This section is divided into three subsections: the first reports and analyzes the estimated parameters of the physical model; in the second, the relative errors of the estimated parameters for different error sources are computed and analyzed; and in the last section, the aging of the valve is analyzed over different production cycles.
4.1. Physical Model
Table 1 presents the results of the parameter estimation problem that were defined in Equations (
20) and (
22) in order to obtain the input–output system model described in Equation (
17). We carried out three different parameter estimations. The first one was given by solving the standard least squares problem defined in Equation (
20) with Equation (
21). This provides
which estimated
,
, and
in Equation (
18). Then, Equation (
17) becomes
with
where
,
, and
are found in Equation (
19). In the second case,
is obtained by solving Equation (
20) with Equation (
21), as in the first case. The difference between the first and second cases is that, for the second case, we considered certain non-linear terms as having
where
and
The third and last case correspond to solving Equation (
22), which gave
. Then, Equation (
17) becomes
where
and
and
are not considered in Equation (
32) since their absolute values were
when solving Equations (
22) and (
23).
, which is the parameter for the cross-term in
and
, are ignored because during operation,
when
and
when
; see
Figure 3b.
is ignored because
is highly correlated with
and
, and the MSE for the training and test data shown in
Table 1 do not change when it is considered. We did not consider higher-order terms in Equations (
14) and (
15) because their absolute values were found to be
when solving the mathematical optimization problem defined in Equations (
22) and (
23).
In
Table 1, the estimated parameter values obtained for the parameters defined in Equations (
26), (
28), and (
31) are found. For
, we have no agreement between
and the physics of the system defined in Equation (
15). From a systems theory perspective, this parameter value implies that the poles of (
17) were
and
, where the last one relates to a pole with
Hz, which is not possible when looking at
Figure 3a,b. The
and
signs are in agreement with the physics; however, when assuming that both pilot valves (I) and (II) are equal to each other, there seems to be an anomaly in the valve. After a discussion with maintenance engineers at the manufacturing plant, a manual calibration of the springs in the pilot valves was performed after the maintenance of the valve. In
, the values of
and
are different in sign than the signs defined in Equation (
15). Considering that the non-linear terms of the input–output model in Equation (
28) increased the conditioning number of
by two orders of magnitude, this could be the reason we obtained these parameter values because of the poor excitation of the signals (see
Figure 3a,b).
gives values in agreement with the signs defined in (
15) regarding the physics of the valve. As for
, there are differences between the second and third elements of the estimated parameter vector.
Figure 4a shows a comparison between
and
, and this is achieved using validation data, i.e., signals from a different production time than the ones used for estimating
.
was computed by using Equation (
31), where
, which is described in Equation (
32), had the parameter values shown in the last column of
Table 1. This case corresponds to a non-linear input-output model with parameters estimated by solving the convex optimization problem defined in Equations (
22) and (
23). The signals
,
, and
x in
Figure 4 show that the valve operated under the same conditions as for the signals that were used for estimating the parameters (see
Figure 3a,b). We can see that
and
overlapped, and that these results are seen in the error signal (see
Figure 4c). There is a small offset between both signals when the main valve’s spindle was at rest, i.e., between
and
and between
and
(
s). In addition, the error signal shows a process that is close to white noise, with a small offset when
. The error signal seems to have a component with a 0.09 s period, i.e., three samples. The error signal agrees with the MSE values shown in
Table 1. The difference between the error signal when
and
agrees with the difference in the parameter values
and
, as for the difference between
, which is negligible, and
.
In addition to analyzing the modeling results with cross-validation and inspecting the poles, we performed a dimensional analysis and checked the size of the parameter values.
Table 2 shows the units of some of the parameters in Equation (
14). These parameter units can be derived by using Equation (
15), and by knowing that
and
have the units
,
and
have the units
;
,
, and
have the units
; and
and
have the units
. The parameter elements of
in Equation (
18) are dimensionless. The parameter elements of
,
, and
in Equation (
26), Equation (
28), and Equation (
31), respectively, are also dimensionless. This finding agrees with the normalized units of the output
x, as well as the inputs
and
.
Table 3 shows the values of the valve related to its mechanical, hydraulic, electronic, and physical dimension characteristics. These values were used to compute the intervals for the estimated parameter vector
defined in Equation (
31). The interval for the mass of the spindle
M was given by maintenance engineers. The damping coefficient
B was obtained from the knowledge of the poles of the system, as discussed in
Section 4.1. The physical
and
dimensions were obtained from the manufacturer drawings of similar valves.
is related to the flow through the pilot valves, and it was obtained from the manufacturer’s flow graphs.
and
were selected based on the measured signals (see
Figure 3).
Table 4 shows the computed intervals from the values and intervals in
Table 3, as well as in Equation (
4) to Equation (
18). We can see that the obtained parameter values for
are within the intervals. In
Table 3, the intervals correspond up to an order of magnitude; furthermore, the interval in
Table 3 is up to three orders of magnitude, which is due the products used to compute the elements of
(see Equation (
9), Equation (
10), Equation (
12), and Equation (
15)). The Reynolds number could be calculated as in Equation (
16) in [
21] for the device under study, which depends on the volumetric flow, the orifice area, the spool diameter, and the kinematic viscosity. The spindle diameter is known from measurements, while the volumetric flow rate and the orifice area of the valve are known from the manufacturer spreadsheet. Different values of the kinematic viscosity of water for different temperatures are considered. The Reynolds number was calculated considering
(see
Figure 3a). The computed interval for the Reynolds number of the main valve, where the flow goes from (VIII) to (IX) (see
Figure 1), is from
to
. This Reynolds number range indicates that the flow through the valve is likely to be turbulent since it is greater than 4000. The presence of bends and valves, among others, promotes turbulent flow in the the hydraulic circuit of the hydraulic systems [
11].
4.2. Modeling Errors
The relative errors in the elements of
were estimated from the known error sources. The different sources are listed in the second column of
Table 5. First, we explain how the relative errors were estimated for each error source. Then, we describe the relative errors obtained from each error source.
The rows (a), (b), and (c) give the relative errors from the offset in the measured data in , , and x. The offset changes between the cycles. The errors in were calculated for a cycle with a negligible offset. Positive and negative offset values of were added to , and we determined the parameter values for . A straight line was fitted for each versus . The estimated errors were calculated with , which was the highest offset in the measured signal . The errors due to the offset and were calculated in the same way.
The relative errors from the noise in the measured data in , , and x are given in rows (d), (e), and (f). A noise signal with a variance of was added to , and we estimated the parameter values for . A straight line was fitted for each versus . A was used to calculate , which was also the highest observed in the measured signals. The errors due to noise in and x were calculated in the same way.
In order to estimate the relative errors from the Euler forward method, as shown in row (g) in
Table 5, a continuous and discrete time model of the system were compared. The continuous time model uses Equations (
14) and (
15) with input signals
and
as a combination of ramp and step functions similar to the measured
and
, respectively (see
Figure 4b). An analytical output signal was derived by solving the differential Equation (
14) with the conventional Laplace transform method. A discrete-time output signal was calculated after the discretization of the differential equation by the Euler forward method (see (
14)–(
17)). When comparing the discrete-time to the analytical output, an error signal was obtained. The variance of that signal was determined and used to generate different realizations of the error signal. These realizations were added to the measured output signal
x, and the parameters
for
were identified.
was determined as the difference from the case with no added error signals.
The rows (h), (i), and (j) provide the relative errors caused by the approximations of Equations (
9), (
10), and (
12). The remainder term of a Taylor polynomial of order 2 was estimated for each approximation and included in Equation (
14). We used
Pa based on the ratio of the areas
and
, as shown in
Figure 2, to estimate the remainder term of Equations (
9) and (
10), as well as to calculate
for
(which is given in row (h)).
were used to estimate the remainder term of Equation (
12), and this was based on the signals in
Figure 4b and the obtained relative errors that are given in row (i). The relative errors from the product of the remainder term of each approximation included in Equation (
14) are given in row (j).
Row (k) gives the relative errors from the approximation that the fluid was incompressible in Equation (
4). The model output obtained from Equation (
1) to Equation (
19) with that of a Matlab/Simulink model were compared, which modeled water as compressible (details are given in
Supplementary Materials). An error signal was calculated as the difference in the model outputs. The mean and variance of that signal were estimated and used to generate different realizations that were added to the measured output signal
x. The parameters
for
were estimated.
were determined as the difference from the case with no added error signal.
Table 5 provides the relative errors. The relative errors in
,
, and
are bigger than that of
from offsets in
,
, and
x (see rows (a), (b), and (c) in
Table 5). These relative errors could be reduced by decreasing the offset levels in the calibration process. In rows (d), (e), and (f), the relative errors of
and
are bigger than those of
and
. The sensitivity of the non-linear term and the poor data of
caused these relative errors. Sensors with lower additive noise could be used to reduce these relative errors. Regarding the discretization of Equation (
14) (see row (g)), large relative errors for
,
, and
were obtained when compared with
. These relative errors came from the fast changes in
x relative to the sampling time (0.03 s) at the initial and final times of the steps in the input signals. One could reduce the sampling time or use more advanced discretization methods to minimize these relative errors. The relative errors of
and
are found to be bigger than those of the other elements in
for the approximations of Equations (
9), (
10), and (
12) (see rows (h), (i), and (j)). The approximations of Equations (
9) and (
10) could be avoided by having a sensor for
, as well as by estimating
(see Equations (
8), (
9), and (
10)). One cannot avoid the approximation of Equation (
12) by using more sensors, but one could collect further informative data to estimate more of the parameters of the Taylor polynomial that was used in the approximation (see
and
in
Figure 4b). The largest relative errors of the elements in
are due to the incompressibility approximation. The assumption of water being incompressible could be avoided by having a sensor for
and estimating
and
. The final row provides the total relative error of the parameters
for
. They were calculated as the sum of the relative errors of
from rows (a) to (k). The largest relative error is
, which is the parameter that described the nonlinear effect of
.
4.3. Aging
This section evaluates a potential use of the system model described in Equation (
31). The elements of
in Equation (
31) were estimated for different production cycles (see Equation (
22)). The elements of the estimated parameter vectors are denoted as
, where
and
is the total number of cycles.
A linear trend in
for the cycles
is observed (see
Figure 5). We fitted the linear regression model
where
and
are the least square estimators, and
is the estimated
at cycle
c. The confidence intervals of
were calculated using Equation (13.17) in [
39], and this was achieved by assuming the residuals of Equation (
34) were white Gaussian noise. In addition, we considered a confidence level of 95% by estimating
using Equation (13.16) in [
39]. The estimated relative error of 6.5% in
was similar to but slightly smaller than the 95% intervals shown in
Figure 5. Thus, the scattering around the trend line was explained by the different error sources summarized in
Table 5. The trend line reveals the aging of the valve; see
in
Figure 5.
Figure 6 shows the values of the other elements of
for
. We can see that
was constant over the cycles, and that it stays at the limits of the constraints defined in
Section 4.1. The scattering in
and
shows an increasing trend with the number of cycles. The estimated parameter values of
are bigger in magnitude than
for
. The total estimated errors of
and
, as given in
Table 5, describe most of the scattering of the estimated parameters
and
, respectively.