closed-loop transfer functions 1.introduction 2.stirred tank heating system 3.closed-loop block...
TRANSCRIPT
Closed-Loop Transfer Functions
1. Introduction
2. Stirred tank heating system
3. Closed-loop block diagrams
4. Closed-loop transfer functions
5. Simulink example
Introduction
Block diagrams» Convenient tool to represent closed-loop systems
» Also used to represent control systems in Simulink
Closed-loop transfer functions» Transfer function between any two signals in a
closed-loop system
» Usually involve setpoint or disturbance as the closed-loop input and the controlled output as the closed-loop output
» Conveniently derived from block diagram
» Can be derived automatically in Simulink
» Used to analyze closed-loop stability and compute closed-loop responses
Stirred Tank Blending System
Control objective» Drive outlet composition (x) to
setpoint (xsp) by manipulating pure stream flow rate (w2) despite disturbances in flow rate (w1) and composition (x1) of other feed stream
Control system» Measure x with composition
analyzer (AT)» Perform calculation with
composition controller (AC)» Convert controller output to
pneumatic signal with current-pressure converter (I/P) to drive valve
Blending Process Model
Mass balances for constant volume
Linearized model
Transfer function model
),,()()()(
0
212211
2211
2121
wxxfV
xxwxxw
dt
dxwxxwxw
dt
Vxdwwwwww
V
wxxwxw
dt
dx
'2
'11
' )1('
)(1
)(1
)(1
)1()(
1)( '
22'
11'
2'1
1' sWs
KsX
s
KsW
swV
wxsX
swV
wwsX
Control System Components
Composition analyzer – assume first-order dynamics
Controller – assume PI controller
I/P converter – assume negligible dynamics
Control System Components cont.
Control valve – assume first-order dynamics
Entire blending system
Closed-Loop Block Diagrams
Gp(s) – process transfer function
Gd(s) – disturbance transfer function
Gv(s) – valve transfer function Gc(s) – controller transfer
function Gm(s) – measurement transfer
function Km – measurement gain
Y(s) – controlled output U(s) – manipulated input D(s) – disturbance input P(s) – controller output E(s) – error signal Ysp(s) – setpoint Ym(s) – measurement
Transfer Function for Setpoint Changes
mpvc
pvcm
sp
mspmcvpcvp
mspmmsp
cvpvppudu
GGGG
GGGK
Y
Y
YGYKGGGEGGGY
YGYKYYE
EGGGPGGUGYYYY
1
~
Transfer Function for Disturbance Changes
mpvc
d
dmcvpcvp
mmmsp
dcvpdvpdpdu
GGGG
G
D
Y
DGYGGGGEGGGY
YGYYYE
DGEGGGDGPGGDGUGYYY
1
~
Simultaneous Changes
Principle of superposition
Open-loop transfer function» Obtained by multiplying all transfer functions
in feedback loop
DGGGG
GY
GGGG
GGGKY
mpvc
dsp
mpvc
pvcm
11
DG
GY
G
GGGKY
GGGGG
OL
dsp
OL
pvcm
mpvcOL
11
General Method
Closed-loop transfer function
» Z = any variable in feedback system» Zi = any input variable in feedback system Z and Zi
» f = product of all transfer functions between Z and Zi
» e = product of all transfer functions in feedback loop
Setpoint change
Disturbance change
e
f
iZ
Z
1
OLmpvcepvcmf GGGGGGGGK
OLmpvcedf GGGGGG
Closed-Loop Transfer Function Example
Simulink Example
>> gp=tf([6.37],[5 1]);
>> kv=0.0103;
>> kip=0.12;
>> km=50;
>> gc=tf([2.5 5],[0.5 0]);
>> gcl=gp/(1+gc*kv*gp*km)
Disturbance transfer function:
15.93 s^2 + 3.185 s
-----------------------------------
12.5 s^3 + 46.01 s^2 + 90.72 s + 16.4
Tank
6.37
5s+1Setpoint
0
PID Controller
PID
Level
y
Kv
0.0103
Km1
50
Km
50
Kip
0.12
Inlet flow
0.05
Add1Add