in this paper can be applied to a filtered signal. The ‘3σ range’ can be made much narrower with the cost of increase in computation time.

Robust quantitative approach can also be adopted for outlier detection if it is of high importance. The 3σ rule may break down at the contamination level greater than 10%. Other statistical outlier detection rules like the Hampel identifier can be used which breaks down at the contamination level greater than 50%⁽²⁵⁾.

The threshold level µ ± 3σ can be changed to ± 3S and S can be expressed as

where x_i is the sample data, $\stackrel{v}{x}$ is the sample median and the whole numerator part of the equation (16) is the median absolute deviation (MAD) i.e. median of the absolute deviations from the data’s median.