class 05 “block diagram, error and pid...

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 combination

blocks in cascade

the output X(s) of the first block is

the input of the second block.


)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


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


)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


)s(G1)s(G2

thus:

)s(G)s(G 21 ⋅

blocks in cascade


)s(G1 )s(G2

)s(G)s(G 21 ⋅

and hence:

blocks in cascade


summing point

)s(C)s(B)s(A)s(Y ++=


error detector

)s(B)s(R)s(E −=


Error

feedback

feedback

information from the output Y(s) is reintroduced in the

input, after comparing with the reference signal R(s).


unit feedback

)s(Y)s(R)s(E −=)s(E)s(G)s(Y ⋅=


[ ]Y(s) G(s) R(s) Y(s)= ⋅ −

E(s)

unit feedback


Y(s) G(s) R(s) G(s)Y(s)= ⋅ −

unit feedback


[ ]Y(s) 1 G(s) R(s)G(s)⋅ + =

unit feedback


)s(G1

)s(G

)s(R

)s(Y

+=

unit feedback


)s(H)s(G1

)s(G

)s(R

)s(Y

⋅+=

non unit feedback


)s(H)s(G1

)s(G

⋅+

)s(H)s(G1

)s(G

)s(R

)s(Y

⋅+=

non unit feedback


the unit feedback

Note that

corresponds to

non unit feedback

with

1)s(H =


)s(H)s(G1

)s(G

⋅+

1)s(H =

unit feedback


)s(G1

)s(G

+

)s(G1

)s(G

)s(R

)s(Y

+=

unit feedback


Example 1:

)4s(s

5)s(G

+=

5s4s

5

)4s(s

51

)4s(s

5

)s(G1

)s(G

)s(R

)s(Y2 ++

=

++

+=+

=


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

+=


Example 2:

)4s(s

5)s(G

+=

)5s12s7s(

)3s(5

)s(R

)s(Y23 +++

+=

)3s(

1)s(H

+=


feedback with tachometers

2K

1K

s

1

)FJs(

K

+

tachometer

position sensor


for servo systems with velocity feedback


2K

1K

s

1

)FJs(

K

+

)FJs(

KK1

)FJs(

K

)s(G1

++

+=

tachometer

position sensor

)s(G



2K

s

1)(sG

position sensor

)FJs(

KK1

)FJs(

K

)s(G1

++

+=)s(G



2K

s

1)(sG

position sensor

)FJs(

KK1

)FJs(

K

)s(G1

++

+=



2K

s

1)(sG

11 KKFJs

K

)FJs(

KK1

)FJs(

K

)s(G++

=

++

+=

position sensor



2K

s

1)(sG

1KKFJs

K)s(G

++=

position sensor



2K

s

1

1KKFJs

K

++

position sensor

1KKFJs

K)s(G

++=



2K

s

1

1KKFJs

K

++

2

1

1

Ks

1

)KKF(Js

K1

s

1

)KKF(Js

K

)s(R

)s(Y

⋅⋅++

+

⋅++=

position sensor



2K

s

1

1KKFJs

K

++

sensor de posição

21

2 KKs)KKF(Js

K

)s(R

)s(Y

+++=

2K

s

1

1KKFJs

K

++



21

2 KKs)KKF(Js

K

+++21

2 KKs)KKF(Js

K

+++



feedback with tachometers(another approach)

2K

1K

s

1

)FJs(

K

+

tachometer

position sensor



2K

sK1

s

1

)FJs(

K

+

tachometer

position sensor



2K

sK1

s

1

)FJs(

K

+

21 KsK +



21 KsK +

s

1

)FJs(

K

+

s)FJs(

)KsK(K1

s

1

)FJs(

K

)s(R

)s(Y

21

+++

⋅+=



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



21

2 KKs)KKF(Js

K

+++21

2 KKs)KKF(Js

K

+++

21

2 KKs)KKF(Js

K

)s(R

)s(Y

+++=


which is the same result obtained earlier, with the first approach


Error

For an open loop system (that is, with no feedback)

the definition of error is:

E(s) = R(s) – Y(s)


Errorunit feedback


E(s) R(s) Y(s)= −here, the

error is also:

E(s) R(s) B(s)= −

non unit feedback

Error

the errorbecomes:


E(s) R(s) Y(s)H(s)= −

that is:

error:

B(s)


E(s) R(s) G(s)E(s)H(s)= −

Y(s) G(s) E(s)= ⋅Y(s)

but:

thus:


[ ]E(s) 1 G(s)H(s) R(s)+ =

hence:


[ ]R(s)

E(s)1 G(s)H(s)

=+

and the expression for the ‘error’ is:


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)


Initial Value Theorem:

(FVT))s(Xslim)(x0s

⋅=∞→

)s(Xslim)0(xs

⋅=∞→

Final Value Theorem:

(IVT)


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

−=+−=

+−⋅=

⋅=∞

→

→

→


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:


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)


Steady state error:

[ ])s(H)s(G1

)s(Rslime

0sss

+⋅=

→


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

+=


hence,

[ ] )s(R)s(H)s(G1

)s(G)s(Y ⋅

+=

[ ] )s(R)s(H)s(G1

)s(Gslim

)s(Yslimy

0s

0sss

⋅+

⋅=

=⋅=

→

→



[ ])s(H)s(G1

)s(R)s(Gslim)s(Yslimy

0s0sss +

=⋅=→→

[ ])s(H)s(G1

)s(R)s(Gslimy

0sss +

=→

and then,



)1s(

2)s(G

+=

Example 5:


)1s(

2

+

)1s(

2)s(G

+=

r(t) = unit step

)t(u)t(r 1=input r(t):

Example 5:


(continued)

)1s(

2

+

s

1)s(R =

r(t) = unit step



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



)1s(

2

+



)1s(

2

+

yss = 2/3



)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



)1s(

2

+

ess = 1/3



)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”


)1s(

2)s(G

+=

Example 6:


(continued)

)1s(

2

+

)1s(

2)s(G

+=

Example 6:

r(t) = unit step

(again)



(continued)

Example 6:

)1s(

2

+

)1s(

K2

+


(continued)

)1s(

2

+

Example 6:

s

1)s(R =

r(t) = unit step



(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

+


(continued)

Example 6:

1)1K2(

K2yss ≅

+=

)1s(

2

+

∞→→ Kwhen1yss

output in steady state yss

or

if K is large enough


(continued)

Example 6:

)1s(

2

+


(continued)

Example 6:

yss ≈ 1

)1s(

2

+


(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

+


(continued)

Example 6:

0)1K2(

1ess ≅

+=

)1s(

2

+

∞→→ Kwhen0ess

error in steady state ess

or

if K is large enough


(continued)

Example 6:

ess → 0

)1s(

2

+


(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


(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”


‘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


(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


(continued)

Using K1 + K2 s, we get a

“proportional derivative controller”

or a “type PD controller”

There are several types of controllers


‘PD’ controller

Using K1 + K2/s, we get a

“proportional integral controller”

or a “type PI controller”


‘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’


‘PID’ controller

Thank you!Felippe de Souza

[email protected]

Departamento de Engenharia Eletromecânica

class 05 “block diagram, error and pid...

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