Speed Control of PMSM using Backstepping Method

Speed Control of PMSM using Backstepping Method

Rini George

Dept.of Electrical & Electronics Engineering Mar Baselios College Of Engineering & Technology

Thiruvananthapuram, India

Arun. S. Mathew

Dept. of Electrical & Electronics Engineering Mar Baselios College Of Engineering & Technology

Thiruvananthapuram, India

AbstractThe Permanent Magnet Synchronous motor is a rotating electric machine where the stator is a classic three- phase stator like that of an induction motor and the rotor has permanent magnets. The speed controller of PMSM is designed using backstepping control. The Back stepping control is a systematic and recursive design methodology for nonlinear feedback control.A PMSM is modeled in the dqspace and an inverter is used for driving the motor. The switching pulses for inverter are generated using Space Vector Modulation (SPWM) method. The effectiveness of the proposed control scheme is verified using simulation.

IndexTermsPMSM, dq modeling, Space vector pulse width modulation(SPWM), Backstepping control.

I.INTRODUCTION

Permanent magnet synchronous motors(PMSM) are widely used in low to medium power applications where torque density and efficiency are critical. The high efficiency, high steady state, torque density and simple controller of the PM motor drives compared to the induction motor drives make them a good alternative in certain applications. Field Oriented Control is the most popular control method for PMSM. They are applied by attaching a rotor reference frame to the rotor via the Park transform. In dealing with dc quantities PI loop can be used for regulation. Due to the electrical quantities and coupling between the motor speed PMSM model is nonlinear.

The field oriented technique (vector control) is one of the most effective control techniques for AC machines in variable speed applications[1]. The FOC method decouple the torque and flux so that each can be controlled separately. A PMSM has better dyanamic performance capabilities under vector control. But the performance is sensitive to parameter variation. So much attention has been given for identifying the changes in parameters of PMSM while the motor is in normal operation. This led to some research in PMSM vector control a;gorithm using non linear control theory[2].

Several nonlinear control techniques have been introduced in the last two decades due to new developments in nonlinear control theory. Backstepping is one of the nonlinear control methods that have been applied to the AC machines. Backstepping is a recursive and systematic design methodology for nonlinear feedback control. First the design of a first-order subsystem is performed, then an additional state is considered and the design for the second-order system is performed for incremental orders until the whole system is

controlled. This is best suitable for strict feedback system. It guarantees global regulation and tracking of the linearizablenon linear feedback systems. It helps in accommodating nonlinearities and uncertainties and retain nonlinearities. The idea of backstepping control is to select some state variables as control inputs for lower order subsystems for the overall system. Each backstepping stage results into a new control design expressed in terms of control designs from the previous stage. When the procedure terminates, a feedback design of true control input results and achieves the original design objective by virtue of a Lyapunov function.[1]-[3].

In this paper, a backstepping control design based on FOC for speed control is proposed. The controller is designed such that it is robust to parameter uncertainties and nonlinearities. The paper is organized as follows. Section 2 shows the mathematical modeling of PMSM. Section 3 shows design of speed controller using backstepping control. Section 4 shows the simulation results. Finally some conclusions are drawn in section 4.

1. MODELING OF PMSM

   Dynamic modeling of PMSM is useful in simulation of PMSM. It is derived using a two phase motor in direct and quadrature (dq) axes. This is also due to the conceptual simplicity with only one set of two windings. The rotor of PMSM has only magnets, but no windings. The dynamic model is used to determine the instantaneous effects of varying voltages/currents, stator frequencies and torque disturbance on the machine and drive systems. The magnets of PMSM are assumed as current or voltage source concentrating all its flux on one axis. A two phase PMSM stator with windings and rotor with PMs is shown in Fig.1.

   The d and q axes stator voltages are derived as the sum of derivative of the flux linkages and resistive voltage drops in the respective windings as

   = + (1)

   = + + (2)

   d = Ldid+ :direct axis flux linkage

   q = Lqiq:quadrature axis flux linkage where,

   id ,iq= d and q axis component of the armature current

   vd,vq= d and q axis component of terminal voltage = flux linkage constant

   R =armature resistance

   Ld,Lq=d and q axis components of armature inductance

   = electrical angular velocity

   It is assumed the parameters are known and invariant. By choosing variable states as [idiq]T. The objective is to regulate the speed to its reference value. Defining the tracking errors as

   r = r r (16)

   = (17)

   = (18)

   r d q

   where* is the desired mechanical rotor speed and i *,i * are the target currents .

   The complete error dynamic system is given as

   =

   +

   (19)

   =

   (20)

   = 3

   (21)

   2

   =

   (22)

   Fig.1. Two phase PMSM

   1. Stabilization of the Mechanical Subsystem

      We begin the design by defining a control Lyapunov function based solely on the output of mechanical subsystem defined as

      = 1 2

      + 1 2

      (23)

      This transformation also assumes that the magnetic circuit

      1 2

      2

      is linear and the back emf and inductance variations are sinusoidal quantities.

      where Kr is a positive design gain. Evaluating the time derivative of V1

      The electromagnetic torque for a p pole machine is

      =

      +

      = 3 (

      + (

      )

      ) (3)

      1

      2 2

      =

      [3

      +

      ] (24)

      The mechanical torque equation is

      2

      where,

      = +

      +

      (4)

      Assume( ) = 0. We treat iq as a virtual input to the motor speed dynamics and backstep through an integrator into the electrical dynamics governing the torque producing

      = damping coefficient

      = load torque

      J= rotor moment of inertia

      currents. Introducing an additional gain 0and the choosing the stabilization gain as

      + +

      m=magnetic flux

      The dq frame model of a PMSM is given as

      =

      3

      2

      (25)

      =

      +

      will make the mechanical subsystem stable about the origin.

      This is done by substituting for in (24) and evaluating 1

      (10)

      =

      1

      = 2 0 (26)

      (11)

      So the above expression is negative semidefinite and hence the mechanical subsystem is asymptotically stable.

   2. Stabilization of Electrical subsystem

   = 3

   + 3 (

   )

   2

   (12)

   2

   To analyze the stability of the whole system we introduce a Lyapunov function

   Since the direct and quadrature inductances are equal, it

   2 = 1 + 1 2 + 1

   2 (27)

   simplifies the dynamics to

   =

   2

   +

   2

   +

   (28)

   =

   +

   (13)

   2 2

   =

   (14)

   (25) is differentiated as

   ( +)

   =

   (29)

   3

   3

   =

   (15)

   2

   2

   The transformed system in state errors is given by

2. Backstepping controller

= + (30)

=

2

(

+ ) +

3

2

((

+ 3 ) (

)(31)

3

= 3

2

(32)

2

=

(33)

The expressions (30)-(33) require the following control laws

= + + (

+ ) ( 3)

2

(34)

=

(35)

Finally (28) becomes

=

2 (+) 2 (+) 2

Fig 2. Response of rotor speed( =100 rpm)

2

(36)

r

is negative semi definite.

1. SIMULATION RESULTS

   To prove the effectiveness of the proposed control scheme, simulations were carried out using Matlab/Simulink software. The machine parameters are given in Table I. The plant is started on a constant load of 25 N-m and a reference speed of 100 rpm. Also a reference speed of 1200 rpm is given and the effectiveness of backstepping controller is checked.

   Sl.No.	Parameter	Value
   1	Resistance (Rs)	1.4
   2	Direct axis inductance(Ld)	0.0066H
   3	Quadrature axis inductance(Lq)	0.0058 H
   4	Magnetic flux (m)	0.1546 wb
   5	Moment of Inertia(J)	0.00176 kgm2
   6	Viscous friction(B)	0.00038818(Nm/(rad/s))
   7	No. of poles(p)	6

   Table I. The parameters of the PMSM

   Fig3. Response of rotor speed(r=1200 rpm)

   With r=1200 rpm, there is large overshoot and take more time to track the reference speed. But the backstepping can handle the parameter variation without much variation in speed.

   Response characteristics	Speed(rpm)
   	100	1200
   Rise time(tr)	0.3s	0.38s
   Peak time(tp)	0	0.7s
   Settling time(ts)	0.6s	1.1s
   Peak overshoot(Mp)	0	51.66%

   The actual speed converges with the reference speed in less time with less steady state error(r=100 rpm). But there is oscillations in the responses of the PMSM. In the case of

   r=1200 rpm, there is a large overshoot and take more time to settle down.

   Table II. Performance characteristics of the controller

2. CONCLUSION

In this paper, a backstepping controller is presented in order to accommodate the nonlinearities and uncertainties. The design of backstepping control for the speed control of a PMSM has been done. The virtual control states of the PMSM drive have been identified using recursive method and stabilizing laws are developed using Lyapunov stability theory. The proposed controller has been analysed using MATLAB/Simulink software. The simulation results show its effectiveness at tracking a reference speed under parameter uncertainties and nonlinearities. But there are oscillations in response which can be reduced by adjusting the parameters of the plant.

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