Automatic Control System

V.

Performance of steady-state

plantcontroller

Terms of feedback control

reference input element

reference signal

comparing elementor error detector

error signal

compensator or control task

action signal

feedback signal

actuator

transmitter

manipulated variable

disturbance variable

controlled variable

block model of the plant

Working points in steady-state

U

YM Y

UMIn steady-state the value of the action signal U,manipulated UM, and controlled Y variables,and the feedback signal YM are constant.

W

Y

UM

U

YM

In steady-state the output of a loop’selement is the input of the next block.The steady-state characteristics of the elements must be fitted each other. The disturbance w variable’s deviationof the average value changes the plant’s steady-state characteristic curve.It’s important: A new steady-state can be achieved by the system.

W0

Choosing steady-state characteristics of controller

The steady-state characteristic of the plant, transmitter and the actuator is given by the technologies. We can

choose only the steady-state characteristic of the controller.

If higher action signal belongs to higher feedback signal then the characteristic is direct.If lower action signal belongs to higher feedback signal then the characteristic is reverse.

The dynamic behavior of the controller eliminates the steady-state error..

Y

UM

U

YM

W0

In this case there isn’t a new steady-state value of system.

Choosing steady-state characteristics of controller

A bad choice of controller steady-state characteristics also causes an unstable control loop.

Y

UM

U

YM

Sometimes the technologies requirements need a reverse characteristics of the actuator. Sometimes the steady-state characteristics of technologies is concave and not convex.

W0

In this case there is a new steady-state value of system.

Transfer functions of feedback loop

)s(yM

)s(uGP(s)

)s(e )s(d )s(yGR(s)

GW(s)

GA(s)GC(s)

GT(s)

)s(r

TPAC

PACyr GGGG1

GGG

r

y)s(G

TPACer GGGG1

1

r

e)s(G

)s(w

TPAC

PWyw GGGG1

GG

w

y)s(G

TPAC

TPWew GGGG1

GGG1

w

e)s(G

System type of feedback loop

n

1kk

m

1jj

i0

TPAC0

)sT1(

)s1(

s

KGGGG)s(G

System type is the number of the pure integral lag in the loop. The possible values are 0, 1, 2. K0 is the loop gain.

Opened-loop transfer function, Bode form.

)s(yM

)s(uGP(s)

)s(e )s(d )s(yGR(s)

GW(s)

GA(s)GC(s)

GT(s)

)s(r

)s(w

A ship wants to turn to the port. The continuous movement of the steering wheel turns the ship. The surface of the ship against the river backwash also continuously increase. How much is the deflection error if r(t)=0.2*t*1(t) is the changes of the reference signal and w(t)=0.1*t*1(t) is the changes of the disturbance variable. The domains are: Y = ± 80 [°], R = 4 – 20 [mA].

)s(e

)s(w

)s(y)s(r

s

s

1.01

15

s1

1

s2.01

1

s5

1

5.0

Examplev

s5.01

2.1

The steady-state error of closed loop control

er ewt s 0lim e(t) lim s{G (s)r(s) G (s)w(s)}

s

1

s2.011

s5.01

s11

s5.012.1

s1.01s1

51

)1(s2.01

1)1(5.0

ss

2.0

s2.011

s5.01

s11

s5.012.1

s1.01s1

51

1slim

20s

)s(e

)s(w

)s(y)s(r

s

s

1.01

15

s1

1

s2.01

1

s5

1

5.0

s5.01

2.1

0.2 1 0.1 0.5 0.6If s 0, then 0.1mA

5 1.2 5 1.2 6 6 60.5 0.5

The steady-state error of closed loop control

x0,1 mA

20 4 mA 80 80

The size of the error might be appropriate, but we do not know anything about the quality characteristics.

0,1x 160 1

16

)s(e

)s(w

)s(y)s(r

s

s

1.01

15

s1

1

s2.01

1

s5

1

5.0

s5.01

2.1

Setpoint tracking

The setpoint tracking depends on:• Order of the feedback system.• The loop‘s gain (K0).• The form of the reference signal

The setpoint tracking means that the controlled variable (y(s)) is able to follow the prescribed value by the changes of the reference signal (r(s)) after the transient has died.We assume that the disturbance variable (W0(s)) equals the designed value (w(s)=0).We can examine this with the error transfer function:

in

1kk

m

1jj

i0

er

)sT1(

)s1(

s

K1

1)s(G

Setpoint tracking

1(t)

t 1(t)

Standard signal

Order of the feedback system

0

1

K

0

1

K 1t

t

0 21

0

0

0

0

1

K2t *1(t)

t

Disturbance suppression

The disturbance suppression depends on:• Order of the feedback system.• The loop‘s gain (K0).• The form of the reference signal.• The attack point of the disturbance variable.

The disturbance suppression means that the controlled variable (y(s)) is able to hold the prescribed value by the changes of the disturbance variable (w(s)) after the transient has died. We assume that the reference signal (R0(s)) equals the working point value (r(s)=0).We can examine this with the disturbance-error transfer function:

in

1kk

m

1jj

i0

TPWew

)sT1(

)s1(

sK

1

)s(G)s(G)s(G)s(G

Disturbance suppression

1(t)

t 1(t)

Standard signal

Order of the feedback system

0

1

K

0

1

K 1t

t

0 21

0

0

0

0

1

K

Entering point of disturbance

0 010 21

0

1

K

0

0

1

K

0

1

K

2t *1(t)

t