3.3.1. Selection of Fitness Function
The controller optimization of PMSM should follow the principle of smaller overshoot and stronger robustness. In order to ensure the small overshoot of the system, the fitness function should include the error part of the speed, and in order to ensure the strong robustness, the fitness function should increase the error part of the observed value and the actual value. i.e.,:
In the actual working process of PMSM, the variation range of inertia is small and relatively slow, while the change of actual load torque is more intense and frequent, which has greater impact on the system. Therefore, this paper only optimizes the system speed tracking and torque observation effect, then (40) is rewritten as follows:
According to (41), the optimization problem of AIBC with differential term for PMSM is a typical multi-objective optimization problem. In order to reduce the difficulty of optimization, according to the above methods, this multi-objective optimization problem is transformed into a single-objective optimization problem. The selection of and are discussed below:
(1) Selection of
PMSM speed control problem is essentially a tracking problem, that is, to control the motor speed, so that it can track the input reference speed signal. The performance evaluation indexes of the system are static error, maximum overshoot and adjustment time. PMSM control system should reduce the steady-state error and the adjustment time of the system on the premise of ensuring a small overshoot. However, the measurement and determination of these three performance indexes are difficult. In practical engineering applications, the following error integral functions are often selected as the indexes to measure the control system:
(a) integral squared error:
(b) integral squared error and time:
(c) integral squared error and squared time:
(d) integral absolute error:
(e) integral absolute error and time:
(f) integral absolute error and squared time:
The above integral formulas have different emphases as performance indexes. The 6 integral formulas can be basically divided into two types: the first type is the integral form of the product of error and time, such as (43), (44), (46) and (47); the second type is the integral form of error only, such as (42) and (45). The first kind of integral form contains the form of the product of error and time, which makes the initial error have little influence on the integral result, but with the passage of time, the error has more and more influence on the result. Therefore, this kind of integral formula has better effect on reducing transition time and steady-state error. For the second kind of integral form, the effect of error on the output results does not change with time, so this kind of integral formula is more effective in shortening the rising time, but less effective in reducing the adjustment process.
In order to study the influence of the above performance indexes on the optimization results, a second-order system is taken as the control object, six integral formulas are used as fitness functions, and single-objective AWPSO is used to optimize the parameters of the controller. The values of
and
are 0.5, the population size
is 50, and the maximum number of iterations
is 500. The simulation results are shown in
Table 1. In
Table 1, when fitness functions are ISE and IAE (the second type of performance indexes), the rise time is shorter than other performance indexes. The adjustment time and steady-state error are better than those of the second performance indexes when using other performance indexes (the first kind of performance indexes), which is consistent with the previous analysis. However, as shown in
Table 1, if these optimization indexes are directly used, the overshoot of the optimized control system is still large. To solve this problem, this article improves the performance of the controller and reduces the overshoot of the system by adding penalty coefficient to the fitness function.
ISE and ITAE are selected for analysis in two types of performance indexes, and then (42) is changed to:
where
is the penalty coefficient. Similarly, (46) is changed to:
Table 2 gives the different results of optimization with the different
values. The simulation parameters are the same as those used in
Table 1. In
Table 2, for ISE, the maximum overshoot decreases with the increase of
value, while the steady-state error increases. For ITAE, with the increase of
value, the maximum overshoot decreases greatly. When
, the maximum overshoot is less than 0.1%, which is an ideal value.
According to
Table 2, the fitness function of the speed error of the backstepping controller is as follows:
(2) Selection of
In order to ensure the anti-disturbance ability of the system, the load observation value should be able to track the change of the actual. Therefore, the function form of load observation error is the same as that of the speed error. The expression is as follows:
(3) Selection of fitness function for single-objective optimization
According to the above method, the AIBC optimization problem is transformed into a single-objective optimization problem. The population size of AWPSO is 50, and the execution time is 500. The fitness function of multi-objective AWPSO is expressed as (41). The weighted value of
is calculated as 0.6798, and the weighted value of
is calculated as 0.3202, then the fitness function of single-objective optimization is as follows:
3.3.2. AWPSO Parameters Setting
(1) Location range of particles
According to (29) and (30), the optimal values of , , , , , and in AIBC need to be searched. Therefore, in AWPSO, each particle is 8-dimensional, representing , , , , , , and parameters, respectively. The location range of each dimension can be estimated according to the following method.
Assuming that the system can track the change of moment of inertia well, when the motor is started,
may have a maximum value
. At the same time, the speed error and the d axis current error should also be positive. Then:
where
is the motor rated voltage. The q axis current error is large when the motor starts. In order to make the estimated
have a larger range of value,
, where
is the rated current of the motor. Then the range of
is:
Similarly, the range of other parameters can be estimated. i.e.,:
where
is the system control cycle, and
is the rated torque of the motor.
(2) Inertial weight and acceleration factor
Inertial weights
and acceleration factors
,
have significant effect on the performance of PSO, and their values are discussed in different references. The inertia weight of AWPSO used in this article can be adjusted randomly, and the acceleration factor increases with the number of iterations. In order to test the influence of different
and
values on the optimization results, this article uses AWPSO to optimize the functions of Sphere Model and Schwefel’s Problem 2.22 [
42,
43]. Because of the randomness of AWPSO’s optimization results, the average of 20 optimization results is taken, the number of population is 200, the number of optimization cycles is 2000, and the dimension is 5 [
44,
45,
46]. The expression of the test function is as follows:
(2)
f2: Schwefel’s Problem 2.22
The final results are shown in
Table 3 and
Table 4. When
,
, AWPSO has better accuracy and convergence ability. Therefore, in this article, the value of
is 0.5, and
is 0.5.