3.1. Eso-Based Composite Control Scheme
The
q-axis current loop represented by (
7) is an ideal model. An actual PMSM servo system may have unmodeled dynamics and parametric uncertainties. Thus, (
7) should be modified as
where
;
represents the effect of the unmodeled dynamics and uncertainties;
, which is regarded as the lumped disturbance. By defining
and introducing an extended state
, (
10) can be represented as
According to the ADRC strategy, an ESO is applied to estimate the state and lumped disturbance of the control system. Thus, a commonly used linear ESO is derived for system (
11) [
29],
where
and
are the estimates of
and
h, respectively; and
and
are the observer gains. When adopting a bandwidth design method, the observer gains are set as
,
, where
represents the bandwidth of the ESO.
According to the estimated lumped disturbance, the control law is designed as
where
represents the output of the speed controller. Therefore, the
q-axis current loop with ESO-based compensation can be represented as
Figure 1.
Assume that the lumped uncertainties satisfy
. If the ESO is suitably designed, the estimation error of the ESO,
will converge to zero. Then, by combining (
10) and (
13), the
q-axis current loop with the ESO-based feedforward compensation can be approximated as an integrator:
Combining (
14) and (
8), under the slowly-varying disturbance torque, the PMSM speed servo plant can be reduced to a double-integrator model:
Thus, with the ESO-based feedforward compensation, the tracking error of the motor velocity caused by the slowly-varying disturbance torque can be completely eliminated.
A
controller is adopted as the speed controller. The
controller can be represented as
where
and
are the proportional and derivative gains, respectively; and
represents the differential operator with real-number order
,
(0, 2). Taking the deviation between the reference and actual angular velocity as the input, the output of the speed controller can be represented as
where
is the reference velocity.
By combining (
15) and (
17), the transfer function of the closed-loop system with the
controller and the double-integrator plant can be obtained:
where
c represents the term
.
3.3. Optimal Controller Design
Optimal control is to select a permissible control law that allows the control system to achieve the optimal performance, which is often quantized by a loss function [
31]. In this paper, the
controller is designed to ensure the optimal step response performance of the servo system, under the given design specifications (
,
). Suppose that
and
are given. Two equations can be derived:
where
and
represent the phase of the plant model and the
controller, respectively. Thus, if the derivative order
is determined,
and
can be calculated according to (
25) and (
26).
The performance of the
controller is quantified by the following loss function.
where
;
is the controller’s output. Thus, the first term represents the integrated time absolute error (ITAE) of the motor speed, and the second term represents the energy consumption of the control signal.
and
are the weights used to balance the requirements of the tracking performance and energy consumption [
32].
An improved DE algorithm was implemented in
Matlab to determine the derivative order
. At first, the population is initialized by randomly selecting
N values of
to be the individuals. Then, the
controller corresponding to each individual (
) is obtained by solving (
25) and (
26), according to the design specifications (
,
).
Secondly, when the population is generated, several individuals are randomly selected as the target individuals according to a given mutation rate. Taking advantage of the adaptive parameter strategies [
33], an adaptive mutation rate is proposed to enhance the efficiency of the algorithm. The mutation rate
in the
mth iteration for the
ith individual is defined as
where
is the initial mutation rate;
is the upper limit of the iteration count;
is the fitness of the
ith individual;
and
are the largest and smallest fitness values of the current population, respectively; and
is a factor used to keep
larger than zero. The term
is used to tune the basic mutation rate in each iteration: at the beginning of the iteration, it is close to zero and then the basic mutation rate is close to
, suitable for keeping the individual diversity and avoiding the over-expansion of the local optimal individuals. Late in the iteration process, it decreases to close to
and the basic mutation rate is close to
, suitable for protecting the potential global optimal individual from being mutated. In addition, the term
is used to tune the mutation rate of each individual: an individual with less fitness has a larger mutation rate to explore the searching space, whereas one with more fitness has a smaller mutation rate.
Once a target individual is selected, a mutated individual will be generated by adding a difference vector to the target individual:
where
represents the mutated individual,
represents the target individual,
and
are two randomly selected individuals and
k is a scaler coefficient. In addition, the
controllers corresponding to the mutated individuals are also obtained.
Thirdly, the step response simulations are implemented on the
Simulink platform, which was constructed according to the closed-loop system shown in
Figure 2. Customized constraints can be introduced into the simulation modules. In this study, the output saturation was introduced according to the amplitude limitation of the actuator. The
controller corresponding to each individual was used as the speed controller. The fitness of each individual is defined as the reciprocal of the loss function of the control system, i.e.,
.
Finally, a comparison is performed between the mutated and target individuals: the one with greater fitness will be selected in the population, while the other one will be abandoned. The termination condition of the optimization is determined as follows: if the variation of the population’s average fitness in the latest 10 iterations is smaller than a threshold , or the iteration count reaches the upper limit, the optimization will be ended.
According to (
24) and (
26), the parameters of the PMSM and
q-axis current controller have no effect on the derivative order of the optimal
controller. Therefore, a universal look-up table of the derivative order with respect to different design specifications can be established to simplify the controller design process. The look-up table is established within the ranges of
and
. The range of
was set as 30 to 80 rad/s. In addition, the range of
was set as
to
, in accordance with the general selection of the phase margin in engineering applications [
26,
34]. According to the ranges of
and
, several values were selected uniformly, obtaining the sequences of
(30 rad/s, 35 rad/s, …, 80 rad/s) and
(30
, 35
, …, 60
). Therefore, the design specification pairs (
,
) could be constructed by combining the selected values of
and
in sequence. Taking the double-integrator model (
24) as the controlled plant, the
controller corresponding to each design specification pair was obtained by applying the DE algorithm. The parameters of the DE algorithm were selected as follows:
N = 20,
= 100,
= 0.1,
k = 0.5,
= 0.0001. The look-up table of the derivative order
for different design specifications is presented in
Table 1.
If the design specifications are located within the grids of the look-up table, the derivative order can be estimated applying a commonly used bilinear interpolation method [
35]. As shown in
Figure 3, the derivative order of the target point (
,
) is estimated based on the four nearest rectangular points surrounding the target point, using the following formula.
where
The effectiveness of the look-up table was verified by the estimation error tests. Several
controllers were designed as the test samples using the DE algorithm, based on the design specifications constructed by the elements in the sequences of
and
: (35 rad/s, 45rad/s, …, 75rad/s) and (35
, 45
, 55
). An estimation error was defined as (
32) to quantify the deviation between the derivative order obtained using the optimization method and that estimated by the look-up table:
where
and
represent the derivative orders obtained using the DE algorithm and the look-up table, respectively. The estimation errors corresponding to different design specifications are listed in
Table 2. It can be observed that all the estimation errors are smaller than 0.1%. Thus, the derivative orders estimated by the look-up table can be regarded as the optimal values.
Remark 2. In actual applications, a PMSM speed servo model can be converted into a double-integrator model by the ESO-based scheme. Then, the derivative order of the controller can be estimated by the universal look-up table. The remaining parameters of the controller can be calculated according to (25) and (26). In this way, the controller of the servo system can be designed without optimization, which may be suitable for actual applications.