1. DC Motor Driving an Inertial Load

2. 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

4. State Space model

5. 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.m^2;
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)

6. Matlab output a = x1 x2 x1 -4 -0.03 x2 0.75 -10 b = u1 x1 2 x2 0 c =y
d = y

7. SS to TF or ZPK representation
>> sys_tf = tf(sys_dc)
Transfer function:
1.5
s^ s
>> sys_zpk = zpk(sys_dc)
Zero/pole/gain:
(s+4.004) (s+9.996)

8. 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.

9. 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/(s^2+14*s+40.02)
Transfer function:
1.5
s^ s
sys_tf = tf(1.5,[ ])

10. 4 ways to enter system model
sys_zpk = zpk([],[ ], 1.5)
Zero/pole/gain:
1.5
(s+9.996) (s+4.004)

11. Liquid Level System Qi H Qo 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

12. Conservation of matter:Q
Conservation of matter:
Qo is dependent on the “head” H
const. coeff. H

13. ∴
This is nonlinear.
To find eq. points, set derivative=0
To linearize: let
where

14. Substitute into eq on top:
use

15. 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:

16. Two tank system:

17. In eq pt: all flow=same 1 3 2 4

19. Op Amp circuit example Note: ip1=0, ∴vp1=vo=vA & vB=vp2=0
Let vC1 & vC2 be s.v., vo output.

20. KCL at A:
vo is not s.v. nor input, use
vo=vC2

21. KCL at B:
vo1 not s.v. nor input,
vo1=vA+vC1=vn1+vC1
=vp1+vC1=vo+vC1
=vC2+vC1

22. Output eq:

23. Modeling Types of systems electric mechanical electromechanical
Types of models I/O o.d.e. models
Transfer Function
state space models

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

25. 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

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

27. 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

28. 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

29. into : Y(s)=C(sI-A)-1BU(s)+DU(s) Y(s)=[C(sI-A)-1B+D] U(s)
H(s)= D+C(sI-A)-1B
is the T.F. from u to y
from 2 1

30. Example

32. >> [n,d]=ss2tf(A,B,C,D) n = 0 3.0000 1.0000 d = 1 3 2
>> [A,B,C,D]=tf2ss(n,d)
A =
B = 1
C =
D =
>> tf(n,d)
Transfer function:
s^2 + 2 s + 3
s^3 + 4 s^2 + 5 s + 6
In Matlab:
>> A=[0 1;-2 -3];
>> B=[0;1];
>> C=[1 3];
>> D=[0];
>> [n,d]=ss2tf(A,B,C,D)
n =
d =