Transcript Chapter 3 mathematical Modeling of Dynamic Systems Modeling Mathematical models are developed from: physical laws chemical laws biological laws economic laws etc Mathematical models take the form of: differential.

Chapter 3 mathematical
Modeling of Dynamic Systems
Modeling
Mathematical models are developed from:
physical laws
chemical laws
biological laws
economic laws
etc
Mathematical models take the form of:
differential equations
transfer functions
state equations
block diagrams
signal flow graphs
etc
Mathematical models are used to analyze dynamic characteristics
and to design control systems
Models
Simplicity vs Accuracy
lumped vs distributed parameter models
linear vs nonlinear models
time-invariant vs time varying models
As simple as possible with the required accuracy
Transfer Function and
Impulse-Response Function
Transfer Function
Convolution Integral
Impulse-Response Function
Convolution Integral
Y ( s )  G ( s ) R( s )
y(t )  
 t
y(t )  ?
ILT
convolution
g (t   )r ( )d  
 0
 t
 0
g ( )r (t   )d
Multiplication in the frequency domain
corresponds to
convolution in the time domain
Convolution
Laplace
Transformation
Time domain functions
g(t), r(t)
Frequency Domain
functions, G(s), R(s)
Time
domain
convolution
Frequency
domain product
Time domain
y(t)=conv(g(t),r(t))
Frequency domain
Y(s)=G(s)R(s)
Inverse
Laplace
Transform
Impulse Response Function
The response of a differential equation to an impulse (delta function)
input is the Impulse Response Function of that differential equation.
Y (s)  G(s) R(s)
For an impulse input
r (t )   (t ) R(s)  1
The Laplace transform of the impulse response is
Y (s)  G(s)1  G ( s )
= transfer function
The impulse response is the inverse Laplace transform of the transfer
function
g (t )
Impulse Response, Convolution,
and solution to DEs
The output of a system due to the input r(t) is the convolution of the
input function and the impulse response of the system.
y(t) = conv(g(t),r(t))
There is an intimate relationship between the impulse response of a
system and the response of the system to any other input.
Automatic Control Systems
Block Diagrams
Signals
Blocks and Transfer functions
Summing point
Branch point
Block diagram of a closed loop system
Open-loop transfer function and feedforward transfer function
Closed-loop transfer function
Automatic Controllers
Industrial Controllers
On-off, P, I, PI, PD, PID
Disturbances
Block diagram reduction
Start by doing what’s necessary,
then what’s possible, and
suddenly you are doing the impossible.
-St. Francis of Assisi
Modeling review
Mechanical systems
Electrical systems
I hope to come back to this topic later
Skip 3-5, 3-9, 3-10
x(t ) 
1

x(t )  kr (t )
x(t )  2n x(t )  n2 x(t )  kr (t )
Mechanical examples?
Electrical examples?
Thermal examples?
Transfer function?
Time constant
Mechanical examples?
Electrical examples?
Thermal examples?
Transfer function?
Undamped natural frequency
damping ratio
Transfer function
Input signal r (t )
Output signal y(t )
Transfer function
 Y ( s) 
G( s )  

R
(
s
)

 i .c . s  0
Block diagram
R(s)
Equation
Y (s)  G(s) R(s)
G(s)
Y(s)
Block Diagram reduction
L(s)
R(s)
GR(s)
+ E(s)
C(s)
U(s)
G1(s)
+
+
Y(s)
G2(s)
+
+
H(s)
T ?
Y /N ?
Y /L?
E/RS ?
N(s)
GR ( s )  1
H (s)  1
G1 ( s )  a /( s  b)
Open loop TF?
Feedforward TF?
Closed loop TF?
G2 ( s )  K /( Ms  Bs )
2
Steady-State
Error
E ( s )  Y ( s )  R( s )
or
E ( s)  H ( s)Y ( s)  GR ( s) R( s)
Steady-state means the output looks like the input.
Step inputs produce step (constant) outputs
Sinusoidal inputs produce sinusoidal outputs
Ramp inputs produce ramp outputs
X inputs produce X outputs
Steady-state means that the transients have died out.
The output has (at least) two terms,
steady-state (mathematically looks like the input)
transient (decays to zero)
sometimes other terms (normally a bad situation)
Computing SS error
steady  state  error  ess  lim  e(t ) 
t 
To compute E(s):
•inverse Laplace transform to get e(t)
•take limit as t goes to infinity.
•Steady state error only exists if the limit exists.
Chapter 3 Problems
A
1-5
6-10, 12,13
11
14-18
19-22
23
24-26
B
1-8
9
10-12
13-18
19-24
25
26-27
Comments
Block diagram reduction
State space
State space to transfer function
Mechanical models
Electrical models
Electro mechanical model
Signal flow graph
27
28-29
linearization
Assignments
Ungraded homework: A1-5
Graded homework: B1-7
Test 1: Solving DEs via Laplace transforms, Block diagrams.
Laplace transforms
definition
2 formulas for functions
derivatives
Euler’s identities
ALGEBRA
CALCULUS
Complex numbers/algebra