Fractional order integral is introduced into active disturbance rejection controller (ADRC) to establish the structure of fractional order proportional integral controller (FPI). Fractional order ADRC (FADRC) is designed by replacing the nonlinear state error feedback control law using nonlinear function combination in ADRC with FPI, which can combine the high performance of ADRC estimating disturbances with the characteristics of fractional order calculus more really describing the physical object and spreading the stable region of the system parameters. The proposed FADRC is applied to permanent magnet synchronous motor (PMSM) speed servo system in order to improve robustness of system against the disturbances. Compared with ADRC, simulation results verify that the proposed control method has given very good robust results and fast speed tracking performance.

Permanent magnet synchronous motor (PMSM) is increasingly used in high precise AC servo driving control system due to its simple structures, high efficiency, low inertia, high power density and high torque-current ratio. PMSM is a nonlinear, strong coupling, and multivariable controlled object, which makes the system become nonlinear system with strong coupling [

Fractional order calculus extends integer order ones to nonintegral order, which can model various real materials more adequately than integer order ones. This study introduces fractional order calculus into ADRC and establishes the structure of fractional order proportional integral controller. Fractional order ADRC is designed by replacing the nonlinear state error feedback control law using nonlinear function combination in ADRC with FPI, which can combine the high performance of ADRC estimating disturbances with the characteristics of fractional order calculus more really describing the physical object and spreading the stable region of the system parameters. The proposed FADRC is applied to permanent magnet synchronous motor speed servo system in order to improve robustness of system against the disturbances. Compared with ADRC, simulation results verify that the proposed control method has given very good robust results and fast speed tracking performance.

On the basis of assumptions that the stator windings generate sinusoidal magnetic field, air gap is uniform and saturation is negligible. With reference to synchronous rotating reference frame, the dynamical rotate speed equation of PMSM may be expressed as follows:

To facilitate the design of ADRC, the rotate speed equation can be expressed as

For the second-order system, type structure of ADRC includes tracking differentiator (TD), extended state observer (ESO), nonlinear state error feedback control law (NLSEF), and compensated control reference [

ESO

NLSEF

Disturbance compensation

Fractional order calculus is a basic subject with mathematics to study differential coefficient and integral of random order, which can be expressed in calculus operator of fractional order and integral order [

Fractional order calculus extends the order number of calculus to fraction. Nonsmooth feedback of NLSEF in ADRC is replaced with the nonlinear fractional order PI controller, which combines the high performance of ADRC estimating disturbances with the characteristics of fractional order calculus more really describing the physical object and spreading the stable region of the system parameters. The structure of fractional order PI controller can be established by putting the

The block diagram of fractional order ADRC.

By using Laplace transform, transfer function for

In order to obtain discrete mathematical model of numerical control to be used, it is required that fractional order integral operator is approximately discretized. Using Euler rule transform method, the fractional order system can be discretized into irrational equation of

The block diagram of PMSM speed servo system is shown in Figure

The block diagram of PMSM speed servo system.

The proposed method is investigated by means of simulations. The Matlab/Simulink environment is used for the simulations. The data of the PMSM are as follows: the nominal power is 1.8 kW; the nominal dc-link voltage is 72 V; the nominal torque is 5 N·m; the number of poles is 4; the moment of inertia is 0.012 kg·m^{2}; the stator resistance is 0.09 Ω; the direct-axis inductance and quadrature-axis inductance are 0.4 mH and 0.5 mH, respectively; sampling period of current loop is set to 2

Figure

Speed response to impulse load disturbance.

Quadrature-axis currents of FADRC and ADRC are shown in Figure

Figures

Speed response by different inertia (

Speed response by different inertia (

A nonlinear feedback active rejection disturbance controlled permanent magnet synchronous motor drive is proposed. In this system fractional order integral is introduced to active rejection disturbance control to replace the nonlinear function in NLSEF, which can efficiently combine the high performance of active rejection disturbance technique observing disturbances with the characteristics of fractional calculus more really describing the physical object and spreading the stable region of the system parameters. It is applied to the permanent magnet synchronous drive. Simulation results show the feasibility and effectiveness of the proposed controller.