A fuzzy multiresolution wavelet neural network (FMWNN) controller with dynamic compensation (DC) is proposed to address the complexities of the electric load simulator (ELS). The FMWNN acts as a main torque tracking controller, which takes full advantage of the merits of an ideal sliding mode, fuzzy rules, and multiresolution WNN. The fuzzy algorithm is used to dynamically adjust the weights of the WNN and effectively accelerate the convergence rate. In addition, the DC controller is designed to greatly decrease the effect of the approximation error and guarantee the system stability in the sense of the Lyapunov theory. Finally, the proposed algorithms are carried out on the semiphysical simulation platform, the precision and superiority of which are comparatively verified based on the simulation results.
During the past few years, load simulators are widely applied to the guns, aircrafts, ships, and so forth, which provide great help for simulating the real-time variable loads. On the basis of different load types, the load simulators can be mainly divided into electrohydraulic load simulators [
Neural networks (NN) have the advantages of parallel computation, approximation, self-learning, and fault tolerance [
With the above-mentioned motivations, this paper proposes a fuzzy multiresolution wavelet neural network controller with dynamic compensation (DCFMWNN) to address the complexities of the ELS. Section
The AC permanent magnet synchronous motor (PMSM) is applied to the ELS for the gun control system. Though there are many merits in the PMSM, like the high ratio of the torque and inertia, the rapid and precise response, the uncertainties, and self-coupling disturbance result in negative impacts. In addition, the relationship between the control current and the output torque in PMSM is also not simple linear, which is different from the direct loading style. In combination with the working principle and practical working conditions of the ELS, the time-varying nonlinear system of the ELS can be simply described [
According to the existing problem, the control objective is to find a control law so that the state trajectory
And a sliding surface is defined as follows:
In combination with (
If the system parameters in (
Since
The ideal controller
The block diagram of the DCFMWNN controller.
By virtue of different resolutions, the multiresolution analysis (MRA) can decompose a signal into components spanned by the scaling and wavelet basis functions. Any finite energy function can be represented as [
With the advantages of easy computation and analysis, the fuzzy controllers with many adaptive and intelligent rules have been widely applied to complex nonlinear systems. Considering the multi-input single output (MISO) system, Takagi and Sugeno provide that the output of fuzzy rules is obtained by the linear combination of the inputs [
The orthogonal wavelet function is usually used as the excitation function of neurons in the neural network, and the wavelet decomposition is used to analyse the data. The networks are trained step by step on the basis of multiresolution analysis. The output of FMWNN controller can be represented as
The structure of the FMWNN.
According to the property of the universal function approximation, it implies that there exists an expansion of (
Substituting (
Then, the derivative of
To prove the stability of the DCFMWNN controller, define a Lyapunov function candidate in the following form [
Taking the derivative of the Lyapunov function yields
In order to further simplify
Thus, substituting (
Assume
Since
In combination with (
As a result, the stability of the control system can be guaranteed.
In order to demonstrate the feasibility and effectiveness of the DCFMWNN controller, the designed reference commands, with different algorithms, like the convergence analysis of the MWNN and FMWNN, the step response of the FMWNN and DCFMWNN, and the sinusoidal tracking of the FMWNN and DCFMWNN, are applied to the simulation platform of the ELS shown in Figure
The simulation platform of the ELS.
The following simulations are carried out in an Intel Core i5 CPU with 3.2 GHz rate, 4 GB RAM, and 64-bit operating system. The parameters of the actual gun control system are as follows: the rotary inertia of the turret is 7000 kg·m2, the total friction torque
The parameters of the torque motor.
Parameters | Values |
---|---|
Rated power |
2.8 kw |
Biggest locked-rotor torque |
19.8 N |
Continuous locked-rotor torque |
12 N |
Continuous current |
80 A |
Rated speed |
1200 r/min |
Coefficient of electromagnetic |
2.6 N·m/A |
Coefficient of counter |
9.6 V/krpm |
Equivalent inductance |
0.0036 H |
Equivalent resistance |
1.2 Ω |
Rotational inertia |
0.038 kg·m2 |
Viscous friction coefficient |
0.22 N·m·s/rad |
The superiority between the proposed MWNN and FMWNN controller is illustrated by analysing the convergence epochs, so the absolute error (
A comparison of the convergence epochs for the MWNN and FMWNN controller.
The step response with external disturbance is evaluated in Figure
A comparison of step response with external disturbance.
In order to compare the superiority of the FMWNN and DCFMWNN, the reference command of the torque motor is usually chosen as
The sinusoidal tracking with
Figure
In order to further analyse the stability and precision of the desired outputs with different frequencies and amplitude, the reference commands of the torque motor are selected as
The dynamic simulation results of the sinusoidal tracking.
The desired reference command | Mean amplitude error (%) | Mean phase error (°) | ||
---|---|---|---|---|
FMWNN | DCFMWNN | FMWNN | DCFMWNN | |
|
9.45 | 6.82 | −9.6325 | −9.0483 |
|
7.28 | 5.76 | −7.0729 | −5.2541 |
|
6.51 | 4.25 | −5.3263 | −4.1036 |
|
5.36 | 3.53 | −3.2541 | −2.0527 |
|
4.12 | 2.53 | −2.2692 | −0.9235 |
The sinusoidal tracking with
The sinusoidal tracking with
The sinusoidal tracking with
The sinusoidal tracking with
The mean amplitude and mean phase errors are the mean absolute values and angle deviations between the desired reference command and the real simulation outputs, which show that the smaller the reference command amplitudes and the higher the reference command frequencies (within certain limits), the worse the control performance. However, what we need is to remember that the control effectiveness is relatively poorer when the direction of the torque motor is changed, which plays a negative effect on the overall performance and needs to be delved into in the future.
This paper has successfully investigated the ELS via DCFMWNN controller on the simulation platform of the ELS. The FMWNN controller has salient merits of free model, effective search, and computation ability with few chattering phenomena, and the DC controller is introduced to eliminate the effect of the approximation error and guarantee the system stability in the sense of Lyapunov theory. Furthermore, the simulation results are promising and reveal that the proposed controller is able to successfully damp the interarea nonlinearities and maintain system precision and robustness.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are grateful to the reviewers for their valuable comments. The authors appreciate the partial financial support from the National Science Foundation of China under Grant 51305205.