Dynamic property evaluation of tri-cyclic servo system

3.1. Simulation model building

The tri-cyclic servo system consists of a current loop, velocity loop and position loop which were built based on the open loop transfer model of the electric machine [8]. And in the position loop, a proportional (P) controller is always used, while in the current loop and the velocity loop, it is a proportional-integral (PI) controller. Moreover, in the physical system, a low-pass filter that can be expressed by first order inertial link is used in the current loop to restrain the disturbance to the servo system. The low-pass filter in the current loop will cause the problem of control delay, the same first order inertial link shall be used before the current loop to make the feedback signal and the given signal transmission at the same time. Analogously, a feedback low-pass filter and pre-filter are also used in the velocity loop. Therefore, took the most frequently used permanent magnet synchronous electric machine (which the open loop transfer model was shown in Fig. 1) as an example, where, $L_q$ is the winding inductance, $R$ is the winding resistance, $K_e$ is the back-EMF coefficient, $K_r$ is the moment coefficient, $J$ is the rotational inertia of electric machine, $T_L$ is the torque of load, $B$ is the viscous friction coefficient, ${\omega }_r$ is the angular velocity of the rotor. The simulation model tri-cyclic servo system can be built which was shown in Fig. 2.

Fig. 1Open loop transfer model of permanent magnet synchronous electric machine

Fig. 2Simulation model tri-cyclic servo system

3.2. Tuning of control parameters

According to the simulation model shown in Fig. 2, there were 5 control parameters to be tuned. The GA had been proved better in tuning control parameters than the traditional method [9], and in this paper, it was used to tune the control parameters for the tri-cyclic servo system. The tuning step was:

(1) Computed the objective function value.

According to the dynamic property evaluation function mentioned in Section 2, the objective function value was defined as:

(2) Encoded for gene.

The tuning of control parameters consists in finding a solution for the optimization of a continuous function, and a gene encoded with the binary system was complicated with low accuracy as compared to the one encoded with a real number. Therefore, the one encoded with a real number was used in this paper.

(3) Selection operation.

In the selection operation, a higher survival probability was made for the individual that had a bigger $F$ computed with Eq. (2). Hypothesis $f_i$ was the objective function values of individual $i$ and $\mathrm{\Sigma }{f}_i$was the accumulated total values of the objective function of all the individuals, the duplication ability of individual $i$ was $f_i/\mathrm{\Sigma }{f}_i$.

(4) Crossover and mutation operation.

Basic arithmetic crossover and mutation operation were used in this paper. And a high mutation constant was selected in the mutation operation to reflect the global search ability of the genetic algorithm. Moreover, the best individual is not used in both the crossover and the mutation operations to ensure the global convergence of GA.

(5) Convergence analysis.

In the traditional convergence analysis, the optimal searching result or the number of generations was used to determine whether convergence conditions are satisfied or not. However, in this paper, the optimal objective function value for the simulation model could not be certified in advance. Therefore, GA searching ended with the number of generations reach to the set numbers.

According to the tuning step for control parameters, a tri-cyclic servo system with its parameter indexes shown in Table 1 was tuned.

The value of the evaluation function ($J$) was 1308 while $n$ was 3 and $\omega$ was 100 after tune, and the dynamic following property was shown in Fig. 3.

Fig. 3Dynamic following property of tuned tri-cyclic servo system

4.1. Experiment design for data acquisition

The accuracy of the evaluation model for the dynamic property based on BP network was decided by the data acquisition method. The orthogonality principle was utilized in the mathematical statistics, and the representative data, which displayed the relations between the electric parameters, were chosen, and the dynamic property could guarantee the accuracy of the evaluation model. The simulation data distribute in the data space uniformly and it is orthogonal with each other by using the optimization Latin square design method [10]. For the parameters $V$, $C$, $R_a$, $L_a$, $\beta$ and $T_{sf}$ shown in Table 2, they were obvious relevant with the dynamic property, the evaluation model for the relationships between these 6 parameters and the dynamic property was studied. The number of sample data groups used for building the evaluation model was set as 50. Therefore, according to the optimization Latin square design method, the dynamic property based on the simulation model and tuning steps was computed, the 50 groups sample data was shown in Table 2 while $n$ was 5 and $\omega$ was 100.

4.2. Evaluation model building based on BP network

As a widely used network, a 3-layers BP network is able to approximate all the continuous functions in any accuracy [11]. And it was used in this paper to build the evaluation model. In the BP network training process, the number of input neurons was 6 as per the parameters $V$, $C$, $R_a$, $L_a$, $\beta$ and $T_{sf}$ shown in Table 3, while the output neuron was $J$ also shown in Table 3. The number of hidden layer neurons was set as 10 according to the empirical formula:

where, $y$ was the number of hidden layer neurons, $m$ was the number of input neurons, $n$ was the number of the output neurons, $a$ was an integer which was varied from 0 to 10.

By comparing the result of substantive BP training experiments, the transfer function between input layer and hidden layer was chosen to be sigmoid, and between hidden layer and output layer it was a purelin function. The training function was training, and the learning rate was set as 0.01, the training error was 10^-5. 45 groups of sample data were used to train, and after training with 52986 times, the error reached to 9.9928×10^-6. The error varied with training process was shown in Fig. 4.

Fig. 4Error varied with BP network training process

According to the result of BP network training, the evaluation model was built as:

where, $J_p$was the evaluation value of the dynamic property, $P_{ij}$ and $b_j$ were the weight and threshold between input layer and hidden layer, $Q_j$ and $k$ were the weight and threshold between hidden layer and output layer, $X_i$ was the values of input parameters.

There were 5 groups of sample data that had not been used in the BP network training. They were used to verify the evaluation model. As shown in Fig. 5, the error between the evaluation values and the actual result of these 5 groups sample data was low, and it proved the evaluation model could be used to study the relations between the electric parameters and dynamic property of a tri-cyclic servo system.

Fig. 5Comparison of evaluation values and actual result