Bài giảng Control system design - Chapter III: State variable models - Nguyễn Công Phương

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

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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 d2 y()() t dy t M b  ky()() t  u t friction dt2 dt b dy() t k x1( t ) y ( t ), x 2 ( t )  dt Mass M dx 1  x dx  dt 2 M2  bx  kx  u() t   y(t) u(t) dt 2 1 dx b k 1  2  x  x  u  dt M2 M 1 M dvC dx1 1 1 ic C  u() t  i L  x  u() t dt  dt C2 C i L   L  di dx2 1 R L vC C R vo L  RiLC  v   x1  x 2 dt dt L L    u() t iC vo Ri L () t   x1 vCL, x 2  i vo () t Rx2 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) xaxax1 111122  ...  axbu 1n n  111  ...  bu 1 m m  xaxax2 211  222 ...  axbu 2n n  211  ...  bu 2 m m    xaxaxn n1 1  n 2 2 ...  axbu nn n  n 1 1  ...  bu nm m x1   a 11 a 12 a 1n   x 1        b11 b 1m   u 2  d x2 a 21 a 22 a 2n x 2                 dt                      bn1  b nm   u m  xn   a n1 a n 2  a nn   x n  x  Ax  Bu y Cx  Du t t x()exp(t A t )(0) x  exp[( A t  ) Bu () r d  Φ ()(0) t x  Φ ( t   ) Bu ()  d  0  0 X()[s s I  A ](0)[1 x s I  A ]  1 BU () s sites.google.com/site/ncpdhbkhn 7 The State Differential Equation (2) dx 1 1 1  x  u() t  dt C2 C  dx1 R  2 x  x  dt L1 L 2  iL L    v v  C C R o vo () t Rx2   u() t iC  1   0   1  C   x   x  C u() t   1 R     0   LL    y 0 Rx sites.google.com/site/ncpdhbkhn 8 The State Differential Equation (3) q p k2 k1 M a u  f  f u 1 1 spring damp M2 M1 M1 p  u  k 1()() p  q  b 1 p   q  b2 b1 Mp1  bp 1   kp 1  u  kq 1  bq 1  Mqkpq2 1()()   bpq 1     kqbq 2  2  Mq2 ()() k 1  kq 2  b 1  bqkpbp 2   1  1  x1  p x3 x 1  p   ,  x2  q x4 x 2  q   b1 k 11 k 1 b 1 x3  p   p   p  u  q  q   MMMMM1 1 1 1 1    k1 k 2 b 1  b 2 k 1 b 1 x4  q   q  q   p  p   MMMM2 2 2 2 sites.google.com/site/ncpdhbkhn 9 The State Differential Equation (4)  b1 k 11 k 1 b 1 x3  p   p   p  u  q  q   MMMMM1 1 1 1 1   k1 k 2 b 1  b 2 k 1 b 1 x4  q   q  q   p  p   MMMM2 2 2 2 x1  p x3 x 1  p   ,  x2  q x4 x 2  q   k1 k 1 b 1 b 1 1 x3  x 1  x 2  x 3  x 4  u  MMMMM1 1 1 1 1    k1 k 1 k 2 b 1 b 1  b 2 x4 x 1  x 2  x 3  x 4  MMMM2 2 2 2 sites.google.com/site/ncpdhbkhn 10 The State Differential Equation (5)  k1 k 1 b 1 b 1 1 x3  x 1  x 2  x 3  x 4  u  MMMMM1 1 1 1 1   k1 k 1 k 2 b 1 b 1  b 2 x4 x 1  x 2  x 3  x 4  MMMM2 2 2 2 0 0 1 0    0  x  p  0 0 0 1 1   0        x2 q k1 k 1 b 1 b 1  x     , A    , B  1  x  p  MMMM 3 1 1 1 1  M      1  x4  q  k1 k 1 k 2 b 1 b 1  b 2     0  MMMM2 2 2 2  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 k 1 b 1 b 1 1 x3  x 1  x 2  x 3  x 4  u  MMMMM1 1 1 1 1   k1 k 1 k 2 b 1 b 1  b 2 x4 x 1  x 2  x 3  x 4  MMMM2 2 2 2 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) i L dx1 1 1  L    x2  u() t dt C C vC C R vo  dx1 R    2 u() t iC  x1  x 2  dt L L R  1 1 1  L  C s L R vo () t Rx2 ?  U() s V() s o 1/ s X1 X 2 Vo () s R/( LC ) 1 G() s    U() s s2 ( R / L ) s  1/( LC ) C R L U() s 1 1 X1 1 () 1 X V() s 2 R o C () L s s 1 sites.google.com/site/ncpdhbkhn C 14 Signal – Flow Graph & Block Diagram Models (2) m m1 Y() s bm s b m1 s ...  b 1 s  b 0 G(), s n n1 n  m U() s s an1 s ...  a 1 s  a 0 (n  m )  ( n  m  1)  ( n  1)  n bm s b m1 s ...  b 1 s  b 0 s  1  (n  1)  n s an1 s ...  a 1 s  a 0 s P k k Sum of the forward-path factor N  1 L 1 sum of the feedback loop factors q1 q sites.google.com/site/ncpdhbkhn 15 Signal – Flow Graph Ex. 1 & Block Diagram Models (3) 4 Y() s b0 b 0 s G() s  4 3 2   1  2  3  4 U() s s as3  as 2  asa 1  01  as 3  as 2  as 1  as 0 4 3 2 ()()()s  as3  as 2  asaYs 1  0  bUs 0 dyb4(/)(/)(/)(/) dyb 3 dyb 2 dyb 0 a 0  a 0  a 0  a(/) y b  u dt43 dt 3 2 dt 2 1dt 0 0 1 1 1 1 X x1 y/ b 0 1 s 4 s s s b0 U() s Y() s x2 x 1  y  / b 0 X 3 X 2 X1 a3 a2 a x3 x 2  y/ b 0 1 x4 x 3   y/ b 0 a0 X X X X 1 4 1 3 1 2 1 1 1 Y() s U() s b0 () () s s s s s a3 a () 2 a () 1 a sites.google.com/site/ncpdhbkhn 0 16 Signal – Flow Graph Ex. 1 & Block Diagram Models (4) 4 Y() s b0 b 0 s G() s  4 3 2   1  2  3  4 U() s s as3  as 2  asa 1  01  as 3  as 2  as 1  as 0 dyb4(/)(/)(/)(/) dyb 3 dyb 2 dyb 0a 0  a 0  a 0  a(/) y b  u dt43 dt 3 2 dt 2 1dt 0 0 x1 y/ b 0 x2 x 1  y  / b 0 x3 x 2  y/ b 0 x4 x 3   y/ b 0 x4   a 0 x 1  a 1 x 2  a 2 x 3  a 3 x 4  u y b0 x 1 sites.google.com/site/ncpdhbkhn 17 Signal – Flow Graph Ex. 1 & Block Diagram Models (5) 4 Y() s b0 b 0 s G() s  4 3 2   1  2  3  4 U() s s as3  as 2  asa 1  01  as 3  as 2  as 1  as 0 x4  a 0 x 1  a 1 x 2  a 2 x 3  a 3 x 4  u y b0 x 1 x1 0 0 0 0   x 1  0         x20 0 0 0 x 2 0        u() t x  Ax  B u x3 0 0 0 0   x 3  0         x4 a0  a 1  a 2  a 3   x 4  1  x1  x  2  y( t )Cx   b0 0 0 0 x3    x4  sites.google.com/site/ncpdhbkhn 18 Signal – Flow Graph Ex. 1 & Block Diagram Models (6) 4 Y() s b0 b 0 s G() s  4 3 2   1  2  3  4 U() s s as3  as 2  asa 1  01  as 3  as 2  as 1  as 0 1 1 1 1 X 1 s 4 s s s b0 U() s Y() s X 3 X 2 X1 a3 a2 a1 a0 P Y( s )k k Sum of the forward-path factor G() s  N  U() s 1 L 1 sumofthefeedbackloopfactors q1 q X X X X 1 4 1 3 1 2 1 1 Y() s U() s b0 () () s s s s a3 a () 2 a () 1 a sites.google.com/site/ncpdhbkhn 0 19 Signal – Flow Graph Ex. 2 & Block Diagram Models (7) 3 2 1  2  3  4 Y() s bs3 bs 2  bsb 1  0 bs 3  bs 2  bs 1  bs 0 G() s  4 3 2   1  2  3  4 U() s s as3  as 2  asa 1  01  as 3  as 2  as 1  as 0 P Y( s )k k Sum of the forward-path factor G() s  N  U() s 1 L 1 sumofthefeedbackloopfactors q1 q b3 1 b2 1 X 4 b1 s 1/ s 1/ s 1/ s b0 U() s Y() s X 3 X 2 X1 a3 a2 a1 a0 sites.google.com/site/ncpdhbkhn 20 Signal – Flow Graph Ex. 2 & Block Diagram Models (8) 3 2 1  2  3  4 Y() s bs3 bs 2  bsb 1  0 bs 3  bs 2  bs 1  bs 0 G() s  4 3 2   1  2  3  4 U() s s as3  as 2  asa 1  01  as 3  as 2  as 1  as 0 b3 1 b2 1 X 4 b1 s 1/ s 1/ s 1/ s b0 U() s Y() s X 3 X 2 X1 a3 a2 a1 a0 X1 X 2 / s  X2 X 3 / s  X3 X 4 / s  X4()/ UaX  3 4  aX 2 3  aX 1 2  aX 0 1 s Y b0 X 1  b 1 X 2  b 2 X 3  b 3 X 4 sites.google.com/site/ncpdhbkhn 21 Signal – Flow Graph Ex. 2 & Block Diagram Models (9) 3 2 1  2  3  4 Y() s bs3 bs 2  bsb 1  0 bs 3  bs 2  bs 1  bs 0 G() s  4 3 2   1  2  3  4 U() s s as3  as 2  asa 1  01  as 3  as 2  as 1  as 0 X1 X 2 / s sX1 X 2   X2 X 3 / s sX2 X 3   X3 X 4 / s  sX3 X 4 X()/ UaX   aX  aX  aX s sX() U  a X  a X  a X  a X  4 3 4 2 3 1 2 0 1  4 3 4 2 3 1 2 0 1 Y b0 X 1  b 1 X 2  b 2 X 3  b 3 X 4 Y b0 X 1  b 1 X 2  b 2 X 3  b 3 X 4  x1 0 1 0 0   x 1  0          d x20 0 1 0 x 2 0 x x         u() t 2 1   dt x3 0 0 0 1   x 3  0  x3 x 2           x4 a0  a 1  a 2  a 3   x 4  1    x4 x 3    x   1  x4 u  a 3 x 4  a 2 x 3  a 1 x 2  a 0 x 1   x  2   y b0 x 1  b 1 x 2  b 2 x 3  b 3 x 4  y() t  b0 b 1 b 2 b 3   x3      x4  sites.google.com/site/ncpdhbkhn 22 Signal – Flow Graph Ex. 2 & Block Diagram Models (10) 3 2 Y() s b3 s b 2 s  b 1 s  b 0 G() s   4 3 2 U() s s a3 s  a 2 s  a 1 s  a 0 3 2 Y()() sb3 s b 2 s  b 1 s  b 0 Z s G(). s   4 3 2 U()() ss a3 s  a 2 s  a 1 s  a 0 Z s  d3 z d 2 z dz 3 2  y b33  b 2 2  b 1  b 0 z Ys()()() bs3  bs 2  bsbZs 1  0  dt dt dt     Us()()() s4  as 3  as 2  asaZs  d4 z d 3 z d 2 z dz  3 2 1 0 u  a  a  a  a z  dt43 dt 3 2 dt 2 1dt 0 x2 x 1 x1  z   x3 x 2 x2 x 1  z    x4 x 3 x x   z  3 2 x  u  a x  a x  a x  a x   4 3 4 2 3 1 2 0 1 x4 x 3   z  y b0 x 1  b 1 x 2  b 2 x 3  b 3 x 4 sites.google.com/site/ncpdhbkhn 23 Signal – Flow Graph Ex. 2 & Block Diagram Models (11) 3 2 Y() s b3 s b 2 s  b 1 s  b 0 b3 G() s   4 3 2 U() s s a3 s  a 2 s  a 1 s  a 0 1 b2 1 X 4 b1 s 1/ s 1/ s 1/ s b0 x2 x 1 U() s Y() s  a X 3 X 2 X1 x3 x 2 3  a2  a1 x4 x 3 a x u  a x  a x  a x  a x 0  4 3 4 2 3 1 2 0 1 phase variable canonical form  y b0 x 1  b 1 x 2  b 2 x 3  b 3 x 4 b3 b2 b1 X X X X 1 4 1 3 1 2 1 1 Y() s U() s b0 () () s s s s a3 a () 2 a () 1 a sites.google.com/site/ncpdhbkhn 0 24 Signal – Flow Graph Ex. 2 & Block Diagram Models (12) 3 2 Y() s b3 s b 2 s  b 1 s  b 0 b3 G() s   4 3 2 U() s s a3 s  a 2 s  a 1 s  a 0 1 b2 1 X 4 b1 s 1/ s 1/ s 1/ s b0 U() s Y() s X 3 X 2 X1 a3 a2 a1 a0 phase variable canonical form b3 b2 X x1 X b1 1/ s 1/ s 2 1 1/ s 1 1 U() s Y() s 1/ s 1 1 b0 x4 X 4 x3 X 3 x2 a2 a1 a3 a0 input feedforward canonical form sites.google.com/site/ncpdhbkhn 25 Signal – Flow Graph Ex. 2 & Block Diagram Models (13) 3 2 Y() s b3 s b 2 s  b 1 s  b 0 G() s   4 3 2 U() s s a3 s  a 2 s  a 1 s  a 0 x1  a 3 x 1  x 2  b 3 u  a31 0 0   b 3    a0 1 0   b  x2  a 2 x 1  x 3  b 2 u dx 2 2   x   u() t  a0 0 1   b  x3  a 1 x 1  x 4  b 1 u   dt 1 1     x  a x  b u  a0 0 0 b  4 0 1 0  0   0   y x1  y( t ) 1 0 0 0x   0 u ( t ) b3 b2 X x1 X b1 1/ s 1/ s 2 1 1/ s 1 1 U() s Y() s 1/ s 1 1 b0 x4 X 4 x3 X 3 x2 a2 a1 a3 a0 input feedforward canonical form sites.google.com/site/ncpdhbkhn 26 Signal – Flow Graph Ex. 2 & Block Diagram Models (14) 3 2 Y() s b3 s b 2 s  b 1 s  b 0 G() s   4 3 2 U() s s a3 s  a 2 s  a 1 s  a 0 b3 b2 X 2 x1 X b1 1/ s 1/ s 1 1/ s 1 1 U() s Y() s 1/ s 1 1 b0 x4 X 4 x3 X 3 x2 a2 a1 a3 a0 b3 b2 b1 x X x X x X x X 4 1 4 3 1 3 2 1 2 1 1 1 U() s b0 Y() s () s () s () s () s a3 a2 a1 a 0 sites.google.com/site/ncpdhbkhn 27 Signal – Flow Graph Ex. 3 & Block Diagram Models (15) Y() s G U() s 3(s 1)( s  2) Y() s T() s   G() s  s( s 3)( s  4) U( s ) 1 G () 3s2  9 s  6  s310 s 2  21 s  6 3s1 9 s  2  6 s  3  1 10s1  21 s  2  6 s  3 X1 X2 G () H X1 G X 2 1 GH sites.google.com/site/ncpdhbkhn 28 Signal – Flow Graph Ex. 3 & Block Diagram Models (15) U() s 3(s 1)( s  2) Y() s 3s1 9 s  2  6 s  3 G() s  T() s  s( s 3)( s  4) 1 10s1  21 s  2  6 s  3 () 3 1 1/ s 1/ s 1/ s 9 6 U() s Y() s X X X 10 3 2 1 21 6 1  2  3  4 b3 s b 2 s  b 1 s  b 0 s T() s  1  2  3  4 1a3 s  a 2 s  a 1 s  a 0 s b3 1 b2 1 X 4 b1 s 1/ s 1/ s 1/ s b0 U() s Y() s X 3 X 2 X1 a3 a2 a1 sites.google.com/site/ncpdhbkhn 29 a0 Signal – Flow Graph Ex. 3 & Block Diagram Models (16) U() s 3(s 1)( s  2) Y() s 3s1 9 s  2  6 s  3 G() s  T() s  s( s 3)( s  4) 1 10s1  21 s  2  6 s  3 () 3 1 1/ s 1/ s 1/ s 9 6 U() s Y() s X X X 10 3 2 1 21 6 3 9 X X X 1 3 1 2 1 1 Y() s U() s 6 () s s s 10 () 21 6 sites.google.com/site/ncpdhbkhn 30 Signal – Flow Graph Ex. 3 & Block Diagram Models (17) U() s 3(s 1)( s  2) Y() s 3s1 9 s  2  6 s  3 G() s  T() s  s( s 3)( s  4) 1 10s1  21 s  2  6 s  3 () 3 1 1/ s 1/ s 1/ s 9 6 U() s Y() s X X X 10 3 2 1 21 6  x1  0 1 0   x 1   0   d         x  x x0 0 1 x  0 u ( t ) 2 1 dt 2     2      x3 x 2  x3  6 21  10   x 3   1     x3 u 10 x 3  21 x 2  6 x 1  x1   y6 x  9 x  3 x  y( t )  6 9 3 x   1 2 3  2   x3  sites.google.com/site/ncpdhbkhn 31 Signal – Flow Graph Ex. 3 & Block Diagram Models (18) U() s 3(s 1)( s  2) Y() s 3s1 9 s  2  6 s  3 G() s  T() s  s( s 3)( s  4) 1 10s1  21 s  2  6 s  3 () 3 1 1/ s 1/ s 1/ s 9 6 U() s Y() s X X X 10 3 2 1 21 6 3 X x X 2 x X1 9 3 2 1/ s 1 1 1/ s 1 U() s Y() s 1/ s 1 6 x3 21 6 10 x1  10 x 1  x 2  3 u  10 1 0   3   dx     x2 21 x 1  x 3  9 u   21 0 1x  9u ( t )   dt       6 0 0   6  x3 6 x 1  6 u         y x1  y( t ) 1 0 0x   0 u ( t ) sites.google.com/site/ncpdhbkhn 32 Signal – Flow Graph Ex. 3 & Block Diagram Models (19) U() s 3(s 1)( s  2) Y() s 3s1 9 s  2  6 s  3 G() s  T() s  s( s 3)( s  4) 1 10s1  21 s  2  6 s  3 () 3 1 1/ s 1/ s 1/ s 9 6 U() s Y() s X X X 10 3 2 1 21 6 3 X x X 2 x X1 9 3 2 1/ s 1 1 1/ s 1 U() s Y() s 1/ s 1 6 x3 21 6 10 3 9 x X x X x X 3 1 3 2 1 2 1 1 1 U() s 6 Y() s () s () s () s 10 21 6 sites.google.com/site/ncpdhbkhn 33 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 34 Alternative Signal – Flow Graph Ex. 1 & Block Diagram Models (1) Field Field s 1 voltage 1 current 6 Velocity R() s 5 Y() s s  5 U() s s  2 I() s s  3  3 6 0   0       Controller Motor & load x 0  2  20 x  5r ( t )       0 0 5   1    y( t )  1 0 0x 5 X X X 3 1 1/ s 2 6 1/ s 1 1 R() s Y() s 1 1/ s 5 U() s I() s 5 2 3 5 X X 1 3 U() s 1 I() s 1 1 R() s 5 Y() s X 6 () s () s 2 () s 5 2 3 sites.google.com/site/ncpdhbkhn 35 Alternative Signal – Flow Graph Ex. 1 & Block Diagram Models (2) Field Field s 1 voltage 1 current 6 Velocity R() s 5 Y() s s  5 U() s s  2 I() s s  3 Controller Motor & load Y() s 30(1) s   20  10 30 T() s      R()( s s 5)( s  2)( s  3) s  5 s  2 s  3 X 1 1 20 1/ s X1 () s 1 20 5 5 () X () 1 X 2 1 2 R() s Y() s R() s 10 Y() s 1/ s 10 () s 2 Diagonal 1 30 2 1/ s canonical X 3 X form 1 3 30 3 () s sites.google.com/site/ncpdhbkhn 3 36 Alternative Signal – Flow Graph Ex. 1 & Block Diagram Models (3) Field Field s 1 voltage 1 current 6 Velocity R() s 5 Y() s s  5 U() s s  2 I() s s  3 x1 5 x 1  r ( t )  Controller Motor & load x2 2 x 2  r ( t )  x3 3 x 3  r ( t )   y( t )  20 x1  10 x 2  30 x 3 X 1 1 20  5 0 0   1  s  () x0  2 0  x   1  r ( t )      5   () 0 0 3   1  X ()      1 2  R() s 10 Y() s  y( t )  20  10 30x () s Diagonal 2 canonical X 1 3 form 30 () s sites.google.com/site/ncpdhbkhn 3 37 Alternative Signal – Flow Graph Ex. 2 & Block Diagram Models (4) x1  x 1   x 2  u 1() t x1     0   x 1   1 0  d         u1() t  x  x   x  u() t x   0 x  0 1 2 1 2 2 dt 2     2    u() t  x x   x 2  3 1 2 x3    0   x 3   0 0   () U 1 1  s X1 1 1/ s X1 () U 1     1/ s 1 X 3  X  3 s 1 1/ s U2  X 2 X  U 1 2 2  () s  sites.google.com/site/ncpdhbkhn 38 Alternative Signal – Flow Graph Ex. 3 & Block Diagram Models (5) my My  ml  u( t )  0 m 2 mly  ml  mgl   0 mg  l (,,,)(,,,)x1 x 2 x 3 x 4  y y   u() t Mx2 mlx  4  u( t )  0 M   x lx   gx  0  2 4 3 y() t x1 x 2  mg 1 x1 0 1 0 0   x 1   0  x2  x 3  u() t         MM d x20 0mg / M 0 x 2 1/ M           u x3 x 4 dt x3 0 0 0 1   x 3   0          g 1 x4 0 0g / l 0   x 4   1/( Ml )  x  x  u() t  4l 3 Ml sites.google.com/site/ncpdhbkhn 39 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 40 The Transfer Function from the State Equation (1) x  Ax  Bu   yCx  D u sX()()() s AX s  B U s   Y()()() sCX s  D U s ()()()sIAXB  s  U s XIAB()[]()s  s  1 U s  ΦB()()s U s Y()[()]() s CΦBD s  U s Y() s G()() s  CΦBD s  U() s sites.google.com/site/ncpdhbkhn 41 The Transfer Function Ex. from the State Equation (2)  1 i L   L   0  1   C   vC C R vo x   x C u  Ax  B u   1 R    u() t  0  iC  LL    y0 Rx  Cx 1   1  0 s s 0  CC    []sIA       0 s  1RR  1     s   LLLL    RR 1      1  s     s    1 1LCLC  1   ΦIA()[]s  s       2 R 1 1 ()s  1  s s  s s L LC LL    sites.google.com/site/ncpdhbkhn 42 The Transfer Function Ex. from the State Equation (3)  1 i L   L   0  1   C   vC C R vo x   x C u  Ax  B u   1 R    u() t  0  iC  LL    y0 Rx  Cx R  s  L 1  1    R  1  G( s )   0 R ()()s C  s C  s        1 LC  Φ()s    1 s  0  ()s 1    s L()() s  s  L  R/( LC )  R 1 Y() s s2  s  G()() s CΦBD s  L LC U() s sites.google.com/site/ncpdhbkhn 43 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 44 The Time Response & the State Transition Matrix (1) t x()t exp( A t )(0) x  exp[ A ( t  ) Bu () r d  0 t Φ(t ) x (0)  Φ ( t  ) Bu (  ) d  0 Φ(t): the state transition matrix sites.google.com/site/ncpdhbkhn 45 The Time Response Ex. & the State Transition Matrix (2)  1 i L   L   0  1   C   vC C R vo x   x C u  Ax  B u   1 R    u() t  0  iC  LL    y0 Rx  Cx 0 2   2  RLC3,1,0.5   ABC   ,    ,10    1 3   0  1 1s3  2  1  s  3  2  ΦIA()[]s s  2      s3 s  2 1s ()s  1 s  (2et e 2 t ) (  2 e  t  2 e  2 t )  Φ()t  t 2 t  t  2 t  (e e ) (  e  2 e )  sites.google.com/site/ncpdhbkhn 46 The Time Response Ex. & the State Transition Matrix (3)  1 i L   L   0  1   C   vC C R vo x   x C u  Ax  B u   1 R    u() t  0  iC  LL    y0 Rx  Cx (2et e 2 t ) (  2 e  t  2 e  2 t )  Φ()t  t 2 t  t  2 t  (e e ) (  e  2 e )  t x()tΦ ()(0) t x  Φ ( t  ) Bu ()  d  0 x() t  1  e2t  x(0) x (0)1,()0  u t  1 Φ () t  1 2     2t  x2 () t  1  e  sites.google.com/site/ncpdhbkhn 47 The Time Response Ex. & the State Transition Matrix (3) iL L 2t    x1() t  1  e v Φ()t    vC C R o     2t  x2 () t  1  e   u() t iC 1 1 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 (t) (t) (t) 1 0.5 1 0.5 2 0.5 x x x 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 t t x1(t) sites.google.com/site/ncpdhbkhn 48 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 49 Analysis of State Variable Models Ex. Using Control Design Software Y( s ) 3 s2  9 s  6 T() s   R() s s310 s 2  21 s  6 sites.google.com/site/ncpdhbkhn 50

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