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

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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,...

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

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 2
State Variable Models 1. The State Variables of a Dynamic System 2. The State Differential Equation 3. Signal – Flow Graph & Block Diagram Models 4. Alternative Signal – Flow Graph & Block Diagram Models 5. The Transfer Function from the State Equation 6. The Time Response & the State Transition Matrix 7. Analysis of State Variable Models Using Control Design Software sites.google.com/site/ncpdhbkhn 3
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 4
The State Variables of a Dynamic System (2) Wall d 2 y (t ) dy (t ) M 2  b  ky (t )  u (t ) friction dt dt b k dy (t ) x1 (t )  y (t ), x2 (t )  dt Mass M  dx1   x2 dx  dt  M 2  bx2  kx1  u (t )   y(t) u(t) dt  dx2   b x  k x  1 u  dt M 2 M 1 M ic  C dvC  u ( t )  iL  dx1 1 1  dt   x 2  u (t ) dt C C iL L    di L L   RiL  vC  dx2  1 x  R x vC vo  C R dt dt L 1 L 2    u (t ) iC vo  RiL (t )  x1  vC , x2  iL  v (t )  Rx  o 2 sites.google.com/site/ncpdhbkhn 5
State Variable Models 1. The State Variables of a Dynamic System 2. The State Differential Equation 3. Signal – Flow Graph & Block Diagram Models 4. Alternative Signal – Flow Graph & Block Diagram Models 5. The Transfer Function from the State Equation 6. The Time Response & the State Transition Matrix 7. Analysis of State Variable Models Using Control Design Software sites.google.com/site/ncpdhbkhn 6
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  2 21 1 22 2 2n n 21 1 2m m    xn  an1 x1  an 2 x2  ...  ann xn  bn1u1  ...  bnm um  x1   a11 a12  a1n   x1         b11  b1m   u2   a 2 n x2  d  x2   a21 a22            dt                     bn1  bnm   um   xn   an1 an 2  ann   xn   x  Ax  Bu y  Cx  Du t t x (t )  exp( At )x (0)   exp[A(t   )Bu(r)d  Φ(t )x(0)   Φ(t   )Bu( )d 0 0 X ( s )  [ sI  A ]1 x (0)[ sI  A ]1 BU( s ) sites.google.com/site/ncpdhbkhn 7
The State Differential Equation (2)  dx1 1 1  dt   x 2  u (t ) C C   dx2  1 x  R x  dt L 1 L 2  iL L    vC C R vo  v (t )  Rx   o 2 u (t )  iC   1  0 C 1  x    x   C  u (t )  1  R     L  0 L   y   0 R  x sites.google.com/site/ncpdhbkhn 8
The State Differential Equation (3) q p k2 k1 M 1a1  u  f spring  f damp M2 M1 u  M 1  p  u  k1 ( p  q )  b1 ( p  q ) b2 b1  M 1  p  b1 p  k1 p  u  k1q  b1q M 2 q  k1 ( p  q)  b1 ( p  q )  k2 q  b2 q  M 2 q  ( k1  k2 ) q  (b1  b2 ) q  k1 p  b1 p  x1  p  x3  x1  p  ,   x2  q  x4  x2  q   b1  k1 1 k1 b1  3x  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 sites.google.com/site/ncpdhbkhn 9
The State Differential Equation (4)   b1  k1 1 k1 b1  3x  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  x3  x1  p  ,   x2  q  x4  x2  q  k1 k1 b1 b1 1 x  3   x1  x 2  x3  x 4  u  M1 M1 M1 M1 M1   x  k1 x  k1  k2 x  b1 x  b1  b2 x  4 M 2 1 M2 2 M2 3 M2 4 sites.google.com/site/ncpdhbkhn 10
The State Differential Equation (5)  k1 k1 b1 b1 1  3x   x1  x 2  x3  x 4  u  M1 M1 M1 M1 M1   x4  k1 x1  k1  k2 x2  b1 x3  b1  b2 x4  M2 M2 M2 M2  0 0 1 0   0   0   x1   p   0 0 1   0  x  q  k k1 b b1    x   2     , A   1  1 , B   1   x3   p   M1 M1 M1 M1  M       k1  1  x4   q  k1  k2 b1 b1  b2   M     0   2 M2 M2 M2   x  Ax  Bu y  p  x1  1 0 0 0 x  Cx sites.google.com/site/ncpdhbkhn 11
The State Differential Equation (6) q p k2 k1 u M2 M1 b2 b1  k1 k1 b1 b1 1 x  3   x1  x 2  x3  x 4  u  M1 M1 M1 M1 M1   x  k1 x  k1  k2 x  b1 x  b1  b2 x  4 M 2 1 M2 2 M2 3 M2 4 q p k2 q k1 ( q  p ) k1 ( p  q) M2 M2 u b2 q b1 ( q  p ) b1 ( p  q ) sites.google.com/site/ncpdhbkhn 12
State Variable Models 1. The State Variables of a Dynamic System 2. The State Differential Equation 3. Signal – Flow Graph & Block Diagram Models 4. Alternative Signal – Flow Graph & Block Diagram Models 5. The Transfer Function from the State Equation 6. The Time Response & the State Transition Matrix 7. Analysis of State Variable Models Using Control Design Software sites.google.com/site/ncpdhbkhn 13
Signal – Flow Graph & Block Diagram Models (1)  dx1 1 1 iL L    dt   C x2  C u (t ) vC C R vo    dx2  1 x  R x u (t ) iC   dt L 1 L 2   1 1 1  R  v (t )  Rx L ?  o 2 U ( s) C s L 1/ s R Vo ( s ) X1 X2 Vo ( s ) R /( LC ) G( s)   2  1 U ( s ) s  ( R / L) s  1/( LC ) C R L U ( s) 1 1 X1 1 ( ) 1 X2 Vo ( s ) R C ( ) L s s 1 sites.google.com/site/ncpdhbkhn C 14
Signal – Flow Graph & Block Diagram Models (2) Y ( s ) bm s m  bm 1s m 1  ...  b1s  b0 G( s)   n n 1 , nm U ( s) s  an 1s  ...  a1s  a0 bm s  ( n m )  bm 1s  ( n  m 1)  ...  b1s  ( n 1)  b0 s  n  s  an 1s 1  ...  a1s  ( n 1)  a0 s  n   P k k  Sum of the forward-path factor 1  L 1  sum of the feedback loop factors N q 1 q sites.google.com/site/ncpdhbkhn 15
Signal – Flow Graph Ex. 1 & Block Diagram Models (3) Y ( s) b0 b0 s 4 G( s)   4  U ( s ) s  a3 s  a2 s  a1s  a0 1  a3 s 1  a2 s 2  a1s 3  a0 s 4 3 2  ( s 4  a3 s 3  a2 s 2  a1s  a0 )Y ( s )  b0U ( s ) d 4 ( y / b0 ) d 3 ( y / b0 ) d 2 ( y / b0 ) d ( y / b0 )  4  a 3 3  a 2 2  a1  a0 ( y / b0 )  u dt dt dt dt 1 1 1 1 x1  y / b0 1 s X4 s s s b0 U ( s) x2  x1  y / b0 Y ( s) a X3 X2 X1 3  a2 x3  x2   y / b0 a1 x4  x3   y / b0 a0 X4 X3 X2 X1 1 1 1 1 1 Y ( s) U ( s) b0 ( ) ( ) s s s s s a3 a2 ( ) a1 ( ) sites.google.com/site/ncpdhbkhn a0 16
Signal – Flow Graph Ex. 1 & Block Diagram Models (4) Y ( s) b0 b0 s 4 G( s)   4  U ( s ) s  a3 s  a2 s  a1s  a0 1  a3s 1  a2 s 2  a1s 3  a0 s 4 3 2 d 4 ( y / b0 ) d 3 ( y / b0 ) d 2 ( y / b0 ) d ( y / b0 )  a 3  a 2  a1  a0 ( y / b0 )  u dt 4 dt 3 dt 2 dt x1  y / b0 x2  x1  y / b0 x3  x2   y / b0 x4  x3   y / b0  x 4   a0 x1  a1 x2  a2 x3  a3 x4  u y  b0 x1 sites.google.com/site/ncpdhbkhn 17
Signal – Flow Graph Ex. 1 & Block Diagram Models (5) Y ( s) b0 b0 s 4 G( s)   4  U ( s ) s  a3 s  a2 s  a1s  a0 1  a3s 1  a2 s 2  a1s 3  a0 s 4 3 2 x4   a0 x1  a1 x2  a2 x3  a3 x4  u y  b0 x1  x1   0 0 0 0   x1   0  x   0 0 0 0   x2   0    2         u (t )  x  Ax  Bu  x3   0 0 0 0   x3   0         x 4    a0  a1  a2  a3   x4   1   x1  x  y (t )  Cx   b0 0 0 0  2   x3     x4  sites.google.com/site/ncpdhbkhn 18
Signal – Flow Graph Ex. 1 & Block Diagram Models (6) Y ( s) b0 b0 s 4 G( s)   4  U ( s ) s  a3 s  a2 s  a1s  a0 1  a3 s 1  a2 s 2  a1s 3  a0 s 4 3 2 1 1 1 1 1 s X4 s s s b0 U ( s) Y ( s) a3 X3 X2 X1  a2 a1 a0 G( s)  Y ( s)  P k k  Sum of the forward-path factor  L 1  sum of the feedback loop factors N U ( s) 1  q 1 q X4 X3 X2 X1 1 1 1 1 Y ( s) U ( s) b0 ( ) ( ) s s s s a3 a2 ( ) a1 ( ) sites.google.com/site/ncpdhbkhn a0 19
Signal – Flow Graph Ex. 2 & Block Diagram Models (7) Y ( s) b3 s 3  b2 s 2  b1s  b0 b3s 1  b2 s 2  b1s 3  b0 s 4 G( s)   4  U ( s ) s  a3 s  a2 s  a1s  a0 1  a3s 1  a2 s 2  a1s 3  a0 s 4 3 2 G( s)  Y ( s)  P k k  Sum of the forward-path factor  L 1  sum of the feedback loop factors N U ( s) 1  q 1 q b3 1 b2 1 s X4 1/ s 1/ s 1/ s b1 b 0 U ( s) Y ( s) a3 X3 X2 X1  a2 a1 a0 sites.google.com/site/ncpdhbkhn 20