- Email: [email protected]

Design of a PID-like compound fuzzy logic controller Dragan D. Kukolja, Slobodan B. Kuzmanovic! a, Emil Levib,* a

Faculty of Engineering, University of Novi Sad, Trg Dositeja Obradovica 6, 21000 Novi Sad, Yugoslavia b School of Engineering, Liverpool John Moores University, Byrom Street, Liverpool L3 3AF, UK Received 1 July 1999; accepted 1 December 2001

Abstract The paper describes a novel method for the design of a fuzzy logic controller (FLC) with near-optimal performance for a variety of operating conditions. The approach is based on the analysis of the system behaviour in the error state-space. The ﬁnal control structure, in a form of a compound FLC, is arrived at in two stages. The ﬁrst stage encompasses design and tuning of a PID-like fuzzy controller. The second stage consists of placing an additional fuzzy controller, of a structure similar to that of the ﬁrst one, in parallel with the PID-like fuzzy controller designed in the ﬁrst stage. The resulting compound controller is characterised with high performance in the wide range of operating conditions, and with small number of parameters that can be adjusted using simple optimisation methods. The controller is developed and tested for a plant comprising a vector controlled induction motor drive. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Fuzzy control design; Phase-plane analysis; PID controller; Speed control; Electric motor drives

1. Introduction Conventional PID controllers are characterised with simple structure and simple design procedures. They enable good control performance and are therefore widely applied in industry. However, in a number of cases, such as those when parameter variations take place and/or when disturbances are present, control system based on a fuzzy logic controller (FLC) may be a better choice. In general, two approaches to the fuzzy controller design can be identiﬁed. The ﬁrst one is of heuristic nature. It is based on qualitative knowledge and experience of an expert, that is used in order to achieve a given control objective. ‘Trial-and-error’ approach is then usually applied in the analysis of the most appropriate domains for input/output fuzzy variables and in construction of the knowledge base (Li and Lau, 1989; Hang et al., 1996). A more general approach often utilises phase space (Li and Gatland, 1996) or response behaviour (Li, 1997) as a mean of connecting the process dynamics and the rule base. The second, deterministic approach is based on a more or less systematic methodology for identiﬁcation of the structure and/or parameters of a fuzzy controller (Galithet and Foulloy, 1995; Li, 1997). The major shortcoming of the existing methods of FLC design is that in general large number of parameters has to be tuned (Hu et al., 1999). With this in mind, an attempt is made here to develop a design method that requires only general knowledge of the plant behaviour and that leads to an FLC with a very small number of parameters. The paper describes a novel method of FLC design and tuning, that enables high performance of the system in closed loop operation. The proposed concept is characterised with the following features: *

*

*

The approach is based on systematic analysis of the system behaviour in the error state-space. The control rules are determined in this way. The computations during inference stage are simpliﬁed using an appropriate form of the zero-order Takagi-Sugeno model. The FLC, developed in this paper, is fully characterised with a very small number of parameters. As a consequence, parameter tuning and optimisation can be performed using simple methods. *Corresponding author. Tel.: +44-151-231-2257; fax: +44-151-298-2624. E-mail address: [email protected] (E. Levi).

0952-1976/01/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 5 2 - 1 9 7 6 ( 0 2 ) 0 0 0 1 4 - 3

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The complexity of the controller for implementation purposes is regarded as acceptable. Processing speed of the proposed fuzzy control algorithm enables its implementation for control of moderately fast processes using standard micro-controllers. Described method is developed and tested on the model of a vector controlled induction motor drive.

The approach to FLC design and tuning encompasses two stages. The ﬁrst stage consists of designing of the PIDlike FLC and tuning of its parameters. A speciﬁc operating point is utilised at this stage. Well-known methods of design of conventional controllers are used, in order to arrive at good initial values of the parameters that are necessary for subsequent ﬁne tuning (optimisation) of the FLC parameters. Fine tuning of the very small number of FLC parameters is done by a simple optimisation method. The second stage of the design aims at improving the performance of the system in the wide range of operating conditions. The structure of the FLC is expanded by addition of new segments: additional control action and additional knowledge base. The newly introduced segments correspond to the second, parallel FLC. Parameters of these new segments of the control structure are optimised as well, with the aim of improving disturbance rejection properties of the system and response to small step excitation. The paper is organised as follows. The second section describes the complete design procedure of a PID-like FLC. Optimal parameter tuning of this FLC is explained in the third section. Reasons for addition of the second FLC, its design and parameter tuning are elaborated in the fourth section. Complete tuning procedure and control algorithm of the compound fuzzy logic control system are summarised in section ﬁve. The sixth section illustrates application of the proposed method in design of the control system for a high performance, vector controlled induction motor drive.

2. Design of a PID-like fuzzy logic controller 2.1. Selection of the input and output variables and their fuzzy sets The most frequent approach to the design of a PID-like FLCPID relies on the second order FLC (Qiao and Mizumoto, 1996; Hu et al., 1999). As the basic idea here is to design an FLC with as small number of parameters as possible, a co-ordinate transformation of input variables, error eðkÞ and change of error DeðkÞ; is applied ﬁrst. New variables rðkÞ and jðkÞ (module and phase, respectively), obtained in this way, are taken as FLC inputs. The FLC structure is shown in Fig. 1. Background behind the co-ordinate transformation is discussed next. When rule base of an FLC is being designed, an operator’s knowledge of a speciﬁc process or an expert’s general knowledge of process dynamics can be of paramount importance. The result of the design procedure is often a diagonal form FLC (Palmer et al., 1996). Magnitude of the output of this controller depends on the distance of the error vector e in the fuzzy plane from the diagonal. The diagonal divides phase plane into two semi-planes with positive and negative control actions, respectively. For a second order system whose inputs are (e; e’), this diagonal can be deﬁned with s ¼ le þ e’ ¼ 0; where l is the slope of a straight line. States located around the diagonal s ¼ 0 are of special interest as control action changes sign in this region. Consider a diagonal form FLC with two inputs, e and De; and let the control action uDIAG be deﬁned with fuzzy terms About Zero (Z), Negative Small (NS), Negative Big (NB), Positive Small (PS), Positive Big (PB). Approximate distribution of these regions in phase plane e-aDe could have the appearance shown in Fig. 2 (coefﬁcient a represents scaling factor between input fuzzy variables, error e and change of error De). Presence of the diagonal can be explained by existence of a sliding line, known in sliding mode control (SMC) (Palmer et al., 1996). According to the SMC approach, control generates such a control law that state vector e is driven towards s ¼ 0: Similarity between diagonal form FLC and SMC has enabled formulation of the concept of sliding mode FLC (SMFLC) (Palmer et al., 1996). Control law of a SMFLC can be shortly stated as u ¼ kF ðjsjÞ sgnðsÞ; where kF ðÞ is kF ðe; e’; lÞ under conditions 0pkF ðÞp þ umax ¼ kF max ; jsj is the perpendicular distance of error vector e from s ¼ 0; and sgnðÞ is the signum function. In the case of a diagonal form FLC, the control action is a non-linear mapping function of a number of arguments (i.e. uDIAG ¼ kðe; e’; lÞ sgnðsÞ). In contrast to this, control law of an SMFLC is function of a single argument, the

e ∆e

α

ρ ϕ

Fuzzy logic controller

∆u

z ---z-1 KD

Fig. 1. PID-like FLC structure.

Σ

uFLCPID

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α∆e

PB Z PS

e

NS Z NB

s =0

Fig. 2. The fuzzy control regions in the phase plane of the diagonal form FLC.

(b)

(a)

(c) Fig. 3. Fuzzy sets of input and output variables. (a) Modules, (b) Phase, (c) Control Action Change.

distance of error vector e from the sliding line. Design of the FLC is therefore in this paper based on co-ordinate transformation of eðkÞ and DeðkÞ into polar co-ordinates rðkÞ and jðkÞ; as already indicated in Fig. 1. Transformation is deﬁned with the following equations: rðkÞ ¼ jeðkÞ þ ja DeðkÞj;

ð1Þ

jðkÞ ¼ argðeðkÞ þ ja DeðkÞÞ;

ð2Þ

where a is the scaling factor between input fuzzy variables, error and change of error, and is one of the parameters of the proposed FLC that will require tuning. A simple control rule base is a desirable feature of an FLC, characterised with an incremental control law. With this in mind, only two fuzzy sets are assigned to each of the two input variables, Module and Phase. The fuzzy sets of the input fuzzy variable Module rðkÞ; namely Zero (Z) and Large (L), are characterised with triangular membership functions (Fig. 3a). Parameter RM (Module Maximum) is such that if rðkÞXRM holds true, then rðkÞ completely

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belongs to the fuzzy set Large (mL ðrðkÞÞ ¼ 1). Fuzzy sets of the variable Module rðkÞ satisfy the property of orthogonality (i.e. sum of membership functions in any point of the universe of discourse always equals one). In the case of a diagonal form FLC the ﬁrst quadrant of e a De plane contains positive, while the third quadrant contains negative control actions. The second and the fourth quadrants usually contain control actions of variable sign and low intensity. It is for this reason that the input variable Phase jðkÞ is deﬁned with only two fuzzy sets with triangular membership functions, that correspond to the ﬁrst and the third quadrant (Fig. 3b). The number of fuzzy sets is thus kept as low as possible, in accordance with the original idea. The fuzzy sets are deﬁned as orthogonal sets and associated membership functions are named as follows: First Quadrant (FQ)-mFQ ðjÞ and Third Quadrant (TQ)mTQ ðjÞ. Change of control action DuðkÞ is selected as output of the FLC (incremental control law). Three fuzzy sets are assigned to the output fuzzy variable Control Action Change DuðkÞ; namely Positive Control (PC), Zero Control (ZC) and Negative Control (NC). Singleton functions, positioned in points þDUM ; 0 and DUM ; are adopted as their membership functions (Fig. 3c). The maximum change of control action is DUM (DU Maximum). The major beneﬁt of the applied co-ordinate transformation is believed to be the strong correlation between this approach to the description of the system behaviour and the SMC concept. Consequently, a description of sufﬁcient quality for both input variables with only two fuzzy sets becomes possible. The number of parameters that have to be tuned is therefore very small. Additional beneﬁt is that number of rules in the rule base is at most four, as discussed in the next sub-section. These features, combined with the use of zero-order Takagi-Sugeno FLC, enable derivation of a very simple expression for the FLCs control action (Section 2.3). 2.2. Fuzzy rule base and inference process of the FLCPID As both input fuzzy variables are described with two fuzzy sets each, the rule base can have at most four fuzzy rules. The ﬁrst rule, given in linguistic form, is ð3Þ

If Module is Zero and Phase is in the First Quadrant;then Control Action Change is Zero:

The complete rule base is given in the compact form in Table 1. The task of the inference operation is to perform selection of the fuzzy rules that are to be ﬁred and to ﬁnd the value of the output fuzzy variable for the given values of the input fuzzy variables. This task is performed on the basis of belonging of input fuzzy variables to their respective fuzzy sets and on the basis of the rule base. Inference process is based on zero-order Takagi-Sugeno model with max-product composition, with each rule’s consequent being speciﬁed by a fuzzy singleton. Application of the algebraic product enables simpliﬁcation of the control law in subsequent derivation. Consequently, ﬁring strength of a rule directly yields value of the consequent membership function. The following then holds true for the rule (3) mZC ðDuðkÞÞ ¼ mZ ðrðkÞÞmFQ ðjðkÞÞ:

ð4Þ

Inference operation takes place for individual control actions delivered from fuzzy rule base. 2.3. Control action law of the FLCPID Crisp control action is obtained by defuzziﬁcation of the output fuzzy variable. Centre of gravity method (Klir and Yuan, 1995) is used P mU ðDuðkÞÞui ; ð5Þ DuðkÞ ¼ Pi i i mUi ðDuðkÞÞ where Ui is the fuzzy set of the output fuzzy variable in the ith fuzzy rule, while ui is the value of the output fuzzy variable for the particular fuzzy set.

Table 1 Rule base of the FLCPID Phase ðjðkÞÞ

First Quadrant (FQ)

Third Quadrant (TQ)

Module ðrðkÞÞ Zero (Z) Large (L)

Zero (ZC) Positive (PC)

Zero (ZC) Negative (NC)

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Substitution of (4) into (5), with subsequent application of all the fuzzy rules deﬁned in Table 1, yields after simple manipulations the following: DUM mL ðrðkÞÞ mFQ ðjðkÞÞ mTQ ðjðkÞÞ : DuðkÞ ¼ ð6Þ mZ ðrðkÞÞ þ mL ðrðkÞÞ mFQ ðjðkÞÞ þ mTQ ðjðkÞÞ Recall that membership functions of fuzzy sets rðkÞ and jðkÞ are deﬁned in such a way that property of orthogonality is always satisﬁed. Denominator of (6) is therefore always equal to one, regardless of the number of membership functions deﬁned in the universe of discourse of fuzzy variables rðkÞ and jðkÞ (numerator of (6) will, of course, change if different number of membership functions is selected). Omission of the denominator in (6) (since it is equal to one) and substitution mFQ ðjðkÞÞ ¼ 1 mTQ ðjðkÞÞ in the numerator reduce expression (6) to the following form: DuðkÞ ¼ DUM mL ðrðkÞÞ½1 2mTQ ðjðkÞÞ :

ð7Þ

Eq. (7) shows that the output DuðkÞ is restricted to interval [DUM ; þDUM ]. As indicated earlier, DUM is the maximum value of the control action change. Due to the fact that linear functions are selected as membership functions (rather then Gaussian, generalised bell or alike functions), expression (7) can be signiﬁcantly simpliﬁed by means of the following transformation of the product DUM mL ðrðkÞÞ: DUM mL ðrðkÞ ¼ DUM minfrðkÞ=RM ; 1g ¼ minfDUM rðkÞ=RM ; DUM g:

ð8Þ

Adopting that RM is larger than every rðkÞ and introducing a new parameter UR ¼ DUM =RM ; (8) can be further written as DUM mL ðrðkÞÞ ¼ minfDUM rðkÞ=RM ; DUM g ¼ DUM rðkÞ=RM ¼ UR rðkÞ:

ð9Þ

Finally, substitution of (9) into (7) yields DuðkÞ ¼ UR rðkÞ½1 2mTQ ðjðkÞÞ :

ð10Þ

Eq. (10) is a modiﬁed version of Eq. (7), valid when limiting of the change of control action DUM is not present. Expression for the control law of the of FLCPID follows directly from FLCPID structure, shown in Fig. 1: uFLCPID ðkÞ ¼ KD DuðkÞ þ uðkÞ:

ð11Þ

Eqs. (11), (10), (1) and (2) completely deﬁne the input/output variables of the FLCPID. Only three parameters, a; UR and KD ; are present in these equations. Tuning of the FLCPID is achieved by tuning of these three parameters.

3. Tuning of the FLCPID In order to avoid complex analysis of the system with FLC, which is in essence non-linear, initial parameter tuning of the FLCPID can be based on parameters of a conventional PID controller. Control law of the conventional PID controller (uCPID) can be stated in a form that corresponds to Eqs. (11) and (10) as uCPID ðkÞ ¼ C2 DuC ðkÞ þ uC ðkÞ;

ð12Þ

where DuC ðkÞ is DuC ðkÞ ¼ C0 eðkÞ þ C1 DeðkÞ:

ð13Þ

If conventional PID controller is represented in the usual way as X uCPID ðkÞ ¼ KP ½eðkÞ þ ðT=T1 Þ eðkÞ þ ðTD =TÞDeðkÞ ;

ð14Þ

then correlation between constants C0 ; C1 and C2 of (12) and (13) and constants of proportional, integral and differential action of (14) follows in the form KP ¼ C1 þ C2 C0 ;

T=T1 ¼ C0 =ðC1 þ C2 C0 Þ;

TD =T ¼ C2 C1 =ðC1 þ C2 C0 Þ:

ð15Þ

Tuning of parameters of the conventional PID controller can be done using the well-known Ziegler–Nichols method KP ¼ 1:2TOB =t;

1=T1 ¼ 1=2t;

TD ¼ t=2;

ð16Þ

where TOB ; t and T are time constant, time delay and sampling time, respectively. Finally, from (15) and (16) it follows that C0 ¼ 0:6TOB T=t2 ;

C1 ¼ 0:6TOB =t;

C2 ¼ t=T:

ð17Þ

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In order to obtain similar behaviour of the system with FLCPID and with conventional PID controller, DuC ðkÞ and DuðkÞ are equated on co-ordinate axes of the error state-space, that is when eðkÞ ¼ 0 or DeðkÞ ¼ 0: This leads to DuC ðkÞEDuðkÞ and uC ðkÞEuðkÞ so that resulting control action is approximately the same, i.e. uCPID ðkÞEuFLCPID ðkÞ: Under these conditions it follows from (13) that DuC ðkÞ ¼ C1 DeðkÞ for eðkÞ ¼ 0

ð18Þ

DuC ðkÞ ¼ C0 eðkÞ for DeðkÞ ¼ 0:

ð19Þ

and Control law of the FLC, given with (10), can be re-written as pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ DuðkÞ ¼ UR e2 ðkÞ þ a2 De2 ðkÞ½1 2mTQ ðjðkÞÞ :

ð20Þ

Conditions eðkÞ ¼ 0 and DeðkÞ ¼ 0 lead to expressions (21) and (22) (values of mTQ ðjðkÞ), required in the derivation, follow from Fig. 3b) DuðkÞ ¼ UR aDeðkÞ0:5

for eðkÞ ¼ 0

ð21Þ

for DeðkÞ ¼ 0:

ð22Þ

and DuðkÞ ¼ UR eðkÞ0:5

By equating control action of the PID controller and FLCPID for eðkÞ ¼ 0; one gets from (18) and (21) the following correlation UR a0:5 ¼ C1 :

ð23Þ

Similarly, by equating the action of the PID controller and FLCPID when DeðkÞ ¼ 0; one obtains from (19) and (22) the following: UR 0:5 ¼ C0 :

ð24Þ

Required parameters of the FLCPID, a; UR and KD ; are calculated from (17), (23) and (24): a ¼ t=T;

UR ¼ 1:2TOB T=t2 ;

KD ¼ t=T:

ð25Þ

These parameter values give similar behaviour of the system with FLCPID and with PID controller with respect to eðkÞ and DeðkÞ: Further improvement in characteristics of the system with FLCPID, for a given performance index, is possible by optimisation of the parameters a; UR and KD : The performance index that is to be minimised, has to be selected on the basis of the speciﬁed criteria and properties of the plant under consideration. Section 6, where the developed procedure is applied to the speed control problem of a vector controlled induction machine, gives explicit expressions for the selected performance indices. As only three parameters are to be tuned, simple static optimisation method such as Nelder–Mead (Simplex) (Nelder and Mead, 1965) can be used. Nelder–Mead method is chosen because it is an efﬁcient pattern search method, that works especially well when the number of variables is less than ﬁve or six. The method is a non-regular simplex method, because its vertices do not have to be equally spaced (Nelder and Mead, 1965). Optimisation is performed off-line. It should be noted that the choice of the Ziegler–Nichols method for initial tuning of the conventional PID controller is not a requirement. Any other method of conventional PID controller tuning, that is appropriate for the process under consideration, can be selected instead. Tuning of conventional PID controller according to a different method changes only the initial values of the conventional PID controller gains in (16) and, consequently, the initial values of the PID-like FLCPID parameters in (25). The procedure however remains unaffected. PID controller and the PID-like FLCPID controller described so far are of similar complexity since each require tuning of three parameters. As will be shown in Section 6, the two controllers yield very similar response for the operating condition for which the tuning is performed (Fig. 8) and essentially identical response once when the parameters are optimised (Fig. 9 and Table 4). However, for all the operating conditions that differ from the one used for the controller design, PID-like FLCPID controller offers an improvement in the response when compared to the PID controller (Figs. 10–13 and Tables 5–8).

4. The second, additional FLC If signiﬁcant disturbances are present, performance of the system with either conventional or FLC can signiﬁcantly deteriorate (Hang et al., 1996). As a way of overcoming this problem, an additional FLC (FLCADD) is placed in

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parallel with the existing FLCPID and its role is to compensate disturbances. By connecting these two controllers into the control structure shown in Fig. 4, a compound FLC is formed (FLCCMP). Results of the simulation study, presented in Section 6, show that the introduction of the FLCADD does improve the performance of the system in the presence of disturbance, when compared to the performance obtainable with conventional PID controller and the PID-like FLC. As indicated in Fig. 4, control action uA ðkÞ of the FLCADD directly enters the ﬁnal control action; simultaneously, this control action multiplied with a constant K inﬂuences the incremental control action (action of FLCPID) as well. Control action uA ðkÞ should reduce the overshoot and settling time during the disturbance. The additional FLC (FLCADD) and the PID-like FLC (FLCPID) are characterised with the following identical features: * * *

*

The input variables of FLCPID and FLCADD are the same, namely Module rðkÞ and Phase jðkÞ: The internal structure of the two controllers is the same. The same phase plane is under consideration (where value of the parameter a was determined during tuning of the FLCPID). The inference process and defuzziﬁcation are the same.

These identical characteristics signiﬁcantly simplify the resulting structure of the FLCCMP. The two FLCs differ in the following: * * * *

the universe of discourse of the FLCADD, fuzzy sets of the input variable jðkÞ; the rule base, control action uA of the FLCADD.

In what follows only those steps in the design of the FLCADD that differ from corresponding steps already described in conjunction with FLCPID design are described. 4.1. Fuzzy sets of input variable Phase jðkÞ and the rule base Parallel FLC is introduced with the idea of improving the system performance under conditions different from rated. At the same time the performance under rated conditions should be affected to a minimum extent. The action of the FLCADD should therefore cover as much as possible those regimes that are insufﬁciently covered by the action of the FLCPID. Analysis is illustrated using a vector controlled induction motor drive as an example (the same plant is used for development and testing of the complete fuzzy logic control system later on). The additional FLC is in this case required to improve the performance of the drive for small reference speed settings and when load torque changes (disturbance rejection). Design of the FLCADD is based on a comparison of the system behaviour for three transients. These are the application of the rated speed reference in a step-wise manner, step-wise change in loading conditions,

Fig. 4. Control structure of the FLCCMP.

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and application of a very small speed reference. Fig. 5 depicts the system behaviour in the error state-space (e aDe plane) when rated speed (i.e. one relative unit) speed reference is applied under no-load conditions. Curve (1) corresponds to change of speed from zero to rated (acceleration), while curve (2) is valid for change of speed from rated to zero (braking). Fig. 6 illustrates system behaviour in e aDe plane for the following operating conditions: change of speed from 0% to 10% of the rated (curve A1); change of speed from 10% of the rated to zero (curve A2); change of load torque from zero to rated (step loading, curve B1) and change of load torque from rated to zero (step unloading, curve B2). FLCPID is tuned for the nominal operating condition, for which the trajectory is dominantly situated in the II and the IV quadrants of the phase plane (Fig. 5). In contrast to this, as evidenced by Fig. 6 for the cases of disturbance, the FLCADD covers the other segments of the phase plane. Figs. 5 and 6 show that there are not signiﬁcant overlaps in the regions of action of the two controllers. Analysis of the last two ﬁgures reveals that curves that correspond to the rated speed reference tend quicker towards the origin and pass through plane segments that approximately correspond to the second and the fourth quadrant. From Fig. 6 it follows that behaviour of the system, for conditions other than rated speed reference application, include segments that approximately correspond to the ﬁrst and the third quadrant in the e aDe plane. In contrast to segments positioned left-up and right-down from boundary lines in Figs. 5 and 6, segments that are positioned leftdown and right-up contain predominantly parts of trajectories of the disturbance state. Approximate boundary lines, that separate those segments of the plane, are shown in Figs. 6 and 7 as dashed lines. For example, change of speed reference from 0% to 10% of the rated speed leads to a speed response with overshoot (see Fig. 10b) and trajectory then passes through left-down segment of the plane (trajectory A1 in Fig. 6a). Existence of overshoot is indicated by parts of the trajectory in these two segments for all the other cases as well. It therefore follows that two segments ‘rightup’ and ‘left-down’ of the error state-space are parts of the plane in which control needs to be improved by means of

Fig. 5. Behaviour of the system in error state-space for rated speed reference application (acceleration and braking).

(a)

(b)

Fig. 6. Behaviour of the system in the error state-space for application of a speed reference equal to 10% of the rated speed (a) and for application and removal of the rated load torque (b).

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(a)

793

(b)

Fig. 7. Fuzzy sets of input fuzzy variable Phase for FLCADD (Overshoot Up—mOU ðjÞ; Overshoot Down—mOD ðjÞ; Good Positive—mOkP ðjÞ; and Good Negative—mOkN ðjÞ). (a) Connection between regions of action and fuzzy sets jðkÞ of the FLCADD, (b) membership functions of fuzzy sets jðkÞ of the FLCADD. Table 2 Rule base of the FLCADD jðkÞ

Overshoot Up

Overshoot Down

Good Positive

Good Negative

rðkÞ Zero (Z) Large (L)

Zero Negative

Zero Positive

Zero Zero

Zero Zero

the FLCADD. These segments are identiﬁed in Fig. 7a as Overshoot Up (OU) and Overshoot Down (OD). Fuzzy sets of the variable jðkÞ; that correspond to these areas, are characterised with symmetrical triangular form of membership functions (Fig. 7b). Fuzzy sets with trapezoidal membership functions Good Positive (OkP) and Good Negative (OkN) of the variable jðkÞ deﬁne areas that are not supposed to be covered by the action of the FLCADD. As in all the previous cases, fuzzy sets of the variable jðkÞ have property of orthogonality. Fuzzy sets can be assigned to the output fuzzy variable Additonal Control Action uA ðkÞ in a similar way as it was done for the output fuzzy variable Control Action Change Du of the FLCPID: Positive, Zero and Negative. However, singleton sets Positive and Negative are now positioned in points þUM and UM ; where UM represents maximum value of the control action uA ðkÞ: Action of the FLCADD is deﬁned with a simple rule base given in Table 2. As FLCADD delivers control action in Overshoot Up and Overshoot Down segments of the phase plane (Fig. 7a), it uses only the left half of the Table 2. In those regions FLCADD reduces overshoot by its control action. 4.2. Control law of the additional FLC Control law of the additional FLCADD is obtained from the expression for defuzziﬁcation of the control variable using centre of gravity method (5), on the basis of the rule base given in Table 2 and fuzzy operation of the algebraic product type, uA ðkÞ ¼

ðUM ÞmL ðrðkÞÞmOU ðjðkÞÞ þ UM mL ðrðkÞÞmOD ðjðkÞÞ : ½mZ ðrðkÞÞ þ mL ðjðkÞÞ mOU ðjðkÞÞ þ mOD ðjðkÞÞ þ mOkP ðjðkÞÞ þ mOkN ðjðkÞÞ

ð26Þ

Taking into account that the fuzzy sets of fuzzy variables rðkÞ and jðkÞ are orthogonal in any point of their respective universe of discourse, denominator in (26) equals one. Hence ð27Þ uA ðkÞ ¼ UM mL ðrðkÞÞ mOD ðjðkÞÞ mOU ðjðkÞÞ : As indicated earlier, UM is the maximum value of the control action uA ðkÞ: By performing the same modiﬁcation as in the case of FLCPID (i.e. introducing RM as the largest value of the fuzzy input variable rðkÞ; see evaluation of (8)–(10)), one obtains ð28Þ uA ðkÞ ¼ UA rðkÞ mOD ðjðkÞÞ mOU ðjðkÞÞ ;

794

D.D. Kukolj et al. / Engineering Applications of Artificial Intelligence 14 (2001) 785–803

where UA ¼ UM =RM : The output of the FLCADD, uA ðkÞ; is the additional control action of the compound FLC composed of the FLCPID and FLCADD. 4.3. Tuning of the additional FLC Tuning of the FLCADD is performed by optimisation of the values of parameters w; UA and K; while the value assigned to parameter a is the one obtained during tuning of the FLCPID. Tuning of parameters w; UA and K can be performed using Nelder–Mead method of static optimisation, as the case was with FLCPID. As FLCADD is added with the idea of improving the behaviour of the system in the presence of disturbances, it is necessary to deﬁne an adequate form of the performance index, for which optimisation of parameters of the FLCADD is to be done. As already noted at the end of Section 3, performance indices can be speciﬁed only ones when the plant under consideration is introduced. More detailed discussion of the adopted performance indices is therefore provided in Section 6. It should be noted that the optimisation of the parameters of the PID-like FLC and the additional FLC is performed in sequence. Parameters of the PID-like FLC are optimised ﬁrst, in the absence of the additional FLC. Optimisation of the additional FLC is performed next, using the already optimised parameters of the PID-like FLC. It is recognised that such an approach decreases the probability of obtaining the true optimal values for all the parameters. However, the advantage of this approach is that a simple optimisation procedure, such as Nelder–Mead method, can be used. It is, of course, possible to optimise all the six parameters simultaneously, by using, for example, genetic algorithms. Use of a complex optimisation procedure (such as genetic algorithms) would be however against the main idea of this paper, which is development of a simple FLC with a small number of parameters, so that parameter tuning and optimisation can be performed using simple methods.

5. Summary of the tuning procedure and control algorithm of the compound FLC The complete developed tuning procedure of the compound FLCCMP is executed in two stages: FLCPID is tuned at ﬁrst, this being followed by tuning of the FLCADD. Parameter tuning procedure of the FLCPID has the goal of achieving as fast response as possible, with minimum overshoot, for the system rated operating conditions. Addition of FLCADD is optional and serves the purpose of improving the performance of the system in the wide range of operating conditions. 1. Parameter tuning procedure of the FLCPID consists of two steps: 1.1. Determination of initial values of the FLCPID parameters on the basis of (25): a; UR and KD : 1.2. Optimisation of values of parameters a; UR and KD with respect to the deﬁned performance criterion (Nelder– Mead (Simplex) method is used here). 2. Parameter tuning procedure for FLCADD consists of two steps: 2.1. Deﬁnition of expanded performance criterion, by addition of the disturbance related terms, with respect to which FLCADD is expected to improve performance achievable by FLCPID. 2.2. Optimisation of values of parameters w; UA and K; on the basis of expanded performance criterion (Nelder– Mead (Simplex) method is used here). Control algorithm of the FLCCMP is summarised in Table 3.

6. Illustration of the method: vector controlled induction motor drive Development and application of FLC in electric drives has gained considerably attention in recent years (Vas, 1999). The described design procedure for the compound FLCCMP is therefore applied to the problem of speed control in a rotor ﬂux oriented (vector controlled) induction motor drive. This is a typical plant in which operating conditions vary to a great extent, due to variation of the load torque (disturbance) and due to variation of the speed set-point in variable speed operation (Vas, 1999). Application of the compound FLC structure FLCCMP is justiﬁed on these grounds (of course, if there is a single set-point and there are no parameter variations in the process, FLCADD is not needed and FLCPID is sufﬁcient). Due to the analogy that exists between a rotor ﬂux oriented induction machine and other types of high performance electric drives (DC motor drives, permanent magnet synchronous motor drives, etc.), the results of the study are directly applicable to any other high performance electric drive. The standard constant parameter motor model (Vas, 1998) is used for motor representation in simulations. The vector control scheme relies on the indirect feed-forward method of rotor ﬂux oriented control. Two controllers in

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Table 3 Control algorithm of the FLCCMP

1. 2. 3. 4. 5.

Description of the calculation

Formulae

Input co-ordinate transformation Membership functions of the input variable j(k) Output control action of the FLCPID Output control action of the FLCADD Resultant control action of the FLCCMP

eðkÞ and DeðkÞ-rðkÞ and jðkÞ mTQ ðjðkÞÞ; mOD ðjðkÞÞ; mOU ðjðkÞÞ DuðkÞ ¼ UR rðkÞ½1 2mTQ ðjðkÞÞ uA ðkÞ ¼ UA rðkÞ½mOD ðjðkÞÞ mOU ðjðkÞÞ uðkÞ ¼ uE ðkÞ þ KD DuðkÞ þ uA ðkÞ; uE ðkÞ ¼ uE ðk 1Þ þ DuðkÞ þ KuA ðkÞ

cascade connection are used for the motor control along the q-axis. Speed controller sets the reference value of the stator q-axis current component (outer control loop), while q-axis current controller sets the reference q-axis stator voltage component (inner control loop). Stator d-axis current reference is a constant for operation in the base speed region (from zero speed up to 1 r.u. speed). For speed higher than rated (1 r.u.), rotor ﬂux reference is decreased in inverse proportion to the motor speed, so that the d-axis current reference is a variable. The d-axis current controller sets the reference d-axis stator voltage. The plant behaves as a non-linear control object for operation at speeds higher than rated. As time constants of the speed and current control loop differ considerably, two controllers are tuned independently. Speed is regarded as constant for the purpose of current controller tuning, while current control loop is regarded as ideal for the purpose of speed controller tuning. Speed control loop is designed with three different speed controllers— conventional PID controller, PID-like FLCPID, and compound FLCCMP. Conventional PI controller is used for the current control loops. Initial setting of the current controller parameters is therefore performed using Chien-HronesReswick method (Chien et al., 1952), for periodic response. Parameters of the current controllers are determined as KP ¼ 0:0337 and T=TI ¼ 0:2259 by studying the current control loop only and assuming that speed is constant and equal to zero. Detailed description of the plant model and of the control system, used in simulations, is provided in Appendix A. 6.1. Tuning of parameters of the FLCPID for the vector controlled induction motor drive Speed control aims at achieving as good as possible speed response to the application of the step rated speed reference (i.e. step reference equal to one in relative units—1 r.u.) with zero load torque. As illustrated in Section 5, parameter tuning of the FLCPID can be executed in two steps: (i) Determination of initial values for parameters of the FLCPID. Values of parameters of the conventional PID controller, determined initially by Ziegler–Nichols method, are KP ¼ 12:892; T=TI ¼ 0:104 and TD =T ¼ 2:4: The following values are then obtained from (25) for FLCPID: a ¼ 4:8; UR ¼ 2:6866 and KD ¼ 4:8: Comparison of speed responses obtained with these two speed controllers shows that they are very similar (Fig. 8). Both yield an overshoot of o15%, while speed error becomes o0.1% after 1 s from the step speed reference application. (ii) Fine tuning of the FLCPID parameters, with the aim of minimisation of the adopted performance criterion that characterises system behaviour. Performance criterion is selected as reduction in the duration of the speed response transient, for unity step speed reference application (applied at instant t0 ¼ 0:2 s) under no-load conditions. With the idea of shortening the transient duration, that is at present 1 s, performance index In is selected as In ¼

0:6 X 0:2

jej þ 20

1:5 X

jej:

ð29Þ

0:61

To enable a fair comparison of the controller performance, optimisation of the conventional PID controller is done as well. Values of controllers’ parameters and values of the two terms of the performance index In ; before and after optimisation, are summarised in Table 4. Performance index of the optimised FLCPID is signiﬁcantly lower than the corresponding value for initial design and is the same as the performance index of the optimised PID controller. Speed response that follows application of the rated step speed command (applied at 0.2 s), obtained with optimised FLCPID and with optimised PID controller, is compared in Fig. 9. Optimisation of the FLCPID controller parameters has reduced both the overshoot and the duration of the transient. Duration of the time interval that yields speed error of o0.1% is reduced from 1 to 0.4 s. Overshoot is reduced to o0.5%. As already noted at the end of Section 3, optimised PID controller and optimised FLCPID controller yield essentially identical performance for the case of the rated speed command application. This is regarded as beneﬁcial for the subsequent comparisons of controllers’

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Fig. 8. Speed response to rated step speed reference with initial PID and initial FLCPID speed controller parameter values.

Table 4 Values of controller parameters and of performance index before and after optimisation (rated step excitation, time domain traces in Figs. 8 and 9) Performance indices

PID controller KP ¼ 12:892 T=TI ¼ 0:104 TD =T ¼ 2:4

FLCPID UR ¼ 2:687 a ¼ 4:8 KD ¼ 4:8

Optimised PID controller KP ¼ 33:042 T=TI ¼ 0:082 TD =T ¼ 2:783

Optimised FLCPID UR ¼ 5:898 a ¼ 7:867 KD ¼ 4:314

FLCCMP w ¼ 1:842 rad UA ¼ 7:155 K ¼ 0:737

0:2 jej

16.912 (100%)

16.924 (100%)

16.162 (96%)

16.163 (96%)

16.163 (96%)

P1:5

38.143 (100%)

30.016 (79%)

0.006 (o1%)

0.006 (o1%)

0.006 (o1%)

55.055 (100%)

46.94 (85%)

16.168 (29%)

16.17 (29%)

16.17 (29%)

P0:6 20

In ¼

0:61 jej

P0:6

0:2 jej

þ 20

P1:5

0:61 jej

Fig. 9. Speed response to rated step speed reference with optimised PID speed controller, with optimised FLCPID and with FLCCMP (all speed responses are overlapped).

performance for other operating conditions. Indeed, a fair comparison is possible only when the controllers that are to be compared are tuned in such a way that the response for the design point is the same or at least as similar as possible (Ibrahim and Levi, 1998). As shown next, FLCPID controller is characterised with better performance for operating regimes that differ from the one used in the controller design. System behaviour is examined for two more cases: (a) change of loading condition and (b) small step change of the speed reference. Proﬁle of the speed reference is shown in Fig. 10a (step 10% speed reference change, no-load conditions). Proﬁle of the load torque change, at rated speed reference, is displayed in Fig. 11a (step change from zero to rated load torque at 0.2 s, followed by step change from rated to zero at 0.6 s).

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(a)

797

(b)

Fig. 10. Response of the system to step speed command of 0.1 r.u. (a) Proﬁle of the reference speed change, (b) response of the system with PID controller, FLCPID and FLCCMP.

(a)

(b)

Fig. 11. Disturbance rejection behaviour of the system. (a) Proﬁle of the load torque change, (b) response of the system with PID controller, FLCPID and FLCCMP.

Decrease in the applied step speed reference leads to deterioration of the drive response regardless of the type of the speed controller. Overshoot and duration of the transient increase. If the speed reference is 10% of the rated speed, better response is obtained with FLCPID (see Table 5). Fig. 10b illustrates speed responses for that case, obtained with optimised PID controller and with optimised FLCPID (dashed line—PID controller, solid line—FLCPID). Disturbance rejection behaviour of the system is somewhat better with FLCPID than with PID controller: speed deviation is smaller, while settling time is the same. Fig. 11b compares responses to load application and removal, of the proﬁle shown in Fig. 11a, obtained with the two optimised speed controllers (dashed line—PID controller, solid line—FLCPID). Presented results indicate that optimisation of parameters of the FLCPID, for rated step excitation, enables achievement of the same quality of response obtainable with optimised conventional PID controller. Response to application of a small step excitation is found to be better with FLCPID (Table 5, columns 1–2). FLCPID leads to smaller speed deviation than the conventional PID controller when load varies in a step-wise manner (Table 6, columns 1–2).

6.2. Tuning of parameters of the FLCADD for the vector controlled induction motor drive Tuning of parameters of the additional FLC (FLCADD) for the drive is performed using (a) very small reference speed step change, and (b) step change of load torque. Proﬁles are illustrated in Fig. 12. Performance index, selected

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798

Table 5 Values of components of the performance index for the system with PID controller, FLCPID and FLCCMP (small step excitation, time domain traces in Fig. 10) Performance indices

Optimised PID controller

FLCPID

FLCCMP

15(max(o)0.1)

0.844 (100%)

0.839 (99%)

0.839 (99%)

P0:35

0.592 (100%)

0.54 (91%)

0.486 (82%)

0.268 (100%)

0.241 (90%)

0.085 (32%)

0:2

jej

P 5 0:60 0:36 jej

Table 6 Values of components of the performance index for the system with PID controller, FLCPID and FLCCMP (disturbance rejection, time domain traces in Fig. 11) Performance indices

Optimised PID controller

FLCPID

FLCCMP

15(1-min(o))

0.71 (100%)

0.675 (95%)

0.637 (90%)

0.719 (100%)

0.57 (79%)

0.363 (50%)

0.275 (100%)

0.291 (106%)

0.032 (12%)

P0:35 0:2

jej þ

P0:75 0:6

jej

P P1 5( 0:6 0:36 jej þ 0:76 jej)

Fig. 12. Proﬁles of reference and disturbance used for tuning of the FLCADD.

for these transients, is given with I ¼ In þ I1 þ I2 ; I1 ¼

0:35 X

jej þ 5

0:2

I2 ¼

0:75 X 0:6

0:6 X 0:36

jej þ 5

1 X 0:76

jej þ 15 max ðoÞ 0:1 ; 0:2C0:6

ð30Þ

jej þ 15 0:1 minðoÞ : 0:6C1

Performance index contains, apart from In of (29), two more parts: I1 that is related to small step excitation; and I2 that is related to the load change. Both of these parts contain three terms. The ﬁrst two aim at minimisation of the speed error and reduction of the transient duration (especially the second term). The third term attempts to reduce the

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overshoot and maximum speed excursion due to change in loading conditions. Static optimisation using Nelder–Mead method, in conjunction with the performance index I of (30), yields the following values of parameters of the FLCADD: w ¼ 1:842 rad, UA ¼ 7:1548 and K ¼ 0:7373: Introduction of the FLCADD and tuning of its parameters could adversely affect system behaviour in the operating state for which FLCPID was designed. This is however not the case here. The reason is that, as already emphasised, the overlap between the regions of action of the two controllers in the phase plane is very small. The additional FLC therefore does not adversely affect system performance, as the values of individual terms of the performance index are not altered by introduction of the FLCADD (see FLCCMP column in Table 4). This is conﬁrmed with the speed response, obtained with FLCCMP, that is included in Fig. 9. Consequences of application of the FLCCMP are much more evident for transients other than rated speed reference application under no-load conditions. Figs. 10b and 11b contain speed response (bold traces), obtained with the FLCCMP, for the relevant proﬁles of speed reference and load torque change (Figs. 10a and 11a). Improvement in the speed response in both cases is noticeable. Tables 5 and 6 contain comparison of values of the parts I1 and I2 of the performance index (30) for these two transients, respectively. Values are given for the system with optimised PID controller, optimised FLCPID and optimised FLCCMP. Each component on its own shows that application of the FLCCMP signiﬁcantly improves performance of the system, except the overshoot for small step excitations (Table 5, ﬁrst row) which is not altered. 6.3. Operation of the drive at speeds higher than rated For operation of the drive above 1 r.u. speed rotor ﬂux reference has to be varied in inverse proportion to the speed. The motor consequently behaves as a non-linear control object. The operation of all the three controllers (optimised PID controller, optimised FLCPID and optimised FLCCMP) is therefore further investigated by simulating two more transients. The ﬁrst transient is the response to the application of a step speed command equal to 2 r.u. under no-load conditions. Previous steady state is the operation at 1 r.u. speed with zero load torque (ﬁnal steady state of Fig. 9). Time domain traces for this transient are shown in Fig. 13a and the corresponding values of the performance index (29) and its components are given in Table 7. The second transient is the application of a step load torque and its subsequent removal during operation with constant speed command of 2 r.u. The load torque proﬁle is identical to the one shown in Fig. 11a, except that the load torque value is 0.5 r.u. rather than 1 r.u. The results for this transient are given in Fig. 13b (time domain traces) and Table 8 (performance index (30) and its individual components). Fig. 13 and Tables 7 and 8 exhibit the same trends as those already observed in conjunction with Figs. 10b and 11b and Tables 5 and 6. Optimised PID-like FLC offers a slight improvement in the response, when compared to the optimised conventional PID controller. A further improvement is achieved by using the compound FLC. The improvement in the FLCCMP response over the one obtained with FLCPID is more signiﬁcant for the case of the load torque application. This is to be expected, since the additional FLC covers by its action the regions of the phase plane related to the disturbance rejection properties of the drive.

(a)

(b)

Fig. 13. Responses of the non-linear system with optimised controllers (PID controller, FLCPID and FLCCMP). (a) Response of the system to speed command stepping from 1 to 2 r.u. (b) Disturbance rejection behaviour of the system at 2 r.u. speed.

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800

Table 7 Values of components of the performance index for the system with optimised PID controller, FLCPID and FLCCMP (step reference speed change from 1 to 2 r.u., time domain traces in Fig. 13a) Performance indices P0:6 0:2 jej 20

P1:5

In ¼

0:61

jej

P0:6 0:2

jej þ 20

P1:5

0:61

jej

Optimised PID controller

FLCPID

FLCCMP

18.286 (100%)

18.286 (100%)

18.286 (100%)

5.282 (100%)

4.787 (91%)

4.530 (86%)

23.568 (100%)

23.073 (98%)

22.816 (97%)

Table 8 Values of components of the performance index for the system with optimised PID controller, FLCPID and FLCCMP (disturbance rejection at 2 r.u. speed, time domain traces in Fig. 13b) Performance indices

Optimised PID controller

FLCPID

FLCCMP

15(1-min(o))

0.566 (100%)

0.521 (92%)

0.459 (81%)

P0:35

0:2 jej þ 0:6 jej P

P1 0:6 5 0:36 jej þ 0:76 jej

0.376 (100%)

0.296 (79%)

0.180 (48%)

0.264 (100%)

0.149 (57%)

0.120 (46%)

S

1.206 (100%)

0.966 (80%)

0.759 (63%)

P0:75

Comparison of results obtained with FLCCMP for the two considered values of the disturbance (load torque of 1 r.u. in Fig. 11b and load torque of 0.5 r.u. in Fig. 13b) indicates that the additional FLC can successfully cope with a variety of different disturbance values, despite its very simple structure. Although FLCADD parameters were tuned for 1 r.u. load torque (Fig. 12), FLCCMP exhibits very goof disturbance rejection properties for the load torque value of 0.5 r.u. as well.

7. Conclusion The paper presents a novel approach to the design of a near-optimal FLC with robust characteristics. Final fuzzy control structure and its parameters are arrived at in two stages. The ﬁrst stage consists of transformation of the input variables into polar co-ordinates of the error state-space, convenient selection of their fuzzy sets, and calculation of the control action of the PID-like fuzzy controller. Characteristics of the conventional PID controller are used to determine initial parameter values for this fuzzy controller. Further tuning is performed using optimisation. The second stage aims at improving the system response over a wide range of operating conditions and therefore introduces the second fuzzy controller, placed in parallel to the ﬁrst one. Region of action of the second controller is deﬁned in the error state-space and synthesis of the controller is performed, using the procedure of the ﬁrst stage. Tuning of the second controller asks for optimisation of a small number of parameters. The resulting fuzzy controller structure is characterised with the following features: (1) high performance over a wide range of operating conditions; (2) simplicity of the design procedure; (3) a small number of parameters has to be tuned using simple optimisation method; (4) overall complexity of the ﬁnal compound fuzzy controller structure and the achievable performance are in good balance.

Appendix A. Simulation model of the vector controlled induction motor drive A block diagram of the drive under consideration is shown in Fig. 14. The three-phase squirrel-cage induction motor (block IM in Fig. 14) is represented for simulation purposes with the constant parameter d–q axis model in the

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Controller

801

Power stage 3-phase, 380 V, 50 Hz

Diode rectifier C Vector va*,vb*,vc* P controller W Fig.15 M

firing signals

PWM - VSI va

ia

vb

vc

ib IM ω P

Speed sensor

Fig. 14. Schematic representation of a vector controlled induction motor drive.

reference frame dictated by the control system (Vas, 1998) dc dc vds ¼ Rs ids þ ds o r cqs 0 ¼ Rr idr þ dr ðo r oÞcqr dt dt dcqs dcqr þ or cds 0 ¼ Rr iqr þ þ ðo r oÞcdr ; vqs ¼ Rs iqs þ dt dt cds ¼ Ls ids þ Lm idr

cdr ¼ Lr idr þ Lm ids

cqs ¼ Ls iqs þ Lm iqr

cqr ¼ Lr iqr þ Lm iqs ;

ðA:1Þ

ðA:2Þ

3 Lm Te ¼ P ðcdr iqs cqr ids Þ 2 Lr J do : ðA:3Þ Te TL ¼ P dt Symbols v; i; T and c stand for voltage, current, torque (e=of the motor; L=load torque) and ﬂux linkage, respectively. Indices s and r apply to stator and rotor, respectively, indices d and q identify d–q axis components, and index m denotes magnetising inductance. Parameters of the machine are the resistances (Rs and Rr ), inductances (Ls ; Lr ; Lm ), inertia (J) and number of pole pairs (P). Electrical angular speed of rotor rotation is o; while or is the angular speed of rotation of the rotor ﬂux space vector (asterisk * indicates that this speed in the machine model is equal to the speed calculated by the control system). Correlation between actual phase variables (a; b; c) and corresponding d–q axis quantities is governed by ids ¼ ð2=3Þðia cos f r þ ib cosðf r 2p=3Þ þ ic cosðf r 4p=3ÞÞ iqs ¼ ð2=3Þðia sin f r þ ib sinðf r 2p=3Þ þ ic sinðf r 4p=3ÞÞ v a ¼ v ds cos f r v qs sin f r v b ¼ v ds cosðf r 2p=3Þ v qs sinðf r 2p=3Þ ðA:4Þ v c ¼ v ds cosðf r 4p=3Þ v qs sinðf r 4p=3Þ; R where fR ¼ or dt: All the variables with an asterisk apply to the control system output, while the corresponding variables without the asterisk are those of the motor model. The induction motor is fed from a pulse width modulated voltage source inverter (block PWM VSI in Fig. 14), whose ﬁring signals are obtained from the pulse width modulator (block PWM

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802

ψr*=ψrn

∆ids

ids*

vds*

va*

PI

1/Lm

2

−

σLs

− ed vb* jφr*

ω*

Lsψr* Lm

∆iqs

iqs* Speed c.

e eq vqs*

vc*

PI

−

−

3 φr*

ωr*

ia

Lm T r ψ r*

i ds

2 −jφr*

ω sl* ωr* ω

ib

e

i qs

3

∫

φr*

ic ω

Fig. 15. Conﬁguration of the vector controller of Fig. 14 (indirect feed-forward rotor ﬂux oriented control with current controllers operating in the rotating reference frame).

in Fig. 14). The inverter is treated here as an ideal source of variable magnitude, variable frequency sinusoidal voltages, so that for simulation purposes v a ¼ va v ds ¼ vds

v b ¼ vb

v c ¼ vc

v qs ¼ vqs :

ðA:5Þ

The vector controller of Fig. 14 is illustrated in Fig. 15. Its inputs are the measured speed of rotation and the two measured phase currents (the third phase current equals ic ¼ ðia þ ib Þ). So-called indirect feed-forward method of rotor ﬂux oriented control is applied, with current control performed in the rotating reference frame. Fig. 15 shows the conﬁguration of the control system for operation in the base speed region (with speed values between 0 and 1 r.u.), so that the rotor ﬂux reference is constant and equal to rated (index n) at all times. For operation above the rated speed rotor ﬂux reference is decreased in inverse proportion to the speed of rotation. Hence the rotor ﬂux reference, c r ; and the associated stator d-axis current reference, ids ; become c r ¼ crn ids

1 oðr:u:Þ

! 1 dc r ¼ cr þ Tr : Lm dt

ðA:6Þ

A couple of new symbols are introduced in Fig. 15 and (A.6). The meaning of these is as follows: Tr ¼ Lr =Rr is the rotor time constant, sR¼ 1 L2m =ðLs LRr Þ is the total leakage coefﬁcient of the machine, o sl ¼ o r o is the angular slip frequency, and f r ¼ ðo sl þ oÞ dt ¼ o r dt is the instantaneous position of the rotor ﬂux space vector. So-called decoupling voltages ed and eq are calculated using ed ¼ sLs iqs o r ; eq ¼ ðLs =Lm Þc r o r for operation at speeds below rated (i.e. oo1 r:u:). For operation at speeds higher than rated, decoupling voltages are Lm dc r ed ¼ þ sLs o r iqs ; Lr dt Lm eq ¼ o c þ sLs o r ids : Lr r r

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References Chien, K.L., Hrones, J.A., Reswick, J.B., 1952. On the automatic control of generalised passive systems. Transactions of the ASME 74, 175–185. Galithet, S., Foulloy, L., 1995. Fuzzy controllers: synthesis and equivalencies. IEEE Transactions on Fuzzy Systems 3 (2), 140–147. Hang, C.C., Ho, W.K., Lee, T.H., 1996. Knowledge based PID control: heuristics and implementation. In: Gupta, M.M., Sinha, N.K. (Eds.), Intelligent Control Systems—Theory and Applications. IEEE Press, Piscataway, NJ, pp. 345–382. Hu, B., Mann, G.K.I., Gosine, R.G., 1999. New methodology for analytical and optimal design of fuzzy PID controllers. IEEE Transactions on Fuzzy Systems 7 (5), 521–539. Ibrahim, Z., Levi, E., 1998. A detailed comparative analysis of fuzzy logic and PI speed control in high performance drives. In: Proceedings of the IEE Seventh International Conference Power Electronics and Variable Speed Drives PEVD. IEE (Conf. Pub. No. 456), London, pp. 638–643. Klir, G.J., Yuan, B., 1995. Fuzzy Sets and Fuzzy Systems—Theory and Applications. Prentice Hall, Englewood Cliffs, NJ. Li, H.X., 1997. A comparative design and tuning for conventional fuzzy control. IEEE Transactions on Systems, Man, and Cybernetics-B 27 (5), 884–889. Li, W., 1997. A method for design of a hybrid neuro-fuzzy control system based on behaviour modelling. IEEE Transactions on Fuzzy Systems 5 (1), 128–137. Li, H.X, Gatland, H.B., 1996. A new methodology for designing a fuzzy logic controller. IEEE Transactions on Systems, Man, and Cybernetics 25 (3), 505–512. Li, Y.F., Lau, C.C., 1989. Development of fuzzy algorithms for servo systems. IEEE Control Systems Magazine 9 (3), 65–72. Nelder, J.A., Mead, R., 1965. A Simplex method for function minimisation. Computer Journal 7, 308–313. Palmer, R., Driankov, D., Hellendoorn, H., 1996. Model Based Fuzzy Control. Springer, Berlin. Qiao, W.Z., Mizumoto, M., 1996. PID type fuzzy controller and parameters adaptive method. Fuzzy Sets Systems 78, 23–35. Vas, P., 1998. Sensorless Vector and Direct Torque Control. Oxford University Press, Oxford. Vas, P., 1999. Artiﬁcial-Intelligence-Based Electrical Machines and Drives. Oxford University Press, New York. Dragan D. Kukolj received his Dipl. Ing., M.Sc. and Ph.D. degrees in control engineering in 1982, 1988, and 1993, respectively, all from the University of Novi Sad, Novi Sad, Yugoslavia. He is currently an Associate Professor of Control Theory in the Department of Computing and Automatic Control, Faculty of Engineering, University of Novi Sad. His research interests include artiﬁcial intelligence techniques and their application in control systems, signal processing and data mining. Dr. Kukolj has published over 85 papers in journals and conference proceedings. He is a member of the IEEE and of a number of national societies. Slobodan B. Kuzmanovic received his Dipl. Ing. degree in Computer Engineering and Systems from the University of Novi Sad, Yugoslavia, in 1998. He is currently working on his M.Sc. in the Department of Computing and Automatic Control, Faculty of Engineering, University of Novi Sad. His research interests are in the areas of artiﬁcial intelligence techniques and their application in data mining and control systems. Emil Levi graduated from the University of Novi Sad in 1982 and received his M.Sc. and Ph.D. degrees from the University of Belgrade in 1986 and 1990, respectively. He was with the Department of Electrical Engineering, Faculty of Engineering, University of Novi Sad, Yugoslavia from 1982 till 1992. In 1992 he joined Liverpool John Moores University, Liverpool, UK, where he currently holds the post of a Professor of Electric Machines and Drives. His main areas of research interest are modelling and simulation of electric machines, control of high performance electric drives and application of artiﬁcial intelligence techniques in electric drive control. Dr. Levi has published over 130 papers, including more than 30 articles in eminent journals. He is a Senior member of the IEEE.