Deadbeat Response Design _8th lecture

Assist. Prof. Dr. Khalaf S. Gaeid
Electrical Engineering Department
Tikrit University
gaeidkhalaf@gmail.com
+9647703057076
Deadbeat response design

1. Design of digital control systems with dead beat response
2. Deadbeat response design when the system poles and
zeros are inside the unit circle
3. Physical realizability of Dc(z) :
4. Deadbeat response design when some of the poles and
zeros are on or outside the unit circle
5.Sampled data control system with Dead beat response
6.Examples
Contents

1. Design of digital control systems with dead beat
response
In case of a continuous time controlled process, the output
c(t) is a function of time t and the dead beat response design,
based on cancellation of stable poles and zeros, may lead to
inter sampling ripples in the output.
The reason behind this is since the process zeros are
cancelled by controller poles, the continuous dynamics are
excited by the input and are not affected by feed back.

The strategy of designing dead beat response for a sampled
data system with the process plant transfer function Gh0Gp(z)
having at least one zero is not to cancel the zeros, whether
they are inside or outside the unit circle.
We must distinguish between the designs of deadbeat
response for a digital control system, where all the
components are subject to only digital data, and a sampled
data control system, where both continuous and discrete
components are present.
An all digital control system is shown in Figure 1.

The transfer function of the digital plant is given by
Let us assume that the transfer function of the cascade
digital controller is

Thus the open loop transfer function becomes
And we get the closed loop transfer function as
Thus for unit step input, the output comes out to be

Thus, the output c(k) represents a unit step response where k
starts from 1, that is one sample later.
In other words, c(k) reaches the desired steady state value 1,
in one sampling period with out any overshoot and stays
there for ever.
This type of response is known as dead beat response.
One should note that if Gp(z) was the result of sampling a
continuous data system, the Dc (z) does not guarantee that no
ripples occur between two sampling instants in constant
output c(t).

2. Deadbeat response design when the system poles and
zeros are inside the unit circle
Design criteria:
1. The system must have a zero steady state error at
sampling instants.
2. The time to reach final output must be finite and minimum.
3. The controller should be physically realizable, i.e., it should
be causal.
We can write from Figure 1,

Let us assume
where,
N: positive integer
A(z): polynomial in z - 1 with no zeros at z =1.
For unit step signal A(z) = 1 and N =1.
For unit ramp signal A(z) = Tz - 1 and N =1.
To achieve zero steady state error

Since A(z) does not contain any zero at z =1, necessary
condition for zero steady state error is that 1 - M(z) should
contain (1 - z - 1) N as a factor, i.e.,
Or,

where,
F(z) is a polynomial in z - 1 .
Q(z) is a polynomial in z.
Substituting M(z) in the expression of E(z), E(z) = A(z)F(z) . Since A(z)
and F(z) are both polynomials of z - 1, E(z) will have a finite number of
terms in the power series in the inverse power of z, i.e., the error will
go to zero in a finite number of sampling periods.
3. Physical realizability of Dc(z) :
Physical realizability condition on Dc(z) imposes constraints
on the form of M(z). Let

where, n and k are the excess poles over zeros of Gp(z) and
M(z) respectively. This implies
For Dc(z) to be realizable, i.e., excess of poles over
zeros for M(z) must be at least equal to excess of poles over
zeros for Gp(z).
Thus, if Gp(z) does not have poles or zeros outside the unit
circle, then M(z) should have the following forms.1. Step input :

2. Ramp input:
Try to prove the above as an exercise problem.
Example 1: Let us consider the earlier example where
When the input is a step function,

Hence the output sequence follows the input after one
sampling instant. Figure 2 shows the output response.
Figure 2: Deadbeat response of an all digital control system for unit step input
When the input is a ramp function

We can conclude from the above expression that the output
sequence follows the input after 2 sampling periods which is
shown in Figure 3.
Figure 3: Deadbeat response of an all digital control system for unit ramp
input

Example 2:
When the input is a step function,
Hence the output sequence follows the input after two
sampling instants.

There are some practical issues in deadbeat response
design.
Dead beat response design depends on the cancellation of
poles and zeros of the plant transfer function.
If the poles are on or outside the unit circle, imperfect
cancellation may lead to instability.
Thus, for practical constraints, one should not attempt to
cancel poles which are on or outside the unit circle.
4. Deadbeat response design when some of the poles and
zeros are on or outside the unit circle
Let the plant transfer function be

where, K and L are the number of zeros and poles on or
outside the unit circle and B(z) is a rational transfer function
in z -1 with poles and zeros inside the unit circle. This implies
Since we should not cancel poles or zeros which are on or
outside unit circle by the controller Dc(z), we have to choose
M(z) such that these get cancelled out.
Thus M(z) must contain the factors

and ( 1- M(z)) must contain the factors

P equals either the order of the poles of R(z) or the order of
poles of Gp(z) at z = 1 which ever is greater.
Truncation depends on the following.
1.The order of poles of M(z) and ( 1- M(z)) must be equal.
2.Total number of unknowns must be equal to the order of
M(z) so that they can be solved independently.
Example 1: Let us consider the plant transfer function as

For Unit Step Input:
Gp(z) has a zero at -2.8 and pole at z = 1. Therefore M(z) must
contain the term 1 + 2.8 z -1 and ( 1- M(z)) should contain 1 - z
-1 .
Gp(z) has two more poles than zeros. This implies
Since minimum order of M(z) is 3, we have 3 unknowns in
total. Combining the 2 equations

Putting the expressions of M(z) and ( 1- M(z)) in the controller
eqn.

One should note that poles on or outside the unit circle are
still present in the output expression.

c(kT) tracks the unit step perfectly after 3 sampling periods
(first term has a non unity coefficient). The output response
is shown in Figure 1.
Figure 1: Deadbeat Response of The System in Example 1
If Gp(z) did not have any poles or zeros on or outside the unit
circle it would take two sampling periods to track the step
input when Gp(z) has two more poles than zeros.

Example 2: Let us consider the plant transfer function as
For Unit Step Input:
Gp(z) has a zero at -2.8 and two poles at z = 1. The number of
poles exceeds the number of zeros by one.
M(z) must contain the term 1 + 2.8 z -1 and 1- M(z) should
contain (1- z -1 )2 . This implies

Combining the 2 equations and equating the like powers of z
-1,
The solutions of the above equations are m1 = 0.72 , m2 = -
0.457 and a1 = 1.28 . Thus

Putting the expressions of M(z) and 1- M(z) in the controller
equation
The output response is plotted in Figure 2.

Figure 2: Deadbeat Response of The System in Example 2

Note that although c(kT) tracks the unit step perfectly after 3
sampling periods, the maximum overshoot is 128 percent.
This is because of the fact that the digital plant is a type 2
system, hence a deadbeat response without overshoot
cannot be obtained for a unit step input.
Thus one can conclude that it is not always possible to
design dead beat response without any over shoot.

5.Sampled data control system with Dead beat response
In case of a continuous time controlled process, the output
c(t) is a function of time t and the dead beat response design,
based on cancellation of stable poles and zeros, may lead to
inter sampling ripples in the output.
The reason behind this is since the process zeros are
cancelled by controller poles, the continuous dynamics are
excited by the input and are not affected by feed back.
The strategy of designing dead beat response for a sampled
data system with the process plant transfer function Gh0 Gp
(z) having at least one zero is not to cancel the zeros, whether
they are inside or outside the unit circle.

H.P Sirisena gave a mathematical formulation and analysis
to dead beat response.
then according to Sirisena the digital controller for ripple free
dead beat response to step input is
Following example will illustrate the design procedure

Example 1:Let us consider a sampled data system as shown
in Fig. 1, where,
Figure 1: A sampled data control system
If we design Dc(z) without bothering about the inter sample
ripples then

This implies that the output response is deadbeat only at
sampling instants. However, the true output c(t) has inter
sampling ripples which makes the system response as
shown in Figure 2.
Figure 2: Rippled output response for Example 1
Thus the system takes forever to reach its steady state. The
necessary and sufficient condition for c(t) to track a unit step
input in finite time is

for finite N and all the higher derivatives should equal to zero.
Let
Taking z -transform,

where A1 is a constant. Unit step response of W(z) will not go
to zero in finite time since poles ofare not all at z = 0.
If we now apply the condition that zero of Gh0 Gp (z) at z = - 0.9
should not be cancelled by Dc(z) , then

where A2 and A3 are constants. This implies that the dead
beat response reaches the steady state after two sampling
periods.
To show that the output response is indeed deadbeat, we
derive the z-transform of w(t) as
Thus c(t) will actually reach its steady state in two sampling
periods with no inter sample ripples which is shown in Figure
3.
Figure 3: Ripple free deadbeat response for Example 1

Example 2: Consider the plant transfer function
as
If we apply the condition that zeros of at z = - 0.2
and z = - 2.8 should not be cancelled by Dc(z) , then

While considering M(z) and 1- M(z), following points should
be kept in mind
1. M(z) should contain all the zeros of
2. The number of poles over zeros of M(z) should be at least
equal to that of which is 2 in this case.
3. (1- M(z) )must include the term 1- z -1.
4. The orders of M(z) and 1- M(z) should be same and should
equal the number of unknown coefficients.

Solving for the coefficients of M(z) and 1- M(z) , we get
The solutions of the above are m1 = 0.219 , a1 = 1, a2 = 0.781
and a3 = 0.123 . The closed loop transfer function is
The transfer function of the digital controller is obtained as

The output for a unit step input is written as
Thus the output response c(kT) reaches the steady state in 4
sampling instants. This is one more sampling instant than the
previous example where we considered the plant to be all
digital.This implies that for sampled data control system, the dead
beat response c(t) reaches the steady state after three
sampling periods but inter sample ripples occur. After four
sampling instants the inter sample ripples disappear.

To show that the output response is indeed deadbeat, we
derive the z-transform of w(t) which will come out to be
where A1, A2 are constants.
thus the derivative of c(t) is zero for , which implies
that the step response reaches the steady state in 4 sampling
instants with no inter sample ripples, as shown in Figure 4.
Figure 4: Ripple free deadbeat response for Example 2

Deadbeat Response Design _8th lecture

More Related Content

What's hot

Similar to Deadbeat Response Design _8th lecture

Recently uploaded

In this document

Deadbeat Response Design _8th lecture