DC Motor Driving an Inertial Load

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DC Motor Driving an Inertial Load
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• 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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State Space model
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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)
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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
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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)
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• 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.

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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)
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4 ways to enter system model
sys_zpk zpk(,-9.996 -4.004,
1.5) Zero/pole/gain
1.5 ------------------------- (s9.996) (s4.004)
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Liquid Level System

• Qi input flow rate
• Qo output flow rate
• H liquid level in tank
• A cross section of tank
• V volume of liquid in tank
• V AH

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• Conservation of matter
• Qo is dependent on the head H
• const. coeff.

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• ?
• This is nonlinear.
• To find eq. points, set derivative0
• ?
• To linearize let
• where

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• Substitute into eq on top
• use

0
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• Output flow
• The quantity R is the called the
  resistance of the valve and A is also denoted as
  C is called the capacitance of the tank.
• Then
• Note

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Two tank system
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In eq pt all flowsame
1
3
2
4
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• Exercise
• 1)get component block diag form 3, 4
• 2)put all four pieces to form block diag.
• 3)get T.F. from qi to q2

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Note ip10, ?vp1vovA vBvp20 Let vC1 vC2
be s.v., vo output.
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KCL at A
vo is not s.v. nor input, use vovC2
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KCL at B
0
vo1 not s.v. nor input,
vo1vAvC1vn1vC1 vp1vC1vovC1
vC2vC1
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Output eq
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Modeling

• Types of systems electric
• mechanical
• 
  electromechanical
• Types of models I/O o.d.e. models
• state space models

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

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• 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

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

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• Then I/O model in L.T. domain becomes
• This is the T.F. model of the system.
• ?T.F.
• or
• i.e. output L.T. is eq. to input L.T. with gain
  H(s)

denote
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• 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
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• 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

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1
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Example
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• 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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• 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