Equivalence between Fuzzy PID Controllers and Conventional PID Controllers

Transcription

1 applied sciences Article Eqivalence between Fzzy PID Controllers Conventional PID Controllers Chn-Tang Chao, Nana Starna, Jing-Shian Chio * Chi-Jo Wang Department of Electrical Engineering, Sorn Taiwan niversity of Science Technology,, Nan-Tai St., Yongang District, Tainan City 7005, Taiwan; tang@stst.ed.tw (C.-T.C.); da320208@stst.ed.tw (N.S.); chijo@stst.ed.tw (C.-J.W.) * Correspondence: jschio@stst.ed.tw; Tel.: ; Fax: Academic Editor: Hng-Y Wang Received: 25 March 207; Accepted: 6 May 207; Pblished: 2 Jne 207 Abstract: This paper proposes eqivalence between fzzy Proportional-Integral-Derivative (PID) controllers conventional PID controllers. A well-designed conventional PID controller, with help of proposed method, can be rapidly transformed to an eqivalent fzzy logic controller (FLC) by observing defining operating ranges of inpt/otpt of controller. Frrmore, nowledge base of proposed eqivalent fzzy PID controller is represented as a cbe fzzy associative memory (FAM), instead of a combination of PD-type PI-type FLCs in most research. Simlation reslts show feasibility of proposed techniqe, both in continos discrete time. Since design techniqes of conventional linear PID controllers have matred, y can act as preliminary expert nowledge for nonlinear FLCs designs. Based on proposed eqivalence relationship, designer can frr tne membership fnctions of fzzy variables in control rles to exhibit nonlinearity of a FLC yield more satisfactory system responses in an efficient way. Keywords: eqivalence; conventional PID controller; fzzy logic PID controller. Introdction Proportional-Integral-Derivative (PID) controllers are widely sed in indstrial process control. The three-mode controller contains a proportional, an integral, a derivative term. The poplarity of a PID controller can be attribted to its good performance fnctional simplicity, which allows engineers to operate it in a simple straightforward manner. For example, three controller gains can be chosen independently by an engineer, based on one s experience or throgh some simple selection methods sch as classical tning rles proposed by Ziegler-Nichols []. For simplicity of controller design, a PI or PD controller are also poplar for practical applications. A PI controller can add damping to a system redce steady-state error, bt yields penalized rise time settling time. A PD controller also adds damping reliably predicts redces large overshoots, bt does not improve steady-state error. Ths, for complete design considerations, a PID controller shold be employed to obtain a desirable system response in settling time, steady-state error, overshoot. On or h, since Lotfi Zadeh rediscovered promoted fzziness in 965, sbseqent two fzzy inference techniqes proposed by Mamdani [2] Sgeno [3] have inspired research in fzzy logic controllers (FLC). The heristic fzzy rles, which reflect experience of hman experts, can be applied to plants that are difficlt to model mamatically. The most common FLCs are PI-type or PD-type controllers [4 6], which possess same characteristics as traditional PI or PD controllers, respectively. Moreover, y exhibit sperior applicability compared with traditional PI or PD controllers [7]. Appl. Sci. 207, 7, 53; doi:0.3390/app

2 Appl. Sci. 207, 7, 53 2 of 2 The FLC commonly otperforms corresponding PI, PD, or PID controller becase a FLC is a nonlinear controller, while a PI, PD, or PID controller is linear. This raises eqivalence problem between a fzzy PID controller a conventional PID controller. It is well nown that design of fzzy rles for a FLC reqires expert nowledge, those who are silled in conventional PID controller design are n qalified as experts. For a well-tned conventional PID controller, design wor is saved by replacing it with an eqivalent linear FLC, n improving performance over a conventional PID controller by slightly modifying fzzy rles. Moon [8] revealed that when a PI controller is given, an FLC otpt is identical to that of PI controller by sing specified fzzy logic operations. However, Moon s design is limited to PI controllers, design procedre is not clear enogh. Several stdies have investigated fzzy PID controller strctres, by taing different combinations of fzzy PID strctral elements [9 ]. This involves a large nmber of parameters in defining fzzy rle base. Manian et al. [2] presented a design for an eqivalent fzzy PID controller from conventional PID controller, bt tning procedre was too complicated reslting FLC was not prely linear according to control srface view of stdy. Therefore, eqivalence problem between different systems is very crcial to many research fields [3]. The objective of stdy is to extend significant reslts derived by Moon [8] by examining eqivalence relationship design procedre between a traditional PID controller its corresponding eqivalent FLC. This research proposes an eqivalent fzzy PID controller which has a simple PID strctre design with a 3-dimensional fzzy rle table, instead of combination of different fzzy PID strctral elements or a hybrid controller strctre [4]. Moreover, to achieve optimal control performance for a FLC, some artificial intelligent techniqes sch as Genetic Algorithm Neral Networ are efficient approaches [5,6]. This inspires s in ftre to propose nonlinear factors for tning membership fnctions to develop an optimal fzzy PID controller design with less parameters. This stdy presents eqivalent fzzy PID controller design (Section 2), followed by simlation reslts of Matlab/Simlin for verifying proposed design (Section 3). Finally, conclding remars implementation isses (Section 4) are discssed. 2. The Eqivalent Fzzy PID Controller Design The fzzy PID controller design proposed is eqivalent to a conventional PID controller, is derived from eqivalence eqations. First, for a conventional PID controller, eqation for otpt (t) in time domain is (t) = K P e(t) + K I de(t) e(t)dt + K D, () dt where controller provides a proportional term, an integration term, a derivative term. The otpt (t) three inpts e(t), e(t), ė(t) can be thoght as fzzy variables in FLC design. It is assmed that operating ranges for (t), e(t), e(t), ė(t) are OR = [ a, a ], OR e = [ a e, a e ], OR i = [ a i, a i ], OR d = [ a d, a d ], respectively. Figre shows membership fnctions for graphically defining for fzzy variables. As shown in Figre, m fzzy sets are eqally-spaced trianglar-shaped for each inpt fzzy variable e(t), e(t), or ė(t). On or h, otpt fzzy variable (t) is fzzified by 3m 2 singleton membership fnctions. Let e, i, d, denote center of fzzy sets E, I, D,, respectively, so that we obtain following eqations e = (2 m ) a e, i m = (2 m ) a m i, d = (2 m ) a m d, m (2) = (2 3m + ) a, (3m 2). (3) 3(m )

3 Appl. Sci. 207, 7, 53 3 of 2 We frr define distance between + as = +, (4) which will be sed in later eqation simplification. Based on above fzzy variables definition, expression of antecedent (IF) conseqent (THEN) for each fzzy rle is defined as IF e(t) is E i e(t) is I j ė(t) is D THEN (t) is l, (5) where three inpt fzzy variables e(t), e(t), ė(t) are taen into consideration simltaneosly. In proposed linear fzzy PID design, overall fzzy rles for three-by-one system can be represented by sliced cbe fzzy associative memory (FAM), as shown in Figre 2. Frrmore, we have following eqation related to Eqation (5) l = i + j + 2. (6) Appl. Sci. 207, 7, 53 3 of 3 E E ( m + )/2 E m I I ( m+ )/2 I m e a e et () em D D ( m + )/2 D m a e i a i et () (3m )/2 i m a i 3m 2 d et () dm t () 3m 2 a d a d a a, et () Figre Figre. Graphical. Graphical definition of of membership fnctions for for fzzy variables, et () e(t), Appl. Sci. 207, 7, 53, et () e(t), ė(t),, 4 of (t). 3 t (). As shown in Figre, m fzzy sets are eqally-spaced m 2m trianglar-shaped 3m 2 for each inpt fzzy variable et (), et (), or et (). On or h, otpt fzzy variable t () is e et () fzzified by 3m 2 singleton membership fnctions. Let e, i, d, m denote center of 2m m fzzy sets E, I, D,, respectively, eso that we obtain following eqations m m et () e (2 m ) (2 mm ) (2 m ) e = m m ae, i = a + i 2 i, d = m + i ad, m (2) m m m e m (2 3m+ ) i = a, + (3 m+ m i 2). e (3) 3( m ) m e We frr define distance m m between 2m + as Δ = +, (4) m which will be sed in later eqation simplification. Based on above fzzy variables definition, expression of antecedent (IF) conseqent m (THEN) for each fzzy rle is defined as e IF et ( ) is Ei et ( ) is I j et ( ) is D THENt ( ) is l, (5) Figre Figre 2. Sliced 2. Sliced cbe cbe fzzy fzzy associative associative memory (FAM) representation representation of nowledge of nowledge base. base. where Sbseqently, three inpt verification fzzy variables of et proposed (), et () design, is done et () by are applying taen into consideration Sgeno-style simltaneosly. inference, reslting In proposed controller linear otpt fzzy t () PID for design, controller overall inpts fzzy et (), rles et () for, three-by-one et () can be system calclated can by be carrying represented ot by an aggregation sliced cbe of fzzy form associative memory (FAM), as shown in Figre 2. Frrmore, we have following eqation related to Eqation (5) l ( μ E(e(t)) μ ( e(t)) l= i+ j+ (e(t))) i I μ j D 2. (6) e

4 Appl. Sci. 207, 7, 53 4 of 2 Sbseqently, verification of proposed design is done by applying Sgeno-style inference, reslting controller otpt (t) for controller inpts e(t), e(t), ė(t) can be calclated by carrying ot an aggregation of form (t) = l (µ Ei (e(t)) µ Ij ( e(t)) µ D (ė(t))) (µ Ei (e(t)) µ Ij ( e(t)) µ D (ė(t))), i, j, m, l = i + j + 2, (7) where prodct operation rle is sed for fzzy logic implications center of gravity (COG) is applied for defzzification process. It is determined that re is at most, eight rles to be fired for any controller inpts e(t), e(t), ė(t). To clarify, consider crisp inpt e(t) corresponding to membership fnctions E i E i+ to degrees of p p, respectively. Similarly, consider that e(t) maps membership fnctions I j I j+ to degrees of q q, respectively. Also, it is assmed that ė(t) has degrees of r r with respect to membership fnctions D D +. Based on above assmption, membership degrees p, q, r can be described as p = e i+ e(t), q = i j+ e(t), r = d + ė(t) (8) e i+ e i i j+ i j d + d The fired eight rles are listed below, Figre 3 is an illstration for se eight rles. Appl. Sci. 207, 7, 53. IF e(t) is E i 5 of 3 e(t) is I j ė(t) is D THEN (t) is i+j+ 2 IF e()is t E e()is t I e ()is t D THEN ()is t IF e()is t E e()is t I e ()is t D THEN ()is t IF e()is t E e()is t I e ()is t D THEN ()is t IF e()is t E e()is t I e ()is t D THEN ()is t IFet ( ) is E et ( ) is I et ( ) is D THENt ( ) is 2. IF e(t) is E i e(t) is I j ė(t) is D + THEN (t) is i+j+ 4. i j+ + i+ j+ 3. IF e(t) is E i e(t) is I j+ ė(t) is D THEN (t) is i+j+ 5. i+ j i+ j+ 4. IF e(t) is E i e(t) is I j+ ė(t) is D + THEN (t) is i+j+ 5. IF e(t) is E i+ e(t) is I j ė(t) is D THEN (t) is i+j+ 6. i+ j + i+ j+ 6. IF e(t) is E i+ e(t) is I j ė(t) is D + THEN (t) is i+j+ 7. i+ j+ i+ j+ 7. IF e(t) is E i+ e(t) is I j+ ė(t) is D THEN (t) is i+j+ 8. IF e(t) is E i+ e(t) is I j+ ė(t) is D + THEN (t) is i+j++ 8. i+ j+ + i+ j+ + E i E i + D () r ( p) ( p) I j ( q) i + j + 2 i + j + I j + ( q) i+ j+ E i D + ( r) ( p) ( q) I j + ( q) i + j + E ( p) i + i + j + i+ j+ + Figre3. 3. The The fired eight rles. I j i + j + i + j + nm (t) nm t ( ) As shown in Eqation (7), (7), crisp crisp otpt otpt (t) t can () becan evalated be evalated as (t) as = t () = by taing by den (t) den t ( ) weighted average of eight rles conseqents. Ths, denominator of (t) with 8 terms will be taing finally redced weighted to, as average shownof in Eqation eight rles (9). conseqents. Ths, denominator of t () with 8 terms will be finally redced to, as shown in Eqation (9). den (t) = pqr + pq( r) + p( q)r + p( q)( r) + ( p)qr den ( t) = pqr + pq( r) p( q) r + p( q)( r) + ( p)qr +( p)q( r) + ( p)( q)r + ( p)( q)( r) (9) = + ( p) q( r) + ( p)( q) r + ( p)( q)( r) (9) On or h, = nominator part of (t) with 8 terms is obtained by On or h, nominator part of t () with 8 terms is obtained by nm (t) = pqr i+j+ 2 + pq( r) i+j+ + p( q)r i+j+ + p( q)( r) i+j+ nm t ( ) = pqr +( p)qr i+ j+ 2 + i+j+ pq( + r )( p)q( i+ j+ + p ( r) i+j+ qr ) + ( i+ j+ + p)( p q)r )( i+j+ r ), (0) i+ j+ +( p)( q)( r) i+j++ +( pqr ) i+ j+ + ( pq ) ( r ) i+ j+ + ( p)( qr ) i+ j+, (0) + ( p)( q)( r) i+ j+ + which can also be confirmed in Figre 3. In order to simplify Eqation (0), we se a method of applying Eqation (4) defining x = i + j +. Then Eqation (0) can be redced to

5 Appl. Sci. 207, 7, 53 5 of 2 which can also be confirmed in Figre 3. In order to simplify Eqation (0), we se a method of applying Eqation (4) defining x = i+j+. Then Eqation (0) can be redced to nm (t) = x + ( p q r) = i+j++ p q r = (2(i+j++ p q r) 3m+), () a 3(m ) where Eqation (3) is also applied. By sbstitting Eqation (2) into Eqation (8), we can rewrite membership degrees p, q, r as p = (2i m+)a e (m )e(t) 2a e, q = (2j m+)a i (m ) e(t) 2a, i r = (2 m+)a d (m )ė(t) 2a d. With Eqation (2) sbstitted into Eqation (), nm (t) is finally obtained as follows (2) nm (t) = a (m )e(t) [2i + 2j + 2 3m (m ) a e 2ia e a e + (m )a e a e + (m ) e(t) a i 2ja i a i + (m )a i a i + (m )ė(t) a d = a 3a e e(t) + a 3a i e(t) + a 3a d ė(t). 2a d a d + (m )a d a d ] (3) Ths, crisp otpt (t) of proposed linear FLC is given by (t) = a e(t) + a 3a e 3a i which implies a linear PID controller with e(t)dt + a de(t), (4) 3a d dt K P = a 3a e, K I = a 3a i, K D = a 3a d. (5) Eqation (5) shows that if a FLC design is based on fzzy nowledge from Figre 2 defzzification process in Eqation (7), n it will yield a linear PID controller reslting PID parameters have no relation with m, nmber of membership fnctions, bt is strongly correlated to operating ranges of control inpt/otpt. With derived important eqivalence reslt, designer can obtain a FLC design prototype based on a conventional PID controller design. In practical application, a FLC will be finished by digital implementation. When considering a digital PID controllers, eqation for otpt [n] at each sampling time will be [n] = K P e[n] + K I T s e[n] + K D e[n] e[n ] T s, (6) where T s is sampling time. In sbseqent section, performance of FLC implemented in digital form is verified. 3. Simlation Reslts In this section, proposed eqivalence relationship is verified by se of Matlab/Simlin. A three-order controlled plant is employed with transfer fnction [7], which is shown below P(s) = s s s +. (7) A conventional PID controller design for P(s) with K P =.2, K I = 0.36, K D =, which was simlated by Simlin is shown in Figre 4, PID controller can simltaneosly improve system responses in rise time, settling time, steady-state error, overshoot. The magnitde of

6 Appl. Sci. 207, 7, 53 6 of 2 inpt Appl. step Sci. signal 207, 7, is 53 set as 5, reslting error signal, error integral, error derivative, control 7 of signal, 3 Appl. Sci. 207, 7, 53 7 of 3 system otpt are shown in Figre 5 (red line). Figre 4. The Proportional-Integral-Derivative (PID)-controlled system in Simlin. Figre Figre The The Proportional-Integral-Derivative Proportional-Integral-Derivative (PID)-controlled (PID)-controlled system system in Simlin. in Simlin. Figre 5. The (a) step inpt; (b) error signal; (c) error integral; (d) error derivative; (e) control signal Figre 5. The (a) step inpt; (b) error signal; (c) error integral; (d) error derivative; (e) control signal (f) Figre system5. otpt The (a) with step PID inpt; controller, (b) error signal; eqivalent (f) system otpt with PID controller, (c) error fzzy eqivalent integral; logic (d) controller fzzy error logic derivative; (FLC), controller (e) (FLC), control eqivalent signal FLC in discrete eqivalent (f) system form. FLC otpt in discrete with form. PID controller, eqivalent fzzy logic controller (FLC), eqivalent FLC in discrete form.

7 Appl. Sci. 207, 7, 53 8 of 3 Sbseqently, we bilt eqivalent FLC based on above conventional PID controller design according to derived eqivalence eqation. By observing system responses et (), Appl. Sci. 207, 7, 53 7 of 2 et (), et (), t () with above conventional PID controller in Figre 5 (red line), operating ranges OR e, OR i, OR d, OR can be defined in accordance with Eqation (5), as Sbseqently, procedre below we bilt shows. eqivalent FLC based on above conventional PID controller design according to derived eqivalence eqation. By observing system responses e(t), e(t), ė(t),. (t) OR ewith is set as above [ aeconventional, ae] = [ 5,5], PID which controller is range in Figre for et 5 () (red. line), operating ranges OR e, OR 2. i, OR d, OR can be defined in accordance with Eqation (5), as procedre below shows. is set as[ a, a] = [ 8,8] to satisfy K P = OR e i is is set set as as[ [ a a e, i, a e i] ] = [[ 5, 6.67,6.67] 5], which is to satisfy range for K I e(t). = OR is [ a, ] [ 8, 8] to satisfy P.2. d is set as[ ad, ad] = [ 6,6] to satisfy K D =. 3. OR i is set as [ a i, a i ] = [ 6.67, 6.67] to satisfy K I = For OR this d is set case, as [ a it shows d, a d ] = good [ 6, reslts 6] to satisfy in defining K D =. operating ranges. Frrmore, this is not a limitation as re are for parameters a e, a, a i, a d for adjstment to satisfy three For this case, it shows good reslts in defining operating ranges. Frrmore, this is not control parameters K a limitation as re are P, K for parameters I, K a e, D in Eqation (5). The Fzzy Logic Designer in a, a i, a d for adjstment to satisfy three control parameters Matlab/Simlin K P, K I was, applied K D in Eqation for eqivalent (5). TheFLC Fzzy design Logic Designer simlation. in Matlab/Simlin Figre 6 shows was applied feedbac for control eqivalent strctre. FLC design simlation. Figre 6 shows feedbac control strctre. Figre 6. The eqivalent FLC-controlled system in Simlin. The FIS Type of of FLC design in in Figre 6 shold be be set set as as Sgeno. Based on above operating ranges of for fzzy variables, corresponding membership fnctions fnctions can can be be defined defined by by Figre Figre. The. The parameter parameter m mwas wasset setas as5, 5, reslting in 5 fzzy sets, which are eqally-spaced trianglar-shaped, for each inpt inpt fzzy fzzy variable variable e(t), et () e(t),, or et () ė(t)., or On et () or. On h, or otpt h, fzzy variable (t) is fzzified by 3 singleton membership fnctions with singleton vales 8, 5, 2, otpt..., 3, fzzy 0, 3, variable..., 2, 5, t () is 8. fzzified Figre 7 by shows 3 singleton settings membership of all membership fnctions with fnctions singleton in vales Matlab 8, environment. 5, 2,, 3, The0, 25 3,, fzzy 2, 5, rles are8. defined Figre according 7 shows to settings nowledge of all basemembership in Figre 2, fnctions reslting Matlab system environment. inpt responses The 25 arefzzy shown rles in Figre are defined 5 (green according line). However, to nowledge it is fond that base in system Figre 2, responses with reslting a green system line cannot inpt be examined, responses which are shown is de in to Figre overlap 5 (green of system line). responses However, in it is fond red line. that In Figre system 5, responses green lines with (responses a green line bycannot eqivalent be examined, FLC) were which plotted is de prior to tooverlap red lines of system (responses responses by in PID controller). red line. In OnFigre contrary, 5, green if we plot lines system (responses responses by by eqivalent PID controller FLC) were (redplotted line) first, prior itto will trn red lines ot that (responses all red by responses PID controller). are covered On by contrary, latter green if we responses plot system in responses eqivalent by FLC. PID This controller verifies (red proposed line) first, eqivalence it will trn relationship ot that all between red responses PID controller are covered by eqivalent latter green FLC. responses Figre 8 frr in eqivalent shows linearity FLC. This ofverifies eqivalent proposed fzzy PID eqivalence controller relationship with control between srface PID viewcontroller nder e(t) = eqivalent FLC. Figre 8 frr shows linearity For eqivalent of eqivalent FLC-controlled fzzy PID system controller in Figre with 6, control Matlab/Simlin srface view finishes nder et simlation () = in a manner of a continos-time system. Alternatively, for practical applications, a FLC cold be implemented in a discrete form. Therefore, to enhance applicability of this research, Sample Hold nit Zero Order Hold nit, which can be sed to model A/D (analog-to-digital) D/A (digital-to-analog) converters, are added to mae a discretized FLC, as shown in Figre 9. It mst be noted that Fzzy Logic Controller in Figres 6 9 are identical. So, in Figre 9, gain (K) in Discrete-Time Integrator or Discrete Derivative does not denote K P or K I, is set as. The system inpt reslting responses (ble line) are shown in Figre 5, with sampling period T s

8 Appl. Sci. 207, 7, 53 8 of 2 set as 0. s. Simlation reslts show eqivalent FLC implemented in a discrete form have similar Appl. Sci. 207, 7, 53 9 of 3 responses compared with conventional PID controller. If sampling period T s is set to a smaller vale, closer of two responses will be obtained. Appl. Sci. 207, 7, 53 9 of 3 (a) (a) (b) (b) (c) (d) (c) (d) Figre 7. Membership fnctions for fzzy variables (a) et () ; (b) Figre Membership fnctions fnctions for fzzy for fzzy variables variables (a) et () (a) e(t); ; (b) (b) et () et ();(c) et ();(d) t () in Matlab. e(t); (c) et ė(t); ; (d) (d) t (t) () in in Matlab. ; (c) () Figre 8. The control srface view of eqivalent fzzy PID controller ( e(t) = 0.36). Figre 8. The control srface view of eqivalent fzzy PID controller ( Figre 8. The control srface view of eqivalent fzzy PID controller ( et () et () = 0.36). = 0.36). For eqivalent FLC-controlled system in Figre 6, Matlab/Simlin finishes simlation For eqivalent FLC-controlled system in Figre 6, Matlab/Simlin finishes simlation in a manner of a continos-time system. Alternatively, for practical applications, a FLC cold be in a manner of a continos-time system. Alternatively, for practical applications, a FLC cold be implemented implemented in in a a discrete discrete form. form. Therefore, Therefore, to to enhance enhance applicability applicability of this of this research, research, Sample Sample Hold Hold nit nit Zero Zero Order Order Hold Hold nit, nit, which which can can be be sed sed to model to model A/D A/D (analog-to-digital) (analog-to-digital) D/A D/A (digital-to-analog) (digital-to-analog) converters, converters, are are added added to to mae mae a discretized a discretized FLC, FLC, as shown as shown in Figre in Figre 9. It 9. mst It mst be be noted noted that that Fzzy Fzzy Logic Logic Controller Controller in in Figres Figres are 9 are identical. identical. So, So, in Figre in Figre 9, 9, gain gain (K) (K) in in Discrete-Time Integrator or or Discrete Discrete Derivative Derivative does does not not denote denote K K P or P or K I, K I, is set is as set. as. The system inpt reslting responses (ble (ble line) line) are are shown shown in Figre in Figre 5, with 5, with sampling sampling period period

9 Appl. Sci. 207, 7, 53 0 of 3 Appl. Sci. 207, 7, 53 0 of 3 set as 0. s. Simlation reslts show eqivalent FLC implemented in discrete form have T s set as 0. s. Simlation reslts show eqivalent FLC implemented in a discrete form have similar responses compared with conventional PID controller. If sampling period similar responses compared with conventional PID controller. If sampling period T is is set set to to a smaller smaller vale, vale, closer closer of of two two responses responses will will be be obtained. obtained. Appl. Sci. 207, 7, 53 s 9 of 2 Figre The Theeqivalent FLC-controlled system in in discrete form. We We slightly adjsted membership fnctions (MFs) of eqivalent FLC design in Figre 7. slightly adjsted membership fnctions (MFs) of eqivalent FLC design in Figre It It cold be fond that MFs are no longer eqally-spaced, which implies that FLC has become cold be fond that MFs are no longer eqally-spaced, which implies that FLC has become nonlinear, as shown in Figre 0. The 3 singleton vales for otpt fzzy variable nonlinear, as shown in Figre 0. The 3 3singleton vales for for otpt fzzy variable t (t) () are are 8, 8, 6.9, 5.05, 2.44, 2.44, 9.05, 9.05, 9.05, 4.9, 4.9, 4.9, 0, 0, 4.9, 4.9, 0, 4.9, 9.05, 9.05, 9.05, 2.44, 2.44, 2.44, 5.05, 5.05, 5.05, 6.9, 6.9, 6.9, Figre Figre shows that control srface view is is no longer a plane. The reslting error signal, error integral, error derivative, control signal, system otpt are shown in Figre 2 (ble line), can be compared with previos PID controller, eqivalent FLC (red lines for both). (a) (a) (b) (b) (c) (c) (d) (d) Figre 0. Membership fnctions adjstment for fzzy variables (a) et () (b) et () (c) Figre 0. Membership fnctions adjstment for fzzy variables (a) e(t); et () (b); (b) e(t); et (c) () ė(t); ; (c)(d) et () (t) ; (d) in Matlab. in Matlab. (d) t () in Matlab.

10 Appl. Appl. Appl. Sci. Sci. Sci. 207, 207, 7, 53 7, 53 0 of of 3 of 23 Figre Figre.. The The control control srface srface view of of eqivalent fzzy fzzypid PIDcontroller ( ( et () e(t) = = 0.36) 0.36) nder nder MFs Figre MFs adjstment.. The control srface view of eqivalent fzzy PID controller ( et () = 0.36) nder MFs adjstment. Figre 2. The (a) step inpt; (b) error signal; (c) error integral; (d) error derivative; (e) control signal Figre 2. The (a) step inpt; (b) error signal; (c) error integral; (d) error derivative; (e) control (f) system otpt with PID controller, eqivalent FLC, eqivalent FLC nder signal (f) system otpt with PID controller, eqivalent FLC, eqivalent FLC nder MFs adjstment. MFs adjstment. Figre 2. The (a) step inpt; (b) error signal; (c) error integral; (d) error derivative; (e) control signal (f) system otpt with PID controller, eqivalent FLC, eqivalent FLC nder MFs adjstment.

11 Appl. Sci. 207, 7, 53 of 2 Table smmarizes response performance, inclding rise time (Tr), settling time (Ts), percentage overshoot (P.O.), steady-state error (E ss ), of three controllers in Figre 2. Which controller has better performance cannot be determined since performance criterion is not defined. Bt based on this stdy, a fzzy PID controller may otperform a conventional PID controller qicly by fine-tning MFs of fzzy variables. Table. Response performance of different controllers. Controller Tr (s) Ts (s) ±5% P.O. (%) E ss PID, FPID FPID with adjstment We have fond that some learning-based techniqes or evoltionary algorithms have been applied in optimal FLC design [4 6]. Experienced researchers shold agree on importance of setting initial vales or weights in learning system, which will greatly inflence learning reslts convergence speed. With proposed eqivalence relationship, one can easily qicly obtain a fzzy PID controller throgh a conventional PID controller design, n derived eqivalent FLC can be set as one of initial designs. This process will reslt in optimal FLC design in an efficient way. 4. Conclsions This paper proposed clearly identified eqivalence relationship between a conventional PID controller a FLC. The derived eqivalence eqation is straightforward, so a well-designed conventional PID controller can be easily transformed to an eqivalent FLC by simply defining inpt/otpt operating ranges following Sgeno-style inference. The nowledge base for eqivalent FLC can be bilt by a cbe FAM, instead of combination strctre of PI-type or PD-type FLCs. Simlation reslts demonstrate effectiveness of proposed approach, where system responses with conventional PID controller or eqivalent FLC are similar. Moreover, eqivalent FLC implemented in discrete form was also provided simlated, with comparable system responses to original conventional PID controller. Based on reslt of this stdy, matre design reslts of traditional PID controllers can be applied as prior nowledge for an FLC design. Sbseqently, a nonlinear FLC can otperform a traditional linear PID controller by changing fzzy rle design or fzzy membership fnctions. The proposed eqivalent FLC can be set as initial design for some learning-based techniqes or evoltionary algorithms, which may achieve optimal FLC design considerably improve convergence speed. Or ftre wor will involve developing an optimal fzzy PID controller with a simpler strctre fewer parameters, which will be designed to be more efficient for practical applications. Acnowledgments: The athors wold lie to express sincere thans to reviewers for ir invalable comments sggestions. This wor is spported by Ministry of Science Technology, Taiwan, nder Grant no. MOST E Athor Contribtions: Jing-Shian Chio Chn-Tang Chao developed methodology drafted manscript. Moreover, Nana Starna Chi-Jo Wang implemented Matlab/Simlin simlations. The athors approved final manscript. Conflicts of Interest: The athors declare no conflict of interest. References. Ziegler, J.G.; Nichols, N.B. Optimm settings for atomatic controllers. Trans. ASME 942, 64, [CrossRef] 2. Mamdani, E.H.; Assilian, S. An experiment in lingistic synsis with a fzzy logic controller. Int. J. Man Mach. Std. 975, 7, 3. [CrossRef]

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