3.1. Low-Power PET Model of UV LEDs
Since the core of a UV LED is a P-N junction [
31], its V-I characteristics can be approximately expressed by the Shockley equation:
where
i is the forward current,
u is the forward voltage,
is the reverse saturation current,
q is the amount of electronic charge,
k is the Boltzmann constant,
T is the thermodynamic temperature, and
n is a constant (ideal factor). From Equation (
1), the relationship between
u and
i can be obtained, namely:
UV LEDs are very sensitive to voltage changes, so they are usually driven by currents. The relationship between electrical power
P and
i can be obtained from Equation (
2). However, since
is very small,
P can be approximately expressed as follows:
When
i is small (i.e., low power operation in CMSs), the logarithmic term of
i in Equation (
3) can be expanded by the Taylor series, and the approximate expression of the first two terms can be obtained as follows:
Among them, the third and second terms of
i are much smaller than the first term, so
P can be further approximated as a linear function of
i, and a small constant
b is added for correction; that is:
Assuming that
is the optical power, it has the following relationship with
P:
where
is the luminous efficiency that represents the ratio of electrical power converted into light output. Previous studies have shown that it is mainly affected by thermal characteristics and will decrease linearly with a constant coefficient
as the junction temperature
increases [
17,
18], namely:
Since UV LEDs are packaged with TO-39,
is not easy to measure, so
cannot be obtained directly by Equation (
7). As the case temperature
will be monitored by temperature sensors in CMSs,
can be replaced by measurable
.
In the actual working process of UV LEDs, a large part of
P is used for heating in addition to luminescence, resulting in an increase of
. The relationship between the thermal power
and
P is as follows:
where
is the thermal efficiency, and
if other small losses are ignored.
To maintain the proper operating temperature of UV LEDs, it is necessary to use a heat sink to enclose the device in an actual CMS, as shown in
Figure 2a. Moreover, to facilitate the analysis, we abstracted a simplified model from this real thermal structure as shown in
Figure 2b, where
is the ambient temperature,
and
, respectively, represent the junction-to-case thermal resistance and the thermal resistance of the heat sink, while other thermal conductors are ignored due to their small thermal resistance.
Thus,
can be expressed as follows:
We bring Equations (
9) into Equation (
7):
We combine Equations (
6), (
8), and (
10):
Since CMSs only require uW-level output, the electrical power of UV LEDs will be in the order of mW, so the quadratic term of
P in Equation (
11) is much smaller than the first term, and it can be simplified as follows:
where
c is a constant to correct the error caused by neglecting the quadratic term in Equation (
11). In addition, due to the difference between the calibration operating point and the actual operating point, a certain constant error may be introduced; thus, a constant correction term
d may be added to improve the accuracy of modeling.
Equation (
12) provides an approximate relationship between optical power and two measurable independent variables, namely the driving current and case temperature. Moreover, only two linear curves are needed for modeling in practical applications, which is very convenient for UV LEDs to be calibrated periodically to correct the accurate discharge without interrupting the operation of CMSs, based only on the data collected by the CMS in real time. Moreover, when an independent variable is fixed (typically the case temperature is kept constant), it becomes a traditional linear model.
3.2. Design of Fuzzy Adaptive PID Controller
Effective optical power control of UV LEDs is essential for stable and reliable charge management, especially in the presence of uncertainties caused by thermal effects and attenuation effects, to ensure accurate discharge and low noise levels [
32]. Moreover, fuzzy adaptive PID control can meet these requirements well. However, the dynamic performance and steady-state performance of a control system are essentially irreconcilable trade-offs [
33]. To strike a balance between them, we propose an optical power control system using the FA_PID algorithm with a switch, which is shown in
Figure 3.
This system uses a two-dimensional fuzzy controller, which converts the optical power control error e and error rate into fuzzy quantities and generates the fuzzy quantities of controller parameter changes suitable for the current system state adjustment in the inference machine based on the rules in the rule base. Then, these fuzzy quantities are transformed into accurate quantities through the defuzzifier by the centroid method. The quantization factors, , , and , are then used to quantify them into , , and , respectively, which are sent to the PID controller. By combining these values with the initial parameters , , and , the self-tuning of the PID controller is achieved.
The addition of a fuzzy controller is identified as the main factor responsible for reducing the dynamic performance. To balance the dynamic performance and steady-state performance, a switch with threshold
is introduced to coordinate the working state of the controller at different stages. During the stable stage (
), the fuzzy controller sends its output to the PID controller through the switch, significantly improving the steady-state performance. In the rising stage (
), the switch disconnects the fuzzy controller, and the PID controller generates a strong control output to meet the dynamic performance requirements. Moreover, by adjusting
, the proposed system can achieve comprehensive optimization of the dynamic performance and steady-state performance. Therefore, the above controller can be described as follows:
where
,
, and
.
The membership function design is crucial for the above controller design, while the sharpness of the membership function determines the resolution and sensitivity of the controller [
34]. In order to achieve better control, it is recommended to use low-resolution fuzzy sets in areas with larger errors and high-resolution fuzzy sets in areas with smaller errors. Based on the balance between control effect and computational complexity, we divided all input and output variables into seven fuzzy subsets ([NB, NM, NS, ZO, PS, PM, PB]), and used the same membership function shown in
Figure 4.
The quality of a fuzzy controller depends largely on the quality of the control rules. Based on the operational experience of CMSs and the existing PID controller parameter tuning experience [
35,
36], the following fuzzy control rules have been established:
When e is large, in order to eliminate the error as soon as possible and prevent the differential supersaturation that may be caused by the excessive moment of e, takes a large value, and and take a small value or zero.
When e is small, in order to further eliminate the error and prevent the oscillation caused by excessive overshoot, should be reduced, should be small, and should be moderate to ensure the response speed of the system.
When e is very small, in order to eliminate the static error and avoid oscillation near the set value, continues to decrease, remains unchanged or slightly larger, and can be slightly larger.
The magnitude of indicates the rate of error changes. The larger the , the smaller the , the larger the , and vice versa.
When e and have the same sign, the controlled variable deviates from the given value direction, and the control action should be strengthened to make the error change in the direction of reduction. Thus, a larger and a smaller should be taken, and should not be too large.
When e and have different signs, the controlled variable changes in the direction close to the given value, so when e is large, take a smaller or zero to accelerate the dynamic process.
Based on the above rules and fuzzy set division, a rule base consisting of 49 rules was designed (presented in
Table 1,
Table 2 and
Table 3) by using the conditional statements “if
E and
then
,
,
”.
The input–output relationship surfaces of the fuzzy controller designed by the above steps are shown in
Figure 5. The different colors reflect the magnitude of the values, with warmer colors indicating higher values and cooler colors indicating lower values.