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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 inputoutput relationship for the DAC.
The zeroorder hold (ZOH) is the mathematical model that
allows modeling the conventional digitaltoanalog 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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