class 05 “block diagram, error and pid...
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Class 05
“Block Diagram, error
and PID controllers”
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
Single block
Black box
outputinput
Black box
outputinput
)s(G
)s(X)s(G)s(Y ⋅=or:
Transfer Function:
)s(X
)s(Y)s(G =
OUTPUT = T. F. X INPUT
that is,
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
Block combination
blocks in cascade
the output X(s) of the first block is
the input of the second block.
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
)s(G1)s(G2
hence,
)s(R
)s(X)s(G1 =
)s(X
)s(Y)s(G 2 =
)s(G1)s(G2
blocks in cascade
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
that is, )s(R)s(G)s(X 1 ⋅=
)s(X)s(G)s(Y 2 ⋅=
)s(G1)s(G2
OUTPUT = T.F. X INPUT
OUTPUT = T.F. X INPUT
for the 1º block
for the 2º block
blocks in cascade
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
)s(G1)s(G2
hence:
or,
)s(G)s(G)s(R
)s(Y21 ⋅=
)s(R)s(G)s(G)s(Y 21 ⋅⋅=
blocks in cascade
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
)s(G1)s(G2
thus:
)s(G)s(G 21 ⋅
blocks in cascade
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
)s(G1 )s(G2
)s(G)s(G 21 ⋅
and hence:
blocks in cascade
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
summing point
)s(C)s(B)s(A)s(Y ++=
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
error detector
)s(B)s(R)s(E −=
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
Error
feedback
feedback
information from the output Y(s) is reintroduced in the
input, after comparing with the reference signal R(s).
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
unit feedback
)s(Y)s(R)s(E −=)s(E)s(G)s(Y ⋅=
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
[ ]Y(s) G(s) R(s) Y(s)= ⋅ −
E(s)
unit feedback
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
Y(s) G(s) R(s) G(s)Y(s)= ⋅ −
unit feedback
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
[ ]Y(s) 1 G(s) R(s)G(s)⋅ + =
unit feedback
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
)s(G1
)s(G
)s(R
)s(Y
+=
unit feedback
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
)s(H)s(G1
)s(G
)s(R
)s(Y
⋅+=
non unit feedback
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
)s(H)s(G1
)s(G
⋅+
)s(H)s(G1
)s(G
)s(R
)s(Y
⋅+=
non unit feedback
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
the unit feedback
Note that
corresponds to
non unit feedback
with
1)s(H =
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
)s(H)s(G1
)s(G
⋅+
1)s(H =
unit feedback
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
)s(G1
)s(G
+
)s(G1
)s(G
)s(R
)s(Y
+=
unit feedback
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
Example 1:
)4s(s
5)s(G
+=
5s4s
5
)4s(s
51
)4s(s
5
)s(G1
)s(G
)s(R
)s(Y2 ++
=
++
+=+
=
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
Example 2:
)4s(s
5)s(G
+=
)3s(
1
)4s(s
51
)4s(s
5
G(s)H(s)1
G(s)
)s(R
)s(Y
+⋅
++
+=+
=
)3s(
1)s(H
+=
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
Example 2:
)4s(s
5)s(G
+=
)5s12s7s(
)3s(5
)s(R
)s(Y23 +++
+=
)3s(
1)s(H
+=
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
feedback with tachometers
2K
1K
s
1
)FJs(
K
+
tachometer
position sensor
feedback with tachometers
for servo systems with velocity feedback
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
2K
1K
s
1
)FJs(
K
+
)FJs(
KK1
)FJs(
K
)s(G1
++
+=
tachometer
position sensor
)s(G
feedback with tachometers
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
2K
s
1)(sG
position sensor
)FJs(
KK1
)FJs(
K
)s(G1
++
+=)s(G
feedback with tachometers
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
2K
s
1)(sG
position sensor
)FJs(
KK1
)FJs(
K
)s(G1
++
+=
feedback with tachometers
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
2K
s
1)(sG
11 KKFJs
K
)FJs(
KK1
)FJs(
K
)s(G++
=
++
+=
position sensor
feedback with tachometers
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
2K
s
1)(sG
1KKFJs
K)s(G
++=
position sensor
feedback with tachometers
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
2K
s
1
1KKFJs
K
++
position sensor
1KKFJs
K)s(G
++=
feedback with tachometers
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
2K
s
1
1KKFJs
K
++
2
1
1
Ks
1
)KKF(Js
K1
s
1
)KKF(Js
K
)s(R
)s(Y
⋅⋅++
+
⋅++=
position sensor
feedback with tachometers
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
2K
s
1
1KKFJs
K
++
sensor de posição
21
2 KKs)KKF(Js
K
)s(R
)s(Y
+++=
2K
s
1
1KKFJs
K
++
feedback with tachometers
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
21
2 KKs)KKF(Js
K
+++21
2 KKs)KKF(Js
K
+++
feedback with tachometers
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
feedback with tachometers(another approach)
2K
1K
s
1
)FJs(
K
+
tachometer
position sensor
feedback with tachometers
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
2K
sK1
s
1
)FJs(
K
+
tachometer
position sensor
feedback with tachometers
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
2K
sK1
s
1
)FJs(
K
+
21 KsK +
feedback with tachometers
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
21 KsK +
s
1
)FJs(
K
+
s)FJs(
)KsK(K1
s
1
)FJs(
K
)s(R
)s(Y
21
+++
⋅+=
feedback with tachometers
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
21 KsK +
s
1
)FJs(
K
+
21
2 KKs)KKF(Js
K
)s(R
)s(Y
+++=
which is the same result obtained earlier, with the first approach
feedback with tachometers
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
21
2 KKs)KKF(Js
K
+++21
2 KKs)KKF(Js
K
+++
21
2 KKs)KKF(Js
K
)s(R
)s(Y
+++=
feedback with tachometers
which is the same result obtained earlier, with the first approach
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
Error
Error
For an open loop system (that is, with no feedback)
the definition of error is:
E(s) = R(s) – Y(s)
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
Errorunit feedback
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
E(s) R(s) Y(s)= −here, the
error is also:
E(s) R(s) B(s)= −
non unit feedback
Error
the errorbecomes:
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
E(s) R(s) Y(s)H(s)= −
that is:
error:
B(s)
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
E(s) R(s) G(s)E(s)H(s)= −
Y(s) G(s) E(s)= ⋅Y(s)
but:
thus:
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
[ ]E(s) 1 G(s)H(s) R(s)+ =
hence:
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
[ ]R(s)
E(s)1 G(s)H(s)
=+
and the expression for the ‘error’ is:
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
Steady state error
Steady state output
Initial Value Theorem:
(FVT))s(Xslim)t(xlim)(x0st
⋅==∞→∞→
)s(Xslim)t(xlim)0(xs0t
⋅==∞→→
Final Value Theorem:
(IVT)
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
Initial Value Theorem:
(FVT))s(Xslim)(x0s
⋅=∞→
)s(Xslim)0(xs
⋅=∞→
Final Value Theorem:
(IVT)
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
Example 3:
)5s(s
2s3)s(Y
+−=
3
)5s(
2s3lim
)5s(s
2s3slim
)s(Yslim)0(y
s
s
s
=+−=
+−⋅=
⋅=
∞→
∞→
∞→
IVT FVT
5/2
)5s(
2s3lim
)5s(s
2s3slim
)s(Yslim)(y
0s
0s
0s
−=+−=
+−⋅=
⋅=∞
→
→
→
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
3s4s
s2)s(E
2 +−=
2
3s4s
s2lim
)s(Eslim)0(e
2
2
s
s
=+−
=
⋅=
∞→
∞→
IVT
0
3s4s
s2lim
)s(Eslim)(e
2
2
0s
0s
=+−
=
⋅=∞
→
→
FVT
Example 4:
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
Steady state error:
[ ]sss 0 s 0
s R(s)e lim s E(s) lim
1 G(s)H(s)→ →
⋅= ⋅ =+
sst s 0
e lim e(t) lim s E(s)→∞ →
= = ⋅
sst
e lim e(t)→∞
=
(FVT)
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
Steady state error:
[ ])s(H)s(G1
)s(Rslime
0sss
+⋅=
→
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
Steady state output:
(FVT)
)t(ylimyt
ss ∞→=
)s(Yslim)t(ylimy0st
ss ⋅==→∞→
However,
we know that)s(H)s(G1
)s(G
)s(R
)s(Y
+=
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
hence,
[ ] )s(R)s(H)s(G1
)s(G)s(Y ⋅
+=
[ ] )s(R)s(H)s(G1
)s(Gslim
)s(Yslimy
0s
0sss
⋅+
⋅=
=⋅=
→
→
Steady state output:
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
[ ])s(H)s(G1
)s(R)s(Gslim)s(Yslimy
0s0sss +
=⋅=→→
[ ])s(H)s(G1
)s(R)s(Gslimy
0sss +
=→
and then,
Steady state output:
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
)1s(
2)s(G
+=
Example 5:
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
)1s(
2
+
)1s(
2)s(G
+=
r(t) = unit step
)t(u)t(r 1=input r(t):
Example 5:
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
(continued)
)1s(
2
+
s
1)s(R =
r(t) = unit step
)t(u)t(r 1=input r(t):
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
Example 5:(continued)
)1s(
2
+
[ ]
)3s(s
2
s
1
)1s(
21
)1s(
2
)s(R)s(H)s(G1
)s(G)s(Y
+=
⋅
++
+=
⋅+
=
s
1)s(R =
3/2
)3s(s
2slim
)s(Yslimy
0s
0sss
=+
⋅=
=⋅=
→
→
output y(t)
output in
steady state yss
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
Example 5:(continued)
)1s(
2
+
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
Example 5:(continued)
)1s(
2
+
yss = 2/3
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
Example 5:(continued)
)1s(
2
+
[ ]
)3s(s
)1s(
)1s(
21
s/1
)s(H)s(G1
)s(R)s(E
++=
++
=
+=
s
1)s(R =
3/1
)3s(s
)1s(slim
)s(Eslime
0s
0sss
=++⋅=
=⋅=
→
→
error e(t)
error in
steady state ess
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
Example 5:(continued)
)1s(
2
+
ess = 1/3
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
Example 5:(continued)
)1s(
2)s(G
+=
Example 6:
the same as in the
previous example
But now we have K, namely,
a “proportional controller”
or the “type P controller”
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
)1s(
2)s(G
+=
Example 6:
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
(continued)
)1s(
2
+
)1s(
2)s(G
+=
Example 6:
r(t) = unit step
(again)
)t(u)t(r 1=input r(t):
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
(continued)
Example 6:
)1s(
2
+
)1s(
K2
+
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
(continued)
)1s(
2
+
Example 6:
s
1)s(R =
r(t) = unit step
)t(u)t(r 1=input r(t):
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
(continued)
Example 6:
[ ]
)12(
2
1
)1(
21
)1(
2
)()()(1
)()(
++=
⋅
++
+=
⋅+
=
Kss
K
s
s
K
s
K
sRsHsG
sGsY
s
1)s(R =
)1K2(
K2
)1K2s(s
K2slim
)s(Yslimy
0s
0sss
+=
++⋅=
=⋅=
→
→
output y(t)
output in
steady state yss
)1s(
2
+
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
(continued)
Example 6:
1)1K2(
K2yss ≅
+=
)1s(
2
+
∞→→ Kwhen1yss
output in steady state yss
or
if K is large enough
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
(continued)
Example 6:
)1s(
2
+
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
(continued)
Example 6:
yss ≈ 1
)1s(
2
+
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
(continued)
Example 6:
[ ]
)1K2s(s
)1s(
)1s(
K21
s/1
)s(H)s(G1
)s(R)s(E
+++=
++
=
+=
s
1)s(R =
)1K2(
1
)1K2s(s
)1s(slim
)s(Eslime
0s
0sss
+=
+++⋅=
=⋅=
→
→
error e(t)error in
steady state ess
)1s(
2
+
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
(continued)
Example 6:
0)1K2(
1ess ≅
+=
)1s(
2
+
∞→→ Kwhen0ess
error in steady state ess
or
if K is large enough
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
(continued)
Example 6:
ess → 0
)1s(
2
+
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
(continued)
Example 6:
100
101,0
)1K2(
1ess =<
+=
)1s(
2
+
which is the value that we should adjust K such that the
steady state error
ess < 0,01
)1K2(100 +<
2/99K > 5,49K >
we should choose K > 49,5
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
(continued)
)1s(
2)s(G
+=
Example 7:
the same as in the
previous example
Using K/s, we get a
“integral controller” or a
“type I controller”
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
‘I’ type
controller
Example 7:
[ ]
)K2ss(
)1s(
)1s(s
K21
s/1
)s(H)s(G1
)s(R)s(E
2 +++=
++
=
+=
s
1)s(R =
0
)K2ss(
)1s(slim
)s(Eslime
20s
0sss
=++
+⋅=
=⋅=
→
→
error e(t)
error in
steady state ess
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
(continued)
Example 7:
thus, now it is possible, for example, to adjust K such that
the steady state error is
ess = 0
we should choose K > 0
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
(continued)
Using K1 + K2 s, we get a
“proportional derivative controller”
or a “type PD controller”
There are several types of controllers
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
‘PD’ controller
Using K1 + K2/s, we get a
“proportional integral controller”
or a “type PI controller”
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
‘PI’ controller
There are several types of controllers
Using K1 + K2/s + K3s, we get a
“proportional integral derivative controller”
or a “type PID controller”
The most general case is a ‘PID controller’
Block Diagrams & error______________________________________________________________________________________________________________________________________________________________________________________
‘PID’ controller
Thank you!Felippe de Souza
felippe@ubi.pt
Departamento de Engenharia Eletromecânica
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