Automatic Control SystemII.

Block diagram model

Modelling dynamical systems

Engineers use models which are based upon mathematical relationships between two variables. We can define the mathematical equations:•Measuring the responses of the built process (black model)•Using the basic physical principles (grey model). In order to simplification of mathematical model the small effects are neglected and idealised relationships are assumed.

Developing a new technology or a new construction nowadays it’s very helpful applying computer aided simulation technique.This technique is very cost effective, because one can create a model from the physical principles without building of process.

LTI (Linear Time Invariant) modelThe all physical system are non-linear and their parameters

change during a long time.

The engineers in practice use the superposition’s method.

x(t) y(t)

x(j) y(j)1 1

2 2

1 2 1 2

x (t) y (t)

x (t) y (t)

x (t) x (t) y (t) y (t)

First One defines the input and output signal range.

In this range if an arbitrary input signal energize the block and the superposition is satisfied and the error smaller than a specified error, than the block is linear.

The steady-state characteristics and the dynamic behavior

min

max min

WP1

X(t){dim} X {dim}100 X(t)%

X {dim} X {dim}

X(t) X x(t)

100

0

x(t)

y(t)

t

t

The steady-state characteristic. When the transient’s signals have died a new working point WP2 is defined in the steady-state characteristic.

The dynamic behavior is describe by differential equation or transfer function in frequency domain.

WP2

WP1

100

0

min

max min

WP1

Y(t){dim} Y {dim}100 Y(t)%

Y {dim} Y {dim}

Y(t) Y y(t)

Transfer function in frequency domain

Amplitude gain:

Phase shift:

( )( )( ) ( )

( ) jout

in

y jG j A e

x j

( ) 20lg ( )a dB A

Im ( )( )

Re ( )

G jarctg

G j

( ) ( )A G j

The graphical representation of transfer function

• The M-α curves: The amplitude gain M(ω) in the frequency domain. In the previous page M(ω) was signed, like A(ω) The phase shift α(ω) in the frequency domain. In the previous page α(ω) was signed, like φ(ω)!

• A Nyquist diagram: The transfer function G(jω) is shown on the complex plane.

• A Bode diagram: Based on the M-α curves. The frequency is in logaritmic scale and instead of A(ω) amplitude gain is:

• A Nichols diagram: The horizontal axis is φ(ω) phase shift and the vertical axis is The a(ω) dB.

( ) 20lg ( )a dB A

The basic transfer functionIn the time domain is the

differential equitation

y(t) Ax(t)

y(s) 1

x(s) 1 sT

dy(t)

T y(t) x(t)dt

y(s)A

x(s)

22

2

d y(t) dy(t)T 2 T y(t) x(t)

dt dt 2 2

y(s) 1

x(s) 1 s2 T s T

i

dy(t)T x(t)

dt

i

y(s) 1

x(s) sT

d

dx(t)y(t) T

dt d

y(s)sT

x(s)

y(t) 1(t )x(t ) sy(s)e

x(s)

In the frequency domain is the transfer function

Block representationActuating path of signals and variables

One input and one output block represents the context between the the output and input signals or variables in time or frequency domain

Summing junction

Take-off point (The same signal actuate both path)

G1 G2 G1G2

G1

G2

G1+G2

G1

G2

21

1

1 GG

G

P proportional

Step response Bode diagram

t

I Integral

Step response Bode diagram

D differential

Step response Bode diagram

The step response is an Dirac delta, which isn’t shown

PT1 first order system

Step response Bode diagram

PT2 second order system

Step response Bode diagram

PH delay

Step response Bode diagram

IT1 integral and first order in cascade

Step response Bode diagram

DT1 differential and first order in cascade

Step response Bode diagram

PI proportional and integral in parallel

Step response Bode diagram

PDT1

Step response Bode diagram

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

Block diagram manipulation

G1

G1 G2 G1G2

G2

G1+G2

G1

G2

21

1

1 GG

G

G1

G1

G1

G1 G1

1

1

G

G1

G1

G1

G1 G1

1

1

G

Block diagram reduction example

G5

G2

G7

G6

G1 G3

G4

)(sx )(sy

765317641532

7417531

1)(

)()(

GGGGGGGGGGGG

GGGGGGG

sx

sysG