基於增益與相位餘量之PID控制器調整方法－利用模糊類神經網路

1

  PID  

Tuning of PID Controllers Based on Gain and Phase Margin Specifications Using Fuzzy Neural Network
NSC87-2218-E009-026  86/08/01- 87/7/31     
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(Keywords: PID Control, Fuzzy neural network, Gain margin, Phase margin)

In this project, we will give an effective PID tuning method using fuzzy neural network based on gain and phase margin specifications (FNGP). We will use the fuzzy neural networks to determine the parameters of PID controllers. Because of the analytical design method to achieve the specified gain and phase margins is not available to date.

In this study, we will use a fuzzy neural modeling method to identify the relationship for different gain and phase margin specifications. So that neither numerical methods nor graphical methods have to be used. This will make it easy and effective to tune the controller parameters to have the specified robustness and performance.

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 EKÞß+,-./0-.(GPM)78Æ ‰@ABFG;p˜™àáâN=Š‹ŒKC DELM;À‰@AãÐãä6FG;Æ ½G=p屈LM{|CDGÊH æ Ì ç è é ê 6 $ % & '( ) * (Fuzzy neural network);ëìw;í‰@«'()*; ՉD«'()*6†î婢@«$% f;Iàáðwñ$%&'()*Hò d'()*6óôõöJ÷#Jøù## ú´Óûd$%E6üôýöþ5# þ5i
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‡;^:Yϱ6LMHR5 C. C. Hang ú [4]w$“Ë•Y–O;J–X>?w FGƉ@«do#J6FG;Iàá >?@«‘do6FG;IFG‰<=$ %&'()*K{|CD PID LM;^PQ àá678H  rs@fEw y;Œ 1.‰ @«D6 *fHOVWf   E6 !" M (Transfer function) g# ñ ) (s GP UGC(s);m+,-./0-.678ñ m m A ,φ H^$ʉ+,-./0-.6VW , )] ( ) ( arg[G_c jw_p G_p jw_p =−π (1) ) ( ) ( 1 p p p c m jw G jw G A = (2) ) ( ) ( g p g c jw G jw G =1 (3) π φ_m=arg[G_c(jw_g)G_p(jw_g)]+ (4)
 wp       (phase crossover

frequency) _wg       (gain crossover frequency)  PI   ) 1 1 ( ) ( I c c sT K s G = + KCTI   Ls dp dp d d d d nq nq n n n n p p e s w s w s w s w s w s w K s G − + + + + + + = ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) ( 2 2 1 1 2 2 1 1 L L _.
 wni, wdi L !" #$(zero)% $(pole)&'() *+,-./0 1(loop transfer function)

Ls dp dp d d d d I nq nq n n n n I p c p c _{sT w s w s ws} e s w s w s w sT K K s G s G − + + + + + + + = ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 )( 1 ( ) ( ) ( ₂ 2 1 1 2 2 1 1 L L . 2345678(1-4) L w w w n w w n T wp I + p n + + q p nq − p

+_tan−_{( ) tan}−_{( ) tan}−_{( )} 2 1 1 1 1 1 1 _L π 0 ) ( tan ) ( tan ) ( tan 1 2 1 2 1 1 1 − − − = − − − − dp p p d p d pw d ww d w w w d L ,(5) dp dp p d d p d d p nq nq p n n p I p I p p c m w w w w w w w w w w T w T w K K A ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( 1 2 2 2 2 2 2 1 2 1 2 2 2 1 2 1 2 2 2 + + + + + + = L L _{, (6)} nq nq g n n g I g dp dp g d d g d d g I g p c w w w w T w w w w w w w T w K K ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( 2 2 1 2 1 2 2 2 2 2 2 2 2 2 1 2 1 2 + + + − + + = L L _{, (7)} L w w w n w w n T wg I g n q g nq g m= + + + + − − − − ) ( tan ) ( tan ) ( tan 2 1 1 1 1 1 1 _L π φ ) ( tan ) ( tan ) ( tan 1 2 1 2 1 1 1 wgwd d wgwd dp wgwdp d − ₋ − _{− −} − − L . (8)
   (Am,φm)(5)-(8)  !"#$%&'() PI *+,-.# (KC,TI)/0123(wp,wg)456789 :;<=#(arctan function)>?&@ABC &D&E 5FGHI&JK(coupled):;<= #@LMNI'AOP(5)-(8)&Q RSTUVW"XYZ[\]^_() `a=#(5)-(8)bc 2dc 2 5efghE ic 2 7>W"XYZ[\]^5B Cjk]^lmnbc 3E[\]^oBk p Linguistic term layerqrsXYtEV W"uv=#5wx=#(bell function)Eoy kTzXY%{|}~BC node € ‚ƒzEo„kp;tVoykr…€ ;tEo†k!"‡ˆ‰%€Š‹ƒzE ojk{p&XYtWŒ%ƒz‡Ž' r…‘E i’“”•–V!"OP‡ˆ‰ %(linear least-squares method)€AO’“ (off-line learning)—'[\]^7-.#$Ei ?   ˜ ™ FNN 5 g B r … š › 5 output=F(I,S),l7 I 5rsœS 5.#E žŸ.# S  ¡¢£¤¥¦iB=# H §'¨©=# HoF 5BOP=#{ !" ‡ˆ‰%_ª«?.#E¬f­AO’ “%® ¯°B®—' M ”•±²³´XP=Yl

3 7 [ ,...., ] , [ ,...., , ,..., ,..., 1,..., ] 1 1 1 0 1 0 2 1 n k k n n T _{P p p p p p p} y y Y = = E ¯°y®!"‡ˆ‰%_µ¶.# P Y X X X P_*=( T ₎−1 T E ¯°„®§"¬·¸6‡ˆ‰% z…. #$ M i i T i i T i i i i i i i i i i i i P P M i x Q x Q x x Q Q Q P x y x Q P P = − = + − = ⋅ − + = + + + + + + + + + + 1 ,...., 1 , 0 , 1 ) ( 1 1 1 1 1 1 1 1 1 1 l7¹º$»5 P0=0,Q0=α⋅I (α 5B¼; #)E(½¾ƒz .¿ÀÁ[5]) NÂW"Ã~œ%(Gradient Method) € O  ’ “ (on-line learning)  § Ä Å = # 2 2 1[ ] I P I y y E = − ˆtl7 y pi2Æ-P r…y pXYZ[\]^ FNGP r…(ÇcI 2.)EÃ~œ% f­b¬®ÈÉI'Ä Å=#-Ã~(gradient) . ) ( ) ( ) ( ) ( ) ( ) ( 1 1 1 1 S k O k e S k y k e S k e k e S E I I I I I I I ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ − = − = = FNN7 Ê.#Ë "¬%{_ÊÌ ( 1) ( ) ( ) ( ) ( ) S E k S k S k S k S I I _∂ ∂ η − + = ∆ + = + l7ηI 5’“3(Learning rate). p͞ PI *+,.#-ÊΠ PID *+,ÊÎ.#% PI *+,Ͻ¾Ð ÑÒÓ ÔÕÐÖ×.Ø.¿ÀÁ[4]E i?!">م% C. C. Hang ڒÓ>€-XÛÜÝÞBßešB à … C. C. Hang ڒÓ>Ù-ÄÅ$á¼â% ãäå"iæçè éê-%{¼¼ë ì?Bí §-LŠîïEXÛ7ðñ =#® ) 1 ( ) ( 0.1 s e s G_p s + = − E c 4 c 5 òó5ôÏ   - X Û Ü Ý d ¯ õ ö å h l   5 ) 45 , 3 ( ) , ( 0 ₌ m m A φ ( , 0)₌(5,60) m m A φ E
 QRSTU÷!"XYZ[\_ª« ø PID *+,.#-ùúE@!"XYZ[\]^ÔûÊÎ PID *+,.#üKôÏ E4?ôýþ
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[1] G. H. Cohen and G. A. Coon, “Theoretical Consideration of Retarded Control,” Trans. ASME, Vol. 75, pp. 827-834, 1953.

[2] C. C. Hang, K. J. Astrom, and W. K. Ho, “Reinements of the Ziegler-Nichols Tuning Formula,” IEE Proc. D, Vol. 138, No. 2, pp.111-118, 1991.

[3] K. J. Astrom, “Automatic Tuning of Simple Regulator with Specification on Phase and Amplitude Margins,” Automatica, Vol. 20, No. 5, pp.645-651, 1984.

[4] W. K. Ho, C. C. Hang, and L. S. Cao, “Tuning of PID Controller Based on Gain and Phase Margin Specification,” Automatica, Vol. 31, No. 3, pp. 497-502, 1995.

[5] T. C. Hsia, System Identification : Least-Squares Methods (New York, Heath, 1977).

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. 1   1 2 0 1 1 1 1 1 2 2 2 2 2 2 2 2 2 π τ τ τ φ π τ + − − = = +₊ = + + = + − − arctan ( ) arctan ( ) , , , arctan ( ) arctan ( ) . w T w w L A K K w T _{w T}w K K w T w w T w T w w L p I p p m c p p I p p I c p g I g g I m g I g g FNGP ( , )K Tc I yp yI e + -( , )Amφm ( $ , $ )K Tc I
2.  lay er 1

layer 2 layer 3 layer 4 layer 5

Am φm A1 An A_{i n+} A2n ∏ ∏ N N ∑ w1 wn2 wn2 w1 _{w f1 1} w f_n2 _n2 yI Am φm A_m φm R R µA1 µAn µAi n+ µA2n
3.   Step response Magni tu de ___ uncom pensated - - - GPM - . - FNGP e s s − + 01 1 . ( ,3 45o) Process Spec. Tim e ( sec)
4.   3dB,45o Step response M agn it ud e Time (sec) e s s − + 01 1 . ( ,5 60o) Process Spec. ___ uncompensated - - - GPM - . - FNGP
5.   5dB,60o     m A o m φ Kc TI wg wp A*m * m φ Am′ φ ′m GPM 3 45 4.908 0.352 5.398 12.78 3.307 40.30 10.2% 10.4% 5 60 3.054 0.541 3.341 13.53 5.990 58.14 19.8% 3.10% FNGP 3 45 6.198 0.562 6.243 13.58 2.978 45.41 0.71% 0.91% 5 60 3.785 0.660 3.921 13.77 5.029 60.03 0.58% 0.07% 1.