Assist. Prof. Dr. Khalaf S. Gaeid
Electrical Engineering Department
Tikrit University
gaeidkhalaf@gmail.com
+9647703057076
Time Response of Discrete
Time Systems

1. Transient response
2. steady state response,
3. Time response parameters of a prototype second order
system.
4. MCQ for both continuous and discrete systems
Contents

Time Response of discrete time systems
Absolute stability is a basic requirement of all control systems. Apart
from that, good relative stability and steady state accuracy are also
required in any control system, whether continuous time or discrete
time.
Transient response corresponds to the system closed loop poles and
steady state response corresponds to the excitation poles or poles of
the input function.

1.Ttransient response specifications
In many practical control systems, the desired performance
characteristics are specified in terms of time domain quantities.
Unit step input is most commonly used in analysis of a system since
it is easy to generate and represent a sufficiently drastic change thus
providing useful information on both transient and steady state
responses.
The transient response of a system depends on the initial conditions.
It is a common practice to consider the system initially at rest.

Consider the digital control system shown in Figure1

Similar to the continuous time case, transient response of a digital
control system can also be characterized by the following.
1. Rise time (tr): Time required for the unit step response to rise
from 0% to 100% of its final value in case of underdamped
system or 10% to 90% of its final value in case of overdamped
system.
2. Delay time (td): Time required for the unit step response to reach
50% of its final value

3. Peak time (tp): Time at which maximum peak occurs.
4. Peak overshoot (Mp): The difference between the maximum peak
and the steady state value of the unit step response.
5. Settling time (ts): Time required for the unit step response to
reach and stay within 2% or 5% of its steady state value.
However since the output response is discrete the calculated
performance measures may be slightly different from the actual
values.

Figure 2 illustrates this. The output has a maximum value cmax
whereas the maximum value of the discrete output is c∗max which is
always less than or equal to c max. If the sampling period is small
enough compared to the oscillations of the response then this
difference will be small otherwise c∗max may be completely erroneous.

2.Steady state error
The steady state performance of a stable control system is measured
by the steady error due to step, ramp or parabolic inputs depending
on the system type. Consider the discrete time system as shown in
Figure 3.

From Figure 2, we can write
We will consider the steady state error at the sampling instants. From final value
theorem

The steady state error of a system with feedback thus depends on the
input signal R(z) and the loop transfer function GH(z).
Type-0 system and position error constant
Systems having a finite nonzero steady state error with a zero order
polynomial input (step input) are called Type-0 systems. The position
error constant for a system is defined for a step input

Type-1 system and velocity error constant
Systems having a finite nonzero steady state error with a first order
polynomial input (ramp input) are called Type-1 systems. The
velocity error constant for a system is defined for a ramp input.

Type-2 system and acceleration error constant
Systems having a finite nonzero steady state error with a second
order polynomial input (parabolic input) are called Type-2 systems.
The acceleration error constant for a system is defined for a
parabolic input.
Table 1 shows the steady state errors for different types of systems
for different inputs

Example 1: Calculate the steady state errors for unit step, unit ramp
and unit parabolic inputs for the system shown in Figure 4.

The time response of a discrete-time linear system is the solution of
the difference equation governing the system.
For the linear time-invariant (LTI) case, the response due to the
initial conditions and the response due to the input can be obtained
separately and then added to obtain the overall response of the
system.
The response due to the input, or the forced response, is the
convolution summation of its input and its response to a unit
impulse. In this section, we derive this result and examine its
implications.

The response of a discrete-time system to a unit impulse is known as
the impulse response sequence. The impulse response sequence can
be used to represent the response of a linear discrete-time system to
an arbitrary input sequence
To derive this relationship, we first represent the input sequence in
terms of discrete impulses as follows:
To derive this relationship, we first represent the input sequence in
terms of discrete impulses as follows:

For a linear system, the principle of superposition applies, and the
system output due to the input is the following sum of impulse
response sequences:

Hence, the output at time k is given by
Example Given the discrete-time system
find the impulse response of the system h(k):
1. From the difference equation
2. Using z-transformation

3. Prototype second order system
The study of a second order system is important because many
higher order system can be approximated by a second order model if
the higher order poles are located so that their contributions to
transient response are negligible

Comparison between continuous time and discrete time systems
The simplified block diagram of a space vehicle control system is
shown in Figure 2.
The objective is to control the attitude in one dimension, say in pitch.
For simplicity vehicle body is considered as a rigid body.
Position c(t) and velocity v(t) are feedback. The open loop transfer
function can be calculated

Now, consider that the continuous data system is subject to sampled
data control as shown in Figure 3.

Dominant Closed Loop Pole Pairs
As in case of s-plane, some of the roots in z-plane have more effects
on the system response than the others. It is important for design
purpose to separate out those roots and give them the name
dominant roots. In s-plane, the roots that are closest to jω axis in the
left plane are the dominant roots because the corresponding time
response has slowest decay. Roots that are for away from jω axis
correspond to fast decaying response.
•In Z-plane dominant roots are those which are inside and closest to
the unit circle whereas insignificant region is near the origin.
•The negative real axis is generally avoided since the corresponding
time response is oscillatory in nature with alternate signs.

In s-plane the insignificant roots can be neglected provided the dc-
gain (0 frequency gain) of the system is adjusted. For example,
10/(s2+ 2s+ 2)(s+ 10)≈1/(s2+ 2s+ 2)
In z-plane, roots near the origin are less significant from the
maximum overshoot and damping point of view

However these roots cannot be completely discarded since the excess
number of poles over zeros has a delay effect in the initial region of
the time response, e.g., adding a pole at z= 0 would not effect the
maximum overshoot or damping but the time response would have
an additional delay of one sampling period.
The proper way of simplifying a higher order system in z-domain is
to replace the poles near origin by Poles at z= 0 which will simplify
the analysis since the Poles at z= 0 correspond to pure time delays

4.CONTROL SYSTEMS QUESTIONS AND ANSWERS
1. Which of the following transfer function will have the greatest maximum overshoot?
a) 9/(s2+2s+9)
b) 16/(s2+2s+16)
c) 25/(s2+2s+25)
d) 36/(s2+2s+36)
2. A system generated by control-systems-questions-answers-time-response-second-order-
systems-i-q2 The ramp component in the forced response will be:
a) t u(t)
b) 2t u(t)
c) 3t u(t)
d) 4t u(t)
3. The system in originally critically damped if the gain is doubled the system will be :
a) Remains same
b) Overdamped
c) Under damped
d) Undamped

4. Let c(t) be the unit step response of a system with transfer function K(s+a)/(s+K).
If c(0+) = 2 and c(∞) = 10, then the values of a and K are respectively.
a) 2 and 10
b) -2 and 10
c) 10 and 2
d) 2 and -10
5. The damping ratio and peak overshoot are measures of:
a) Relative stability
b) Speed of response
c) Steady state error
d) Absolute stability
6. Find the type and order of the system given below:
a) 2,3
b) 2,2
c) 3,3
d) None of the mentioned

7. A system has a complex conjugate root pair of multiplicity two or more in its
characteristic equation. The impulse response of the system will be:
a) A sinusoidal oscillation which decays exponentially; the system is therefore stable
b) A sinusoidal oscillation with a time multiplier ; the system is therefore unstable
c) A sinusoidal oscillation which rises exponentially ; the system is therefore unstable
d) A dc term harmonic oscillation the system therefore becomes limiting stable
8. The forward path transfer function is given by G(s) = 2/s(s+3). Obtain an expression for
unit step response of the system.
a) 1+2e-t+e-2t
b) 1+e-t-2e-2t
c) 1-e-t+2e-2t
d) 1-2e-t+e+2t
9. Find the initial and final values of the following function:
F(s) = 12(s+1)/s(s+2)^2(s+3)
a) 1,∞ b) 0,∞ c) ∞,1 d) 0,1

10. The step response of the system is c(t) = 10+8e-t-4/8e-2t . The gain in time constant form
of transfer function will be:
a) -7
b) 7
c) 7.5
d) -7.5
11. If the system is initially relaxed at time n=0 and memory equals to zero, then the
response of such state is called as:
a) Zero-state response
b) Zero-input response
c) Zero-condition response
d) None of the mentioned

12. Zero-state response is also known as:
a) Free response
b) Forced response
c) Natural response
d) None of the mentioned
13. Zero-input response is also known as Natural or Free response.
a) True
b) False
14. The solution obtained by assuming the input x(n) of the system is zero is:
a) General solution
b) Particular solution
c) Complete solution
d) Homogenous solution

15. What is the homogenous solution of the system described by the first order difference
equation y(n)+ay(n-1)=x(n)?
16. What is the zero-input response of the system described by the homogenous second
order equation y(n)-3y(n-1)-4y(n-2)=0 if the initial conditions are y(-1)=5 and y(-2)=0?
a) (-1)n-1 + (4)n-2
b) (-1)n+1 + (4)n+2
c) (-1)n+1 + (4)n-2
d) None of the mentioned

17. What is the particular solution of the first order difference equation y(n)+ay(n-1)=x(n)
where |a|<1, when the input of the system x(n)=u(n)?
a) 1/(1+a) u(n)
b) 1/(1-a) u(n)
c) 1/(1+a)
d) 1/(1-a)
18. What is the particular solution of the difference equation y(n)= 5/6y(n-1)- 1/6y(n-2)+x(n)
when the forcing function x(n)=2n, n≥0 and zero elsewhere?
a) (1/5) 2n
b) (5/8) 2n
c) (8/5) 2n
d) (5/8) 2-n

19. The total solution of the difference equation is given as:
a) yp(n)-yh(n)
b) yp(n)+yh(n)
c) yh(n)-yp(n)
d) None of the mentioned
20. What is the impulse response of the system described by the second order difference
equation y(n)-3y(n-1)-4y(n-2)=x(n)+2x(n-1)?
a)[-1/5 (-1)n-6/5 (4)n]u(n)
b)[1/5 (-1)n – 6/5 (4)n]u(n)
c)[ 1/5 (-1)n+ 6/5 (4)n]u(n)
d)[- 1/5 (-1)n+ 6/5 (4)n]u(n)

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