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Physical Laws for Mechanical

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Title: Elements of Feedback Control Subject: Weapons Author: Brien W. Dickson Description: 1 Hour Last modified by: Degang J. Chen Created Date: 6/4/1997 12:49:12 PM – PowerPoint PPT presentation

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Title: Physical Laws for Mechanical


1
Physical Laws for Mechanical
2
Example car suspension
3
Car suspension simplified
  • Ignore tire deformation.
  • Suppose y1(t) is measured
  • from equilibrium position
  • when gravity has set in.
  • So gravity is canceled by
  • spring force at eq. pos.
  • ?There are two forces on m

y1(t)
x(t)
4
  • Newtons Law
  • or
  • num
  • den
  • T.F.H(s)
  • or

5
State Space Model
  • For linear motion
  • Define two state variables for each mass
  • x1position, x2 velocity x1-dot x2
  • x2-dot is acc and solve for it from Newtons
  • For angular motion
  • Define two state variables for each rotating
    inertia
  • x1 angle, x2 angular velocity x1-dot x2
  • x2-dot is angular acc and solve for it from
    Eulers law

6
Quarter car suspension
7
u
8
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9
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10
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11
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12
Electromechanical systems
  • Motors
  • DC motors
  • Induction motors
  • Variable reluctance motors
  • Generators
  • Angular position sensors
  • Encoders
  • Tachometers

13
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14
For field control with constant armature current
For armature control with constant field current
15
Armature controlled motor in feedback
16
Get TF from wd to w and Td to w.
17
DC Motor Driving an Inertial Load
18
  • w(t) angular rate of the load, output
  • vapp(t) applied voltage, the input
  • i(t) armature current
  • vemf(t) back emf voltage generated by the motor
    rotation
  • vemf(t) constant motor velocity
  • t(t) mechanical torque generated by the motor
  • t(t) constant armature current

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20
State Space model
21
Matlab
R 2.0 Ohms L 0.5 Henrys Km .015
torque constant Kb .015 emf constant Kf
0.2 Nms J 0.02 kg.m2 A -R/L -Kb/L
Km/J -Kf/J B 1/L 0 C 0 1 D
0 sys_dc ss(A,B,C,D)
22
Matlab output
a x1 x2
x1 -4 -0.03 x2
0.75 -10 b u1
x1 2 x2 0 c
x1 x2 y1
0 1 d u1
y1 0
23
SS to TF or ZPK representation
gtgt sys_tf tf(sys_dc) Transfer function
1.5 ------------------------ s2 14 s
40.02 gtgt sys_zpk zpk(sys_dc) Zero/pole/gain
1.5 ------------------------- (s4.004)
(s9.996)
24
  • Note The state-space representation is best
    suited for numerical computations. For highest
    accuracy, convert to state space prior to
    combining models and avoid the transfer function
    and zero/pole/gain representations, except for
    model specification and inspection.

25
4 ways to enter system model
sys tf(num,den) Transfer function sys
zpk(z,p,k) Zero/pole/gain sys ss(a,b,c,d)
State-space sys frd(response,frequencies)
Frequency response data s tf('s') sys_tf
1.5/(s214s40.02) Transfer function
1.5 ------------------------ s2 14 s
40.02 sys_tf tf(1.5,1 14 40.02)
26
4 ways to enter system model
sys_zpk zpk(,-9.996 -4.004,
1.5) Zero/pole/gain
1.5 ------------------------- (s9.996) (s4.004)
27
Modeling
  • Types of systems electric
  • mechanical

  • electromechanical
  • fluid systems
  • thermal systems
  • Types of models I/O o.d.e. models
  • Transfer Function
  • state space models

28
  • I/O o.d.e. model o.d.e. involving input/output
    only.
  • linear
  • where u input
  • y output

29
  • State space model
  • linear
  • or in some text
  • where u input
  • y output
  • x state vector
  • A,B,C,D, or F,G,H,J are const matrices

30
  • Other types of models
  • Transfer function model (This is I/O model) from
    I/O o.d.e. model, take Laplace transform

31
  • Then I/O ODE model in L.T. domain becomes
  • or

denote
32
ODE or TF to SS
33
  • State space model to T.F. / block diagram
  • s.s.
  • Take L.T.
  • From sX(s)-AX(s)BU(s)
  • sIX(s)-AX(s)BU(s)
  • (sI-A)X(s)BU(s)
  • X(s)(sI-A)-1BU(s)

1
2
1
34
  • into Y(s)C(sI-A)-1BU(s)DU(s)
  • Y(s)C(sI-A)-1BD U(s)
  • H(s) DC(sI-A)-1B
  • is the T.F. from u to y
  • from

2
1
35
Example
36
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37
gtgt n1 2 3d1 4 5 6 gtgt A,B,C,Dtf2ss(n,d)
A -4 -5 -6 1 0 0 0
1 0 B 1 0 0 C 1
2 3 D 0 gtgt tf(n,d) Transfer
function s2 2 s 3 ---------------------
s3 4 s2 5 s 6
  • In Matlab
  • gtgt A0 1-2 -3
  • gtgt B01
  • gtgt C1 3
  • gtgt D0
  • gtgt n,dss2tf(A,B,C,D)
  • n
  • 0 3.0000 1.0000
  • d
  • 1 3 2
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