Design of a Hybrid PID Plus Fuzzy Controller for Speed Control of Induction Motors

T.-J. Ho and L.-Y Yeh

Department of Electrical Engineering Chung Yuan Christian University

Chung–Li, Taiwan e-mail: tjho@dec.ee.cycu.edu.tw

Abstract – In this paper, a Ziegler-Nichols (Z-N) based PID plus fuzzy logic control (FLC) scheme is proposed for speed control of a direct field-oriented induction motor (DFOIM). The Z-N PID is adopted because its parameter values can be chosen using a simple and useful rule of thumb. The FLC is connected to the PID controller for enhancing robust performance in both dynamic transient and steady-state periods. The FLC is developed based on the output of the PID controller, and the output of the FLC is the torque command of the DFCIM. The complete closed-loop speed control scheme is implemented for the laboratory 0.14-hp squirrel-cage induction motor. Experimental results demonstrate that the proposed Z-N PID+FLC scheme can lead to desirable robust speed tracking performance under load torque disturbances.

Key words – speed control, hybrid PID plus fuzzy controller, induction motor, Ziegler-Nichols method.

I. INTRODUCTION In recent years, field-oriented induction machine (FOIM)

drives [1] have been increasingly utilized in motion control applications due to easy implementation and low cost. Besides, they have the advantage of decoupling the torque and flux control, which makes high servo quality achievable. However, the decoupling control feature can be adversely affected by load disturbances and parameter variations in the motor so that the variable-speed tracking performance of an IM is degraded. In general, both conventional PI and PID controllers have the difficulty in making the motor closely follow a reference speed trajectory under torque disturbances. In this regard, an effective and robust speed controller design is needed. In [2]-[8], fuzzy-logic-based intelligent controllers have been proposed for speed control of FOIM drives. Those intelligent controllers are associated with adaptive gains due to fuzzy inference and knowledge base. As a result, they can improve torque disturbance rejections in comparison with best

trial-and-error PI or PID controllers. Nonetheless, no performance advantages of intelligent controllers in combination with a PI or PID controller are investigated in [2]-[8].

Motivated by the successful development and application in [2]-[8], we propose a hybrid PID+fuzzy controller consisting of a PID controller and a fuzzy logic controller (FLC) in a serial arrangement for speed control of FOIM drives, more specifically, direct field-oriented IM (DFOIM) drives. The Ziegler-Nichols (Z-N)) method in [9] is adopted for designing a PID controller (denoted as “the Z-N PID”) because its design rule is simple and systematic. We next design a FLC carrying out fuzzy tuning of the output of the Z-N PID controller to issue adequate torque commands.

Based on a simulation model of the DFOIM drives incorporating the proposed controller, experiments are set up in an Matlable/Simulink environment and implemented in real time using the MRC-6810 analog-to-digital (AD)/ digital-to-analog (DA) servo control card together with a DSP electronic controller. The results show that the incorporation of the proposed controller into the DFOIM drives can yield superior and robust variable-speed tracking performance.

This paper is organized as follows. The mathematical model and control structure of a DFOIM are given in Sec. II. The proposed controller is given in Sec. III. Simulations and experiments are demonstrated in Sec. IV and V.

II. INDUCTION MOTOR AND CONTROL STRUCTURE In this section, we introduce the DFOIM drive shown in

Fig. 1. The dynamics of an induction motor can be described by

synchronously rotating reference frame direct-quadrature (d-q) equations [10] as

1352978-1-4244-5046-6/10/$26.00 c©2010 IEEE

+−−−−+−

−+−+

=

edr

eqr

eds

eqs

rrrremmrerrerrmrem

mmesssememsess

eds

eqs

i

i

i

i

pLRLpLLLpLRLpL

pLLpLRLLpLLpLR

v

v

)()()()(

00

ωωωωωωωω

ωωωω

(1)

eLrmmrmm TTBpJ =++ ωω , (2)

)(4

3 eqr

eds

edr

eqsme iiiiLNT ⋅−⋅= , (3)

rNrm ωω 2= (4)

where the notational superscript “e” stands for the synchronous reference frame; e

dsv , eqsv , e

dsi , eqsi , e

dri , and eqri stand for

the d-axis and the q-axis stator voltages, stator currents and rotor currents; sR , rR , sL and rL denote the resistances and self-inductances of the stator and the rotor; mL denotes the mutual inductance; eT and LT represent the electromagnetic and external force load torques, respectively; mJ and mB arethe rotor inertia and the coefficient of viscous damping, respectively; rω and rmω denote the rotor and motor

mechanical speeds; eω stands for electrical angular velocity; Nis the number of poles of the motor mechanical speed; pstands for the differential operator )( dtd . The notational superscript “s” in Fig. 1 stands for stationary reference frame.

For a DFOIM drive, the flux has to fall entirely on d-axis. Therefore, the q- axis rotor flux e

qrφ is set to zero. The root-locus method [12] is utilized for the design of PI controllers. The controllers PI-1, PI-2, and PI-3 are chosen to ensure that

eqs

eqs ii ≅

*, e

dseds ii ≅

*, and the flux command rφ and the

estimated d-axis rotor flux edrφ̂ satisfies e

drr φφ ˆ≅ , respectively.

The parameters τ and σ are given by r

rRL=τ and

rs

mLL

L21−=σ . To control the speed of the IM, the speed

controller of the DFOIM drive transforms the speed error signal

e into an appropriate electromagnetic torque command *eT .

rNφ34

rφ

e

sdsv

sqsv e

asv

ebsvecsv

easi

ebsi

ecsi

sqsi

sdsi

eqsi

edsi

*eqsi

*edsi

eθ̂

rmω

*rmω

edrφ̂

*eT +

−

+

+

+ +

+

+

−

−−

− eqsv

edsv

sLσ

sLσ

sdrφ̂

sqrφ̂

Figure 1. The block diagram of speed control of a DFOIM

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III. THE PROPOSED HYBRID PID PLUS FUZZY CONTROL

The structure of the proposed controller is shown in Fig. 2. The steps to acquire the Z-N PID [9] for speed control of

the DFOIM in Fig. 1 are given as follows. First, we use a fixed step input rmω and a linear proportional speed controller. The proportional gain of the speed controller is increased until the DFOIM reaches its stability limit. As a result, we obtain the period uT of the critical oscillation at the stability limit of the DFOIM with the critical proportional gain uK . Next, the values of the parameters pK , IT , DT are given by

7.1/uP KK = (5) 2/uI TT = (6) 4ID TT = (7)

where PK is the proportional gain; IT is the integral time and

DT is the derivative time. In the fuzzification process, we only employ three input

membership functions )(N xμ , )(Z xμ and )(P xμ shown in Fig. 3 to map a crisp input to a fuzzy set with a degree of certainty where )(tgx = or )(tgΔ with )()( 1 tfKtg = and

)()( 2 tfKtg Δ=Δ . Those three membership functions are chosen because of their simplicity for computation since a large number of membership functions and rules can cause high computational burden for a fuzzy controller. For any Nx ∈ where N denotes the interval ]0,(−∞ , its corresponding linguistic value is ‘N’. Moreover, for any Px ∈ where P denotes the interval ),0( ∞ ,its corresponding linguistic value is ‘P’. For any Zx ∈ where Z denotes the interval ],[ bb− , its corresponding linguistic value is ‘Z’. The membership functions )(N xμ , )(Z xμ and

)(P xμ are given by

≤<−−−≤

=

otherwise.0

01

)(N xbbx

bxxμ (8)

≤<−

≤<+

=

otherwise.0

0

0-

)(Z bxb

xb

xbb

bx

xμ (9)

≤<≤

=

otherwise.0

01

)(P bxbx

xbxμ (10)

The fuzzy inference engine, based on the input fuzzy sets in combination with the expert’s experience, uses adequate IF-THEN rules in the knowledge base to make decisions and produces an implied output fuzzy set u . For this particular application, the proposed IF-THEN fuzzy rule base is shown in Table 1 and is described as follows:

i. If ∈Δ )(tg N, then btgtgu =Δ ))(),(( . ii. If ∈Δ )(tg P, then btgtgu −=Δ ))(),(( .

iii. If ∈Δ )(tg Z and ∈)(tg N, then btgtgu −=Δ ))(),(( ..

iv. If ∈Δ )(tg Z and ∈)(tg P, then btgtgu =Δ ))(),(( .

v. If ∈Δ )(tg Z and ∈)(tg Z, then 0))(),(( =Δ tgtgu .

Moreover, the Mamdani-type min operation for fuzzy inference is employed in this study.

Fuzz

ifica

tion

Knowledge base

FuzzyInference

Def

uzzi

ficat

ion

One-step delay

+−

Fuzzy Controller

)(te

)(* tTe

)(tf

)(tfΔ

1K

2K

Ziegler Nichols

PID 3K)(* tTeΔ

)(tg

)(tgΔ

)1( −tf

Figure 2 . The block diagram of the proposed controller

1354 2010 5th IEEE Conference on Industrial Electronics and Applicationsis

1

0

Nμ Zμ Pμ

xb5.0− b5.0b− 0 bFigure 3. Membership functions with )(tgx = or )(tgΔ .

Table 1. Fuzzy rule base

u

)(tg

)(tgΔ

In the defuzzification process, we employ the‘center of mass’ defuzzification method for transforming the implied output fuzzy set into a crisp output, and obtain

∈ Δ∈

∈ Δ∈

Δ

×Δ=Δ

))((i ))((jji

))((i ))((jji

*))}(()),((min{

)j,i())}(()),((min{)(

tgFL tgFL

tgFL tgFLe tgtg

utgtgtT

μμ

μμ (12)

where

∉∈∉∈∈∈∈∈

=

.andif}P{andif}N{andif}Z,P{andif}Z,N{

)(

ZaPaZaNaZaPaZaNa

aFL (13)

The output of the fuzzy controller is given by )()( *

3* tTKtT ee Δ⋅= . (14)

IV. SIMULATION RESULTS A computer simulation model of Fig. 1 is developed using

the Matlab/Simulink software. The parameter values of the 0.14-hp squirrel-cage induction motor are given as follows:

17)( =ΩsR , 11)( =ΩrR , 196.0)H( =sL , 196.0)H( =rL ,31088.1)H( −×=mL , 4=N , 42 104.2)( −×=−− scmKgJ m ,

3102.9)( −×=− cmkgBm .Based on the root-locus method and the control objectives

of the PI controllers in Fig. 1, we obtain PI-1

as )65.1581(4162.220s

+ , PI-2 as )64.1751(0613.370s

+ and PI-3

as )019.131(80574.15s

+ . Given a fixed step input rmω rpm, we

obtain the critical gain 2.2=uK and the critical oscillation period 049.0=uT of the DFOIM. From (5)-(7), we get the Z-N

PID as )006125.00245.0

11(29.1 ss

++ . To design the fuzzy control

part of the proposed controller in Fig. 2, we first set 9=b and 12 =K . Then gains 1K and 3K are varied until the desired

system response under no torque disturbance is achieved. In this regard, we get 21 =K and 33 =K . The Simulink Fuzzy Logic Toolbox [13] is employed for fuzzy control simulations.

Fig. 4 shows that the proposed controller performs better than the Z-N PID under the condition that the command speed is increased from 0 to 900 r.p.m and a load disturbance 1.1 N-m is suddenly applied to the shaft at 4.2 sec.

V. EXPERIMENTAL RESULTS Based on the simulation model shown in Fig. 1, the

experimental setup is shown in Fig. 5. The objective of the experiments is to investigate the effectiveness of the proposed speed controller for the Nikki Denso NA21-3F 0.14Hp induction motor. The control system is implemented in real time using the MRC-6810 AD/DA servo control card as the interface between software and hardware.

In Figs. 6-7, we examine the performance of the proposed controller compared to other intelligent controllers in [8],[14]. We set 9=b and select the best trial-and-error parameter values 5.11 =K , 05.02 =K and 33 =K for the FLC under no torque disturbance.

In Fig. 6, the command speed is increased from 0 r.p.m. and reaches 900 r.p.m at 4.25 sec, and then starts decreasing from 900r.p.m at 8.25 sec. In addition, no torque disturbance is applied to the shaft. It shows that the proposed controller outperforms both the NFC[8] and the FLC[14]. The speed tracking response of the FLC yields large fluctuations when the speed command is 900 r.p.m.. The tracking response of the NFC cannot follow the command speed properly when the speed

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command is decreased from 900 to 0 r.p.m.. In Fig. 7, the command speed is the same as that in Fig. 6,

and a load torque disturbance of 1.1N-m is applied to the shaft at the 6th second. It is noted that the proposed controller yields much smaller tracking errors than the FLC and the NFC. In addition, the proposed controller takes shorter time to resume the command speed following than the FLC and the NFC when the load disturbance takes place. Accordingly, it is suggested that the proposed controller has a robust performance.

VI. CONCLUSION In this paper, a novel hybrid modified Z-N PID+FLC-based

speed control of a DFOIM has been presented. The proposed controller has exhibited the combined advantages of a PID controller and a FLC. Specifically, it can improve the stability,

the transient response and load disturbance rejection of speed control of a DFOIM. The complete DFOIM drive incorporating the proposed controller has been implemented in real time using a MRC-6810 AD/DA servo control card for the Nikki Denso NA21-3F 0.14Hp induction motor. The fuzzy logic and only with three membership functions are used for each input and output for low computational burden, which can achieve satisfactory results. Simulation and experiment results have illustrated that the proposed controller scheme has a good and robust tracking performance. As suggested in [15] that a modified Z-N PID can perform better than a Z-N PID, our future effort will focus on how to further improve the performance of the proposed controller herein by incorporating a modified Z-N PID.

Figure 4. Simulation results of the DFOIM using the proposed controller and the Z-N PID under a load disturbance of 1.1 N-m occurring at the 4.2 second.

Figure 5. Experimental setup of speed control of a DFOIM

1356 2010 5th IEEE Conference on Industrial Electronics and Applicationsis

Figure 6. Experimental results of the DFOIM using various speed controllers under no load disturbance.

Figure 7. Experimental results of the DFOIM using various speed controllers under a load disturbance of 1.1N-m occurring at the 6th second.

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