Bài giảng Control system design - Chapter II: Mathematical Models of Systems - Nguyễn Công Phương

Nguyễn Công Phương CONTROL SYSTEM DESIGN Mathematical Models of Systems Contents I. Introduction II. Mathematical Models of Systems III. State Variable Models IV. Feedback Control System Characteristics V. The Performance of Feedback Control Systems VI. The Stability of Linear Feedback Systems VII. The Root Locus Method VIII.Frequency Response Methods IX. Stability in the Frequency Domain X. The Design of Feedback Control Systems XI. The Design of State Variable Feedback Systems XII. Robust Control Systems XIII.Digital Control Systems sites.google.com/site/ncpdhbkhn 2 Mathematical Models of Systems 1. Differential Equations of Physical Systems 2. Linear Approximations of Physical Systems 3. The Laplace Transform 4. The Transfer Function of Linear Systems 5. Block Diagram Models 6. Signal – Flow Graph Models 7. The Simulation of Systems Using Control Design Software sites.google.com/site/ncpdhbkhn 3 Differential Equations of Physical Systems (1) i  v  • Current i: a through-variable • Voltage v: an across-variable sites.google.com/site/ncpdhbkhn 4 Differential Equations of Physical Systems (2) Variable Integrated Variable Integrated through through- across across- System element variable element variable Current, Charge, Voltage, Flux linkage, Electrical i q v λ Mechanical Translational Force, Velocity, Displacement, translational momentum, F v y P Mechanical Angular Angular Angular rotational Torque, T momentum, velocity, displacement, h ω θ Fluid Fluid Pressure volumetric rate Volume, Pressure, momentum, of flow, V P γ Q Thermal Heat flow rate, Heat energy, Temperature, q H T sites.google.com/site/ncpdhbkhn 5 Differential Equations of Physical Systems (3) Inductive storage Electrical inductance: L v: voltage di 1 2 i v21  L E Li i: current dt 2 v v L: inductance 2 1 Translational spring: v: translational velocity 1 dF 1 F 2 k F F: force v21  E  k dt 2 k v v k: translational stiffness 2 1 Rotational spring: ω: angular velocity 1 dT 1 T 2 k F T: torque 21  E  k dt 2 k 2 1 k: rotational stiffness Fluid inertia: I P: Pressure dQ 1 2 Q PI21  E IQ Q: fluid volumetric flow rate dt 2 P2 P1 I: fluid inertance sites.google.com/site/ncpdhbkhn 6 Differential Equations of Physical Systems (4) Capacitive storage Electrical capacitance: C dv21 1 2 i v: voltage; i: current i C E Cv21 C: capacitance dt 2 v2 v1 Translational mass: 2 dv 1 F F v1  v: translational velocity; F: force FM 2 E  M const M: mas; k: translational stiffness dt 2 k v2 Rotational mass:   d2 1 2 T 1 : angular velocity; T: torque TJ EJ  ω 2 J const J: moment of inertia dt 2 2 Fluid capacitance: dP 1 2 Q P: Pressure; C : fluid capacitance QC 21 ECP C f f dt 2 f 21 f Q: fluid volumetric flow rate P2 P1 Thermal capacitance: q dT T1  2 Ct P: Pressure; q: heat flow rate; q Ct ECT t 2 const dt T2 Ct: thermal capacitance sites.google.com/site/ncpdhbkhn 7 Differential Equations of Physical Systems (5) Energy dissipators Electrical resistance: R v v2 i v: voltage; i: current i  21 21 P  v R: resistance R R v2 1 Translational damper: 2 F b v: translational velocity; F: force F bv21 P bv21 b: viscous friction v2 v1 Rotational damper: T b : angular velocity; T: torque T b P b2 ω 21 21   b: viscous friction 2 1 Fluid resistance: 2 R f P21 P Q P: Pressure; R : fluid resistance Q  P  21 f R R Q: fluid volumetric flow rate f f P2 P1 Thermal resistance: T T Rt q P: Pressure; q: heat flow rate; q  21 P  21 Rt Rt Rt: thermal resistance T2 T1 sites.google.com/site/ncpdhbkhn 8 Differential Equations of Physical Systems (6) Ex. 1 R L C Wall friction r() t b k – + Mass y M i() t Force di( t ) 1 t r(t) Ri()()() t L  i t dt  r t dt C 0 d2 y()() t dy t M b  ky()() t  r t dt2 dt dv() t t M  bvt()()()  kvtdtrt  dy() t dt 0 v() t  dt sites.google.com/site/ncpdhbkhn 9 Differential Equations of Physical Systems (7) v1() t iR1 v2 () t Ex. 2 iL iC1 iR2 R1 iC2 L1 k R2 C C1 2 Friction b2 r() t Velocity v (t) M2 2 Friction b iCRR1 i 2  i 1  r() t 1  iRCL1 i 2  i  0 M1 Velocity v1(t)  dv1 v 1 v 1 v 2 C1    r() t Force r(t)  dt R R  2 1  t v1 v 2 dv 2 1 C2  v 2 dt  0 dv1 0 M b v  b v  b v  r() t  R1 dt L 1dt 1 1 2 1 1 2  dv11 1 v 2 C1 v 1  v 1   r() t dv t  dt R2 R 1 R 1 M2  b v  b v  k v dt  0   2 1 2 1 1 2 t dt 0  dv2 1 1 1 C2 v 2  v 1  v 2 dt  0 0  dt R1 R 1 L sites.google.com/site/ncpdhbkhn 10 Mathematical Models of Systems 1. Differential Equations of Physical Systems 2. Linear Approximations of Physical Systems 3. The Laplace Transform 4. The Transfer Function of Linear Systems 5. Block Diagram Models 6. Signal – Flow Graph Models 7. The Simulation of Systems Using Control Design Software sites.google.com/site/ncpdhbkhn 11 Linear Approximations of Physical Systems (1) y x() t y() t System y mx 0 x x() t y() t x() t y() t 1 System 1 System x2 () t y2 () t System kx() t ky() t System x()() t x t y()() t y t 1 2 System 1 2 Superposition Homogeneity sites.google.com/site/ncpdhbkhn 12 Linear Approximations of Physical Systems (2) y y mx  b x() t y() t y1 System y y0 x 0 x0 x1 x y1 mx 1  b  y 0   y  m() x 0   x  b y0 mx 0  b  y  m  x Linear !!! sites.google.com/site/ncpdhbkhn 13 Linear Approximations of Physical Systems (3) y y g() x dg x() t y() t y0 dx System x x0 y( t ) g [ x ( t )] 0 x0 x dg()()x x   d2 g x  x 2  0 0  g( x0 )   2  ... dx x x 1!  dx 2!  0  x x0  dg g()() x  x  x 0dx 0 x x0 y0  m() x  x 0 ()()y  y0  m x  x 0  y  m  x sites.google.com/site/ncpdhbkhn 14 Linear Approximations of Physical Systems (4) x() t y() t System y g( x1 , x 2 ,..., xn )  g g( x10 , x 20 ,..., xn 0 )  ( x 1  x 10 )  x1 x x0 g  g (x2  x 20 )  ...  ( xn  x n 0 ) x2  xn x x0 x x0 sites.google.com/site/ncpdhbkhn 15 Linear Approximations of Physical Systems (5) Ex. T mgLsin sin  T  mgL ()   0 0  0 00;T 0  0 T  mgL(cos0o )(  0 o )  mgL  T   2   doku.php?id=phy141:labs:lab1 0   2 sites.google.com/site/ncpdhbkhn 16 Mathematical Models of Systems 1. Differential Equations of Physical Systems 2. Linear Approximations of Physical Systems 3. The Laplace Transform 4. The Transfer Function of Linear Systems 5. Block Diagram Models 6. Signal – Flow Graph Models 7. The Simulation of Systems Using Control Design Software sites.google.com/site/ncpdhbkhn 17 The Laplace Transform (1)  The Laplace transformation of f(t): F()() s f t est dt 0 1  j  The inverse Laplace transform of F(s): f()() t F s est ds 2 j  j  f(t)  ()t u() t eat t teat sin at cosat 1 1 1 1 a s F(s) 1 2 2 2 2 2 2 s s a s ()s a s a s a sites.google.com/site/ncpdhbkhn 18 The Laplace Transform (2) Property f(t) F(s) 1. Magnitude scaling Af() t AF() s 2. Addition/subtraction f1()() t f 2 t F1()() s F 2 s 1 s  3. Time scaling f() at F   a a  f( t a ) u ( t  a ), a  0 eas F() s 4. Time shifting f( t ) u ( t a ), a  0 eas L[ f ( t a )] 5. Frequency shifting eat f() t F() s a 6. Differentiation dn f()/ t dt n sn F( s ) s n1 f (0)  s n  2 f 1 (0) ...  s o f n  1 (0) 7. Multiplication by t tn f() t ( 1)nd n F ( s ) / ds n  8. Division by t f()/ t t F() d  s t 9. Integration f() d  F()/ s s 0 t 10. Convolution f1()*()()() t f 2 t f 1 f 2 t   d  F()() s F s 0 1 2 11. Final value limf ( t ) limsF ( s ) t s0 sites.google.com/site/ncpdhbkhn 19 The Laplace Transform (3) d2 y()() t dy t Ex. 1 M b  ky()() t  r t dt2 dt r()() t R s y()() t Y s ky()() t  kY s dy() t b b sY( s )  y (0 )  dt   d2 y() t dy  M M s2 Y( s )  sy (0 )  (0  ) dt2 dt  dy  M s2 Y() s  sy (0)  (0)   b sY () s  y (0)    kY () s  R () s dt    dy    R() s M sy (0)  (0)  by (0) dt  p() s ? Y() s     y() t Ms2  bs  k q() s sites.google.com/site/ncpdhbkhn 20 The Laplace Transform (4) Ex. 1 4(s  3) 1.5 Y() s  (s 1)( s  2) 1 KK 0.5 1  2 s1 s  2 0 4(s  3) 4(s  3) ImaginaryPart -0.5 K1    8 (s  1) (s  2) (s  2) -1 s1 s1 -1.5 4(s  3) 4(s  3) K    4 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 2 (s 1) ( s  2) (s  1) Real Part s2 s2 4 3.5 8 4 Y() s   3 s1 s  2 2.5 2 at K Ke  1.5 s a 1 0.5 y( t )  8 et  4 e 2 t 0 0 1 2 3 4 5 sites.google.com/site/ncpdhbkhn 21 The Laplace Transform (5) Ex. 2 15 10 4s  76 5 Y() s  2 s6 s  265 0 ImaginaryPart -5 KK1 2   -10 s3  j 16 s  3  j 16 -15 -30 -25 -20 -15 -10 -5 0 5 10 Real Part 4s  76 K  6 1 -3t o 5.66e cos(16t - 45 ) (s 3  j 16) (s 3  j 16) 5.66e-3t s3  j 16 4 -5.66e-3t o 2 j 2  2.83  45 2 0 y( t )  2  2.83 e3t cos(16 t  45 o ) -2 -4 5.66e3t cos(16 t  45 o ) -6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time sites.google.com/site/ncpdhbkhn 22 Mathematical Models of Systems 1. Differential Equations of Physical Systems 2. Linear Approximations of Physical Systems 3. The Laplace Transform 4. The Transfer Function of Linear Systems 5. Block Diagram Models 6. Signal – Flow Graph Models 7. The Simulation of Systems Using Control Design Software sites.google.com/site/ncpdhbkhn 23 The Transfer Function of Linear Systems (1) R() s Y() s System OutputY ( s ) G() s   InputR ( s ) dn y d n1 y d n  1 r d n  2 r q ...  q y  p  p  ...  p r dtnn1 dt n1 0 n  1 dt n  1 n  2 dt n  2 0 n1 n  2 p() s pn1 s p n  2 s ...  p 0 Y()()()()() s  G s R s  R s  n n1 R s q() s s qn1 s ...  q 0 m()() s p s m()()() s p s n s   R() s   Y()()() s  Y s  Y s q()() s q s q()()() s q s d s 1 2 3 y()()()() t  y t  y t  y t 1 2 3 transient response steady-state reponse sites.google.com/site/ncpdhbkhn 24 The Transfer Function of Linear Ex. 1 Systems (2) d2 y dy dy Given 3  2y  4(),where r t y (0)  1; (0)  0; r () t  1, t  0.Find y ()? t dt2 dt dt dy  1 s2 Y() s sy (0)  (0)  3 sY () s  y (0)  2() Y s  4() R s  4 dt  s [s2 Y () s  s ]3[  sY ()1]2() s   Y s  4/ s s  3 4 KKK1 2  KK3 4 5  Y() s          s23 s  2 s ( s 2  3 s  2) s1 s  2   s s  1 s  2  2 1   2 4 2          s1 s  2   s s  1 s  2  y( t )  (2 et  e 2 t )  (2 e  2 t  4 e  t )  2 e2t 2 e  t  2 sites.google.com/site/ncpdhbkhn 25 The Transfer Function of Linear Ex. 2 Systems (3) Find the transfer function? R R 1 2 i2 vi ii – + vo ii   i2   vi v v i  v v i  0 v i ii     RRRR1 1 1 1   vo v v o  v v o  0 v o i2     RRRR2 2 2 2 v v v R i   o o   2 RR1 2 vi R1 sites.google.com/site/ncpdhbkhn 26 The Transfer Function of Linear Ex. 3 Systems (4) Find the transfer function? k  dv M1 ()() b  b v  b v  r t Friction b2  1dt 1 2 1 1 2 Velocity v2(t)  t M2  dv2 M2 b 1 v 2  b 1 v 1  k v 2 dt  0  0  dt Friction b1 MsVs()()()()() b  bVs  bVs  Rs  1 1 1 2 1 1 2 M Velocity v (t)   V() s 1 1 MsVs( ) bVs ( )  bVs ( )  k 2  0  2 2 1 2 1 1 s Force r(t) (/)()M2 s b 1  k s R s V1() s  2 (M1 s b 1  b 2 )( M 2 s  b 1  k / s )  b 1 2 V1() s M 2 s b 1 s  k Gv () s   2 2 R() s (M1s  b 1  b 2 )( M 2 s  b 1 s  k )  b 1 s 2 X1()()/ s V 1 s sGv () s M 2 s b 1 s  k Gx () s     2 2 R()() s R s s sM[(1s  bbMs 1  2 )( 2  bsk 1  )  bs 1 ] sites.google.com/site/ncpdhbkhn 27 The Transfer Function of Linear Ex. 4 Systems (5) Find the transfer function? Angle θ   Kf i f Tm K1 i a () t Inertia load  K1 Kf i f()() t i a t magnet-dc-motor-or-pmdc-motor/ ()()()K1 Kf I a i f t  K m i f t Tm()() s  K m I f s TL()()() s  T d s  T L s 2 TL ()()() s Js s  bs  s Vf()()() s R f  sL f I f s  ()s K G() s   m Vf( s ) s ( Js b )( L f s  R f ) K/( JL )  m f s( s b / J )( s  Rf / L f ) sites.google.com/site/ncpdhbkhn 28 The Transfer Function of Linear Ex. 4 Systems (8) Find the transfer function? Tm()()() s K1 K f I f s I a s If the armature current Ia is constant  ()s K G() s   m Vf( s ) s ( Js b )( L f s  R f ) K/( JL ) K/( bR )  m f  m f s( s b / J )( s  Rf / L f ) s(f s 1)(  L s  1) Td () s () Vf () s 1 If () s Tm () s 1 ()s 1  ()s Km Rf L s s Js b s TL () s Field Load Field – controlled DC motor sites.google.com/site/ncpdhbkhn 29 The Transfer Function of Linear Ex. 4 Systems (9) Find the transfer function? Tm()()() s K1 K f I f s I a s If the field current If is constant Tm()()()() s  K1 K f I f I a s  K m I a s Va()()()() s R a  sL a I a s  V b s Va()() s V b s Ia () s  Ra sL a Vb()() s K b s 2 TL ()()() s Js s  bs  s  ()s K G() s   m Vsa( ) sJsbLsR [( )( a  a )  KK b m ] Armature – controlled DC motor sites.google.com/site/ncpdhbkhn 30 The Transfer Function of Linear Ex. 4 Systems (10) Find the transfer function? Tm()()() s K1 K f I f s I a s Tm()() s  K m I a s If the field current If is constant V()() s K s Va()() s V b s b b Ia () s  2 Ra sL a TL ()()() s Js s  bs  s  ()s K G() s   m Vsa( ) sJsbLsR [( )( a  a )  KK b m ] Td () s T() s () Va () s Km m 1 ()s 1  ()s R L s Js b s () a a TL () s Armature Load K Back electromotive force b Armature – controlled DC motor sites.google.com/site/ncpdhbkhn 31 Mathematical Models of Systems 1. Differential Equations of Physical Systems 2. Linear Approximations of Physical Systems 3. The Laplace Transform 4. The Transfer Function of Linear Systems 5. Block Diagram Models 6. Signal – Flow Graph Models 7. The Simulation of Systems Using Control Design Software sites.google.com/site/ncpdhbkhn 32 Block Diagram Models (1) V() s K  ()s  ()s K f G() s  m G() s   m s( Js b )( Lf s  R f ) Vf( s ) s ( Js b )( L f s  R f ) R1() s Y1() s Y1()()()()() s G 11 s R 1 s  G 12 s R 2 s System  R2 () s Y2 () s Y2()()()()() s G 21 s R 1 s  G 22 s R 2 s R1() s Y1() s R2 () s System Y2 () s   RJ () s YI () s Y1() s  G11()()()() s G 12 s G 1J s   R 1 s        Y() s G()()()() s G s G s R s 2   21 22 2J   2  Y  GR                 YI () s  GI1()()()() s G I 2 s G IJ s   R J s  sites.google.com/site/ncpdhbkhn 33 Block Diagram Models (2) R1() s Y1() s Y1()()()()() s G 11 s R 1 s  G 12 s R 2 s System  R2 () s Y2 () s Y2()()()()() s G 21 s R 1 s  G 22 s R 2 s R() s Y() s 1 G11(s) 1 R() s 2 G22(s) Y2 () s sites.google.com/site/ncpdhbkhn 34 Block Diagram Models (3) X1 X2 X3 X1 X3 G1(s) G2(s) G1G2 or X1 X3 G2G1 Combining blocks in cascade Moving a summing point behind a block X1 X3 X1 X3 G G () () X2 X2 G sites.google.com/site/ncpdhbkhn 35 Block Diagram Models (4) X1 X2 X1 X2 G G G X2 X2 Moving a pickoff point ahead of a block Moving a pickoff point behind a block X1 X2 X1 X2 G G X X 1 2 1/G sites.google.com/site/ncpdhbkhn 36 Block Diagram Models (5) X1 X3 X1 X3 G G () () X2 1/G X2 Moving a summing point ahead of a block Eliminating a feedback loop X X 1 2 X G X 2 G 1 () 1 GH H sites.google.com/site/ncpdhbkhn 37 Block Diagram Models (6) R(s) Ea(s) Controller Z(s) Actuator U(s) Process Y(s) (–) Gc(s) Ga(s) G(s) Sensor B(s) H(s) Esa ()()()()()() Rs  Bs  Rs  HsYs Y()()() s G s U s  G()()() s Ga s Z s  G()()()() s Ga s G c s E a s Ys()  GsGsGsRs ()a () c ()[()  HsYs ()()] Y() s G()()() s G s G s   a c R()1 s G () s Ga () s G c ()() s H s sites.google.com/site/ncpdhbkhn 38 Block Diagram Models (7) Ex. H2 R(s) () Y(s) G1 G2 G3 G4 () H1 H3 X1 X2 X1 X2 G G X1 X 2 1/G Moving a pickoff point behind a block H2/G4 R(s) () Y(s) G1 G2 G3 G4 () H1 H3 sites.google.com/site/ncpdhbkhn 39 Block Diagram Models (8) Ex. H2/G4 R(s) () Y(s) G1 G2 G3 G4 () H1 H3 X1 X2 X3 X1 X3 G1(s) G2(s) G1G2 Combining blocks in cascade H2/G4 R(s) () Y(s) G1 G2 G3G4 () H1 H3 sites.google.com/site/ncpdhbkhn 40 Block Diagram Models (9) Ex. H2/G4 R(s) () Y(s) G1 G2 G3G4 () H1 H3 X X X G X 2 1 G 2 1 () 1 GH H Eliminating a feedback loop H2/G4 () R(s) GG3 4 Y(s) G1 G2 () 1 GGH3 4 1 H3 sites.google.com/site/ncpdhbkhn 41 Block Diagram Models (10) Ex. H2/G4 () R(s) GG3 4 Y(s) G1 G2 () 1 GGH3 4 1 H3 H2/G4 () R(s) GGG2 3 4 Y(s) G1 () 1 GGH3 4 1 H3 R(s) GGG2 3 4 Y(s) G1 () 1GGHGGH3 4 1  2 3 2 H3 sites.google.com/site/ncpdhbkhn 42 Block Diagram Models (11) Ex. H2 R(s) () Y(s) G1 G2 G3 G4 () H1 H3 R(s) GGG2 3 4 Y(s) G1 () 1GGHGGH3 4 1  2 3 2 H3 Y() s GGGG  1 2 3 4 Rs( ) 1 GGH341  GGH 232  GGGGH 12343 sites.google.com/site/ncpdhbkhn 43 Mathematical Models of Systems 1. Differential Equations of Physical Systems 2. Linear Approximations of Physical Systems 3. The Laplace Transform 4. The Transfer Function of Linear Systems 5. Block Diagram Models 6. Signal – Flow Graph Models 7. The Simulation of Systems Using Control Design Software sites.google.com/site/ncpdhbkhn 44 Signal – Flow Graph Models (1) R(s) Y(s) G(s) Y()()() s G s R s R(s) Y(s) R() s Y() s 1 1 G11() s G11(s) R1(s) Y1(s) G12 () s G21() s R2 () s Y() s 2 R (s) Y (s) G (s) 2 2 22 G22 () s Y1()()()()() s G 11 s R 1 s  G 12 s R 2 s  Y2()()()()() s G 21 s R 1 s  G 22 s R 2 s sites.google.com/site/ncpdhbkhn 45 Signal Flow Graph Models (2) – a11 1 a11 x 1 a 12 x 2  r 1  x 1 R X  1 1 a21 x 1 a 22 x 2  r 2  x 2 a21 a12 (1a ) x  a x  r  11 1 12 2 1  R X  a21 x 1 (1  a 22 ) x 2  r 2 2 2 1 a22  (1a22 ) r 1  a 12 r 2 1  a 22 a 12 x1  r 1  r 2  (1a11 )(1  a 22 )  a 12 a 21      (1a11 ) r 2  a 21 r 1 1  a 11 a 21 x2  r 2  r 1  (1a11 )(1  a 22 )  a 12 a 21    (1 a11 )(1  a 22 )  a 1221 a  1  a 11  a 22  a 1122 a  a 1221 a N  1 LLLLLLn   n m   n m p  ... n1 n , m n , m , p nontouching nontouching sites.google.com/site/ncpdhbkhn 46 Signal – Flow Graph Models (3) N  1 LLLLLLn   n m   n m p  ... n1 n , m n , m , p nontouching nontouching = 1 – (sum of all different loop gains) + (sum of the gain products of all combinations of two nontouching loops) – (sum of the gain products of all combinations of three nontouching loops) + ... a11 1 L1 R1 X1 L a;; L  a a L  a 1 11 2 1221 3 22 a21 L2 a12 (sum of all different loop gains) R2 X 2 = L + L + L = a + a a + a 1 L3 1 2 3 11 12 21 22 a22 (sum of the gain products of all combinations of two nontouching loops) = L1L3 = a11a22   1  (a11  a 22  a 1221 a )  a 1122 a sites.google.com/site/ncpdhbkhn 47 Signal – Flow Graph Models (4) N  1 LLLLLLn   n m   n m p  ... n1 n , m n , m , p nontouching nontouching Y() s Pk k G() s   k R() s  • Pk: gain of kth path from input to output • Δk (cofactor): the determinant Δ with the loop(s) touching the kth path removed. sites.google.com/site/ncpdhbkhn 48 Signal – Flow Graph Models (5) Ex. 1 H2 H3 P  Y() s k k k G() s   L L R() s  1 2 G 1 G4 G2 G3 • Pk: gain of kth path from input to output R() s Y() s • Δ (cofactor): the determinant G G k G 6 7 G8 Δ with the loop(s) touching the 5 kth path removed. L3 L4 PGGGGPGGGG1 1 2 3 4; 2  5 6 7 8 H6 H7 Δ = 1 – (sum of all different loop gains) + (sum of the gain products of all combinations of two nontouching loops) – (sum of the gain products of all combinations of three nontouching loops) + ... LGHLGHLGHLGH1 222;;;  333  664  77  1  (LLLLLLLLLLLL1  2  3  4 )  ( 1314   23  24 ) sites.google.com/site/ncpdhbkhn 49 Signal – Flow Graph Models (6) Ex. 1 H2 H3 P  Y() s k k k G() s   L L R() s  1 2 G 1 G4 PGGGG1 1 2 3 4 G2 G3 PGGGG2 5 6 7 8 R() s Y() s 1 (LLLL ) G6 G7 G   1  2  3  4 G5 8 ()LLLLLLLL1 3  1 4  2 3  2 4 L3 L4 Δk (cofactor): the determinant Δ with the loop(s) touching the kth path removed. H6 H7 1   1  (LL 3  4 ) LL1 2 0 2   1  (LL 1  2 ) LL3 4 0 PP   GGGGLLGGGGLL[1 (  )]  [1  (  )] G() s 1 1 2 2  1234 34 5678 12 1  (LLLLLLLLLLLL1  2  3  4 )  ( 1314   23  24 ) sites.google.com/site/ncpdhbkhn 50 Signal – Flow Graph Models (7) T() s Ex. 2 d T() s () Va () s Km m 1 ()s 1  ()s  ()s  Pk k G() s   k R L s Js b s V() s  () a a TL () s a Armature Load Δ = 1 – (sum of all different loop gains) K + (sum of the gain products of all Back electromotive force b combinations of two nontouching loops)  ()s K – (sum of the gain products of all G() s   m combinations of three nontouching loops) + ... Vsa( ) sJsbLsR [( )( a  a )  KK b m ] LGGH  1 2 Td () s 1 Va () s Tm () s TL () s  ()s  1 LGGH  1  1 2 1 G1 1 1 G2 G3 • P : gain of kth path from input to output k L • Δk (cofactor): the determinant Δ with the loop(s) touching the kth path removed. H Km 1 1 PGGG1 1 2 3 GGGHK1;;; 2  3   b Ra L a s Js  b s     1 1 L0 K 1 1 m    ()s P G G G1 R  L s Js  b s G() s  1 1 1 2 3  a a Va ( s ) 1  G1 G 2 H Km 1 1   Kb Ra L a s Js  b sites.google.com/site/ncpdhbkhn 51 Signal – Flow Graph Models (8) Ex. 3 H2 P  R(s) () Y(s) Y() s k k k G() s   G1 G2 G3 G4 R() s  () H Δ = 1 – (sum of all different loop gains) 1 + (sum of the gain products of all H combinations of two nontouching loops) 3 – (sum of the gain products of all Y() s GGGG combinations of three nontouching loops) + ...  1 2 3 4 Rs( ) 1 GGH341  GGH 232  GGGGH 12343 LGGH1  2 3 2 LGGH2 3 4 1 LGGGGH3  1 2 3 4 3  1 LLL1  2  3 • Pk: gain of kth path from input to output • Δk (cofactor): the determinant Δ with the loop(s) touching the kth path removed. PGGGG1 1 2 3 4 1    1 LLL1, 2 , 3  0 Y() s P GGGG 1 GGGG G() s  1 1  1 2 3 4  1 2 3 4 R( s ) 1  L1  L 2  L 3 1GGHGGHGGGGH341  232  12343 sites.google.com/site/ncpdhbkhn 52 Signal – Flow Graph Models (9) G7 Ex. 4 G8 P   ()s k k k G G G() s   1 G1 2 G4 G5 6 Va () s  R() s Y() s G3 Δ = 1 – (sum of all different loop gains) H4 + (sum of the gain products of all H1 combinations of two nontouching loops) H2 – (sum of the gain products of all combinations of three nontouching loops) + ... H3 LGGGGH1  2 3 4 5 2; LGGH2  5 6 1; LGH3  8 1; LGGH4  2 7 2; LGH5  4 4; LGGGGGGH6  1 2 3 4 5 6 3; LGGGGH7  1 2 6 7 3; LGGGGGH8  1 2 3 4 8 3; 1 (LLLLLLLLLLLLLL12345678 ) ( 575434   ) • Pk: gain of kth path from input to output • Δk (cofactor): the determinant Δ with the loop(s) touching the kth path removed. PGGGGGG1 1 2 3 4 5 6; PGGGG2 1 2 7 6; PGGGGG3 1 2 3 4 8; 1   3    1 2   1  L 5 all loops are zero all loops except L5 are zero Y() s PPP     GGGGGGGGGGLGGGGG1  (1  )   1 G() s   1 1 2 2 3 3  123456 1276 5 12348 R() s  1LLLLLLLLLLLLLL12345678575434   sites.google.com/site/ncpdhbkhn 53 Mathematical Models of Systems 1. Differential Equations of Physical Systems 2. Linear Approximations of Physical Systems 3. The Laplace Transform 4. The Transfer Function of Linear Systems 5. Block Diagram Models 6. Signal – Flow Graph Models 7. The Simulation of Systems Using Control Design Software sites.google.com/site/ncpdhbkhn 54 The Simulation of Systems Using Control Design Software (1) Ex. 1 Wall friction b k Mass d2 y()() t dy t y M b  ky()() t  r t M dt2 dt Force y(0) r(t) nt 2  y( t ) e sinn 1   t   2   1    k b ;   ;   cos1  n M 2 kM sites.google.com/site/ncpdhbkhn 55 The Simulation of Systems Using Control Design Software (2) Ex. 2 4s  76 Y() s  s2 6 s  265 • zplane • roots • poly • conv • polyval • tf • series • parallel • feedback sites.google.com/site/ncpdhbkhn 56 The Simulation of Systems Using Control Design Software (3) Ex. 3 H2 1 () G1() s  R(s) Y(s) s  5 G1 G2 G3 G4 1 () G() s  H1 2 s 1 H3 2s2  7 G() s  3 s2 4 s  4 s 1 G() s  4 s  6 s 1 H() s  1 s  2 H2 ( s ) 3 H3( s ) 1 sites.google.com/site/ncpdhbkhn 57 The Simulation of Systems Using Control Design Software (4) Ex. 3 H2 R(s) () Y(s) G1 G2 G3 G4 () H1 H3 X1 X2 X1 X2 G G X1 X 2 1/G Moving a pickoff point behind a block H2/G4 R(s) () Y(s) G1 G2 G3 G4 () H1 H3 sites.google.com/site/ncpdhbkhn 58 The Simulation of Systems Using Control Design Software (5) Ex. 3 H2/G4 R(s) () Y(s) G1 G2 G3 G4 () H1 H3 X1 X2 X3 X1 X3 G1(s) G2(s) G1G2 Combining blocks in cascade H2/G4 R(s) () Y(s) G1 G2 G3G4 () H1 sites.google.com/site/ncpdhbkhn H3 59 The Simulation of Systems Using Control Design Software (6) Ex. 3 H2/G4 R(s) () Y(s) G1 G2 G3G4 () H1 H3 X X X G X 2 1 G 2 1 () 1 GH H Eliminating a feedback loop H2/G4 () R(s) GG3 4 Y(s) G1 G2 () 1 GGH3 4 1 sites.google.com/site/ncpdhbkhn H3 60 The Simulation of Systems Using Control Design Software (7) Ex. 3 H2/G4 () R(s) GG3 4 Y(s) G1 G2 () 1 GGH3 4 1 H3 H2/G4 () R(s) GGG2 3 4 Y(s) G1 () 1 GGH3 4 1 H3 R(s) GGG2 3 4 Y(s) G1 () 1GGHGGH3 4 1  2 3 2 sites.google.com/site/ncpdhbkhn H3 61 The Simulation of Systems Using Control Design Software (8) Ex. 4 G1() s Td () s G2 () s G3() s  ()s 10 () 1 ()s d 500 () () s 1 2s  0.5 0.1 sites.google.com/site/ncpdhbkhn 62