(201c) Sensor Maintenance Optimization Based On a Prognostic Model

Introduction

Sensors based on an Ion-Selective Electrode (ISE) are
characterized by an intercept (α)
and slope (β) which
link a transmitter output voltage (or electrical current) to the
variable of interest. In this work, we write:

with z the variable of interest and y the measured
voltage.

It is typical that the sensor parameters (α
and β) change over
time, both due to reversible (e.g., fouling) and irreversible (e.g.,
buffer leaking) effects. It is therefore recommended to recalibrate
such sensors on a regular basis [1-3]. Unfortunately, this can be a
time-consuming task as limited efforts have been made to optimize the
effort in maintenance and calibration of ISE sensors
[4]. In this work, a state-space model is formulated which models the
evolution of a sensor's parameters as a continuous
time-varying process. Given this process model, reference
measurements are used to update the parameter estimates by means of
the (linear) Kalman filter. The resulting parameter estimates and the
associated variance-covariance matrices allow prediction of future
parameter estimates and covariances. Based on these predictions,
future reference measurements can be scheduled as soon as the
expected confidence intervals are considered too large.
Methodology
Simulated sensor

A pH sensor is simulated as follows. The intercept
(α) and slope (β)
are joined in a single state vector, x:

(1)

Given this definition, the evolution in time of this
vector is given in the form of a continuous-time process:

(2)

with u(t) an external input, A and B defining the
linear, deterministic parts of the model and v representing the
stochastic input disturbances. This is a general representation of a
Wiener process.

The sensor's parameters are however considered to
evolve independently of their own values and independent of any
deterministic input so that one can simplify this model as follows:

(3)

The resulting model is a special case of the general
Wiener process, namely that of a multivariate Brownian motion. The
input disturbances are considered to be zero mean and to have a
diagonal covariance matrix, meaning that intercept and slope evolve
independently from each other. More concretely, one writes:

(4)

The actual values for x are of course not available
and can only indirectly be estimated from the measured voltages. The
value for the voltage measurement depends on the pH (z), the sensor
parameters (x(t)) and a measurement error (white noise):

(5)

It is assumed that the considered measurements are
taken with a sensor placed in a buffer solution with known pH value
and following steady-state. More concretely, pH buffer solutions are
considered available at 4, 7 and 10, and measurements are taken for
each buffer solution in a very short interval. These intervals are
considered sufficiently small so that one can consider the
measurements as simultaneous. In this case, the more specific
measurement equation is the following:

(6)

The resulting vector y(t) contains the noisy voltage
measurements at buffer values 10, 7 and 4. These are further referred
to as reference measurements. The vector w is a white noise vector,
i.e., zero-mean and with a diagonal variance-covariance matrix:

(7)

The initial values for alpha and beta are set to 4
(intercept 4V at pH=0) and 16/14=1.14 (slope of 16V units over 14 pH
units). This completely specifies the executed sensor model
simulations.
Filtering

The Kalman filter [5] is used to obtain on-line
estimates of the sensor parameters. To this end, prediction
equations are derived by integration of Eq. 3 from a previous time
instant, t(k-1), to the current, t(k) [Harvey]:

(8)

where μ_x(k|k)
and Σ_x(k|k)
are the current state estimate and associated covariance matrix and
μ_x(k+1|k)
and Σ_x(k+1|k)
are the predicted (future) state estimate and associated covariance
matrix. The above equations signify that the expected future
state vector is equal to the current state estimate and the
covariance matrix of the estimates increases linearly with time.

When reference measurements, y(k+1), are present at a
given time, t(k+1), they are used to update the state estimates. For
this, the so called update equations are:

(9)

The update equations are executed whenever a set of
measurements for the buffers is available. In contrast, the
prediction equations can be executed at any time.
Prediction

At any point in time, the above prediction equations
can be used to estimate the future values for the sensor parameters
and associated covariance matrix. In addition, one can also estimate
the expected voltage value and associated variance for any pH value
of interest (z). To this end, one writes:

(10)
Scheduling of reference measurements
Default scheduling

In a first simulation study, reference measurements
are taken at predefined time intervals, namely on a weekly repeated
schedule in which measurements are taken on the first (1), second
(2), third (3) and fifth (5) day. This leads to time intervals of 1,
2 and 3 days between consecutive reference measurements. This is
continued for 10 weeks (70 days). After this time, the sensors
behaviour is simulated for another 4 weeks (until day 98).
Optimized scheduling

In a second simulation study, the lowest predicted
variances for the measurements observed for pH values 5 and 8
(Sigma_y,5, Sigma_y,8) in the first simulation study are used as a
reference. A new measurement is taken for the buffer solutions (pH =
4, 7 10) as soon as the predicted variances are higher than 1.1 times
these reference values.
Results

Due to space
limitation, only results for the optimized scheduling are shown here.
As discussed above, reference measurements are
taken only when the variance of the expected voltage at pH 5 or pH 8
is 1.1 times higher than the minimal value obtained in the first
simulation. Figure 1 shows the simulated parameters as well as the
reference measurements taken with this schedule. In this case, the
total number of reference measurements is 36 while in the default
scheduling simulation (default scheduling) it is 40 (not shown).

Figure
1: Optimized scheduling - Simulated data. The vertical line indicates
the end of the measurement collection. Top: Simulated sensor
parameters. Bottom: Reference voltage measurements for three buffer
solutions. The frequency of measurements is relatively high at the
beginning and is reduced and constant as from the 10^th day
until the last reference measurement (day 70).

Figure 2 and 3 display the Kalman filtering results. These are
similar to the default scheduling simulation except that the
variances in the parameter estimates are reduced faster at the
beginning of the simulated period. This is because the optimized
scheduling results in a higher sampling frequency at the beginning of
the simulated period. After the parameter variances are below the set
threshold, this frequency is reduced.

Figure
2: Optimized scheduling - True sensor parameters (.), estimates (--)
and 3-sigma confidence intervals (-). The vertical line indicates the
end of the measurement collection. The parameter estimates remain
constant while the confidence intervals widen after this time.

Figure
3: Optimized scheduling – Variances of the parameter
estimates. The vertical line indicates the end of the measurement
collection. The uncertainty about the slope (beta) is higher than the
uncertainty about the intercept (alpha). As in the default scheduling
case, the variances are reduced at each reference measurement and
increase linearly over time between reference measurements. The
reduction in variance at the beginning of the simulated period is
faster however compared to the default scheduling case (see Figure
3).
Discussion

In this work, a prognostic state-space model is
proposed to model the time-varying nature of a typical ISE sensor. By
means of such model, reference measurements can be used to estimate
the values of the sensor's characterizing parameters as well as the
predict the future uncertainty in its measurements. Based on such
uncertainty evaluation, it is possible to schedule new reference
measurements so to maintain uncertainty (variance) below a pre-set
level. This approach is tested in a simulated study which suggests
that a reduction of maintenance effort can effectively be bargained
against accuracy. Furthermore, the proposed scheduling effectively
results in higher maintenance efforts when the variances of parameter
estimates are higher.
References

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