Περί αυτοματισμού (3.8

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* . 3.8, 2008 * + (, ): 1 . 1 ....................................................................................... 61.1 ................................................................................................................................................... 61.2 ............................................................................................ 61.3 Laplace .................................................................................................................... 71.3.1 .................................................................................................................................................. 71.3.2 Laplace .......................................................................131.3.3 Laplace ...................................................................141.3.4 Laplace ...............................................................171.3.5 Laplace .......................................................202 ............................................................................................. 222.1 ................................................................................................................222.1.1 .......................................................................................................302.2 .............313 .................................................................................. 323.1 .................................................................................................................323.2 ...............................................................................................................363.2.1 ..............................................................................................................383.3 .................................................................413.3.1 ......................................................................................................423.3.2 ................................................................................................................................................503.3.3 ...............................................................................................................523.3.4 .....................................................................................................................................553.4 ..........................................................................................................563.4.1 ...........................................563.4.2 ..................................................593.4.3 ....................................................................................................................614 ....................................................................................... 644.1 ...............................................................................................................................................644.2 ...............................................................644.2.1 ...............................................................................................654.3 .......................................................................................................................................684.3.1 PID (--) ..........................................................................68 4.3.2 PID .......................................................................................................834.3.3 - ..........................................................................................845 ............................................................................................... 885.1 ...................................................................................................................................896 .......................................................................................... 93 ............................................................................................................ 95 ........................................................................................................ 96 ..........................................................................................................................................................96 ............................................................................................................................................................96 ....................................................................................................... 100 . . . : - 6 1 1.1 ,-, . , ' . ,, , ) ( ) ( ) ( 2 ) (20t f t t t = + + - : (t)=1(t)+2(t) , 1(t) -2(t).f(t), (t) . 1.2 -. . - s., s=+j . 1.1. - . =0 . . G(s) , G j G s G Im Re ) ( + =. . . : - 7 0Im(s)Re(s)s - 1(/ )2 2 + 1.1 s Re, Im . - 4 -., . 1.2. 0 Im(s) Re(s)s1=1+1i 0Im(G) Re(G)G (s1) 1.2 , |.|

\|= Z + = = + =Zs s e s j ss j , ,2 2 s s Z . 1.3 Laplace 1.3.1 Laplace - , -,()., Laplace,- , Cest, s - . , - , . . . : - 8 ) ( ) ( ) ( 2 ) (20t f t t t = + + (1.1)(;)(1.1) - (1.1)., (t)=1(t)+2(t) , 0 ) ( ) ( 2 ) (120 1 1= + + t t t f (t) Fest, F s , 2(t) est, , 20222022022) (22 + += + += = + +s sFets sFFe e e s e sstst st st st - 2(t) , 1(t) . -est.1.3. . . . : - 9 1.3 est f(t), , , f(t)est.. Fourier.M , x(t) , iTns t e t x Ce CTt xstnstn 2, d ) (1) (= ==+ (1.2) . - . . 1.4. . . . : - 10

1.4 Fourier :- . .' , (1.2) , Tt e t x s Ce s C t xi n ssti n snst 2, d ) ( ) () (21) (0//00000=

=

=== = (1.3) T (1.3).0,- ( -), 0. d=ds/i , , i sttsti sssstt e t x s Ce s sC t x= = == = =

=

=d ) ( ) () ( d21) ( (1.4) (1.4) (1.4) Fourier (1.4) Fourier. x(t) , (1.4)s=i. x(t)=x0u(t),-, Fourier. . . . : - 11 x(t), s=+i (1.4), (1.4) : i sttti tt e e t x s C + = = =

=d ) ( ) ( (1.5)(1.5)Laplace.x(t) (1.5) , x(t)=etu(t) Re() + = = j sseT t usTL, =0, | | 0 , ,1) ( ) ( > + = = = j sss U t u L. . Mu(t)eat , | |a sMe t Muat+=) ( L- e. , | |2 2) ( ) (+=st t u L u(t) t>0. . . . : - 17 1.3.4 Laplace , , , n ms as ba s a s a sb s b s b s bs Fnnnmmmm< =+ + + ++ + + +=,) () (......) (0 1110 111 , , Laplace. , , . -F(s).1, 2, ..., n a(s). . . , F(s) , nnkk s c s c s cs F+ ++ += ... ... ) (11 , | |t nt kt n ke c e c e c s F t f + + + + = =... ... ) ( ) (111L ci , n k s F s ck skk..., , 1), ( ) ( = = (1.20)1.1Laplace ) 2 )( 1 (2 4) (+ + +=s s sss F . : F(s) (0, 1, 2). , 3) 1 (2 4 ) ( ) 2 ( 2) 2 (2 4 ) ( ) 1 ( 1) 2 )( 1 (2 4 ) ( 2 231 120 01 =++= + ==++= + ==+ ++= = s s ss F s cs sss F s cs sss sF cs ss ss s , +++ =2312 1) (s s ss Ff(t)=1+2et3e2t

. . . : - 18 . ., , 1 a(s) r, , ( ) I+ = =nr iir s s s a11) ( ) ( F(s) : ( )( )nnrrrr s c s c sc s cs F+ +++ +=++... ... ) (11111(1.21) (1.21) , | |t nt rt rrt n re c e c e t c e c s F t f + + + + + = =++ ... ... ) ( ) (1 1 11111L ci : ( ) | | r k s F ssk rcrk rk rsk..., , 1, ) (dd)! (111=)`= (1.22) (1.20). 1.2 y (t) ( )22) (nns ss Y+= . : Y(s) , ( )23 2 1) (nnscs cscs ++++ =, { }11 ) ( ) (dd ) ( ) ( 12222223= =)` = + = =)`= + = css Y sscss Y s cnsnsnnsnsn nn n , ( )21 1) (nnnss ss Y++ = Laplace : ( ) t e te e t ynt tntn n n + = = 1 1 1 ) (. . . : - 19 . ., , a(s) - i =2 , 1. - (), , tntkt tnnkkn ke c e c e c e c t fs cs cs cs cs F + + + + + =+ ++ ++=... ... ) (... ... ) (1 11 11111 . (). , te c11 . : 1. c1, ) ( ) ( 1 11s F s cs = 2. , | |( )1 11 1 1 21 1 1 1c t e ce e e e e c e c e ctti c i ti c i t t tZ + =+ = + Z Z (1.23)(: c1 ) 1.3 Laplace ) 25 8 (5) (2+ ++=s s sss F . : sci sci scs F2 1 1) 3 4 ( ) 3 4 () ( + +++ += , ) 3 (30124 183 1) 6 )( 3 4 (3 1) ( ) 3 4 ( 25 85) ( 3 41510202iiii i is F i s cs sss sF ci sss+ =+ = = + + ==+ + += = = , (1.23), ( ) ( ) | |311 4 3 151051) ( + + = t e t ft . . . : - 20 ( ) 3 / 1 , 1030111 1 = Z =c c . 1.3.5 Laplace n , ) ( + ... ) ( ) ( ) ( + ... ) ( ) (0) 1 (1) (0) 1 (1) (t r b t r b t r b t y a t y a t ymmmmnnn+ + = + + (1.24) n , ) 1 (00) 1 (0 0 0 0) ( ..., , ) ( , ) ( = = =n ny t y y t y y t y Laplace , ) () () ( ) () () () () () () (s a s cs R s Gs a s cs Rs as bs Y + = + =(1.25) 0 1110 1......) () () (a s a s a sb s b s bs as bs Gnnnmm+ + + ++ + += = (1.26) (1.24). (1.25) : )`+)`= ) () () () () () (1 1s a s cs Rs a s bt y L L(1.27) (1.27) , Laplace -,- . , . (1.27) , )`=) () () () (1s Rs a s bt y L(1.28) , . . (1.24) n=2, m=0, , ) ( ) ( ) ( ) (0 0) 1 (1) 2 (t r b t y a t y a t y = + + , 0 0 0 0) ( , ) ( y t y y t y = =(1.29) Laplace (1.25) , . . . : - 21 (s2+a1s+a0)Y(s)=b0R(s)+(s+a1)y0+0y (s2+a1s+a0)Y(s)=b0R(s)+(s+a1)y0(s)+0y (s) (s)1 Laplace - (t) ( 1.18). 0y (s) 0y (t). (s+a1)y0(s) - sy0(s). , | |) ( ) (dd) (dd) (dd0 01 -0 0s s y ttytty t yt =)` =L (1.29) , | | 0 ) ( ) (), ( ) ( ) ( ) ( ) ( ) (0 0 0 0 0 1 0 0) 1 (1) 2 (= = + + + = + + t y t y t y t y y a t r b t y a t y a t y u(t) , - . :Laplace t=0(.. ). , . . . . : - 22 2 2.1 - - . , - . - 2. - . . , , ) ( ) ( + ... ) ( ) (0) 1 (1) (t r t y a t y a t y annnn= + + , 0111ddddatatannnnnn+ + + = , {y(t)}=r(t) , r(t) y(t) 2