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Digital Control:Summary#4

02/07/2013

Fig.1.Relationship between continuous (differential equations),discrete

(difference equations) and frequency domain representations

Fig.2.Top:block diagram for a digital control system,middle:an ideal

sampler,and bottom zero order hold.

I.MODELING OF DIGITAL CONTROL SYSTEMS

The block diagram of a digital control system is shown in

ﬁgure 2,where

DAC converts numbers calculated by the micro controller

into analog signals

The analog subsystem includes the plant,ampliﬁers,

actuators,etc.

The output of the analog system is measured and con-

verted into a number fed back to the microcontroller.

A.ADC model

We assume that there is no delay and the sampling is

uniform,this implies a ﬁxed sampling rate.These assumptions

are reasonable and accepted for most engineering applications.

The ideal sampler of period T is just a switch.Ideal sampler

implies that the switch closure time is much smaller that

the sampling period.Ideal sampling is also called impulse

sampling because it can modeled as an impulse train as

Fig.3.Top:ZOH,FOH and SOH continuous approximations of digital

signals,and bottom approximation of a rectangular pulse,by a positive step

followed by a negative step.

follows:

T

(t) =

1

∑

k=0

(t kT) (1)

where (t kT) is a delayed impulse.The sampled signal

becomes

f(kT) = f(t)

T

(t)) =

1

∑

k=0

f(t)(t kT) (2)

where f(kT) represents f(t) at sampling time kT.

B.DAC model

Continuous signal reconstruction is achieved by the DAC.

We want to ﬁnd an input-output relationship for the DAC.

The zero-order hold (ZOH) is the mathematical model that

allows modeling the conventional digital-to-analog converter

(DAC).The ZOH reconstructs the analog signal by holding

each sample value for one sampling period:

fu(k)g )u(t) = u(k) for kT t (k +1)T (3)

Zero order hold is the most widely used technique,but ﬁrst

order hold and second order hold are also used.

First order hold uses straight line as shown in ﬁgure 3.

Second order hold uses a parabola as shown in ﬁgure 3.

The transfer function of a zero order hold can be obtained

noting that a rectangular pulse can be represented by a positive

step followed by a negative step (ﬁgure 3–bottom).We already

know that

Lfu(t)g =

1

s

(4)

Using Laplace transform properties,we can write

Lfu(t T)g =

e

sT

s

(5)

Digital Control,spring 2013 L4

Fig.4.

Therefore

G

zoh

(s) =

1 e

sT

s

(6)

And the corresponding frequency response is given by

G

zoh

(j!) =

1 e

j!T

j!

(7)

C.DAC,analog subsystem and ADC

Cascading the DAC,and analog system and ADC appears

frequently in digital controls systems.The goal here is to

derive discrete time transfer function of the entire system.

D.Example

Consider the circuit in ﬁgure 4,the goal is to ﬁnd the digital

transfer function of the system.It is possible to write

G

za

(s) = G(s)G

zoh

(s) = (1 e

sT

)

G(s)

s

(8)

and

g

za

(t) = g(t) g

zoh

(t) (9)

from which it is possible to write

G(z)

za

= (1 z

1

)Z

{

G(s)

s

}

(10)

E.Example

Find G

za

knowing the analog system is given by the circuit

of ﬁgure 4.We have

G

zas

= (1 z

1

)Z

{

1

s +

1

}

(11)

with = R=L.From the table it is possible to write

G

zas

=

z 1

z

z

z e

T=

(12)

G

zas

=

z 1

z e

T

(13)

Fig.5.A unity feedback closed loop system.G

za

stands for the discrete

time transfer function of the ZOH and the analog subsystem

II.CLOSED LOOP TRANSFER FUNCTION AND

CHARACTERISTIC POLYNOMIAL

The characteristics and properties of the closed loop system

play an important role in control studies.Consider the unity

feedback system of ﬁgure 5.The input is R(z) and the output

is denoted by Y (z),C(z) is the digital controller.The goal is

to derive the closed loop transfer function.The error signal is

given by

E(z) = R(z) Y (z) (14)

We also have

Y (z) = C(z)G

za

(z)E(z) (15)

By deﬁnition,the open loop system is C(z)G

za

(z).Substitut-

ing the error by its value in equation (14),we get

Y (z) = C(z)G

za

(z) (R(z) Y (z)) (16)

from which the transfer function is derived

G

cl

(z) =

C(z)G

za

(z)

1 +C(z)G

za

(z)

(17)

The closed loop characteristic equation is given by

1 +C(z)G

za

(z) = 0 (18)

The roots of this equations are called the poles of the closed

loop system.

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