Lecture Control system design: State variable models - Nguyễn Công Phương

This chapter presents the following content: The state variables of a dynamic system, the state differential equation, signal – flow graph & block diagram models, alternative signal – flow graph & block diagram models, the transfer function from the state equation,...

Nguyễn Công Phương CONTROL SYSTEM DESIGN State Variable Models 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 State Variable Models The State Variables of a Dynamic System The State Differential Equation Signal – Flow Graph & Block Diagram Models Alternative Signal – Flow Graph & Block Diagram Models The Transfer Function from the State Equation The Time Response & the State Transition Matrix Analysis of State Variable Models Using Control Design Software sites.google.com/site/ncpdhbkhn The State Variables of a Dynamic System (1) • The state of a system is a set of variables whose values, together with the input signals & the equations describing the dynamics, will provide the future state & output of the system • The state variables describe the present configuration of a system & can be used to determine the future response, given the excitation inputs & the equations describing the dynamics sites.google.com/site/ncpdhbkhn The State Variables of a Dynamic System (2) d y (t ) dy (t ) M  b  ky (t )  u (t ) dt dt dy (t ) x1 (t )  y (t ), x2 (t )  dt Wall friction b  dx1  x2  dx  dt  M  bx2  kx1  u (t )   dt  dx2   b x  k x  u  dt M M M ic  C dvC  u ( t )  iL dt di L L   RiL  vC dt vo  RiL (t ) x1  vC , x2  iL 1  dx1   x  u (t )  dt C C   dx2  x  R x  dt L L  u (t )   v (t )  Rx  o sites.google.com/site/ncpdhbkhn  vC  k Mass M y(t) iL C u(t) L  vo  R iC State Variable Models The State Variables of a Dynamic System The State Differential Equation Signal – Flow Graph & Block Diagram Models Alternative Signal – Flow Graph & Block Diagram Models The Transfer Function from the State Equation The Time Response & the State Transition Matrix Analysis of State Variable Models Using Control Design Software sites.google.com/site/ncpdhbkhn The State Differential Equation (1)  x1  a11 x1  a12 x2   a1n xn  b11u1   b1m um  x  a x  a x   a x  b u   b u  21 22 2n n 21 2m m    xn  an1 x1  an x2   ann xn  bn1u1   bnm um  x1   a11 a12    d  x2   a21 a22         dt     xn   an1 an  a1n   x1   b11  b1m   u2      a n x2                      bn1  bnm   um   ann   xn   x  Ax  Bu y  Cx  Du x (t )  exp( At )x (0)  t t 0  exp[A(t   )Bu(r)d  Φ(t )x(0)   Φ(t   )Bu( )d X ( s )  [ sI  A ]1 x (0)[ sI  A ]1 BU( s ) sites.google.com/site/ncpdhbkhn The State Differential Equation (2) 1  dx1   x  u (t )  dt C C   dx2  x  R x  dt L L    v (t )  Rx  o 1   1  0 C  x    x   C  u (t )    1  R 0   L  L   y   R  x sites.google.com/site/ncpdhbkhn u (t )  vC  iL C L  vo  R iC The State Differential Equation (3) q k1 k2 M 1a1  u  f spring  f damp  M  p  u  k1 ( p  q )  b1 ( p  q ) M2 b2 p u M1 b1  M  p  b1 p  k1 p  u  k1q  b1q M q  k1 ( p  q)  b1 ( p  q )  k2 q  b2 q  M q  ( k1  k2 ) q  (b1  b2 ) q  k1 p  b1 p  x3  x1  p   x4  x2  q b1 k1 k1 b1    x  p   p  p  u  q  q  M1 M1 M1 M1 M1    x4  q   k1  k2 q  b1  b2 q  k1 p  b1 p  M2 M2 M2 M2  x1  p ,   x2  q sites.google.com/site/ncpdhbkhn The State Differential Equation (4) b1 k1 k1 b1    x  p   p  p  u  q  q  M1 M1 M1 M1 M1    x4  q   k1  k2 q  b1  b2 q  k1 p  b1 p  M2 M2 M2 M2  x1  p ,   x2  q  x3  x1  p   x4  x2  q k1 k1 b1 b1  x   x  x  x  x  u  M1 M1 M1 M1 M1    x  k1 x  k1  k2 x  b1 x  b1  b2 x  M M2 M2 M2 sites.google.com/site/ncpdhbkhn 10 Ex Alternative Signal – Flow Graph & Block Diagram Models (2) R( s) s 1 s5 Field voltage Controller T ( s)  1/ s 1/ s Y ( s) 10 2 1/ s 30 X3 Y ( s) X1 20 20 X2 Velocity Y ( s) 30( s  1) 20 10 30     R ( s ) ( s  5)( s  2)( s  3) s  s  s  X1 5 I ( s) s3 Motor & load ( ) s R( s) U ( s) s2 Field current R( s) Diagonal canonical form 3 sites.google.com/site/ncpdhbkhn ( ) s ( ) X2 10 ( ) Y ( s) ( ) s X3 30 36 Ex Alternative Signal – Flow Graph & Block Diagram Models (3)   x1  5 x1  r (t )     x2   x2  r ( t )    x3  3x3  r (t )  y (t )  20 x  10 x  30 x  R( s) s 1 s5 Field voltage Controller   5 0  1    x  1 r (t ) x          0 3 1   y (t )   20 10 30 x  U ( s) s2 Field current I ( s) s3 X1 20 Diagonal canonical form sites.google.com/site/ncpdhbkhn Y ( s) Motor & load ( ) s R( s) Velocity ( ) s ( ) X2 10 ( ) Y ( s) ( ) s X3 30 37 Alternative Signal – Flow Graph & Block Diagram Models (4) Ex x1   x1   x2  u1 (t ) x2   x1   x2  u2 (t ) x3   x1   x2  x1    d    x2      dt  x3       0  x1  1 0  u1 (t )       x2       u2 (t )  0  x3   0  ( ) U1 U1 1/ s   U2 1/ s  X2 X1    s ( ) X1  1/ s  s X3  U2 sites.google.com/site/ncpdhbkhn ( ) s  X2 X3  38 Ex Alternative Signal – Flow Graph & Block Diagram Models (5) My  ml  u (t )  my m mly  ml 2  mgl   mg l ( x1 , x2 , x3 , x4 )  ( y , y ,  , )  Mx2  mlx4  u (t )    x2  lx4  gx3  M u (t ) y (t )  x1  x2   x1  0 mg  x2   x3  u (t )    d  x2    M M    dt  x3  0  x3  x4     x4   g  u (t )  x4  x3  l Ml  0  mg / M 0 g /l sites.google.com/site/ncpdhbkhn 0  x1    0  x2   1/ M      u   x3        0  x4   1/( Ml )  39 State Variable Models The State Variables of a Dynamic System The State Differential Equation Signal – Flow Graph & Block Diagram Models Alternative Signal – Flow Graph & Block Diagram Models The Transfer Function from the State Equation The Time Response & the State Transition Matrix Analysis of State Variable Models Using Control Design Software sites.google.com/site/ncpdhbkhn 40 The Transfer Function from the State Equation (1)  x  Ax  Bu   y  Cx  Du  sX ( s )  AX ( s )  BU ( s )  Y ( s )  CX ( s )  DU ( s )  ( sI  A ) X ( s )  BU ( s )  X ( s )  [ sI  A ]1 BU ( s )  Φ( s )BU ( s )  Y ( s )  [CΦ( s )B  D]U ( s ) Y ( s)  G( s)   CΦ( s )B  D U ( s) sites.google.com/site/ncpdhbkhn 41 Ex The Transfer Function from the State Equation (2) 1    1  0 C   x    x   C  u  Ax  Bu     R  0    L L    y   R  x  Cx   s 0  [ sI  A ]     0 s    L u (t )  vC  iL L C  vo  R iC 1    s   C C     R   1 R s   L  L L  R  s    L 1    Φ( s )  [ sI  A ]  R  s2  s  L LC  L sites.google.com/site/ncpdhbkhn 1  R  s     C L    ( s)  s   L  1  C   s   42 Ex The Transfer Function from the State Equation (3) 1    1  0 C   x    x   C  u  Ax  Bu     R  0    L L    y   R  x  Cx R  s     L  Φ( s )  ( s)   L 1  C   s   u (t )  vC  R  s   L  ( s )  G ( s)  0 R     L ( s )   Y ( s) G( s)   CΦ( s )B  D U ( s) sites.google.com/site/ncpdhbkhn iL C L  vo  R iC  1    C  ( s )   C     s 0  ( s )  R /( LC ) R s  s L LC 43 State Variable Models The State Variables of a Dynamic System The State Differential Equation Signal – Flow Graph & Block Diagram Models Alternative Signal – Flow Graph & Block Diagram Models The Transfer Function from the State Equation The Time Response & the State Transition Matrix Analysis of State Variable Models Using Control Design Software sites.google.com/site/ncpdhbkhn 44 The Time Response & the State Transition Matrix (1) x (t )  exp( At )x (0)  t  exp[A(t   )Bu( r)d  Φ(t )x (0)  t  Φ(t   )Bu( )d Φ(t): the state transition matrix sites.google.com/site/ncpdhbkhn 45 Ex The Time Response & the State Transition Matrix (2) 1    1  0 C   x    x   C  u  Ax  Bu     R  0    L L    y   R  x  Cx u (t )  vC  iL C L  vo  R iC  2  2 R  3, L  1, C  0.5  A   , B    , C  1 0   3  0  s  3 2  1   s  3   Φ( s )  [ sI  A ]     ( s)   s s s  3s     1 (2e  t  e 2t ) ( 2e  t  2e 2t )   Φ(t )    t 2 t t 2 t  ( e  e ) (  e  2e )  sites.google.com/site/ncpdhbkhn 46 Ex The Time Response & the State Transition Matrix (3) 1    1  0 C   x    x   C  u  Ax  Bu     R  0    L L    y   R  x  Cx u (t )  vC  iL C L  vo  R iC (2e  t  e 2t ) ( 2e  t  2e 2t )  Φ(t )    t 2 t t 2 t  ( e  e ) (  e  2e )  x (t )  Φ(t )x (0)  t  Φ(t   )Bu( )d 2 t  x1 (t )  1  e  x1 (0)  x2 (0)  1, u (t )     Φ(t )        t 1  e   x2 ( t )  sites.google.com/site/ncpdhbkhn 47 The Time Response & the State Transition Matrix (3) Ex u (t ) 1 0.9 0.9 0.9 0.8 0.8 0.8 0.7 0.7 0.7 0.6 0.6 0.6 0.5 x2(t) x1(t) x1(t)  x1 (t )  1  e   x (t )   Φ(t ) 1   2t     e    0.5 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0.5 t 1.5 0 0.5 t 1.5 sites.google.com/site/ncpdhbkhn L C  vo  R iC 0.5 0.4 iL  vC  2 t 0 0.2 0.4 0.6 0.8 x1(t) 48 State Variable Models The State Variables of a Dynamic System The State Differential Equation Signal – Flow Graph & Block Diagram Models Alternative Signal – Flow Graph & Block Diagram Models The Transfer Function from the State Equation The Time Response & the State Transition Matrix Analysis of State Variable Models Using Control Design Software sites.google.com/site/ncpdhbkhn 49 Ex Analysis of State Variable Models Using Control Design Software Y ( s) 3s  s  T ( s)   R ( s ) s  10 s  21s  sites.google.com/site/ncpdhbkhn 50 ... Feedback Control Systems XI The Design of State Variable Feedback Systems XII Robust Control Systems XIII.Digital Control Systems sites.google.com/site/ncpdhbkhn State Variable Models The State Variables... 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 State Transition Matrix Analysis of State Variable Models Using Control Design Software sites.google.com/site/ncpdhbkhn The State Variables of a Dynamic System (1) • The state of a system