Fuzzy PDFF-IIR controller for PMSM drive systems

Fuzzy PDFF-IIR controller for PMSM drive systems

Stone Cheng

n

, Chi-Wei Li

Department of Mechanical Engineering, National Chiao Tung University, EE426, 1001 University Road, Hsinchu 300, Taiwan, ROC

a r t i c l e

i n f o

Article history:

Received 3 February 2010 Accepted 8 April 2011 Available online 25 May 2011 Keywords: PMSM PDFF Fuzzy control Servo control Field-oriented control

a b s t r a c t

Pseudo-derivative feedback with feed-forward gain (PDFF) combines the advantages of proportional-integral (PI) and pseudo-derivative feedback (PDF) controllers. However, PDFF responds more slowly to a command than does PI. To increase the speed of response of the PDFF controller, this work presents a PDFF with moving average errors control. A low-pass IIR filter path for errors compensation that accelerates the slow response is added to a PDFF control loop. A fuzzy inferencer is utilized to adjust the feed-forward gain and integral gain of the PDFF controller to allow closed-loop poles of the transfer function to be properly placed to improve load torque disturbance rejection capability. Simulated and experimental results reveal that the response and load disturbance rejection ability of the fuzzy PDFF-IIR controller are better than those of the traditional PDFF controller.

&2011 Elsevier Ltd. All rights reserved.

1. Introduction

Permanent magnet synchronous motor (PMSM) servo drives are important in high-accuracy and high-performance drive systems in industrial motion control applications. In such appli-cations, the classical multi-loop structure, in which each loop is controlled using an appropriate controller, has been extensively utilized. Currently, motor servo drivers in industrial products are based mostly on proportional-integral (PI) controllers. The PI controller has the advantages of a simple algorithm, high relia-bility, high starelia-bility, and excellent command response, but it tends to overshoot and provide a low DC stiffness. However, a favorable response and load disturbance rejection are very important in most motion control systems. The pseudo-derivative feedback with a feed-forward gain (PDFF) controller, which combines the advantages of PI and the pseudo-derivative feed-back (PDF) controller, satisfies the aforementioned requirements. It provides a trade-off between the command response and the DC stiffness by appropriately tuning three constant gains. Furthermore, it avoids velocity overshoot when a motor accel-erates or decelaccel-erates. Accordingly, the PDFF control algorithm provides a flexible solution in various cases of motion control.

Research on PDFF controllers has attracted attention in recent years. Some control design schemes for improving the perfor-mance of PDFF control or applications have been proposed, such

as the papers by Liu and Li (2008), Cheng, Huang, and Chou

(2008), andTsai and Shen (2006, 2007). Furthermore, some robust

PDFF controller products have been commercialized for industrial applications, including the VFD-VE series, introduced by Delta

Electronics in 2007 and the Servostar CD series, introduced by Kollmorgen. Accordingly, the importance of PDFF control in servo systems has gradually increased. Self-tuning adaptive regulator controllers, that take both regulations and observations capabil-ities into account and are able to modify system parameters in order to maintain the desired response of the system, are being increasingly used (Li & Liu, 2009;Mohamed, 2007). However, the complexity of these algorithms is a large barricade to low cost PMSM drives.

The structure of PDFF controller is less responsive than PI algorithm under the same conditions, but allows closed-loop poles of the transfer function to be properly placed to improve

load torque disturbance rejection capability (Romeral &

Chekkouri, 2002; Shen & Tsai, 2006). In this study, a fuzzy

PDFF-IIR controller with improved system response and load disturbance rejection capability is designed and analyzed. The fuzzy PDFF-IIR controller comprises three parts: (1) a PDFF scheme prevents overshoot; (2) a low-pass infinite impulse response (IIR) filter loop adjusts the system response, and (3) a fuzzy controller enhances the load disturbance rejection capacity. Each element of this combination is important and will be detailed. The important contributions of the proposed scheme are as follows:



_{The structure of the fuzzy PDFF-IIR model provides dynamic}

feed-forward action a given drive constraints of stability margins. It also naturally adapts to varying environmental load disturbance conditions and can therefore be generalized to more complex applications. Furthermore, the PDFF and IIR model actions are kept simple for fuzzy inference algorithm to tune the parameters and therefore are easy to implement. The model requires only two gains to be adjusted by fuzzy inferencer over the entire operating range. The results of the Contents lists available atScienceDirect

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Control Engineering Practice

0967-0661/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2011.04.011

n

Corresponding author. Fax: þ886 35720634.

experiment herein demonstrate that the fuzzy PDFF-IIR model is effective for controlling the PMSM system.



The novelty of the proposed strategy is in the combination of PDFF, fuzzy inference, and an IIR filter scheme. The approach takes into account enhanced DC stiffness, the disturbance rejection, and the improvement in the response speed in a simple and highly efficient control law, as revealed by exten-sive experimental results.

2. Two-axis dynamic model of PMSM

Field-oriented control (FOC) is applied on the PMSM in the AC servo system. The field orientation of the PMSM requires that the dynamic equations in the stator axis frame be transformed to the rotor synchronous axis frame (d–q-axis). The nonlinear dynamic d–q model of the PMSM machine at the synchronous reference

frame was derived in an adequate form (Chiasson, 2005). This

model consists of nonlinear differential equations that relate the electrical quantities, currents, fluxes, and voltages, with the mechanical quantities, torque, and speed. The goal of the FOC is to perform real-time control of torque variations demand, control the rotor mechanical speed, and regulate phase currents in order to avoid current spikes during transient phases. To perform these controls, the electrical equations are projected from a 3 phase non-rotating frame into a 2-phase (d, q) system that rotates at the speed of the electrical speed of the rotor. In d–q coordinates, the PMSM electrical model equations are

vds vqs " # ¼ Rs 

o

eLqs

o

eLds Rs " # ids iqs " # þ Lds 0 0 Lqs " # d dtids d dtiqs " # þ 0

o

e

l

f " # Te¼32 Np 2

l

fiqsþ ðLdsLqsÞidsiqs 8 > > > < > > > : ð1Þ where vdsand vqsare the stator voltages of the d and the q axes,

and idsand iqsare the d and the q axes stator currents,

respec-tively, Rs is the stator resistance, Lds and Lqs denote the stator

inductances of the d and the q axes, respectively;

o

e is the

electrical angular velocity of the rotor,

l

fis the flux linkage of

the rotor, Teis the electromagnetic torque of the motor, and Npis

the number of poles of the motor. Therefore, the nonlinear state equations of idsand iqsare given as

dids dt ¼L1dsðvdsRsidsþ

o

eLqsiqsÞ diqs dt ¼ 1 LqsðvqsRsiqs

o

eLdsids

o

e

l

fÞ 8 < : ð2Þ

It can be seen that the electromagnetic torque Tecan be controlled

by regulation of currents idsand iqsin closed loop. The direct-axis

current idscorresponds to the component of stator magnetic field

along the axis of the rotor magnetic field, while the quadrature-axis current iqscorresponds to the orthogonal component. Considering

field-oriented control for PMSM that idsdoes not contribute to the

torque, voltage inputs are designed to guarantee the convergence of idsand iqsto their desired control inputs ids,iqs, by this means the control scheme yields i

ds¼0, the desired electromagnetic torque being directly proportional to the desired current input i

qs. For the currents, tracking control defines the current errors as

eds¼idsids eqs¼iqsiqs (

ð3Þ And the dynamics derived from (2) and (3) are

deds dt ¼  1 LdsðvdsRsidsþ

o

eLqsiqsÞ deqs dt ¼  1 LqsðvqsRsiqs

o

eLdsids

o

e

l

fÞ 8 < : ð4Þ

In order to ensure the convergence of the current errors to zero, the positive definite Lyapunov function is chosen as

V ¼1

2ðe 2

dsþe2qsÞ ð5Þ

The derivative of (5) is computed as _

V ¼ edsU_edsþeqsU_eqs¼eds  1 Lds ðvdsRsidsþ

o

eLqsiqsÞ   þeqs  1 Lqs ðvqsRsiqs

o

eLdsids

o

e

l

fÞ   ð6Þ To guarantee the global asymptotic stability in the current loop, the control voltage of d–q-axis is chosen as

vds¼KdsLdsedsþRsids

o

eLqsiqs vqs¼KqsLqseqsþRsiqsþ

o

eðLdsidsþ

l

fÞ (

ð7Þ where Kdsand Kqsare positive constant gains.

Substituting (7) into (6) the derivative of the Lyapunov func-tion becomes

_

V ¼ ðKdse2dsþKqse2qsÞr0 ð8Þ

Clearly, (8) is only negative semi-definite. Since the PMSM must settle down at ð _eds¼0, _eqs¼0Þ and ðeds¼0, eqs¼0Þ, according to the invariant set theorem (Haddad & Chellaboina, 2008), the system is globally asymptotically stable in the sense of Lyapunov at the origin. It implies that the resulting current closed-loop is asymptotically stable and, hence, the error variables eds and eqs

will converge to zero asymptotically.

However, the control of such drive system is complicated because of the coupling between all control inputs. This non-linear dynamic model should be non-linearized to a non-linear time-invariant dynamic model for the PMSM machine at field orientation. On the other hand, the paper focuses on the motor–load system with low resonant frequency and small load–motor inertia ratio. The influence of the simplification of the two-mass system to the one-mass system on the dynamic property of the closed-loop control structure is small. In this case, the motor torque equation is

o

e¼N₂p

o

m Vqs |{z} q-axis voltage ¼ eqGiq zfflffl}|fflffl{V  qs¼ |fflffl{zfflffl} q-axis control voltage þ Np 2

o

m

l

f |fflfflfflfflffl{zfflfflfflfflffl} q-axis back emfcompensation

Vds |{z} d-axis voltage ¼ edGid zfflffl}|fflffl{V  ds¼ |fflffl{zfflffl} d-axis control voltage Np 2

o

mLqsIqs Iqs¼_ðsL 1 qsþRaÞ Vqs Np 2

o

m

l

f |fflfflfflfflffl{zfflfflfflfflffl} Ke 0 B B @ 1 C C A Te¼3N4p

l

fiqs¼KTiqs¼JmdodtmþBm

o

mTL 8 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > : ð9Þ

where eqis the error of q-axis current, Giqis the controller of

q-axis, s is the Laplace operator, KTis the motor torque constant,

TL represents the torque disturbance, Jm is the moment of

inertia,

o

m is the mechanical angular velocity of the rotor,

and Bmis the viscous friction. Since the magnetic flux from the

permanent magnetic rotor is fixed in relation to the rotor shaft position, the flux position in the d–q co-ordinates can be

determined by the shaft-position sensor. Wallace, Novotny,

ampere attainable is not severely affected by the effect of non-ideal field-oriented operation, indicating that this simple, con-stant parameter field-oriented controller can be applied in some situations without resorting to complex controllers, which compensate for the large flux changes.

3. Design and simulation of control system 3.1. Control scheme

Fig. 1shows the function block diagram of speed controller

and PMSM that adopts the FOC algorithm. The figure presents the fuzzy PDFF-IIR control scheme. Notably, the output of the low-pass IIR is added to the second summing point to enhance the system response. FLC is a fuzzy logic controller that is utilized to adjust the feed-forward gain (Kff) and the integral gain (Ki) of the

PDFF controller, to reduce the disturbance TL.

The linear mathematical model of a PMSM plant transfer

function from voltage command

o

r to motor speed

o

m can be

modeled by a second-order system as GpðsÞ ¼ Kt ðsLssþRaÞðsJmþBmÞ ¼ K1 ðs þ 1=

t

eÞðs þ1=

t

mÞ ð10Þ where

t

mis mechanical time constant, and

t

e is electrical time

constant of the PMSM plant. When high gain closed-loop current control is implemented as a minor-loop control, two poles are pushed apart, so that the mechanical time constant is increased, while the electrical time constant is decreased (7). With high gain current-loop control, the plant can be simplified as a first-order model of

GpðsÞ ¼ K1

t

e s þ 1=

t

m

ð11Þ For this system, a PDFF control structure (3) can be constructed by pole- placement type controller, because this scheme moves the locations of open-loop poles without affecting open-loop zeros, with a feed-forward term adding to the controller, as shown in

Fig. 1. In order to study characteristics of the PDFF controllers, a

set of transfer functions is derived by selecting various input and output points fromFig. 1as shown in the following:

command to output : GcðsÞ ¼o_omðsÞ

rðsÞ ¼

K1teðKiþsKffÞ

s2þ ð1=tmþK1teKfÞs þ K1teKi

command to error : GeðsÞ ¼_oeðsÞ_rðsÞ¼ s2_{þ ð1=t} mþK1teKfK1teKffÞs s2_{þ ð1=t} mþK1teKfÞs þ K1teKi 8 > < > : ð12Þ

Consider state error of unit step and ramp inputs,

steady-state errors ess can be calculated by applying the final value

theorem to (12) as Unit step input : ess¼lim

s-0 s GeðxÞ

s

 

¼0 Ramp input : ess¼_oeðsÞ

rðsÞ¼lims-0 s GeðsÞ s2   ¼1=tmþK1teðKfKffÞ K1teKi 8 > < > : ð13Þ

Since the mechanical time constant of PMSM is large in most motion control applications, the ramp command error will be very small when the feed-forward gain and feedback gain are equal. A feed-forward gain Kffinjects the command ahead of the

integral making the system more responsive to commands. When

the application requires low-frequency stiffness, low Kff gain

allows much higher integral gain without inducing overshoot. Unfortunately, it also makes the system slower in responding to

the command. The experiments of motion control with Kff¼60%

usually give good results. Over the entire range of Kff , the

response increases in a velocity loop controller by about 45% and the DC stiffness declines by about five times. PDFF controller gives flexibility in performance requirements by tuning feed-forward gain and integral gain for best reference tracking response without overshoot, or greatest low-frequency stiffness.

Furthermore, to avoid oscillatory response in Gc(s), the

damp-ing ratio must be

z

Z0.707, which gives the relationship between parameters of the PDFF controller

1:5 Z

z

¼ð1=tmþK1teKfÞ 2 ffiffiffiffiffiffiffiffiffiffiK1teKi p Z0:707 Kir ð1=tmþK1teKfÞ2 2K1te KfZ1:414 ffiffiffiffiffiffiffiffiffiffi K1teKi p 1=tm K1te 8 > > > > < > > > > : ð14Þ

3.2. Low-pass IIR path design

The low-pass IIR path and fuzzy logic controller are designed herein. First, the following two filters structures are considered. Experiments on system at sampling frequency of 5KHz is carried out to implement the IIR filter.

Filter 1: IIR1¼F1ðzÞ ¼

k

cbz1 ð15Þ

Eq. (15) is the transfer function of the first-order IIR, where k, b, and c are three positive tunable parameters. The values of these parameters affect the system response. A larger k corresponds to a

faster system response, because k is equivalent to the gain of IIR1.

Parameters c and b determine the pole of IIR1. When the stable

pole is closer to the unit circle in the z plane, the system response is fast.Fig. 2compares the simulation results for velocity control loops associated with a PDFF controller with those associated

with a PDFF plus IIR1 path controller. The system response is

improved by adding the low-pass IIR path to the traditional PDFF construction. However, a positive output from the IIR filter degrades system response. Additionally, PDFF with only IIR filter cannot reduce the step disturbance, because all areas under the velocity responsive curve are the same. For most motion systems, the area under the velocity response is commonly more impor-tant than the peak excursion, because the area under the velocity curve represents the error in the position of the system.

Fig. 3(a) and (b) shows the effects of parameter k and the pole

location of IIR1, respectively, on the system response. InFig. 3(a),

the pole was set to 2/7 and remained constant. In Fig. 3(b),

parameter k was set to unity and remained constant. Filter 2:

Filter 1 is a simple low-pass IIR filter. Generally, the IIR filter design relies on a common analog filter to yield a digital filter. In

Filter 2, a Butterworth low-pass filter FB(z) is applied to

strengthen the system response. The order N and cutoff frequency

o

c of the filter are parameters that must be designed.

Fig. 4(a) compares the results of the simulation of the velocity

control loops with PDFF and those with PDFF plus a Butterworth IIR controller. Here, the IIR filter’s cutoff frequency is 1 kHz and the order is two to compensate the velocity loop response of 500 Hz bandwidth. The figure shows that a negative output from the IIR filter can accelerate the system response but includes an overshoot. Hence, the design must be revised to reduce the output signal of the filter. The revised design satisfies

F0

BðzÞ ¼ KFBðzÞ ð16Þ

where FB(z) is the original Butterworth filter, F0BðzÞ is the modified Butterworth filter, and K is a constant gain between zero and unity that limits the filter’s output.Fig. 4(b) compares the results of the simulation of the velocity control loops with PDFF with those of PDFF plus the modified Butterworth IIR controller with K¼0.45.

Fig. 5 presents the influence of the order N and cutoff

frequency on the system response. In Fig. 5(a), the cutoff

frequency is set to 200 Hz, 1 kHz, and 3 kHz, and the order is two. The K gain is set to prevent overshoot. Intuitively, increasing the cutoff frequency considerably increases the system response. In

Fig. 5(b), the cutoff frequency and K gain are set to 1 kHz and 0.4,

respectively. A high-order Butterworth filter does not enhance the system response or make it smooth. A high-order filter produces vibration in the response. Consequently, the design should incorpo-rate a lower-order Butterworth filter. In summary of the comparison of filters 1 and 2, IIR1 has a simple structure and is easier to be

PDFF with IIR (minus)

1 0.8

PDFF with IIR (plus)

0.6 speeds (rad/s) 0.4 0.2 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 time (seconds) PDFF with IIR(plus) PDFF PDFF with IIR(minus)

Fig. 2. Comparison of performance of PDFF and PDFF with IIR1controller.

1.2 1 0.8 0.6 0.4 speeds (rad/s) speeds (rad/s) 0.2 0 0 0.005 0.01 0.015 -0.2 time (seconds) 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 0 0.005 0.01 0.015 0.02 0.025 -0.4 time (seconds) pole = 0.2 pole = 0.4 pole = 0.6 pole = 0.8 pole = 1 a=0.5 a=1.5 a=2.5 a=3.5 a=4.5

Fig. 3. System responses with varying parameters: (a) k and (b) pole locations.

1.4 minus 1.2 minus 1 0.8 plus plus 0.6 0.4 speeds (rad/s) 0.2 0 -0.2 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 time (seconds) 1 0.8 0.6 0.4 0.2 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 time (seconds)

with IIR (plus) only PDFF with IIR (minus)

PDFF with modified IIR(minus) PDFF

speeds (rad/s)

Fig. 4. Performance comparisons under the PDFF with Butterworth filter com-pensation: (a) original Butterworth filter and (b) modified Butterworth filter.

implemented than second-order filter. The determination of the parameters in both the filters is similar, with cutoff frequency and gain. Both methods improve the system response but neither reduces the load disturbance rejection. For this reason Ki and Kf

should be set as high as possible. 3.3. Design of fuzzy logic controller

The above simulation results reveal that a controller using PDFF plus IIR improves only the system response and does not attenuate the step load disturbance. In this study, a fuzzy logic controller that exploits the properties of PDFF is designed to address load disturbance. Since PDFF combines the advantages of PI and PDF controllers, it frequently has a feed-forward gain (Kff)

to feedback gain (Kf) ratio of p. When p is one, PDFF and PI are

equivalent; when p is zero, PDFF and PDF are equivalent. PDF is a simple one-degree-of-freedom controller and is less responsive to the reference input than is PI control, but it can provide a higher DC stiffness. Its contribution to the robustness of the system parameters and external disturbance rejection exceeds that of PI control, because DC stiffness is proportional to the integral gain

(Ellis, 1999). However, PI control easily causes a large overshoot

when the integral is high. In PDF control (Ellis & Lorenz, 1999;

Ohm, 1994; Perdikaris, 1999), the entire signal is filtered by

determining the integral. Therefore, the integral gain can be increased without causing overshoot, unlike in PI control.

In order to develop the supervisory system to tune the PDFF controller, some expressions must be carried out



Eq. (13) shows that integral term Kicannot lead to eliminate

the ramp tracking error. It can be only reduced if Kiincreases

greatly.



A feed-forward gain Kffmakes the system more responsive to

commands.



The effect of disturbance to the output is dependent on

damping ratio, a normal choice of damping ratio would be 0.707–1.5.



Pole-zero cancellation tuning should be avoided because of its poor robustness characteristics and slow load torque distur-bance response.

In this investigation, the fuzzy tracking block uses the same Mamdani fuzzy reasoning method and centroid defuzzification approach. The fuzzy logic controller has a single normalized input variable (error signal, E) to simplify the design of the membership functions and two output variables (u1and u2), where u1and u2

are adopted to regulate the feed-forward gain and the integral gain, respectively. Here, one input variable suffices for the application of FLC to electrical system; the use of plant evolution variables, the derivative of the speed error, does not improve controller performance from the results of simulation and experi-ment. The fuzzy PDFF-IIR control algorithm is expressed as iqcðkÞ ¼ u2Ki 1z1  eðkÞ þ u1Kff

o

mðkÞKff

o

mðkÞ þfoðkÞ ð17Þ where e(k)¼

o

m n 

o

m/

o

m n

is the normalized input,

o

m n

refers to the velocity set point at the kth sample time, and

o

mdenotes the

process output at the kth sample time. fo(k) is the IIR output

signal.

The membership functions of the input and output variables that correspond to each fuzzy set are denoted as [NL, NM, NS, ZO, PS, PM, PL] and [Min, S, M, L, MAX], respectively. The member functions that describe the fuzzy input variable E and output

variables u1 and u2 have the same isosceles triangular shape

across the universe of discourse, as shown inFig. 6. The universes of discourse of the input and output variable are each normalized to [ 1, 1], [0, 1], and [1, 5]. The PDFF controller can be regarded as a generalized PI controller. The ratio of Kff/Kf, with a range

between 0 and 1, determines that the PDFF controller is approaching PI or PDF characteristics. Since Kfis set to one, the

universe of discourse of u1is set to [0, 1] to limit the ratio of the

feed-forward gain (Kff) to the feedback gain (Kf) in between zero

and one. Furthermore, the effect of disturbance rejection capability to the output is dependent upon the ratio of Ki/Kf .

The peak magnitude of the disturbance response can be reduced by increasing the Ki/Kf ratio. But as the ratio is increased, a

reduced peak magnitude can only be achieved with the cost of slow settling time. A compromise must be made between opti-mizing peak magnitude and settling.

The main concept that underlies design of the fuzzy rule-based design is that PDFF should exhibit the characteristics of PI control to reach the set point rapidly when the error is large (Pajchrowski

& Zawirski, 2008; Romeral & Chekkouri, 2002). Therefore, the

feed-forward gain of the PDFF controller must be set high, and the integral gain must be set small, to prevent overshoot in this case. However, PDFF control exhibits the characteristics of PDF control when the error is small, in which case, overshooting is unlikely. 1 0.8 0 6 sp ee ds ( ra d/s ) 0.4 0 2 0 0 005 0 01 0 015 0 02 0 025 0 03 0 time (seconds) 1 0.8 0.6 0.4 0.2 0 0.005 0.01 0.015 0.02 0.025 0.03 0 time (seconds) sp ee ds ( ra d/s ) cut freq = 200Hz cut freq = 1kHz cut freq = 3kHz only PDFF N = 2 N = 10 only PDFF

Fig. 5. Variation of system responses with (a) cutoff frequency and (b) order of the filter.

Simultaneously, the integral gain can be set high to increase the load disturbance rejection capacity.Table 1presents the seven-term fuzzy rule table. A fuzzy rule has the form

If E is Ak, then ðu1is Bkand u2is CkÞ, k ¼ 1,2,. . .,N

where E, u1, and u2are the fuzzy input and output variables, as

presented inFig. 7; Ak, Bk, and Ckare fuzzy sets.Fig. 8compares

the original PDFF with IIR with fuzzy PDFF-IIR controllers. Here, the integral gain, feed-forward gain, and feedback gain of the two controllers are set to the same values. The IIR filter is a second-order Butterworth filter with a cutoff frequency of 1 kHZ. The ratio of the feed-forward gain to the feedback gain is 60%, which is commonly used in industrial control.Fig. 8 presents that the rise time of the fuzzy PDFF-IIR is faster than that of PDFF-IIR.

Additionally, the load disturbance rejection capacity of fuzzy PDFF-IIR is better than that of PDFF-IIR control. However, the settling time is too long, and this problem becomes the main issue with fuzzy PDFF-IIR control. Fuzzy PDFF-IIR is similar to PDF control when the error is small, because the integrator dominates PDF control, but the integrator requires some time to accumulate error data to achieve control, which is expected to increase the delays in systems.

4. Experiments on PMSM drive system

Experiments on system at sampling frequency of 20KHz have been carried out to check the effectiveness of the proposed controller. The system consists of three main parts: (i) a PMSM coupled to a load through a torque detector, (ii) the current-regulated PWM motor drive, and (iii) a TMS320C2407 DSP implements the fuzzy PDFF-IIR digital control part for speed controller and current regulator. The measured data stored in the on-board dual port memory is then passed back to the PC host

for performance evaluation. Fig. 9 shows the flowchart of the

fuzzy PDFF-IIR control algorithms in DSP program for PMSM drive. The three control loops use the same interrupt clock to generate the different sampling rate for each loop. PMSMs with

powers of 300 W and 1 kW are tested here. Table 2shows the

parameters of the PMSMs used in the experiment. PDFF-IIR controller conditions during fuzzy tuning procedure are nominal values as shown inFig. 10.

Experiment 1 adopts a 1 kW PMSM; the reference velocity input is a 0.5 Hz square wave; the peak-to-peak voltage of velocity command is 2 V, which corresponds to 400 RPM motor speed, and the offset voltage is 1 V. The PI control algorithm is executed in the current loop and PDFF is used in the velocity loop of the drive system. Fig. 10also describes a detailed data of IIR path to compensate for PDFF. To improve the load disturbance rejection capability, the parameters of PDFF are tuned

to be close to those of PDF control: Kff/Kf is set to 20%.

Fig. 11 presents the step response of a 1 kW PMSM and drive

system with IIR filter compensation in the velocity loop. It can be seen that the closed-loop system has a better speed response after the IIR filter compensation is added. Compared with the speed response without IIR tuning (60.2 ms settling time), the speed response under IIR tuning has a shorter settling time (45.8 ms). The IIR filter compensation can increase the band-width of the system. The peak current feedback is reduced to prevent the flow of the highest current.Fig. 12plots the velocity step response in experiment 2, in which the 300 W PMSM was used in torque mode. In Fig. 12(a), PI control is applied in the current and velocity loops. PI used in the servo system easily causes overshoot. In Fig. 12(b), PI control is used in the current loop, and PDFF without IIR compensation is used in the velocity loop. If PDFF replaces PI, then the overshoot declined

to one below that in Fig. 12(a). Fig. 12(c) reveals that IIR

compensation can accelerate the system response. The experi-mental results indicate that the PDFF with IIR filter compensation demonstrates its superiority in both tracking and regulation conditions. It does not cause overshoot and its bandwidth is not lower than PI, the behavior of the system results is mostly improved.

5. Conclusions

This work developed a fuzzy PDFF-IIR control scheme for improving the response of traditional PDFF controllers. IIR filter compensation is added to the traditional PDFF controller,

Table 1 Fuzzy rule base.

E NL NM NS ZO PS PM PL

u1 Max L M Min M L Max

u2 Min S M Max M S Min

Fig. 7. Shape of output membership function: (a) u1and (b) u2.

1 0.8 0 6 sp ee d s ( rad /s ) 0.4 0 2 0 0 005 0 01 0 015 0 02 0 025 0 time (seconds) without Fuzzy with Fuzzy

increasing its responsiveness to velocity commands. Next, a fuzzy inferencer is applied to compute the feed-forward gain and integral gain of the PDFF controller using a fuzzy rule base design procedure. The proposed control strategy was applied to PMSM drive systems, which was designed without either recursive estimations or model reference, but defining a self-tuning proce-dure by means of fuzzy logic. Typical simulation results and the experiments of loaded PMSM drive machinery show the favorable performance of the controlled system. To demonstrate the versa-tility of the proposed approach, two power motors were tested. A performance comparison with standard PDFF controller has

Fig. 9. Program flowchart of FOC and PDFF-IIR control scheme. Each loop has 50ms execution time.

Table 2

Parameter values of PMSM used in experiment.

Rated power 300 W 1 kW Moment inertia Jm(kg cm2) 0.658 6.37 Armature resistance Rs(O) 8.37 1.82 Inductance Ldsand Lqs(mH) 17.4 10.05 Number of pole NP 8 8 Ke(V/KRPM) 54.9 106.8 KT(N m/A) 0.524 1.02 Rated voltage (V) 107.7 185.3 Rated speed (RPM) 3000 2000

Rated current (A) 2.0 5.16

Rated torque (N m) 0.95 4.782

Mechanical time constant (ms) 1.96 1.11

Electrical time constant (ms) 2.05 5.52

Fig. 10. PDFF-IIR control scheme with nominal values for experiment, where T¼ 0.2 ms sampling period.

been made, confirming the superiority of the fuzzy PDFF-IIR controller. Additional advantage of the proposed controller is that it can be implemented easily to the existing digital signal processor that carries out the supervision without major revision of software structure.

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Fig. 12. 300 W PMSM step response with different controllers: (a) PI, (b) PDFF and (c) PDFF with IIR.