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A novel Fuzzy PID-Variable Structure Adaptive Control is proposed for position tracking of Permanent Magnet Synchronous Motor which will be used in electric extremity exoskeleton robot. This novel control method introduces sliding mode variable structure control on the basis of traditional PID control. The variable structure term is designed according to the sliding mode surface which is designed by system state equation, so it could compensate for the disturbance and uncertainty. Considering the chattering of sliding mode system, the fuzzy inference method is adopted to adjust the parameters of PID adaptively in real time online, which can attenuate chattering and improve control precision and dynamic performance of system correspondingly. In addition, compared with the traditional sliding mode control, this method takes the fuzzy PID control item to replace the equivalent control item of sliding mode variable structure control, which could avoid the control performance reduction resulted from modeling error and parameter error of system. It is proved that this algorithm can converge to the sliding surface and guarantee the stability of system by Lyapunov function. Simulation results show that Fuzzy PID-Variable Structure Adaptive Control enjoys better control precision and dynamic performance compared with traditional control method, and it improves the robustness of system significantly. Finally, the effectiveness and practicability of the algorithm are verified by the method of Rapid Control Prototyping on the semiphysical simulation test bench.

Lower extremity exoskeleton system is a kind intelligent assist robot, which is wearable, combines the operator with machine, and could be used in medical and military fields wildly [

Sliding Mode Cotrol (SMC) is widely used as a nonlinear control method currently. By designing the sliding mode surface reasonably, the control performance of system will not be affected by internal parameter perturbation and external disturbance, and the system will have strong robustness and high control precision. However, the equivalent control term of sliding mode control depends on the accuracy of system model and parameters, while in practical application, only an approximate mathematical model of motor can be obtained. And there are still some uncertain factors, such as parameter error and external interference, which will reduce the performance of the sliding mode control and even cause the instability of system [

As is known, PID control is a simple, easy to implement, and widely used control method. In the face of the traditional SMC the accurate models and parameters of system cannot be obtained. It is natural to think of the combination of sliding mode variable structure control and PID control. In the design of sliding surface, the integral term was introduced to form a sliding surface which was similar to PID structure in [

Since the variable structure control is introduced in the design of controller in this paper, the chattering caused by variable structure control must be considered. References [

In this paper, the Fuzzy PID-Variable Structure Adaptive Control, which is mutual compensation of sliding mode variable structure control and fuzzy PID control, is proposed for trajectory tracking of PMSM used in extremity exoskeleton system. Firstly, this algorithm ensures stability of system by PID control instead of equivalent control. Then, a sliding mode surface based on state equation of system is designed for sliding mode variable structure control which compensates for insensitivity of PID control to parameter perturbations and external disturbances. Finally, the fuzzy reasoning method is added to adjust the parameters of PID adaptively online in real time, which improves robustness of system again and attenuates chattering of sliding mode system as well.

The mathematical model of PMSM in the rotating shaft (

where

PMSM servo system is a three-closed-loop control system based on flux orientation, including position control loop, speed control loop, and current control loop. The three-closed-loop control of PMSM is shown in Figure

Control schematic of PMSM.

In order to design the position loop controller conveniently, it is assumed that the speed control loop, the current control loop, and the inverter are ideal. Therefore, the mathematical model of the PMSM can be simplified as a second-order differential link [

It can be also expressed as

Let

Then (

where

The control target of the PMSM position loop is to enable the output to track the reference value quickly and accurately. Let

The fuzzy PID controller is designed as a system with two-dimensional input and three-dimensional output. The inputs are the error

where

The control principle of fuzzy PID control is shown in Figure

Control schematic of Fuzzy PID.

The core of fuzzy PID is the design of fuzzy inference rules. Firstly, fuzzify the input variables

The fuzzy subset of two-dimensional input variables and three-dimensional output variables is defined as

The same membership function is applied to the input variables and the output variables for real-time calculation and online adjustment. The membership functions

Membership function curve.

The fuzzy rules of

Fuzzy rules table.

| | ||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

| | | | | | | |||||||||||||||

| PS | PM | NB | PS | NB | NB | PB | NB | NB | PM | NB | NM | PM | NB | NS | ZO | NB | ZO | ZO | PM | ZO |

| PB | PS | NB | PB | NM | NB | PB | NM | NM | PS | NM | NS | PS | NM | NS | ZO | NM | ZO | ZO | PS | ZO |

| PM | PS | NM | PM | NM | NM | PM | NM | NS | PS | NM | NS | ZO | NM | ZO | NS | NM | PS | NS | PS | PS |

| PM | ZO | NM | PM | NS | NS | PS | NS | NS | ZO | NS | ZO | NS | NS | PS | NM | NS | NM | NM | ZO | NM |

| PS | ZO | NS | PS | ZO | NS | ZO | ZO | ZO | NS | ZO | PS | NS | ZO | PS | NM | ZO | PM | NM | ZO | PM |

| ZO | PB | ZO | ZO | PS | ZO | NS | PS | PS | NM | PS | PS | NM | PS | PM | NM | PS | PB | NB | PB | PB |

| ZO | PB | ZO | ZO | PM | ZO | NM | PM | PM | NB | PM | PM | NM | PM | PM | NB | PM | PB | NB | PM | PB |

The process of adaptively adjusting PID parameters should follow the following principles:

Finally, the fuzzy quantity of the output variable is defuzzified as the output of the controller by barycenter method as

where

Then, the output of Fuzzy PID controller can be expressed as

According to state equations (

From (

From (

The switch item should be designed as

where

Considering that the equivalent control item in SMC will affect the servo control performance due to modeling error and inaccurate parameters of system, the effect of the equivalent control item in SMC is compensated by PID control item, and then the effect of the nonlinear is compensated by the variable structure control item. Meanwhile, the fuzzy inference method is used to adjust the parameters of PID in real time online, which can attenuate chattering. The fuzzy PID controller can ensure the stability of system, limit the error to a narrow range, and attenuate the chattering of SMC as well. On the other hand, the variable structure control item is used to compensate the effect of nonlinear, which can suppress the effect of parameter perturbation and load disturbance. The schematic diagram of Fuzzy PID-Variable Structure Adaptive Control is shown in Figure

Control schematic of Fuzzy PID-Variable Structure Adaptive Control.

The output of Fuzzy PID-Variable Structure Adaptive Controller is

The controller designed by (

where

In the actual control system, the position target input of the system is bounded, and the perturbation of system parameters and the disturbance of external load will change within a certain range. Thus the assumption in (

For

The control item in (

To construct a control matrix for an

Construct a positive definite Lyapunov function as

In order to guarantee the positive definiteness of the Lyapunov function (

Differentiating

From (

In the process of stability analysis, as there is no integral term in the design of sliding mode surface, the integral term in fuzzy PID can be ignored. Then (

Substituting (

Because of (

Combining (

It can be deduced that

The model of the position-speed-current three-closed-loop control system for PMSM vector control is built based on Matlab/Simulink. The speed loop and the current loop adopt the conventional PID control while the position loop adopts the Fuzzy PID-Variable Structure Adaptive Control which is introduced in this paper. The simulation model is shown in Figure

Simulink model.

The parameters of the simulation model are shown in Table

The parameters of the simulation model.

Parameter | Value | Parameter | Value |
---|---|---|---|

| 2.875Ω | | 700 |

| 0.00153H | | 6 |

| 0.175Wb | | 0.1 |

| 0.0008 | | 268.5 |

| 4 | | 2.8 |

| 1.05 |

Figures

Step response (a).

Step response (b).

Step response (c).

Step response error (a).

Step response error (b).

From Figure

Step response speed under Fuzzy PID-Variable Structure Adaptive Control.

Step response

Figures

Sinusoidal response (a).

Sinusoidal response (b).

Sinusoidal response (c).

Sinusoidal response speed under Fuzzy PID-Variable Structure Adaptive Control.

Sinusoidal Response

By analyzing the step response and sinusoidal response with load disturbance, the Fuzzy PID-Variable Structure Adaptive Control can improve the servo precision, the dynamic performance, and the robustness of system. At the same time, the chattering of variable structure control is also attenuated obviously because the PID parameters can be adjusted online adaptively by fuzzy inference.

The Rapid Control Prototype (RCP) is a kind of semiphysical simulation. After the mathematical simulation of the system control model meets the desired effect, the control algorithm model of the servo system will be extracted individually. Then the Real-Time Driver (RTD) of actual controlled object, feedback original, and drive unit are added to the control algorithm, which will constitute a closed-loop system [

RCP model.

The control system test bench is shown in Figure

The control system test bench.

The control principle of test bench is shown in Figure

The control principle.

The step response performances of conventional PID control and Fuzzy PID-Variable Structure Adaptive Control are compared by a step signal. The step response testing is performed with a constant torque of 5Nm, and the step motion mode is a reciprocating motion with the range of 100° to 200°. The experimental results are shown in Figures

Step response position tracking.

Step response position tracking error.

Step response speed.

Step response original torque data.

Step response torque data after low-pass filter.

It can be seen from Figures

The dynamic response performances of conventional PID control and Fuzzy PID-Variable Structure Adaptive Control are compared by sinusoidal response by a sinusoidal signal (amplitude: 100°, frequency: 2Hz) with a constant torque of 5Nm. The experimental results are shown in Figures

Sinusoidal response position tracking.

Sinusoidal response position tracking error.

Sinusoidal response speed.

Sinusoidal response original torque data.

Sinusoidal response torque data after low-pass filter.

Both conventional PID control and Fuzzy PID-Variable Structure Adaptive Control achieve good performances under the proper parameter as shown Figure

There is no obvious chattering in the response curve of static testing and dynamic testing under Fuzzy PID-Variable Structure Adaptive Control, which indicates that it is feasible to introduce fuzzy inference into PID control to attenuate chattering. Finally, we conclude the advantages of the proposed Fuzzy PID-Variable Structure Adaptive Control method as high precision, good robustness, and simplicity based on experiments.

The Fuzzy PID-Variable Structure Adaptive Control algorithm for the position tracking control of PMSM which will be used in electric extremity exoskeleton robot is proposed. The controller consists of two parts which are mutual compensation: one is a nonlinear part (Variable structure control), and the other is an approximate linear part (fuzzy PID control). Variable structure control has high robustness. Fuzzy PID control is used to compensate the equivalent control item in SMC and attenuate chattering. The simulation results show that the algorithm can improve control accuracy, dynamic performance, and robustness. The chattering of sliding mode system is attenuated to some extent. At the same time, the Fuzzy PID-Variable Structure Adaptive Control algorithm has the advantages of simple structure and easy engineering realization as well. Finally, the effectiveness and practicability of the algorithm are verified based on a semiphysical simulation test bench with the method of RCP.

Undoubtedly, the control algorithm proposed in this paper as the lower level control method can be applied in our designed electric exoskeleton successfully. And it is also the foundation and origin of designing the higher level control method in our later works on this electric extremity exoskeleton.

The simulation and experimental data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the Chinese National Science Foundation (no. 51075017).