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Permanent Magnet Synchronous Motor (PMSM) displays chaotic phenomenon when PMSM in power on or power off. At present, there are many methods to control chaos in PMSM. However, there appears oscillation in course of control chaos in PMSM, which has an effect on practical application. This paper proposes error control based on adaptive backstepping to control chaos in PMSM; an error control item is added in each step virtual control design which has control effect of unknown dynamical error on system. This scheme can eliminate oscillation in course of control chaos. Finally, the simulation results show the effectiveness of theoretical analysis.

Research on PMSM has been going on for many years due to the fact that they have many advantages over the conventional internal combustion engine vehicle, such as independence from petroleum, reliability and quiet [

However there appear phenomena of chaos in PMSM when PMSM in turn on or turn off [

With the development of theory of chaos, there are many methods for control and analysis chaotic system [

Various ways and techniques had been successfully used to control or suppress chaos in PMSM. For example, in 2009, M. Zribi et al. proposed to control chaos in PMSM by instantaneous Lyapunov exponent control algorithm [

In this paper, a scheme is proposed to suppress oscillation in course of control chaos in PMSM. An error control item is added in the each step virtual control design which has control effect of unknown dynamical error on system. This scheme can gain more smoothly chaotic stabilization process and overcome oscillation in course of control chaos in PMSM. At the same time, all the signals in the system are bounded which based on Lyapunov function. This scheme has better transient response by simulation.

The dynamics PMSM, which model base on d-q axis, can be described as follows:

where

The system (1) can be changed into nondimensionalized form as follows:

where

System (2) is smooth air-gap when

Now, for model of PMSM of smooth air-gap (3), research motor without external force which can be considered PMSM no-load running and power fail interrupt, namely,

the parameters value of system (4),

Due to

Set

So system (4) can be change as follows:

To realize stability of system (4), we may add controller to the third equation of system (4), system (4) can be changed as follows:

Definite three error variables:

Step 1: Base on system (7), the first derivative of

Choose the Lyapunov function candidate as:

then the time derivative of

The virtual control

where

(10) that

Step 2: Derivative of

substituting Equation. (8) into Equation (13), the Equation (13) expression is given by

where

Choose the Lyapunov function as follows,

the time derivative of

Choose parameters adaptive rule:

where

Construct the virtual control

where

Base on Young inequality [

so a straightforward calculation produces the following inequality

Step 3: Derivative of

choose

choose the Lyapunov function candidate as

The time derivative of

setting

substituting Equation (25) into Equation (24), we have the following equation.

Similar to

set

inequality can be obtained as follows,

Theorem 1. Consider chaotic system (6) and parameter identification (16), for bounded initial conditions, the following conclusion was established:

(1) All the signals the consistent bounded in chaos system, state error

(2) Reasonable choosing parameters m, n and

Proof: Choose Laypunov function

Equation (29) above both sides by the same

namely

integral of formulas (30) in

For bounded initial conditions

When

So state error

From inequality (31), inequality can be obtained as follows

where

Given constant

Choose

This paper puts forward error control for permanent magnet synchronous motor with uncertain parameter based on adaptive backstepping which can effectively eliminate oscillation during the course of control chaos in PMSM. An error control item is added in the each step virtual control design which has control effect of unknown dynamical error on system. This scheme can gain more smoothly chaotic stabilization process. At the same time, all the signals in the system are bounded base on Lyapunov function. This scheme has better transient response by simulation.

This research is supported by the Sichuan Province Natural Science Foundation of China (Nos. 2014GZX0008, 2016JY0179), the Innovation Group Build Plan for the Universities in Sichuan (No. 15TD0024), the High-level Innovative Talents Plan of Sichuan University of Science and Engineering (2014), the Talents Project of Sichuan University of Science and Engineering (No. 2015RC50), the Cultivation Project of Sichuan University of Science and Engineering (Nos. 2012PY18, 2012PY19, 2012PY20), and the Project of Artificial Intelligence Key Laboratory of Sichuan Province (Nos. 2011RZY05, 2014RYY05, 2015RYY01).

Hua Jiang,Da Lin,Yongchun Liu,Hong Song, (2016) Based on Adaptive Backstepping Error Control for Permanent Magnet Synchronous Motor. Intelligent Control and Automation,07,17-24. doi: 10.4236/ica.2016.72003