^{1}

^{2}

^{1}

^{1}

^{2}

Many of the existing control methods for the permanent magnet synchronous motor (PMSM) either deal with steady state models or consider dynamic models under particular cases. A dynamic model of the PM machine allows powerful control-theoretic techniques such as linearization to be applied to the system. Existing exact feedback linearization of dynamic model of PMSM suffers from singularity issues. In this paper, we propose a quadratic linearization approach for PMSM based on the approximate linearization technique which does not introduce singularities. A MATLAB simulation is used to verify the effectiveness of the linearization technique proposed. Also, to account for higher-order and unmodelled dynamics of PMSM, tuning of the linearizing transformation is proposed and verified using simulation.

Permanent magnet (PM) machines, particularly at low power range, are widely used in the industry because of their high efficiency. They have gained popularity in variable frequency drive applications. The merits of the machine are elimination of field copper loss, higher power density, lower rotor inertia, and a robust construction of the rotor [

Steady state models, such as in [

A dynamic model of a PM machine using direct and quadrature axis variables [

Bodson and Chiasson [

Krener [

In this paper, input and state transformations are derived for a 4-dimensional PM machine model in order to linearize PMSM machine model. The PMSM model is quadratic linearized using the approximate technique. Tuning rules are derived for the linearizing transformations to account for higher-order terms, by back propagation of error between the outputs of quadratic linearized system with a normal form output.

The linearization technique is verified using SIMULINK model which is developed for interior permanent magnet (IPM) machine. The core loss which consists of higher-order terms is included in the SIMULINK model of PMSM, and tuning rules are also simulated. The closed loop response and open loop gain for the system before and after tuning are obtained.

The simulation results after linearization indicate a uniform closed loop response for different reference and load conditions, thus verifying the theory. The simulation results after tuning also verify the effectiveness of tuning.

To summarise the rest of the paper, in Section

Consider a single input control affine system of the form [

In order to linearize the system, a change of coordinates and feedback [

Applying the transformations (

Quadratic linearization involves specialization of the above result for

The PM machine model can be derived [

The model (

The quadratic linearization given in Section

Given the 4-dimensional model of a PM synchronous motor (IPM model) of the form (

The system then reduces to

Applying transformations (

Premultiplying (

Given the parameters ^{2} and

PMSM design using SIMULINK.

Linearization of PMSM.

Prior to linearization, the open loop steady state gain of

Steady state gain of

5 | 2.55 | — |

10 | 4.7709 | 0.44418 |

15 | 6.4706 | 0.33994 |

20 | 7.6082 | 0.22752 |

25 | 8.2567 | 0.1297 |

30 | 8.5326 | 0.05518 |

Steady state gain of

5 | 2.176 | — |

10 | 4.366 | 0.438 |

15 | 6.585 | 0.4438 |

20 | 8.845 | 0.452 |

25 | 11.162 | 0.4634 |

30 | 13.55 | 0.4776 |

We now proceed to show that the nearly constant gain of the linearized model results in a uniform closed loop response on a range of set point and load inputs with a fixed controller. This is in contrast to the case before linearization under the corresponding conditions.

Figures

Time response of angular velocity in closed loop when

Time response of angular velocity in closed loop when

Figures

Time Response of

Time Response of

Figures

Time response of angular velocity in closed loop when

Time response of angular velocity in closed loop when

Figures

Time Response of

Time Response of

The core loss or iron loss, caused by the permanent magnet (PM) flux and armature reaction flux, is a significant component in the total loss of a PMSM, and, thus, it can have a considerable effect on the PMSM modeling and performance prediction.

The net core loss

The mechanical torque equation including core losses is given by

To account for the core loss (

Block diagram for tuning of transformation.

Error (

The error can be written as

Tuning of

Hence,

Tuning of

Updation of

PMSM model including core loss is given in Figure

Steady state gain of

5 | 2.5317 | — |

10 | 4.673 | 0.4286 |

15 | 6.2126 | 0.30792 |

20 | 7.147 | 0.18688 |

25 | 7.594 | 0.0894 |

30 | 7.7001 | 0.02122 |

Steady state gain of

5 | 2.0033 | — |

10 | 4.0264 | 0.40462 |

15 | 6.0925 | 0.41322 |

20 | 8.193 | 0.4201 |

25 | 10.295 | 0.4204 |

30 | 12.434 | 0.4278 |

PMSM model including core loss.

Simulation diagram for controller tuning.

Time response of angular velocity in closed loop when

Time response of angular velocity in closed loop when

Time response of

Time response of

As the PMSM is inherently nonlinear, to design a controller that can provide a predictable uniform performance of the drive under varying operating conditions, it is necessary to linearize the PMSM. In this paper, the dynamic model of a PM synchronous motor involving quadratic nonlinearity is linearized. The technique used is based on the control input extension of Poincare’s work due to Kang and Krener which is in line with the approximate linearization technique of Krener [

The PMSM machine model, together with the state and input transformations, are simulated using SIMULINK. The simulation results show that the quadratic linearizing transformations effectively linearize the system thus, supporting the theory. The simulation verifies that a uniform response under a fixed controller is obtained for the linearized system for variations of reference speed and load conditions, in contrast to the case before linearization.

Further, to account for the core loss, unmodelled dynamics and third- and higher-order nonlinearities, tuning of the transformation parameters is proposed by comparing the output of the linearized system with a normal form output. After tuning, it is shown that the linearized system shows improved linearity in terms of static gain compared to the condition before tuning. Closed loop system response for the linearized system is also shown to be uniform due to the effect of tuning of linearization transformations, under varying reference inputs.

The proposed linearization method can be extended to induction motor and wound synchronous motor models as well [