Transcript Chapter 4

Chapter 5
Transfer Functions and
State Space Models
Overall Course Objectives
• Develop the skills necessary to function as an
industrial process control engineer.
– Skills
•
•
•
•
Tuning loops
Control loop design
Control loop troubleshooting
Command of the terminology
– Fundamental understanding
• Process dynamics
• Feedback control
Transfer Functions
• Provide valuable insight into process
dynamics and the dynamics of feedback
systems.
• Provide a major portion of the terminology
of the process control profession.
Transfer Functions
• Defined as G(s) = Y(s)/U(s)
• Represents a normalized model of a
process, i.e., can be used with any input.
• Y(s) and U(s) are both written in deviation
variable form.
• The form of the transfer function indicates
the dynamic behavior of the process.
Derivation of a Transfer Function
M
dT
dt
 F1 T1  F 2 T 2  ( F1  F 2 ) T
Tˆ  T  T 0
M
dTˆ
dt
Tˆ1  T1  T 0
Tˆ2  T 2  T 0
 F1 Tˆ1  F2 Tˆ2  ( F1  F2 )Tˆ
• Dynamic model of
CST thermal
mixer
• Apply deviation
variables
• Equation in terms
of deviation
variables.
Derivation of a Transfer Function
T (s) 
G (s) 
T (s)
T1 ( s )
F1 T1 ( s )  F 2 T 2 ( s )
M

s  F1  F 2
M

F1
s  F1  F 2

• Apply Laplace transform
to each term considering
that only inlet and outlet
temperatures change.
• Determine the transfer
function for the effect of
inlet temperature changes
on the outlet temperature.
• Note that the response is
first order.
In-Class Exercise: Derive a
Transfer Function
For a level process:
A
dL
dt
 Fin  kL
Fin is the input variable
L is the output variable
D erive the transfer function for this process
Poles of the Transfer Function
Indicate the Dynamic Response
G (s) 
Y (s) 
1
( s  a ) ( s  bs  c ) ( s  d )
2
A
(s  a)
y (t )  A e
 at

B
( s  bs  c )
2
 B e
pt

C
(s  d )
sin(  t )  C  e
dt
• For a, b, c, and d positive constants, transfer
function indicates exponential decay, oscillatory
response, and exponential growth, respectively.
Poles on a Complex Plane
Im
Re
Exponential Decay
Re
y
Im
Time
Damped Sinusoidal
Re
y
Im
Time
Exponentially Growing
Sinusoidal Behavior (Unstable)
Re
y
Im
Time
What Kind of Dynamic Behavior?
Im
Re
Unstable Behavior
• If the output of a process grows without
bound for a bounded input, the process is
referred to a unstable.
• If the real portion of any pole of a transfer
function is positive, the process
corresponding to the transfer function is
unstable.
• If any pole is located in the right half plane,
the process is unstable.
Routh Stability Criterion
a n s  a n 1 s
n
n 1
 ...  a1 s  a 0  0 w here a i  0.
A necessary and sufficient condition for all the
roots of the polynom ial to have negative real parts
is that all the elem ents of the first co lum n of the
R outh array a re positive.
N ote that the stability of a system can be assessed
by applying the R outh stability criterio n to the
denom inator of the system transfer funct ion.
Routh Array for a 3rd Order System
a3


a2

 a 2 a1  a 3 a 0

a
2


a0
a1 

a0


0 

0 
Routh Stability Analysis Example
D eterine if this system is stable:
s  2s  1
2
G p (s) 
s  2 s  3s  9
3
2
R outh A rray: a 3  1; a 2  2; a1  3; a 0  9
 1

2

6  9

2

 9
3

9

 T herefore, this system is unstable
0

0 
In-Class Exercise
D eterm ine the stability of the
system represented by the
follow ing transfer function:
G p (s) 
s8
3s  4 s  6 s  7
3
2
Zeros of a Transfer Function
• The zeros of a transfer functions are the
value of s that render N(s)=0.
• If any of the zeros are positive, an inverse
response is indicated.
• If all the zeros are negative, overshoot can
occur in certain situations
Combining Transfer Functions
• Consider the CST thermal mixer in which a
heater is used to change the inlet
temperature of stream 1 and a temperature
sensor is used to measure the outlet
temperature.
• Assume that heater behaves as a first order
process with a known time constant.
Combining Transfer Functions
Ga (s) 
G (s) 
T1 ( s )
1

H s  1
T1 , spec ( s )
T (s)

T1 ( s )
Gs (s) 
M
F1
s  F1  F 2
Ts ( s )
T (s)

• Transfer function for
the actuator
• Transfer function for

the process
1
s s  1
• Transfer function for
the sensor
Combining Transfer Functions
G oa ( s )  G a ( s ) G ( s ) G s ( s )
G oa ( s ) 
 H
F1
s  1M s  F1  F 2  s s  1
In-Class Exercise: Overall Transfer
Function For a Self-Regulating Level
For a level process (earlier in-class ex ercise):
A
dL
dt
 Fin  kL
Fin is the input variable
L is the output variable
D erive the overall transfer function for this process
Block Diagram Algebra
• Series of transfer functions
• Summation and subtraction
• Divider
Block Diagram Algebra
C (s)
U (s )
D (s )
G 1 (s)
F (s)
G 2 (s)
H (s )
G 3 (s )
Y (s)
E (s )
+
+
Block Diagram Algebra
W e w ant to determ ine Y ( s ) / U ( s ) so w e start w i th
Y (s)  E (s)  H (s)
(sum m ation function)
E ( s ) / C ( s )  G 1 ( s ) G 2 ( s ) (series of tranfer functions)
U (s)  C (s)  F (s)
H (s) / F (s)  G3 (s)
(divider function)
( transfer function definition)
S ubstitute into 1st eqn:
Y ( s )  G 1 ( s ) G 2 ( s )U ( s )  G 3 ( s )U ( s )
R earrange: G O A ( s )  Y ( s ) / U ( s )  G 1 ( s ) G 2 ( s )  G 3 ( s )
In-Class Exercise: Block
Diagram Algebra
E (s)
Kc
 Ds
1/ I s
+
+
C (s)
Solution of In-Class Exercise
For summation
:
 1 
E ( s ) K c  E ( s ) K c D s 
  C (s)
 I s 
Rearrangin
g

D 
 K c 1 

E (s)
I 

C (s)
What if the Process Model is
Nonlinear
• Before transforming to the deviation
variables, linearize the nonlinear equation.
• Transform to the deviation variables.
• Apply Laplace transform to each term in the
equation.
• Collect terms and form the desired transfer
functions.
• Or instead, use Equation 5.7.3.
Transfer Function for a Nonlinear
Process
C onsider a nonlinear first-order O D E
dy
 f ( y, u)
dt
f ( y , u )
E quation 5.7.3 : G ( s ) 
Y (s)
U (s)
u

s
y ,u
f ( y , u )
y
y ,u
Advantages of Equation 5.7.3
• Equation 5.7.3 was derived based on
linearing the nonlinear ODE, applying
deviation variables, applying Laplace
transforms and solving for Y(s)/U(s).
• Therefore, Equation 5.7.3 is much easier to
use than deriving the transfer function for
both linear and nonlinear first-order ODEs.
Application of Equation 5.7.3
C onsider:
dL
dt
Initially,
dL
dt
 Fin  k
 0, L  L 0
From the O D E , Fin (0)  k
 f ( L , Fin )
 Fin
G (s) 
L  f ( L , Fin )
 1;
L 0 , Fin ( 0 )
1
s  1 /(2 L 0 )
L0
 f ( L , Fin )
L

L 0 , Fin ( 0 )
1
2 L0
based on the initial conditions
In-Class Exercise
D eterm ine the transfer function
based on the initial conditions for
dy
dt
 auy  y
2
at t  0, y  y 0 , u  u 0
State Space Models
• State space models are a system of linear
ODEs that approximate a system of
nonlinear ODEs at an operating point.
• Similar to Equation 5.7.3, state space
models can be conveniently generated using
the definitions of the terms in the coefficient
matrices and the nonlinear ODEs.
State Space Models
dx
 Ax + Bu
y = Cx
dt
A , B , C are m atrices; x, u , y are vectors
x  state variables; u  input variables
y  m easured or output variables
a ij 
fi
x j
bij 
x,u
fi
u j
x,u
Poles of a State Space Model
T he poles of a system are equal to
the values of s that satisfy
det  s I  A   0
based on a local linearization.
Overview
• The transfer function of a process shows the
characteristics of its dynamic behavior
assuming a linear representation of the
process.