 # Lecture Control system design: Mathematical models of systems - Nguyễn Công Phương

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1 ## Lecture Control system design: Mathematical models of systems - Nguyễn Công Phương

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Lecture Control system design: Mathematical models of systems presents the following content: Differential equations of physical systems, linear approximations of physical systems, the laplace transform, the transfer function of linear systems, block diagram models, signal – flow graph models, the simulation of systems using control design software.

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## Nội dung Text: Lecture Control system design: Mathematical models of systems - Nguyễn Công Phương

1. Nguyễn Công Phương CONTROL SYSTEM DESIGN Mathematical Models of Systems
2. 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
3. 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
4. Differential Equations of Physical Systems (1) i  v  • Current i: a through-variable • Voltage v: an across-variable sites.google.com/site/ncpdhbkhn 4
5. 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
6. Differential Equations of Physical Systems (3) Inductive storage Electrical inductance: v: voltage di 1 L i i: current v21  L E  Li 2 dt 2 v2 v1 L: inductance Translational spring: v: translational velocity 1 dF 1 F2 k F F: force v21  E k dt 2 k v2 v1 k: translational stiffness Rotational spring: ω: angular velocity 1 dT 1 T2 k F 21  E T: torque k dt 2 k 2 1 k: rotational stiffness Fluid inertia: I dQ 1 2 Q P: Pressure P21  I E  IQ Q: fluid volumetric flow rate dt 2 P2 P1 I: fluid inertance sites.google.com/site/ncpdhbkhn 6
7. Differential Equations of Physical Systems (4) Capacitive storage Electrical capacitance: C dv 1 2 i v: voltage; i: current i  C 21 E  Cv21 C: capacitance dt 2 v2 v1 Translational mass: dv 1 F2 F v1  v: translational velocity; F: force FM 2 E M const M: mas; k: translational stiffness dt 2 k v2 Rotational mass: d 2 1 T 1  ω: angular velocity; T: torque T J E  J22 J J: moment of inertia dt 2 2 const Fluid capacitance: dP21 1 Q P: Pressure; Cf: fluid capacitance Q  Cf E  C f P212 Cf Q: fluid volumetric flow rate dt 2 P2 P1 Thermal capacitance: dT q T1  P: Pressure; q: heat flow rate; q  Ct 2 E  Ct T2 Ct const dt T2 Ct: thermal capacitance sites.google.com/site/ncpdhbkhn 7
8. Differential Equations of Physical Systems (5) Energy dissipators Electrical resistance: 2 R i v v21 v: voltage; i: current i  21 P R R v2 v1 R: resistance Translational damper: v: translational velocity; F: force F  bv21 P  bv21 2 F b b: viscous friction v2 v1 Rotational damper: b T  b21 T ω: angular velocity; T: torque P 2 b21 2 1 b: viscous friction Fluid resistance: Rf Q P21 P212 P: Pressure; Rf: fluid resistance Q P Q: fluid volumetric flow rate Rf Rf P2 P1 Thermal resistance: T21 T21 Rt q P: Pressure; q: heat flow rate; q P Rt: thermal resistance Rt Rt T2 T1 sites.google.com/site/ncpdhbkhn 8
9. 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 dt  C  i(t )dt  r(t ) 0 d 2 y (t ) dy (t ) M 2  b  ky (t )  r (t ) dt dt dv (t ) t dy (t ) M dt  bv (t )  k  v(t )dt  r(t ) 0 v (t )  dt sites.google.com/site/ncpdhbkhn 9
10. Differential Equations of Physical Systems (7) v (t ) i v (t ) R1 2 Ex. 2 1 iL iC1 iR 2 R1 iC 2 L1 k C1 R2 C2 Friction b2 r (t ) M2 Velocity v2(t) Friction b1 iC1  iR 2  iR1  r (t )  iR1  iC 2  iL  0 M1 Velocity v1(t)  dv1 v1 v1  v2 C1 dt  R  R  r (t ) Force r(t)   2 1  v1  v2  C2 dv2  1 v2 dt  0 t M1 dv1  b1v1  b2 v1  b1v2  r (t )  R1 dt L 0  dt  dv1 1 1 v2  1 dt R 1 R 1 R  r (t ) C  v  v  dv t  M 2 2  b1v2  b1v1  k  v dt  0  2 1 1 2 C2 dv2  1 v2  1 v1  1 v2 dt  0 t  dt 0  dt R1 R1 L 0 sites.google.com/site/ncpdhbkhn 10
11. 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
12. Linear Approximations of Physical Systems (1) y x (t ) y (t ) y  mx System 0 x x1 (t ) y1 (t ) x (t ) y (t ) System System x2 ( t ) y2 ( t ) System kx (t ) ky (t ) System x1 (t )  x2 (t ) y1 (t )  y2 (t ) System Superposition Homogeneity sites.google.com/site/ncpdhbkhn 12
13. Linear Approximations of Physical Systems (2) y y  mx  b x (t ) y (t ) y1 System y y0 x 0 x0 x1 x y1  mx1  b  y0  y  m ( x0  x )  b y0  mx0  b  y  m x Linear !!! sites.google.com/site/ncpdhbkhn 13
14. Linear Approximations of Physical Systems (3) y y  g ( x) dg x (t ) y (t ) y0 dx x  x0 System y (t )  g [ x (t )] 0 x0 x  dg ( x  x0 )   d 2 g ( x  x0 ) 2   g ( x0 )    2   ...  dx x  x0 1!   dx 2!   x  x0  dg  g ( x0 )  ( x  x0 ) dx x  x0  y0  m( x  x0 )  ( y  y0 )  m( x  x0 )  y  m x sites.google.com/site/ncpdhbkhn 14
15. Linear Approximations of Physical Systems (4) x (t ) y (t ) System y  g ( x1 , x2 ,..., xn ) g  g ( x10 , x20 ,..., xn 0 )  ( x1  x10 )  x1 x  x0 g g  ( x2  x20 )  ...  ( xn  xn 0 )  x2 x  x0  xn x  x0 sites.google.com/site/ncpdhbkhn 15
16. Linear Approximations of Physical Systems (5) Ex. T  mgL sin   sin   T0  mgL (   0 )    0  0  0; T0  0  T  mgL(cos 0o )(  0o )  mgL T   http://www.ic.sunysb.edu/Class/phy141md/  2  doku.php?id=phy141:labs:lab1 0   2 sites.google.com/site/ncpdhbkhn 16
17. 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
18. The Laplace Transform (1)  The Laplace transformation of f(t): F ( s )   0 f (t )e  st dt 1   j The inverse Laplace transform of F(s): f (t )  2 j   j F ( s )e st ds f(t)  (t ) u (t ) e  at t te  at sin at cos at 1 1 1 1 a s F(s) 1 s sa s2 ( s  a )2 s2  a2 s2  a2 sites.google.com/site/ncpdhbkhn 18
19. 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 )  F2 ( 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 d n f (t ) / dt n s n F ( s )  s n 1 f (0)  s n 2 f 1 (0) ...  s o f n 1 (0) 7. Multiplication by t t n f (t ) ( 1) n d n F ( s ) / ds n  8. Division by t f (t ) / t  s F ( )d  t 9. Integration  0 f ( )d  F ( s) / s t 10. Convolution f1 ( t ) * f 2 ( t )   f 1 (  ) f 2 ( t   ) d  F1 ( s ) F2 ( s ) 0 11. Final value lim f (t ) lim sF ( s ) t  s 0 sites.google.com/site/ncpdhbkhn 19
20. The Laplace Transform (3) Ex. 1 M d 2 y (t )  b dy (t )  ky (t )  r (t ) 2 dt dt r (t )  R ( s ) y (t )  Y ( s )  ky (t )  kY ( s ) dy (t ) b  b  sY ( s )  y (0 )  dt d 2 y (t )  2  dy   M M  s Y ( s )  sy (0 )  (0 )   2 dt dt  dy   M  s 2Y ( s )  sy (0 )  (0 )   b  sY ( s )  y (0 )   kY ( s )  R ( s )  dt   dy  R ( s )  M  sy (0 )  (0 )   by (0 ) ?  Y ( s)   dt   p( s)  y (t ) Ms 2  bs  k q( s ) sites.google.com/site/ncpdhbkhn 20 