generic two degree of freedomlinearandfuzzy

8/13/2019 Generic Two Degree of Freedomlinearandfuzzy

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Journal of the Franklin Institute ] (]]]]) ]]]]]]

Generic two-degree-of-freedom linear and fuzzy

controllers for integral processes

Radu-Emil Precupa,, Stefan Preitla, Emil M. Petriub,Jozsef K. Tarc, Marius L. Tomescud, Claudiu Poznae

aDepartment of Automation and Applied Informatics, Politehnica University of Timisoara, Bd. V. Parvan 2,RO-300223 Timisoara, Romania

bSchool of Information Technology and Engineering, University of Ottawa, 800 King Edward, Ottawa,

ON, Canada K1N 6N5cInstitute of Intelligent Engineering Systems, Budapest Tech Polytechnical Institution, Be csi ut 96/B,

H-1034 Budapest, HungarydFaculty of Computer Science, Aurel Vlaicu University of Arad, Complex Universitar M,

Str. Elena Dragoi 2, RO-310330 Arad, RomaniaeDepartment of Product Design and Robotics, Transilvania University of Brasov, Bd. Eroilor 28,

RO-500036 Brasov, Romania

Received 13 June 2008; received in revised form 16 February 2009; accepted 2 March 2009

Abstract

This paper presents a new framework for the design of generic two-degree-of-freedom (2-DOF),

linear and fuzzy, controllers dedicated to a class of integral processes specific to servo systems. The

first part of the paper presents four 2-DOF linear PI controller structures that are designed using

the Extended Symmetrical Optimum method to ensure the desired overshoot and settling time. The

second part of the paper presents an original design method for 2-DOF TakagiSugeno PI-fuzzy

controllers based on the stability analysis theorem. Experimental results for the speed control of aservo system with variable load illustrate the performance of the new generic control structures.

r 2009 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

Keywords: Fuzzy control; PI and PID control; Servo systems; Tracking; Tuning

ARTICLE IN PRESS

www.elsevier.com/locate/jfranklin

0016-0032/$32.00 r 2009 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

doi:10.1016/j.jfranklin.2009.03.006

Corresponding author. Tel.: +40 256 40 3229; fax: +40 256 40 3214.E-mail address: [email protected] (R.-E. Precup).

Please cite this article as: R.-E. Precup, et al., Generic two-degree-of-freedom linear and fuzzy controllers for

integral processes, J. Franklin Inst. (2009), doi:10.1016/j.jfranklin.2009.03.006
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1. Introduction

Linear PI and PID controllers are used nowadays to control approximately 90% of

industrial processes worldwide [1]. However, the performance of these PI and PID

controllers depends not only on the tuning parameters but also on additionalfunctionalities such as anti-windup, feedforward action and set-point filtering [2].

Two-degree-of-freedom (2-DOF) PI and PID controllers have an advantage over the

1-DOF ones from the point of view of achieving high performance in set-point tracking

and the regulation in the presence of disturbance inputs[35]. But, the main drawback of

2-DOF controllers is that the overshoot reduction is paid by a slower set-point response.

Fuzzy control provides cost-effective nonlinear control solutions to a large number of

industrial applications. It has the following advantages [69]:

the use of linguistic rules and approximate reasoning is a relatively simple and easy way

to solve concrete control problems,

there is no need for sophisticated models and control design tools, it is sometimes the only way to initially approach the control of a complex, uncertain or

even not well defined process.

The introduction of dynamics in the fuzzy controller (FC) structures in the form of PD-,

PI- or PID-fuzzy controllers allows to further improve the control system (CS)

performance [1014]. They are based on the Mamdani model for PD-FCs, PI-FCs and

PID-FCs. The TakagiSugeno model is also used for the design of PD-FCs, PI-FCs and

PID-FCs.There are two main approaches to the design of the PD-FCs, PI-FCs and PID-FCs:

the first one is based on the fact that in some well-stated conditions the approximateequivalence between linear and fuzzy controllers is generally acknowledged[15,16],

the second one relies on the consideration of these FCs as nonlinear PD, PI or PIDcontrollers with variable gains [17,18].

We are using the first approach, for two reasons: (i) the design methods for FCs based

on the merge between the knowledge on conventional linear PI controllers and the

experience of experts in controlling the processes are widely accepted, and (ii) bylinearization around some steady-state operating points, certain classes of processes can be

considered as linear ones with variable parameters. The parameter variance makes the

control of such integral processes a difficult but challenging task and the fuzzy control

represents an attractive solution in this context. This situation is not encountered in case of

other conventional control solutions that give satisfaction only under particular operating

regimes.

We are using the TakagiSugeno FC model [19]because it allows to develop low-cost

automation solutions. The low-cost aim concerns the linear dependence of each rule on the

inputs making them behave as bumpless interpolators between linear controllers [20].

The adopted 2-DOF fuzzy controllers have the following attractive advantages [2125]:

improve the CS performance with respect to the modifications of set-point and loaddisturbance inputs ensured by the FCs,

ARTICLE IN PRESSR.-E. Precup et al. / Journal of the Franklin Institute ] (]]]]) ]]]]]]2


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overcome of the mentioned drawback of 2-DOF controllers to ensure small overshootand settling time.

The new 2-DOF TakagiSugeno PI-FCs presented in this paper provide efficient low-cost

control solutions and offer transparency in controller design and tuning. These controllersare intended to a representative class of integral processes specific to servo systems. The

design method is based on the fuzzification of the 2-DOF linear PI controllers tuned in

terms of a 2-DOF formulation of the Extended Symmetrical Optimum (ESO) method[26].

The proposed new design method for these PI-FCs ensures a systematic design

framework due to

the controller structure, characterized by small number of membership functions andrules,

the design method itself, expressed in terms of relatively simple design steps.The proposed design method has a generality advantage over other TakagiSugeno FC

structures that usually employ combinations ofHN

and FC techniques[2732]to meet the

desired CS performance specifications. The transparency of the proposed design method is

essentially due to the fact that, unlike other fuzzy CS structures[20], it offers an easy to use

connection between the controller tuning parameters and the CS performance indices.

The paper is organized as follows. Section 2 presents the 2-DOF PI CS structures in

relation with the linear control case in terms of the ESO method. Section 3 presents four

2-DOF TakagiSugeno PI-FCs together with their design method highlighting the design

steps that correspond to the design method for 2-DOF linear PI controllers. The design isenabled by an original stability analysis theorem based on Lyapunovs theorem for time-

varying systems referred in [33,34]. Section 4 illustrates the case study dedicated to speed

control of a servo system with variable load. The proposed controllers and design methods

are validated by the inclusion of experimental results for low speed patterns and controllers

implemented as low-cost automation solutions. Section 5 discusses the conclusions.

2. Generic linear control system structures and extended symmetrical optimum method

The class of integral processes considered here is characterized by the transfer function P(s):

Ps kP=s1TSs, (1)wherekPis the controlled process gain and TSis the small time constant or the time constant

corresponding to the sum of parasitic time constants.

This class of integral processes characterizes servo systems encountered in many

practical applications including the speed control systems of hydro power generators,

control systems for electrical drives, position and speed control of mechatronics systems,

mobile robot guidance and control, etc. [3542]. Due to the parameter variance, the

control of this class of systems (1) is a challenging task when very good CS performance

indices are required. One solution to tackle this is the use of PI or PD control algorithms.

Acceptable CS performance indices with respect to the set-point and disturbance inputcan be obtained if the process with the transfer function P(s) is included to the generic

2-DOF PI CS structures presented in Fig. 1, referred to as the set-point filter structure

(Fig. 1(a)), the feedforward structure (Fig. 1(b)), the feedback structure (Fig. 1(c)) and the

ARTICLE IN PRESSR.-E. Precup et al. / Journal of the Franklin Institute ] (]]]]) ]]]]]] 3




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component-separated structure (Fig. 1(d)). The variables in Fig. 1 are: r set-point,

y controlled output,eryorer1y control error,u control signal,r1,u2,u3,u4outputs of blocks F(s), C(s) in Fig. 1(b), C*(s) and CI(s), respectively, d1, d2, d3 load

disturbance input scenarios defined according toFig. 1(e), where the two blocks stand for

P1s kP1=1TSs; P2s kP2=s; kPkP1kP2. (2)The following nomenclature has been used for the blocks in all four 2-DOF PI CS

structures considered:

C(s) is the transfer function of the PI controller in Fig. 1(a) and (b):Cs kC11=Tis, (3)

with kC controller gain and Ti integral time constant, F(s) is the transfer function of the set-point filter in Fig. 1(a):Fs 1 1aTis=1Tis, (4)

with the design parameter a, andar1 to avoid the non-minimum phase character of

the closed-loop CS,

the other transfer functions in Fig. 1(b)(d):CFs kCa; Cs kC1a1=Tis; CPs 1a; CIs 1=Tis; CSs kC.

(5)

These four 2-DOF PI CS structures are equivalent schemes because they ensure the same

overall CS transfer functions, Gy,r(s) with respect to the set-point, Gy,d1(s) with respect to

d1, Gy,d2(s) with respect to d2 andGy,d3(s) with respect to d3:

Gy;r

s

1

a

Tis

1

=m

s

,

Gy;d1s Ti=kCs=ms; Gy;d2s kP2Ti=kC=kPsTSs1=ms,Gy;d3s Ti=kC=kPs2TSs1=ms; ms a3s3 a2s2 a1sa0,a01; a1Ti; a2Ti=kC=kP; a3TiTS=kC=kP. (6)

ARTICLE IN PRESS

Fig. 1. Set-point filter 2-DOF PI control system structure (a), feedforward 2-DOF PI control system structure

(b), feedback 2-DOF PI control system structure (c), component-separated 2-DOF PI control system structure

(d), definition of load disturbance input scenarios (e).

R.-E. Precup et al. / Journal of the Franklin Institute ] (]]]]) ]]]]]]4




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The open-loop transfer function L(s) takes the expression (7) in case of all four 2-DOF

PI CS structures:

Ls CsPs kCkPTis1=s2TiTSs1. (7)For the given class of processes with the transfer function in (1), making use of PI

controllers with the transfer function C(s) tuned in terms of Kesslers Symmetrical

Optimum (SO) method can ensure acceptable CS performance indices [2]. Kesslers SO

conditions concerning the denominator coefficients in (6):

2a0a2a21; 2a1a3a22 (8)were generalized to the following form specific to the ESO method [26]:

ffiffiffib

p a0a2a21;

ffiffiffib

p a1a3a22, (9)

where b represents a design parameter. The PI tuning conditions can be expressed as

kC 1=ffiffiffib

p kPTS; TibTS. (10)

Eq. (9) results in the following optimal expressions of the open-loop and closed-loop

transfer functions:

Ls 1bTSs=bffiffiffib

p T2Ss

21TSs; Gy;rs bTS1as1=mos,Gy;d1s b

ffiffiffib

p kPT

2Ss=mos; Gy;d2s kP2b

ffiffiffib

p T2SsTSs1=mos,

Gy;d3s bffiffiffib

p T2Ss

2TSs1=mos; mos b ffiffiffib

p T3Ss

3 bffiffiffib

p T2Ss

2 bTSs1.

(11)It is fully justified to consider the expressions (11) as optimal ones and the conditions (9)

as optimization ones because the conditions (9) ensure the maximization of the phase

margin in case of kPconst. The symmetry of open-loop Bode plots enables thisapproach; besides it guarantees a minimum desired phase margin in case of variable kP.

The definitions of overshoot and settling time depend on the CS inputs involved. They

are expressed using the general notations sv1 for the percent overshoot, sv1 for the

normalized overshoot, tvs for the normalized settling time, tvs for its normalized value,

where the superscript v stands for the current dynamic regime corresponding to the unit

step modification of the set-point, v

r, and disturbance inputs, v

d1, v

d2 orv

d3:

definitions with respect to the unit step modification of the set-point r:sr1sr1 100irxs1r; irx 1arexpbrx expdrxcr sinerx

fr coserx grexphrx; trsTStrs ; trs minfxtsrj jirx 1j 0:028 xxtsrg,(12)

wherexs1r is the solution to (13):

frer crdrsinerx crer frdrcoserx arbrexpdr brxg

r

hr

expdr

hr

x 0 (13)and the parameters in (12) and (13) have the expressions presented in Table 1with

Db2ffiffiffib

p 340, (14)





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definitions with respect to the unit step modification of the disturbance input d1:sd11 kPTSsd11 ; sd11 100id1xs1d1; id1x ad1expbd1x expdd1xcd1 sined1x

fd1 cosed1x gd1 exphd1x; td1s TStd1s ; td1s minfxtsd1j jid1xj 0:028 xxtsd1g,(15)

where xs1d1 is the solution to (16):

fd1ed1 cd1dd1sined1x cd1ed1 fd1dd1cosed1x ad1bd1 expdd1 bd1xgd1hd1expdd1 hd1x 0 (16)

and the parameters in (15) and (16) have the expressions illustrated in Table 2withD

defined in (14),

definitions with respect to the unit step modification of the disturbance input d2:sd21 kP2TSsd21 ; sd21 100id2xs1d2; id2x ad2expb

d2

x expdd2

xcd2 sined2xfd2 cosed2x gd2exphd2x; td2s TStd2s ; td2s minfxtsd2j jid2xj 0:02 8 xxtsd2g,

(17)

ARTICLE IN PRESS

Table 1

Parameters in (12) and (13).

Parameter bo9 b9 b49

ar

ffiffiffibp a ffiffiffibp 1=3 ffiffiffibp 1 ffiffiffibp a ffiffiffibp 1=3 ffiffiffibp br 1=

ffiffiffib

p 1/3 1=

ffiffiffib

pcr a

ffiffiffib

p =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3b2

ffiffiffib

pq 0 0dr 0:511=

ffiffiffib

p 1/3 0:5

ffiffiffib

p 1 ffiffiffiffiDp = ffiffiffibp

er0:5


ffiffiffib

pq =

ffiffiffib

p 0 0fr 2a

ffiffiffib

p=3

ffiffiffib

p x/3 2 a1b

ffiffiffib

p

ffiffiffib

p ffiffiffiffiD

p =ffiffiffiffiD

p =b3

ffiffiffib

p

ffiffiffib

p ffiffiffiffiD

p 2

ffiffiffiffiD

p

gr 0 3a2x2=18 2 a1bffiffiffib

p

ffiffiffib

p ffiffiffiffiD

p =

ffiffiffiffiD

p =b3

ffiffiffib

p

ffiffiffib

p ffiffiffiffiD

p 2

ffiffiffiffiD

p

hr 0 1/3 0:5ffiffiffib

p 1

ffiffiffiffiD

p =ffiffiffib

p

Table 2

Parameters in (15) and (16)


ad1 b=3ffiffiffib

p 0.5x2 b=3

ffiffiffib

p

bd1 1=ffiffiffib

p 1/3 1=

ffiffiffib

pcd1

b=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3 b2ffiffiffib

pq 0 0dd1 0:511=

ffiffiffib

p 0 0:5

ffiffiffib


ed10:5 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3b2 ffiffiffibpq = ffiffiffibp

0 0

fd1 b=3ffiffiffib

p 0 2b= ffiffiffiffiDp = ffiffiffiffiDp 3 ffiffiffibp

gd1 0 0 2b=ffiffiffiffiD

p = ffiffiffiffiDp 3 ffiffiffibp

hd1 0 0 0:5ffiffiffib






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wherexs1d2 is the solution to

fd2ed2 cd2dd2sined2x cd2ed2 fd2dd2cosed2x ad2bd2expdd2 bd2x

gd2hd2exp

dd2

hd2

x

0 (18)

and the parameters in (17) and (18) have the expressions summarized in Table 3,

definitions with respect to the unit step modification of the disturbance input d3:sd31 sd31 100id3xs1d3; id3x ad3expbd3x expdd3xcd3 sined3x

fd3 cosed3x gd3exphd3x; td3s TStd3s ; td3s minfxtsd3j jid3xj 0:028 xxtsd3g,(19)

wherexs1d3 is the solution to

fd3ed3 cd3dd3sined3x cd3ed3 fd3dd3cosed3x ad3bd3expdd3 bd3xgd3hd3expdd3 hd3x 0 (20)

and the parameters in (19) and (20) have the expressions presented in Table 4.

ARTICLE IN PRESS

Table 3



ad2

b

ffiffiffibp =3 ffiffiffibp x/3 b ffiffiffibp =3 ffiffiffibp bd2 1= ffiffiffibp 1/3 1= ffiffiffibpcd2 ffiffiffi

bp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3b 2ffiffiffib

pq =3

ffiffiffib

p 0 0

dd2 0:511=ffiffiffib

p 1/3 0:5

ffiffiffib


ed20:5


ffiffiffib

pq =

ffiffiffib

p 0 0fd2 b

ffiffiffib

p =3

ffiffiffib

p x2/3

ffiffiffib

p ffiffiffiffiDp ffiffiffibp 1= ffiffiffiffiDp = ffiffiffiffiDp ffiffiffibp 3

gd2 0 0 ffiffiffi

bp

ffiffiffiffiDp ffiffiffibp 1= ffiffiffiffiDp = ffiffiffiffiDp 3 ffiffiffibp hd2 0 0 0:5

ffiffiffib


Table 4



ad3 1ffiffiffib

p =3

ffiffiffib

p x/3 1

ffiffiffib

p =3

ffiffiffib

p

bd3 1=ffiffiffib

p 1/3 1=

ffiffiffib

pcd3 2=3

ffiffiffib

p 0 0

dd3 0:511=ffiffiffib

p 1/3 0:5

ffiffiffib


ed30:5 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3b 2 ffiffiffibpq = ffiffiffibp

0 0

fd3 0 x2/9 ffiffiffiffiDp ffiffiffibp 1= ffiffiffiffiDp = ffiffiffiffiDp ffiffiffibp 3gd3 0 1 ffiffiffiffiDp ffiffiffibp 1= ffiffiffiffiDp = ffiffiffiffiDp 3 ffiffiffibp hd3 0 1/3 0:5

ffiffiffib


R.-E. Precup et al. / Journal of the Franklin Institute ] (]]]]) ]]]]]] 7




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No simultaneous action of the disturbance inputs is supposed as far the expressions

(12)(20) are concerned. In addition, solving (13), (16), (18) and (20) requires numerical

techniques. Several diagrams for the normalized percent overshoot sv1 and normalizedsettling time tvs versus the design parametersa and b have been presented in[24].

The controller tuning of the 2-DOF PI CS structures presented inFig. 1is based on onlytwo design parameters, a and b, when the ESO method is applied.

Eqs. (12)(20) and Tables 14, together with a choice of design parameters a and b

within the domains ar1 and 1obo20, allow modifying the CS performance indices

represented by overshoot and settling time in a transparent manner according to designers

preferences. Since both the desired overshoot and the desired settling time should be

fulfilled, a trade-off to those performance indices can be defined.

To ensure the desired CS performance with respect to both the set-point r and one of the

disturbance inputs (d1,d2ord3) two classes of linear 2-DOF PI controllers can be designed:

the class of 2-DOF PI controllers, referred to as 2-DOF PI-C-r, tuned to ensure thedesired CS behaviour with respect to set-point in terms of the choice of design

parametersa and br and applying the tuning conditions obtained from (10) for bbr:

kC krC 1ffiffiffiffiffibr

p kPTS

; TiTri brTS, (21)

the class of 2-DOF PI controllers, referred to as 2-DOF PI-C-d, tuned to ensure thedesired CS behaviour with respect to disturbance inputs in terms of the choice of design

parametersa and bd and applying the tuning conditions obtained from (10) for b

bd:

kC kdC 1ffiffiffiffiffibd

q kPTS

; TiTdibdTS. (22)

Connecting the linear approach to fuzzy control, as an alternative control solution, the

2-DOF TakagiSugeno PI-FCs and a design method for these controllers will be proposed

in the following section. The 2-DOF TakagiSugeno PI-FCs belong to the class of type III

fuzzy systems[43,44]and blend the linear 2-DOF PI controllers separately designed with

respect to set-point and one of the disturbance inputs.

3. Generic fuzzy controller structures and design method

The 2-DOF TakagiSugeno PI-FCs play the roles of 2-DOF PI controllers in the CS

structures shown in Fig. 1 to improve the CS performance indices. These PI-FCs are

designed starting with the two classes of continuous-time 2-DOF PI controllers designed in

the previous section, 2-DOF PI-C-r and 2-DOF PI-C-d. Next, the dynamics components in

these linear 2-DOF PI controller structures are discretized as follows resulting in two

classes of quasi-continuous digital 2-DOF PI controllers with digital integral (I) and

proportional-integral (PI) components characterized by

the discrete-time equations of the digital PI components to substitute the blocks C(s)and C*(s) inFig. 1(a)(c) in case of 2-DOF PI-C-r:

Duk Durk Du2;k Dur2;k Du3;k Dur3;kKrPDekKrIek, (23)





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the discrete-time equations of the digital PI components to substitute the blocks C(s)and C*(s) inFig. 1(a)(c) in case of 2-DOF PI-C-d:

Duk Dudk Du2;k Dud2;k Du3;k Dud3;kKdPDekKdIek, (24)

where Dek, Duk, Du2,k and Du3,k stand for the change of control error, of control

signal, ofu2 and u3, respectively:

Dek ekek1=h; Duk ukuk1=h; Du2;k u2;ku2:k1=h; Du3;k u3;ku3;k1=h,(25)

h is the sampling period, k is the index of current sampling interval, and the

parameters of the two incremental digital PI controllers (23) and (24) can be calculated

in terms of (26) in case of Tustins method and expressed in a unified manner for the

blocksC(s) and C*(s):

KrPkrC1ah=2Tri; KrI krC=Tri; KdPkdC1ah=2Tdi; KdI kdC=Tdi, (26)with a0 in case of blocks C(s) inFig. 1(a) and (b),

the discrete-time equations of the digital component to substitute the block F(s) inFig. 1(a):

r1;kg1r1;k1d0rkd1rk1, (27)where the parameters can be expressed by Tustins method:

d0 2Tri1a h=2Trih; d1 2Tri1a h=2Trih; g1 2Trih=2Trih,(28)

the discrete-time equations of the digital I components to substitute the block CI(s) inFig. 1(d) in case of 2-DOF PI-C-r:

Du4;k Dur4;k KrPDekKrIek, (29)

the discrete-time equations of the digital I components to substitute the block CI(s) inFig. 1(d) in case of 2-DOF PI-C-d:

Du4;k Dud4;k KdPDekKdIek, (30)

with the parameters

KrPh=2Tri; KrI 1=Tri; KdPh=2Tdi; KdI1=Tdi. (31)

The two classes of I and PI components are fuzzified resulting in four 2-DOF

TakagiSugeno PI-FC structures. These generic fuzzy controller structures are referred to

as: set-point filter 2-DOF PI-FC presented inFig. 2(a) (the fuzziffied version of the linear

controller structure in Fig. 1(a)), feedforward 2-DOF PI-FC illustrated in Fig. 2(b) (the

fuzziffied version of the linear controller structure in Fig. 1(b)), feedback 2-DOF PI-FC

shown inFig. 2(c) (the fuzziffied version of the linear controller structure inFig. 1(c)), and

component-separated 2-DOF PI-FC presented in Fig. 2(d) (the fuzziffied version of thelinear controller structure in Fig. 1(d)).

The key element inFig. 2is the basic four inputs two outputs fuzzy controller (B-FC)

that represents a TakagiSugeno fuzzy system. It makes use of the MAX and MIN





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operators in the inference engine and it employs the weighted sum method for

defuzzification [4547]. The fuzzification is done in terms of the membership functions

illustrated in Fig. 3, where Drk is the change of set-point. Fig. 3 points out the

(positive) tuning parameters of the 2-DOF TakagiSugeno PI-FCs to be determined

by the design method, Se, SDe, SDr, Ss, and contributes to the fulfilment of the low-cost

aim.

Besides the task to elaborate Duk, Du2,k, Du3,kand Du4,k, the block B-FC has the task toobserve the current dynamic regimes of the CS. B-FC calculates the variable sk, with the

linguistic terms ZE and P corresponding to the dynamic regimes caused by the

modification of the disturbance inputs (of type d1, d2 or d3) and set-point r, respectively.

ARTICLE IN PRESS

Fig. 2. Set-point filter 2-DOF PI-fuzzy controller structure (a), feedforward 2-DOF PI-fuzzy controller structure (b),

feedback 2-DOF PI-fuzzy controller structure (c), component-separated 2-DOF PI-fuzzy controller structure (d).

Fig. 3. Input membership functions.





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The variableskcan take two values, skSs40 when modifications ofr occur, andsk0when modifications ofd1, d2 or d3 occur. The variable sk is zero in two situations:

either the system is in steady-state regime and the set-point is constant,

or the set-point is constant and one of the disturbance inputs is variable.In all other situations skSs. This definition of the variable sk allows the simple

characterization of the dynamic regimes mentioned before. Use is made of the definition to

formulate the rule base of B-FC expressed in terms of the decision tables presented in

Tables 5 and 6.

The 2-DOF PI-FCs combine four linear 2-DOF PI controllers separately designed with

respect to the CS inputs, two of them which belong to the class 2-DOF PI-C-r and have the

design parameters b1 and b2, and the other two of them which belong to the class 2-DOF

PI-C-d and have the design parameters b3

andb4

. With this regard the rule consequentsl1

ARTICLE IN PRESS

Table 5

Decision table to calculate Duk, Du2,k, Du3,kand Du4,k.

|Drk|

ZE P

ek ek

N ZE P N ZE P

sk1P Dek P l1 l1 l2 l1 l1 l2ZE l1 l3 l1 l1 l1 l1N l2 l1 l1 l2 l1 l1

ZE Dek P l3 l3 l4 l1 l1 l2ZE l3 l3 l3 l1 l1 l1N l4 l3 l3 l2 l1 l1

Table 6

Decision table to calculate sk.

|Drk|

ZE P

ek ek

N ZE P N ZE P

sk1P Dek P Ss Ss Ss Ss Ss Ss

ZE Ss 0 Ss Ss Ss SsN S

s S

s S

s S

s S

s S

s

ZE Dek P 0 0 0 Ss Ss SsZE 0 0 0 Ss Ss SsN 0 0 0 Ss Ss Ss





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andl2 correspond to 2-DOF PI-C-r, and l3 andl4 correspond to 2-DOF PI-C-d:

l1 Durk Dur2;k Dur3;k Dur4;kKrP1DekKrI1ek,l2

Durk

Dur2;k

Dur3;k

Dur4;k

KrP2Dek

KrI2ek,

l3 Dudk Dud2;k Dud3;k Dud4;kKdP1DekKdI1ek,l4 Dudk Dud2;k Dud3;k Dud4;kKdP2DekKdI2ek, (32)

where the additional subscript (1 and 2) inserted into the digital PI controller parameters

highlights a certain rule consequent.

The rule base to calculate Duk, Du2,k, Du3,kand Du4,kmaking use of the rule consequents

l1, l2, l3 and l4 in Table 5 can be interpreted in terms of four rules for each PI-FC

structure,Ru1, Ru2, Ru3 andRu4:

Ru1 :IF

ek IS N AND Dek IS P AND

jDrk

j IS ZE AND sk

1 IS P

OR

ek IS ZE AND Dek IS P AND jDrkj IS ZE AND sk1 IS P OR . . . ORek IS P AND Dek IS N AND jDrkj IS P AND sk1 IS ZE THENDuk Du2;k Du3;k Du4;kl1,

Ru2 :IF ek IS P AND Dek IS P AND jDrkj IS ZE AND sk1 IS P ORek IS P AND Dek IS P AND jDrkj IS P AND sk1 IS P OR . . . ORek IS N AND Dek IS N AND jDrkj IS P AND sk1 IS ZE THENDuk Du2;k Du3;k Du4;kl2,

Ru3 :IF

ek

IS ZE AND Dek

IS ZE ANDjDr

kj IS ZE AND s

k1 IS P

OR

ek IS N AND Dek IS P AND jDrkj IS ZE AND sk1 IS P OR . . . ORek IS P AND Dek IS N AND jDrkj IS ZE AND sk1 IS ZE THENDuk Du2;k Du3;k Du4;kl3,

Ru4 :IF ek IS P AND Dek IS P AND jDrkj IS ZE AND sk1 IS ZE ORek IS N AND Dek IS N AND jDrkj IS ZE AND sk1 IS ZE THENDuk Du2;k Du3;k Du4;kl4, (33)

with the parameters in the rule consequents calculated according to (21), (22) and (26)

under the conditionsbr b1 in case of l1; br b2 in case of l2; bd b3 in case of l3; bd b4 in case of l4.

(34)

The rule base to calculate the variable skcan be interpreted as the following two rules

valid for each PI-FC structure:

Rs1 :IF ek IS N AND Dek IS P AND jDrkj IS ZE AND sk1 IS P ORek IS ZE ANDDek IS P AND jDrkj IS ZE AND sk1 IS P OR . . . ORek IS P AND Dek ISN AND

jDrk

j IS P AND sk1 IS ZE

THEN sk

Ss

Rs2 :IF ek IS ZE AND Dek IS ZE ANDjDrkj IS ZE AND sk1 IS P ORek IS NAND Dek IS P AND jDrkj IS ZE AND sk1 IS ZE OR . . . ORek IS P ANDDek IS N AND jDrkj IS ZE AND sk1 IS ZE THEN sk0. (35)





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The presence of additional rules with the control laws l2 andl4 in their consequents is

necessary in order to alleviate the overshoot that appears when ekand Dektake the same

sign. This aspect emphasizes:

1. The presence of four linear PI controllers for each fuzzy controller structure, blended bymeans of the PI-FC operating mechanism.

2. The following design recommendations expressed in terms of relations between the

design parameters specific to linear PI controllers:

in case ofd1 andd2 type disturbance inputs:b14b2; b3ob4, (36)

in case ofd3 type disturbance inputs:b

14b

2; b

34b

4, (37)

and the parameters in (28) are calculated according to (21) with br b2.

The proposed design method is dedicated to 2-DOF TakagiSugeno PI-FCs. The method

consists of the following steps presented in unified manner for all four fuzzy controllers:

Step 1: Identify the controlled process and express the simplified mathematical model interms of the transfer function P(s) in (1), specific to integral servo systems.

Step 2: Set the values of the design parametersa,b1,b2,b3andb4of the linear continuous-time 2-DOF PI controller types, 2-DOF PI-C-r and 2-DOF PI-C-d, as function of the

desired/imposed CS performance indices taking into account the recommendations (36)and (37) and the results derived from (12)(20) assisted by Tables 14.

Step 3: Tune the parameters of the linear continuous-time 2-DOF PI controller types,2-DOF PI-C-r and 2-DOF PI-C-d, in terms of (21) for br b1and br b2and (22) forbdb3 and bdb4. Step 4: Set an adequate value of the sampling period, h, accepted by quasi-continuousdigital control and account for the presence of the zero-order hold.

Step 5: Discretize the dynamics components in the linear 2-DOF PI controller structuresand calculate the parameters of the quasi-continuous digital components employing

(26), (28) and (31).

Step 6: Set the value of the parameter Se of the 2-DOF TakagiSugeno PI-FCs inaccordance with the experience of the control systems designer and apply (38) (for the 2-

DOF PI-FCs inFig. 2(a)(c)) or (39) (for the 2-DOF PI-FC inFig. 2(d)) corresponding

to the modal equivalence principle[14]to obtain the value ofSDe:

SDeSe maxbr2fb1 ;b2gbd2fb3 ;b4g

fKrIj=jKrPjj; KdIj=jKdPjjg; j1; 4, (38)

SDeSe maxbr2fb1 ;b2gbd2fb3 ;b4g

fKrI4j=jKrP4jj; KdI4j=jKdP4jjg; j1; 4. (39)

Step 7: Set the values of the other two 2-DOF TakagiSugeno PI-FC parameters, SDrandSs, in terms of (40) that accepts the constant rmaxstep modification ofr and the 2%





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settling time, and makes the difference between dynamic regimes concerning the

modifications of the set-point and disturbance inputs:

SDr0:02rmax; Ss1. (40)

The first three steps in the design method represent the design method for the

continuous-time linear 2-DOF PI controllers. The first five steps correspond to the design

method for the discrete-time linear 2-DOF PI controllers. Eqs. (38)(40) ensure the

approximate equivalence, accepted in Section 1, between the suggested generic structures

of 2-DOF TakagiSugeno PI-FCs and the linear 2-DOF PI controllers.

The stability analysis theorem to be presented as follows will offer valuable information

to set the values of the parameter Se in step 6 of the design method. A stable CS will be

obtained. Accepting the quasi-continuous digital control case the mathematical model of

the controlled process (1) can be transformed generally to the state-space form

_xfx; t bx; tu,xt0 x0, (41)

wherex x1 x2 . . . xnT 2D is the state vector, nAN*, _x _x1 _x2 . . . _xnT isthe derivative ofx with respect to the independent time variable t, f; b :D 0; 1 !Rnare continuous functions in t, fx; t f1x; t f2x; t . . . fnx; t T, bx; t b1x; t b2x; t . . . bnx; t T, T stands for matrix transposition, and the disturbanceis absent. The generality of the problem is not reduced because all controllers have integral

components that deal with the rejection of constant load-type disturbances. The particular

expressions of the variables and functions in (41) become (42) for the given processcharacterized by n2:

x x1 x2T; x1ery; x2 _x1; fx; t x2 x2=TSt; bx; t 0 kP=TST,(42)

where the time-variable character has been introduced to increase the generality when

parametric disturbances can occur making the process belong the class of time-variant

systems.

The rule base in the fuzzy controllers consists ofrBfuzzy control rules. The ith rule is

Rule i :IF x1 IS Xi;1 AND . . . AND xn IS Xi;n THEN uuix; i1; rB; rB2,(43)

where Xi;1;. . .; Xi;n are the fuzzy sets that describe the linguistics terms of the inputvariables,ui(x) is the control signal produced by the ith rule and the function AND is the

MAX operator. ui can be a constant (that is also the case of Mamdani fuzzy controllers

with singleton consequents) or a function depending on the state vector. For the given

inference engine each fuzzy rule generates a certain firing strength

aix

minm X

i;1 x1

;m X

i;2x2

;. . .;m X

i;n xn

; 0

ai

x

1; i

1; rB, (44)

with the assumption

8x2D9aia0; i1; rB. (45)





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For the given defuzzification method the expression of the control signal is

u XrB

i

1

aiui

!,X

rB

i

1

ai

!. (46)

The stability analysis theorem to be presented as follows is based on Lyapunovs theorem

for time-varying systems [33,34]. The theorem ensures sufficient uniform asymptotic

stability conditions for the fuzzy control systems. A Lyapunov function candidate

V :D 0; 1 !R; Vx; t gtxTPx (47)is suggested, where PARn n is a constant positive definite matrix and g:[0,N)-[0,N) is a

continuously differentiable function. The derivative of V with respect to time with the

system being subject to the trajectory (41) is

_Vx; t _gtxTPxgt _xTPxxTP _x _gtxTPxgtfx; tbx; tuTPxgtxTPfx; t bx; tu Fx; t Bx; tu, (48)

with the following notations:

Fx; t gtfx; tTPxgtxTPfx; t _gtxTPx,Bx; t gtbx; tTPxgtxTPbx; t. (49)

The derivative ofVwith respect to time with the system accepted on the trajectory (41)

and calculated for u

uk(x) is

_Vkx; t Fx; t Bx; tukx. (50)Use is made of (50) and the stability analysis theorem is expressed in terms of Theorem 1.

Theorem 1. Let the fuzzy control systems be characterized by one of the four 2-DOF

Takagi Sugeno PI-FCs and the process(41). Let x0ADCRn be an equilibrium point for(41) and V the Lyapunov function candidate (47) such that the conditions (51) and (52) are

fulfilled:

Vx; t

W1

x, (51)

_Vkx; t W2kx, (52)for8k1; rB; 8t0; 8x2D,where W1 and Wk2 are continuous positive definite functionson D. Then x0 will be uniformly asymptotically stable.

Proof. From the hypotheses of Theorem 1,8t0; 8x2D, (50) and (52) result inFx; t Bx; tukx W2kx; k1; rB. (53)

Next (53) is multiplied by ak(x) and the sum is calculated:

Fx; tXrBk1

akx Bx; tXrBk1

akxukx XrBk1

W2kxakx. (54)





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The division byPrB

k1akx leads to

Fx; t Bx; tX

rB

k

1

akxukx" #,

XrB

k

1

akx" #

X

rB

k

1

W2kxakx" #,

XrB

k

1

akx" #

.

(55)

Use is made of (48) and the final result is

_Vx; t XrBk1

W2kxakx" #,XrB

k1akx

" #. (56)

Summarizing, the conditions (51) and (56) fulfil Lyapunovs theorem for time-varying

systems. Therefore, the equilibrium point at the origin will be uniformly asymptotically

stable and the proof is complete. Concluding, Theorem 1 ensures sufficient stability

conditions for the accepted fuzzy control systems and it can be employed in setting the

values ofSe.

4. Experimental results

In order to validate the proposed design methods, we implemented 2-DOF linear PI

controllers and TakagiSugeno PI-FCs for the control of a laboratory DC drive (AMIRA

DR300).

The experimental setup is illustrated in Fig. 4. The DC motor is loaded by a current

controlled DC generator mounted on the same shaft, and the drive has built-in analog

current controllers for both DC machines having rated speed equal to 3000 rpm, ratedpower equal to 30 W, and rated current equal to 2 A. An A/D D/A data converter card is

used as an interface between the digital speed-controller and the DC motor. The speed

sensors are a tacho generator and an additional incremental rotary encoder mounted

on the drive-shaft. The controlled process can be well approximated by the transfer

function P(s) in (1), decomposed according to (2) with kP4900, kP11, kP24900,TS0.035 s.

ARTICLE IN PRESS

Fig. 4. Experimental setup.





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The design steps presented in Section 3 were applied for the PI-FCs and the main

parameters involved in the design of 2-DOF linear PI controllers and TakagiSugeno

PI-FCs are:

the design parameters: a1, b19, b26, b316, b44, the parameters of the continuous-time linear controllers: kC10.0019, Ti10.315 s,kC20.0024, Ti,20.24 s,kC30.0015, Ti30.56 s,kC40.0029, Ti40.14 s, the parameters of the digital PI controllers:h0.005 s, in case of set-point filter 2-DOFPI-FC and feedforward 2-DOF PI-FC: KrP10:0018, KrI10:0062 (corresponding tokC1andTi1),K

rP20:0021,KrI20:0113 (corresponding tokC2andTi2),Kd3P10:0014,

Kd3I1 0:0026 (corresponding tokC3andTi3), Kd3P20:0024,Kd3I2 0:0208 (correspond-ing tokC4andTi4), in case of feedback 2-DOF PI-FC: K

rP1 1:5104,KrI10:0062,

KrP2

2:8

104,KrI2

0:0113 , Kd3P1

6:5

104,Kd3I1

0:0026,Kd3P2

5:2

104,

Kd3I2 0:0208 , in case of component-separated 2-DOF PI-FC: KrP10:0794, KrI13:1746 (corresponding to Ti1), K

rP20:119 , KrI24:7619 (corresponding to Ti2),

Kd3P10:0446, Kd3I1 1:7857 (corresponding to Ti3), Kd3P20:1786, Kd3I2 7:1429

ARTICLE IN PRESS

Fig. 5. Speed response of conventional CS with 2-DOF PI controllers without load (a), forr2500 rpm and 5 speriod of 10% d3 rated load (b).





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(corresponding to Ti4), and the elements without dynamics together with the set-point

filter implemented for bb2, the parameters of B-FC as part of all four fuzzy controllers: Se750,SAr50, Ss1,

the parameterSAeof B-FC:SAe

8695.7 in case of set-point filter 2-DOF PI-FC and of

feedforward 2-DOF PI-FC, SAe 30000 in case of feedback 2-DOF PI-FC and ofcomponent-separated 2-DOF PI-FC.

The four 2-DOF TakagiSugeno PI-fuzzy controllers designed here were tested by real-

time experiments and compared with the conventional 2-DOF controllers designed for

bb2. The speed responses exhibited by the designed linear and fuzzy control systems arepresented in Figs. 59 for low speed patterns. The results are presented with respect to

the modifications of the set-pointrand thed3type load disturbance input (according to the

definition inFig. 1(e)).

The experimental results prove that all fuzzy controllers outperform the linear 2-DOF PI

controllers accounting for the fact that the behaviour of all four CSs with linear 2-DOF PI

ARTICLE IN PRESS

Fig. 6. Speed response of CS with set-point filter 2-DOF PI-FC without load (a), forr2500 rpm and 5 s periodof 10% d3 rated load (b).





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controllers is very close, the control schemes being equivalent as mentioned in Section 1.

The comparison of experimental results proves that the smallest settling time and

overshoot are exhibited by

in case of set-point modifications: the CS with feedforward 2-DOF PI-FC, next the CSwith component-separated 2-DOF PI-FC, the CS with feedback 2-DOF PI-FC and the

CS with set-point filter 2-DOF PI-FC,

in case ofd3type load disturbance input modifications: the CS with feedback 2-DOF PI-FC, next the CS with component-separated 2-DOF PI-FC, the CS with set-point filter

2-DOF PI-FC and the CS with feedforward 2-DOF PI-FC.

5. Conclusions

This paper presents a new framework for the design of generic two-degree-of-freedom

(2-DOF), linear and fuzzy, controllers dedicated to a class of integral processes specific to

ARTICLE IN PRESS

Fig. 7. Speed response of CS with feedforward 2-DOF PI-FC without load (a), forr2500 rpm and 5 s period of10%d3 rated load (b).





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servo systems. These controllers can be implemented as low-cost automation solutions

because:

their relatively simple and transparent design method requires small computational costand ensures easy tuning,

the controller structures are relatively simple.The main advantages of the FC structures with respect to the linear ones are that they

exhibit a bumpless transfer from one linear PI controller to another, and they have a very

good behaviour in both set-point tracking and regulation (for all three load disturbance

types).

The rule base defined in Section 3 has a special formulation that improves the CS

behaviour when ekand Dekhave the same sign. This situation illustrates the downshoot

specific to non-minimum phase systems with right half-plane zeros. Therefore, theapplication areas of the fuzzy controller structures proposed here can be enlarged.

The proposed design methods have been shown to be very effective in set-point tracking

and load disturbance regulation when dealing with control of real-world processes because

ARTICLE IN PRESS

Fig. 8. Speed response of CS with feedback 2-DOF PI-FC without load (a), for r2500 rpm and 5 s period of10%d3 rated load (b).





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the industrial applications involve various integral servo systems. Experimental results

discussed in the paper validate both the design methods and the controller structures in

case of low speed patterns. The FC structures appear to be particularly appropriate forindustrial applications due to their low-cost and compatibility with linear 2-DOF PI

controllers.

The apparent complexity of the proposed, seven-step tuning procedure is in fact reduced

significantly due to (12)(20) andTables 14which connect the performance specifications

and the design parameters. They support the practitioners in the choice of the linear or

fuzzy controllers such that to fulfil the performance specifications.

The four linear control systems are identical. Their fuzzy counterparts are different but

they exhibit similar behaviours for the accepted processes. However all fuzzy control

system structures can be designed relatively easily to fulfil the performance specifications.

The choice of the appropriate fuzzy controllers for a certain process such that the desires/imposed performance indices are obtained represents the practitioners option. They

should account for the most convenient way to insert fuzzy logic into an already

implemented linear 2-DOF controller dealing with that process.

ARTICLE IN PRESS

Fig. 9. Speed response of CS with component-separated 2-DOF PI-FC without load (a), forr2500 rpm and 5speriod of 10% d3 rated load (b).





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To improve the CS performance indices these controller structures can be extended with

minor effort to advanced fuzzy controller structures [10,20,27,45]. Further research will be

focused on fuzzy controllers extending the area of applicability to servo systems with

essential nonlinearities, but together with appropriate analyses including the stability

analysis due to many variables and parameters involved [4852].

Acknowledgements

This work was supported by the Budapest Tech Polytechnical Institution and the

Politehnica University of Timisoara in the framework of the Hungarian-Romanian

Intergovernmental Science & Technology Cooperation Program, and by the CNCSIS &

CNMP of Romania.

References

[1] K.-J. Astrom, T. Hagglund, The future of PID control, Control Eng. Pract. 9 (11) (2001) 11631175.

[2] K.-J. Astrom, T. Hagglund, PID Controllers Theory: Design and Tuning, Instrument Society of America,

Research Triangle Park, NC, 1995.

[3] G. Prashanti, M. Chidambaram, Set-point weighted PID controllers for unstable systems, J. Franklin Inst.

337 (23) (2000) 201215.

[4] M. Araki, H. Taguchi, Two-degree-of-freedom PID controllers, Int. J. Control Automat. Syst. 1 (4) (2003)

401411.

[5] A. Visioli, A new design for a PID plus feedforward controller, J. Process Contr. 14 (4) (2004) 457463.

[6] P. Albertos, Fuzzy logic control: light and shadow, IFAC Newsletter 26 (3) (2002) 12.

[7] M. Oosterom, R. Babuska, Design of a gain-scheduling mechanism for flight control laws by fuzzy clustering,

Control Eng. Pract. 14 (7) (2006) 769781.

[8] M. Sunar, O. Toker, Substructural control of fuzzy nonlinear flexible structures, J. Franklin Inst. 344 (5)

(2007) 646657.

[9] A. Bagis, D. Karaboga, Evolutionary algorithm-based fuzzy PD control of spillway gates of dams,

J. Franklin Inst. 344 (8) (2007) 10391055.

[10] K. Michels, F. Klawonn, R. Kruse, A. Nurnberger, Fuzzy Control: Fundamentals, Stability and Design of

Fuzzy Controllers, Springer-Verlag, Berlin, Heidelberg, New York, 2006.

[11] L. Wen, Y. Hori, Vibration suppression using single neuron-based PI fuzzy controller and fractional-order

disturbance observer, IEEE Trans. Ind. Electr. 54 (1) (2007) 117126.

[12] B.M. Mohan, A. Sinha, Analytical structures for fuzzy PID controllers?, IEEE Trans. Fuzzy Syst. 16 (1)

(2008) 5260.

[13] R. Shahnazi, H.M. Shanechi, N. Pariz, Position control of induction and DC servomotors: a novel adaptivefuzzy PI sliding mode control, IEEE Trans. Energy Conv. 23 (1) (2008) 138147.

[14] R.-E. Precup, S. Preitl, I.J. Rudas, M.L. Tomescu, J.K. Tar, Design and experiments for a class of fuzzy

controlled servo systems, IEEE/ASME Trans. Mechatronics 13 (1) (2008) 2235.

[15] S. Galichet, L. Foulloy, Fuzzy controllers: synthesis and equivalences, IEEE Trans. Fuzzy Syst. 3 (2) (1995)

140148.

[16] J. Jantzen, Foundations of Fuzzy Control, John Wiley & Sons, Chichester, 2007.

[17] H. Ying, Theory and application of a novel Takagi-Sugeno fuzzy PID controller, Inform. Sci. 123 (34)

(2000) 281293.

[18] G.K.I. Mann, R.G. Gosine, Three-dimensional min-max-gravity based fuzzy PID inference analysis and

tuning, Fuzzy Sets Syst. 156 (2) (2005) 300323.

[19] T. Takagi, M. Sugeno, Fuzzy identification of systems and its application to modeling and control, IEEE

Trans. Syst. Man Cyb. 15 (1) (1985) 116132.[20] A. Sala, T.M. Guerra, R. Babuska, Perspectives of fuzzy systems and control, Fuzzy Sets Syst. 156 (3) (2005)

432444.

[21] C.M. Liaw, S.Y. Cheng, Fuzzy two-degrees-of-freedom speed controller for motor drives, IEEE Trans. Ind.

Electr. 42 (2) (1995) 209216.





23/24


24/24

[50] C. Prieur, R. Goebel, A.R. Teel, Hybrid feedback control and robust stabilization of nonlinear systems,

IEEE Trans. Automat. Control 52 (11) (2007) 21032117.

[51] R. Dargahi, A.H.D. Markazi, HN

-optimal digital redesign method, J. Franklin Inst. 344 (5) (2007) 553564.

[52] J.S.-H. Tsai, C.-L. Wei, S.-M. Guo, L.S. Shieh, C.R. Liu, EP-based adaptive tracker with observer and fault

estimator for nonlinear time-varying sampled-data systems against actuator failures, J. Franklin Inst. 345 (5)

(2008) 508535.


generic two degree of freedomlinearandfuzzy

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