Application of Quadratic Linearization for the Control of Permanent Magnet Synchronous Motor

Abstract

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.

1. Introduction

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 [1].

Steady state models, such as in [2], have been used to develop control strategies for the permanent magnet synchronous motor. Vector control [3] enables independent control over the magnitude and angle of the current with respect to the rotor such that instantaneous control over torque is possible. But vector control implemented, in practice [4], assumes a constant flux in the closed loop and uses static models to effect the control scheme. The steady state models used to describe the machine behaviour do not capture explicitly the dynamics of the machine involved. Decoupled control [5], while considering the dynamics of the machine, corresponds to application of particular control strategies for machine control.

A dynamic model of a PM machine using direct and quadrature axis variables [1] allows more powerful general control theories to be brought to bear on the problem of the control of PM machine. One such control-theoretic approach is quadratic linearization of the PM machine model. Linearization of a nonlinear system allows a simple fixed controller to be applied for the control of the linearized system and yet get a uniform closed loop response for different reference and load conditions.

Bodson and Chiasson [6] have designed a controller using differential geometric method based on exact feedback linearization. The main drawback with this method and its variations [7] is that, even if the system is linearizable, the linearizability is subject to certain function of the state being nonsingular in a region of operation of the machine. This puts a constraint on the practical implementation of the control strategy. Zhu et al. [8] provide a static feedback linearization of a PM model (see [8, equation (25)]) which has a simpler quadratic term compared to our model. With the assumption that 𝐿_𝑑=𝐿_𝑞, the solution of quadratic linearization becomes trivial. Our method is more general and is applicable to PMSM where 𝐿_𝑑≠𝐿_𝑞.

Krener [9] formulated an approximate feedback linearization technique based on Taylor series expansion; Kang and Krener [10, 11] extended the work of Poincare [12] on the lines of Krener to consider quadratic linearization of control affine systems. Quadratic linearization can be applied to a PMSM model [1] as the model has a predominant quadratic nonlinearity. But being an approximate technique, higher-order terms are introduced. Consideration of core loss involves fourth-order terms involving product of squares of both current and angular velocity. These together with the stray losses and unmodelled dynamics coupled with the third- and higher-order nonlinearities introduced due to quadratic linearization are best accounted for by tuning the transformations against an actual PM machine on the lines similar to those proposed by Levin and Narendra [13].

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 2, background required for quadratic linearization is given. In Section 3, the model of PM synchronous motor is derived and is reduced to normal form. In Section 4, PMSM model is linearized. In Section 5, a SIMULINK model of PMSM is constructed and simulation is presented to show the effectiveness of the proposed method. In Section 6, tuning of the transformation coefficients is derived by including core loss in the PMSM model and also simulation results are given to verify the effectiveness of tuning. In Section 7, the paper is concluded.

2. Background

Consider a single input control affine system of the form [10]𝑥̇=+𝐴𝑔𝑥+₍₁₎𝐵𝑢+𝑓⁽²⁾(𝑥)+𝑓⁽³⁾(𝑥)+⋯+𝑓^(𝑚)(𝑥)+⋯(𝑥)+𝑔⁽²⁾(𝑥)+⋯+𝑔^(𝑚−1)(𝑥)+⋯,(1) where 𝐴 and 𝐵 are matrices in the controller normal form𝐴=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣0000⋅1000⋅⋅010⋅⋅⋯⋅⋅⋅⋅⋅⋅⋅0010⋅⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦;𝐵=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣001⋅⋅⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,(2)𝐴 is an 𝑛×𝑛 constant matrix and 𝐵 is an 𝑛×1 constant matrix.

𝑥=[𝑥₁𝑥₂⋯𝑥_𝑛]^𝑇 and 𝑢 is a scalar input. 𝑓^(𝑚)(𝑥), 𝑔^(𝑚−1)(𝑥) are vector homogeneous polynomials of order 𝑚 and (𝑚−1), respectively, 𝑚=2,3,….

In order to linearize the system, a change of coordinates and feedback [10, 11] of the following form is considered:𝑢=(1𝑦+=𝛽𝑥(+𝑥)𝜙)𝑣(𝑥+),𝛼(𝑥),((34)) where 𝑦=[𝑦₁𝑦₂⋯𝑦_𝑛]^𝑇, 𝑣 is a scalar, and 𝛼(𝑥), 𝛽(𝑥), and 𝜙(𝑥) are polynomials given by 𝜙𝛼((𝑥𝑥))==𝛼𝜙⁽²⁾₍₂₎(𝑥)+𝜙⁽³⁾(𝑥)+⋯+𝜙^(𝑚)(𝑥)+⋯,𝛽(𝑥)=𝛽₍₁₎(𝑥)+𝛼⁽³⁾(𝑥)+⋯+𝛼^(𝑚)(𝑥)+⋯,(𝑥)+𝛽⁽²⁾(𝑥)+⋯+𝛽^(𝑚−1)(𝑥)+⋯.(5)

Applying the transformations (3) and (4), (1) is reduced to 𝑦̇=𝐴𝑦+𝐵𝑣(6) provided that the following equations, called generalized homological equations [10], are satisfied:−𝐴𝜙^(𝑚)(𝑥)+𝐵𝛼^(𝑚)(𝑥)+𝑓^(𝑚)(𝑥)+𝜕𝜙^(𝑚)(𝑥)𝐵𝛽_(𝑚−1)𝜕𝑥𝐴𝑥=0,(𝑥)𝑣+𝜕𝜙^(𝑚)(𝑥)𝜕𝑥𝐵𝑣+𝑔^{(𝑚−1)}(𝑥)𝑣=0;∀𝑣,(7) where 𝑓^(𝑚)(𝑥)=𝑓^(𝑚)(𝑥) for 𝑚=2 and 𝑓^(𝑚)(𝑥) is expressed in terms of 𝑓^{(𝑚−𝑖)}(𝑥), 𝑖=0,1,2…(𝑚−2) and 𝜙^{(𝑚−𝑗)}(𝑥), 𝑗=1,2…(𝑚−2), 𝑚>2. 𝑔^{(𝑚−1)}(𝑥)=𝑔⁽¹⁾(𝑥), 𝑚=2 and 𝑔^{(𝑚−1)}(𝑥) is expressed in terms of 𝑔^{(𝑚−𝑖)}(𝑥), 𝑖=1,2…(𝑚−1) and 𝜙^{(𝑚−𝑗)}(𝑥), 𝑗=1,2…(𝑚−2),𝑚>2.

Remark 1. Quadratic linearization involves specialization of the above result for 𝑚=2. Consider 𝑥̇=𝐴𝑥+𝐵𝑢+𝑓⁽²⁾(𝑥)+𝑔⁽¹⁾(𝑥)+𝑂⁽³⁾(𝑥,𝑢),(8) where 𝐴 and 𝐵 are in controller normal form [10] and 𝑂⁽³⁾(𝑥,𝑢) corresponds to terms of order 3 or more. Equations (3) and (4) for 𝑚=2 can be written as 𝑢=1𝑦+=𝛽𝑥₍₁₎+𝜙⁽²⁾(𝑥),(9)(𝑥)𝑣+𝛼⁽²⁾(𝑥).(10) Applying (9) and (10), (8) can be written as 𝑦̇=𝐴𝑦+𝐵𝑣+𝑂^(3)(𝑦,𝑣)(11) provided that the homological equations −𝐴𝜙⁽²⁾(𝑥)+𝐵𝛼⁽²⁾(𝑥)+𝑓⁽²⁾(𝑥)+𝜕𝜙⁽²⁾(𝑥)𝐵𝛽₍₁₎𝜕𝑥𝐴𝑥=0,(𝑥)𝑣+𝜕𝜙⁽²⁾(𝑥)𝜕𝑥𝐵𝑣+𝑔⁽¹⁾(𝑥)𝑣=0;∀𝑣(12) can be solved for 𝜙⁽²⁾(𝑥),𝛼⁽²⁾(𝑥), and 𝛽⁽¹⁾(𝑥).

3. Machine Model

The PM machine model can be derived [1, 14, 15] as 𝑥=𝑥₁𝑥̇=𝐴𝑥+𝐵𝑢+𝑓⁽²⁾(𝑥),(13)𝑢=𝑥₂𝑢𝑥₃₁𝑥₄^𝑇=𝜃𝜔_𝑒𝑖_𝑞𝑖_𝑑^𝑇,𝑢₂^𝑇=𝑣_𝑞𝑣_𝑑^𝑇,(14) where 𝑣_𝑞,𝑣_𝑑,𝑖_𝑞, and 𝑖_𝑑 represent the quadrature and direct axis voltages and currents, respectively, and 𝜃 and 𝜔_𝑒 represent rotor position and angular velocity, respectively.𝐴=⎡⎢⎢⎢⎢⎢⎢⎢⎣00101.50𝑝𝜆00−𝜆𝑝𝐽0𝐿_𝑞−𝑅𝐿_𝑞000−0𝑅𝐿_𝑑𝐵=⎡⎢⎢⎢⎢⎢⎢⎢⎣001⎤⎥⎥⎥⎥⎥⎥⎥⎦,𝐿_𝑞0𝐿0001_𝑑⎤⎥⎥⎥⎥⎥⎥⎥⎦,𝑓₍₂₎(𝑥)=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣1.5𝑝𝐿^𝑑0−𝐿_𝑞𝑖_𝑑𝑖_𝑞−𝐿_𝑑𝐽𝑝𝜔_𝑒𝑖_𝑑𝐿_𝑞𝐿_𝑞𝑝𝜔_𝑒𝑖_𝑞𝐿_𝑑⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,(15)λ is the flux induced by the permanent magnet of the rotor in the stator phases. 𝐿_𝑑 and 𝐿_𝑞 are the direct and quadrature inductances, respectively. 𝑅 is the stator resistance, 𝑝 is the number of pole pairs, and 𝐽 is the system moment of inertia.

The model (13) can be reduced, in a standard way using linear change of coordinates and feedback [16], to the Brunovsky form [17] for two inputs as in (16) below (where 𝑥,𝑢,𝐴, and 𝐵 are retained for simplicity of notation):𝑥̇=𝐴𝑥+𝐵𝑢+𝑓⁽²⁾(𝑥),(16) where 𝐴=⎡⎢⎢⎢⎢⎢⎢⎣0000100001000000⎤⎥⎥⎥⎥⎥⎥⎦,𝐵=⎡⎢⎢⎢⎢⎢⎢⎣00100001⎤⎥⎥⎥⎥⎥⎥⎦,𝑓⁽²⁾(𝑥)=⎡⎢⎢⎢⎢⎢⎢⎣𝑘¹0𝑥₃𝑥₄𝑘₂𝑥₂𝑥₄𝑘₃𝑥₂𝑥₃𝑘₁⎤⎥⎥⎥⎥⎥⎥⎦,=1.5𝑝𝐿^𝑑−𝐿_𝑞𝑎₄𝐽𝑎₁,𝑘₂=−𝐿^𝑑𝑝𝑎₁𝑎₄𝐿_𝑞𝑘₃,=𝐿^𝑞𝐶₁₂𝑎₁𝐿_𝑑𝑎₄𝑎₁,=1.5𝑝𝜆𝐽,𝑎⁴=−𝑅𝐿_𝑑,𝐶₁=1𝐿_𝑞.(17)

4. Linearization of Interior Permanent Magnet (IPM) Synchronous Motor

The quadratic linearization given in Section 2 is extended to the case of two inputs in a straightforward way and applied to a PM machine model in this section [14, 15].

Theorem 2. Given the 4-dimensional model of a PM synchronous motor (IPM model) of the form (16), the system can be linearized using the following transformations: 𝑢=𝐼₂𝑦=𝑥+𝜙⁽²⁾(𝑥),(18)+𝛽⁽¹⁾(𝑥)𝑣+𝛼⁽²⁾(𝑥),(19) where 𝑢=[𝑢₁𝑢₂]^𝑇, 𝑣=[𝑣₁𝑣₂]^𝑇, 𝜙⁽²⁾(𝑥)=⎡⎢⎢⎢⎢⎢⎢⎣𝑘₁00𝑥₃𝑥₄𝛼0⎤⎥⎥⎥⎥⎥⎥⎦,(20)₍₂₎(𝑥)=⎡⎢⎢⎣−𝑘²𝑥₂𝑥₄−𝑘₃𝑥₂𝑥₃𝛽⎤⎥⎥⎦,(21)₍₁₎(𝑥)=(−1)𝐵^𝑇𝜕𝜙⁽²⁾(𝑥)𝜕𝑥𝐵=−⎡⎢⎢⎣𝑘¹𝑥₄𝑘₁𝑥₃00⎤⎥⎥⎦,(22) where 𝐼₂ is the identity matrix of order 2.
The system then reduces to 𝑦̇=𝐴𝑦+𝐵𝑣+𝑂⁽³⁾(𝑦,𝑣),(23) where 𝑂⁽³⁾(𝑦,𝑣) represents third- and higher-order nonlinearities.

Proof. Applying transformations (18) and (19) (which correspond to a natural extension of (9) and (10) to two inputs) to (16), the homological equations to be considered can be written as −𝐴𝜙⁽²⁾(𝑥)+𝐵𝛼⁽²⁾(𝑥)+𝑓⁽²⁾(𝑥)+𝜕𝜙⁽²⁾(𝑥)𝐵𝛽₍₁₎𝜕𝑥𝐴𝑥=0,(24)(𝑥)+𝜕𝜙⁽²⁾(𝑥)𝜕𝑥𝐵=0.(25) By choosing 𝜙⁽²⁾(𝑥) and 𝛼⁽²⁾(𝑥) as in (20) and (21), (24) reduces to −⎡⎢⎢⎢⎢⎢⎢⎣𝑘¹0𝑥₃𝑥₄00⎤⎥⎥⎥⎥⎥⎥⎦+⎡⎢⎢⎢⎢⎢⎢⎣−𝑘₂00𝑥₂𝑥₄−𝑘₃𝑥₂𝑥₃⎤⎥⎥⎥⎥⎥⎥⎦+⎡⎢⎢⎢⎢⎢⎢⎣𝑘₁0𝑥₃𝑥₄𝑘₂𝑥₂𝑥₄𝑘₃𝑥₂𝑥₃+⎡⎢⎢⎢⎢⎢⎢⎣000000𝑘₁0000⎤⎥⎥⎥⎥⎥⎥⎦𝑥₄𝑘₁𝑥₃0000⎤⎥⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎢⎣𝑥²𝑥₃00⎤⎥⎥⎥⎥⎥⎥⎦=0.(26)
Premultiplying (25) by 𝐵^𝑇 and noting that 𝐵^𝑇𝐵=𝐼₂, (25) reduces to 𝛽⁽¹⁾(𝑥)=(−1)𝐵^𝑡𝜕𝜙⁽²⁾(𝑥)𝜕𝑥𝐵.(27) Also, substitution of (27) satisfies (25). Hence, the homological equations are satisfied and quadratic linearization is achieved, hence the proof.

5. Experimental Simulation Results

Given the parameters 𝑅=2.875 Ω, 𝐿_𝑞=9 mH, 𝐿_𝑑=7 mH, 𝜔_𝑒=3500 rpm, 𝑝=4, 𝐽=0.0008 kgm² and 𝜆=175 mWb turns, of an actual PM machine, Figure 1 shows the SIMULINK model of the PMSM which is constructedby using speed and torque blocks and control circuit. 𝑣_𝑞 and 𝑣_𝑑 are taken as inputs to the motor. The model is reduced to normal form in a standard way by using the linear transformations. The PM model in Figure 1 is especially configured for the IPM where 𝐿_𝑑≠𝐿_𝑞. 𝑁₁ and 𝑁₂ blocks in Figure 2 include the above linear transformation and the nonlinear transformation (18) and (19) as well.

Figure 1 

PMSM design using SIMULINK.

Figure 2 

Linearization of PMSM.

Prior to linearization, the open loop steady state gain of 𝜔_𝑒 versus 𝑣_𝑞 of the PMSM model is investigated and the results are given in Table 1. In this table, it is observed that the open loop steady state gain of 𝜔_𝑒 versus 𝑣_𝑞 (keeping 𝑣_𝑑 constant) is not constant because of the system nonlinearity. To verify the linearity of the system after linearization, we investigated the variation of its gain of 𝑦₂ (a scaled version of 𝜔_𝑒 as can be seen from (18) and (20)) with input 𝑣₁ (see Figure 2) and the results are given in Table 2. The table reveals that the gain of the system is nearly constant, thus verifying that by applying the homogeneous linearizing transformation, the PMSM model is made nearly linear for the given set of inputs.

Table 1 

Steady state gain of 𝜔_𝑒 versus 𝑣_𝑞 for the system in open loop (prior to linearization).

Table 2 

Steady state gain of 𝑦₂ versus 𝑣₁ for the linearized system in open loop.

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 3 and 4 show the time response of angular velocity 𝜔_𝑒 by closing the loop around PMSM model before linearization when 𝑣_𝑞=5 and 30 units, respectively. It is observed that the dynamic response for 𝑣_𝑞=5 is more oscillatory compared to the case of 𝑣_𝑞=30 with a fixed controller of proportional gain = 50 and integral constant = 2. This is to be expected since the loop gain is higher in the former case with a higher static gain in the motor as can be seen from Table 1.

Figure 3 

Time response of angular velocity in closed loop when 𝑣𝑞=5; 𝑘𝑝=50; 𝑘𝑖=2 (before linearization).

Figure 4 

Time response of angular velocity in closed loop when 𝑣𝑞=30; 𝑘𝑝=50; 𝑘𝑖=2 (before linearization).

Figures 5 and 6 show the time response of 𝑦₂ of the transformed PMSM system (Figure 2) in closed loop when 𝑣₁=5 and 30 units, respectively. It is observed that a uniform output response is obtained in the closed loop after linearization when the reference is varied. Since the static gain in Table 2 is nearly uniform, the loop gain is also nearly constant for the extreme points in the operating range, thus resulting in the uniform dynamic responses as in Figures 5 and 6.

Figure 5 

Time Response of 𝑦2 for the linearized system in closed loop when 𝑣1=5; 𝑘𝑝=50; 𝑘𝑖=2.

Figure 6 

Time Response of 𝑦2 for the linearized system in closed loop when 𝑣1=30; 𝑘𝑝=50; 𝑘𝑖=2.

Figures 7 and 8 show the time response of angular velocity 𝜔_𝑒 of the PMSM model before linearization in closed loop when 𝑣_𝑞=15 units with the load torque 𝑇 varied as 𝑇=0 and 1 unit, respectively. Note that the dynamic responses in Figures 7 and 8 are different, with the response of Figure 8 corresponding to an inverse response (i.e., speed reduces first before increasing). The latter corresponds to a response of a nonminimum phase system.

Figure 7 

Time response of angular velocity in closed loop when 𝑣𝑞=15 units, 𝑇=0 N-m; 𝑘𝑝=50; 𝑘𝑖=2 (before linearizaton).

Figure 8 

Time response of angular velocity in closed loop when 𝑣𝑞=15 units, 𝑇=1 N-m; 𝑘𝑝=50; 𝑘𝑖=2 (before linearization).

Figures 9 and 10 show the time response of 𝑦₂ of the transformed PMSM system in closed loop when 𝑣₁=15 units and 𝑇=0 and 1 unit, respectively. It is observed that a uniform output response is obtained in the closed loop after linearization when load torque is varied. Hence, it is verified that a linearized system gives a uniform closed loop response for the different reference and load conditions used for testing as above.

Figure 9 

Time Response of 𝑦2 for the linearized system in closed loop when 𝑣1=15 units, 𝑇=0 N-m; 𝑘𝑝=50; 𝑘𝑖=2.

Figure 10 

Time Response of 𝑦2 for the linearized system in closed loop when 𝑣1=15 units, 𝑇=1 N-m; 𝑘𝑝=50; 𝑘𝑖=2.

6. Tuning

6.1. Core Loss

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 𝑃_𝑙𝑐 [2] for the machine is computed as follows: 𝑃_𝑙𝑐=1.5𝜔^𝑟2𝐿_𝑞𝐼_𝑞²𝑅_𝑐+1.5𝜔^𝑟2𝜆_𝑎𝑓+𝐿_𝑑𝐼_𝑑²𝑅_𝑐,(28) where 𝑅_𝑐 represents core loss resistance, 𝜆_𝑎𝑓 represents magnet flux linkage and 𝜔_𝑟 denotes rotor electrical speed.

The mechanical torque equation including core losses is given by𝑇_𝑒=2𝑃𝐽𝑑𝑑𝜔𝑡^𝑒+𝑇^𝑙𝑇+𝑇_𝑙𝑐,(29)_𝑙𝑐=𝑃^𝑙𝑐𝜔_𝑟.(30)

6.2. Tuning Formula

To account for the core loss (28), unmodelled dynamics, and higher-order nonlinearities, tuning of the quadratic linearization transformations 𝑁₁ and 𝑁₂ can be done [13]. Figure 11 shows the block diagram for tuning.

Figure 11 

Block diagram for tuning of transformation.

Error (𝐸) can be calculated as 𝐸=𝜀^𝑇𝜀=(𝑦−𝑦)̂^𝑇(𝑦−𝑦)̂^1/2,(31) where 𝜀=(𝜀₁𝜀₂𝜀₃𝜀₄)^𝑇.

The error can be written as 𝐸=𝜀²¹+𝜀²²+𝜀²³+𝜀²⁴^1/2.(32) Since 𝜙⁽²⁾(𝑥) and 𝛽⁽¹⁾(𝑥) are both functions of 𝑘₁, we shall redefine 𝜙⁽²⁾(𝑥)=⎡⎢⎢⎢⎢⎢⎢⎣𝑘₁00𝑥₃𝑥₄𝛽₍₁₎0⎤⎥⎥⎥⎥⎥⎥⎦,(𝑥)=−⎡⎢⎢⎣𝑘^1𝑥₄𝑘^1𝑥₃00⎤⎥⎥⎦(33) so that 𝜙⁽²⁾(𝑥) and 𝛽⁽¹⁾(𝑥) can be independently tuned by tuning 𝑘₁ and 𝑘^1, respectively. 𝛼⁽²⁾(𝑥) is not varied.

6.2.1. Updation of 𝑁₂ Transformation Coefficients

Tuning of 𝑁₂ transformation implies the tuning of 𝜙⁽²⁾(𝑥). As 𝜙⁽²⁾(𝑥) is a function of only 𝑘₁𝑥₃𝑥₄, the coefficient 𝑘₁ has to be updated based on the error between the outputs of quadratic linearized system and normal form. The updation law is derived as follows: Δ𝑘₁=𝜕𝐸𝜕𝑘₁=𝜕𝐸𝜕𝑦𝜕𝜕𝑘𝑦₁.(34) From (31), it is seen that 𝜕𝐸/𝜕𝑦_𝑖=𝜀_𝑖/𝐸, 𝑖=1,2,3,4.

Hence, 𝜕𝐸/𝜕𝑦=[𝜀₁/𝐸𝜀₂/𝐸𝜀₃/𝐸𝜀₄/𝐸],∴Δ𝑘₁=𝜀¹𝐸𝜀𝐸²𝜀𝐸³𝜀𝐸⁴⎡⎢⎢⎢⎢⎢⎢⎣𝑥₃00𝑥₄0⎤⎥⎥⎥⎥⎥⎥⎦=𝜀³𝐸𝑥³𝑥₄.(35) Updation of 𝜙⁽²⁾(𝑥) is done by using the formula 𝑘₁(𝑚)=𝑘₁(𝑚−1)−𝜌₁Δ𝑘₁(𝑚);0<𝜌₁<1,(36) where 𝑚 corresponds to the updating step and 𝜌₁ corresponds to the accelerating factor.

6.2.2. Updation of 𝑁₁ Transformation Coefficients

Tuning of 𝑁₁ transformation is achieved by tuning of 𝛽⁽¹⁾(𝑥). As 𝛽⁽¹⁾(𝑥) is a function of 𝑘^1𝑥₃ and 𝑘^1𝑥₄, the coefficient 𝑘^1 has to be updated based on the error between the outputs of quadratic linearized system and normal form. The updation law is derived as follows: Δ𝑘^1=𝜕𝐸𝜕𝑘^1=𝜕𝐸𝜕𝑦𝜕𝜕𝑥𝑦𝜕𝜕𝑢𝑥₁𝜕𝑢₁𝜕𝑘^1,(37) where 𝜕𝐸𝜕𝑦𝜕𝑦=𝜀𝐸¹𝜀𝐸²𝜀𝐸³𝜀𝐸⁴,𝜕𝑥=⎡⎢⎢⎢⎢⎢⎢⎣1000101+00𝑘_1𝑥₄00𝑘^1𝑥₃0001⎤⎥⎥⎥⎥⎥⎥⎦,𝜕𝑥𝜕𝑢₁=⎡⎢⎢⎢⎢⎢⎢⎢⎣𝑘−²0𝑥1⁴−01𝑘₂𝑥₂⎤⎥⎥⎥⎥⎥⎥⎥⎦,(38) assuming that the steady state of the SIMULINK model is reached within the tuning period: 𝜕𝑢₁𝜕𝑘^1=−𝑣₁𝑥₄−𝑣₂𝑥₃.(39) Thus, Δ𝑘^1=((𝑣₁𝑥₄+𝑣₂𝑥₃)/𝐸)(𝜀₂/𝑘₂𝑥₄+(𝜀₃𝑘^1𝑥₃+𝜀₄)/𝑘₂𝑥₂).

Updation of 𝛽⁽¹⁾(𝑥) is done by using the formula 𝑘^1(𝑚)=𝑘^1(𝑚−1)−𝜌₂Δ𝑘^1(𝑚),0<𝜌₂<1,(40) where 𝑚 corresponds to the updating step and 𝜌₂ corresponds to the accelerating factor.

6.3. Tuning Simulation Results

PMSM model including core loss is given in Figure 12 where the block due to loss torque (30) is also included. The tuning of the transformations is done using the updations for 𝜙⁽²⁾(𝑥) and 𝛽⁽¹⁾(𝑥) as per formulae (36) and (40) as given in Figure 13. The simulation diagram for tuning is done by using memory blocks to store the updated values of 𝜙⁽²⁾(𝑥) and 𝛽⁽¹⁾(𝑥). It is seen that the error after tuning is reduced to 0.01. Table 3 shows the variation of 𝑦₂ versus 𝑣₁ of the PMSM model after linearization including core loss prior to tuning of the coefficients. It is observed that the open loop steady state gain is not constant due to the effect of core loss. Table 4 shows the variation of 𝑦₂ versus 𝑣₁ for the linearized system after incorporating tuning of the transformation coefficients. The table reveals that the gain of the system is nearly constant thus verifying the effectiveness of tuning. Figures 14 and 15 show the closed loop time response of 𝑦₂ of the linearized system including core loss before tuning when 𝑣₁=5 units and 30 units, respectively; 𝑘_𝑝=50; 𝑘_𝑖=2. Figures 16 and 17 show the closed loop time response of 𝑦₂ of the linearized system including core loss after tuning when 𝑣₁=5 units and 30 units, respectively; 𝑘_𝑝=50; 𝑘_𝑖=2. It is observed that the dynamic response before tuning for 𝑣₁=5 units is more oscillatory compared to the case of 𝑣₁=30 units. This is to be expected since the loop gain is higher in the former case with a higher static gain in the plant or motor as can be seen from Table 3. Since the static gain in Table 4 is nearly uniform, the loop gain is also nearly constant for the extreme points in the operating range, thus resulting in the uniform dynamic responses in Figures 16 and 17. Thus, the effectiveness of tuning the linearizing transformations to overcome the unmodelled core loss is verified.

Table 3 

Steady state gain of 𝑦₂ versus 𝑣₁ for the linearized system in open loop before tuning.

Table 4 

Steady state gain of 𝑦₂ versus 𝑣₁ for the linearized system in open loop after tuning.

Figure 12 

PMSM model including core loss.

Figure 13 

Simulation diagram for controller tuning.

Figure 14 

Time response of angular velocity in closed loop when 𝑣𝑞=5 unit; 𝑘𝑝=50; 𝑘𝑖=2 before tuning.

Figure 15 

Time response of angular velocity in closed loop when 𝑣𝑞=30 unit; 𝑘𝑝=50; 𝑘𝑖=2 before tuning.

Figure 16 

Time response of 𝑦2 for the linearized system in closed loop when 𝑣𝑞=5 unit; 𝑘𝑝=50; 𝑘𝑖=2 after tuning.

Figure 17 

Time response of 𝑦2 for the linearized system in closed loop when 𝑣𝑞=30 unit; 𝑘𝑝=50; 𝑘𝑖=2 after tuning.

7. Conclusion

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 [9, 18]. The existing techniques of exact feedback linearization introduce singularities in the system, which may cause difficulties in the implementation of the closed loop control. In the proposed method, the problem of singularities does not arise.

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 [19].