1. Modern Control Systems (MCS)
Ali Raza
Assistant Professor
email: ali_raza@indus.edu.pk
Lecture-23-24
Time Response Discrete Time Control Systems
Steady State Errors
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3. Introduction
โข The time response of a discrete-time linear system is the
solution of the difference equation governing the system.
โข 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.
โข The response due to the input, or the forced response, is
the convolution summation of its input and its response
to a unit impulse.
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4. Example-1
โข Given the discrete-time system
โข Find the impulse response of the system.
โข Taking z-transform
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๐ฆ ๐ + 1 โ 0.5๐ฆ ๐ = ๐ข ๐
Solution
๐ง๐ ๐ง โ 0.5๐ ๐ง = ๐ ๐ง
๐(๐ง)
๐(๐ง)
=
1
๐ง โ 0.5
6. Example-2
โข Given the discrete time system
โข find the system transfer function and its response to a
sampled unit step.
โข The transfer function corresponding to the difference
equation is
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๐ฆ ๐ + 1 โ ๐ฆ ๐ = ๐ข ๐ + 1
Solution
๐ง๐ ๐ง โ ๐ ๐ง = ๐ง๐ ๐ง
๐(๐ง)
๐(๐ง)
=
๐ง
๐ง โ ๐ง
8. Home Work
โข Find the impulse, step and ramp response functions for
the systems governed by the following difference
equations.
1. ๐ฆ ๐ + 1 โ 0.5๐ฆ ๐ = ๐ข ๐
2. ๐ฆ ๐ + 2 โ .01๐ฆ ๐ + 1 + 0.8๐ฆ ๐ = ๐ข(๐)
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9. Final Value Theorem
โข The final value theorem allows us to calculate the limit of a
sequence as k tends to infinity, if one exists, from the z-
transform of the sequence.
โข If one is only interested in the final value of the sequence, this
constitutes a significant short cut.
โข The main pitfall of the theorem is that there are important
cases where the limit does not exist.
โข The two main case are
1. An unbounded sequence
2. An oscillatory sequence
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10. Final Value Theorem
โข If a sequence approaches a constant limit as k tends to
infinity, then the limit is given by
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๐ โ = lim
๐โโ
๐ ๐
๐ โ = lim
๐งโ1
๐ง โ 1
๐ง
๐น ๐ง
๐ โ = lim
๐งโ1
(๐ง โ 1)๐น ๐ง
11. Example-3
โข Verify the final value theorem using the z-transform of a
decaying exponential sequence and its limit as k tends to
infinity.
โข The z-transform of an exponential sequence is
โข Applying final value theorem
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Solution
๐น ๐ง =
๐ง
๐ง โ ๐โ๐๐
๐ โ = lim
๐งโ1
๐ง โ 1
๐ง
๐น ๐ง = lim
๐งโ1
๐ง โ 1
๐ง
๐ง
๐ง โ ๐โ๐๐
๐ โ = 0
12. Example-4
โข Obtain the final value for the sequence whose z-
transform is
โข Applying final value theorem
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Solution
๐ โ = lim
๐งโ1
๐ง โ 1
๐ง
๐ง2
(๐ง โ ๐)
(๐ง โ 1)(๐ง โ ๐)(๐ง โ ๐)
๐น ๐ง =
๐ง2(๐ง โ ๐)
(๐ง โ 1)(๐ง โ ๐)(๐ง โ ๐)
๐ โ =
1 โ ๐
(1 โ ๐)(1 โ ๐)
13. Home work
โข Find the final value of following z-transform functions if it
exists.
1. ๐น(๐ง) =
๐ง
๐ง2โ1.2๐ง+0.2
2. ๐น(๐ง) =
๐ง
๐ง2โ0.3๐ง+2
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14. Steady State Error
โข Consider the unity feedback block diagram shown in
following figure.
โข The error ratio can be calculated as
โข Applying the final value theorem yields the steady-state
error.
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๐ธ(๐ง)
๐ (๐ง)
=
1
1 + ๐บ ๐๐ด๐ ๐ง ๐บ(๐ง)
๐ โ = lim
๐งโ1
๐ง โ 1
๐ง
๐ธ ๐ง
15. Steady state Error
โข As with analog systems, an error constant is associated with
each input (e.g., Position Error constant and Velocity Error
Constant)
โข Type number can be defined for any system from which the
nature of the error constant can be inferred.
โข The type number of the system is the number of unity poles in
the system z-transfer function.
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16. Position Error Constant ๐พ ๐
โข Error of the system is given as
โข Where
โข Therefore, the steady state error due to step input is given as
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๐ธ(๐ง) =
๐ (๐ง)
1 + ๐บ ๐๐ด๐ ๐ง ๐บ(๐ง)
๐ โ = lim
๐งโ1
๐งโ1
๐ง
1
1+๐บ ๐๐ด๐ ๐ง ๐บ(๐ง)
๐ง
๐งโ1
๐ ๐ง =
๐ง
๐ง โ 1
๐ โ = lim
๐งโ1
1
1 + ๐บ ๐๐ด๐ ๐ง ๐บ(๐ง)
17. Position Error Constant ๐พ ๐
โข Position error constant ๐พ ๐ is given as
โข Steady state error can be calculated as
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๐ โ = lim
๐งโ1
1
1 + ๐บ ๐๐ด๐ ๐ง ๐บ(๐ง)
๐พ ๐ = lim
๐งโ1
๐บ ๐๐ด๐ ๐ง ๐บ(๐ง)
๐ โ =
1
1 + ๐พ ๐
18. Velocity Error Constant ๐พ๐ฃ
โข Error of the system is given as
โข Where
โข Therefore, the steady state error due to step input is given as
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๐ธ(๐ง) =
๐ (๐ง)
1 + ๐บ ๐๐ด๐ ๐ง ๐บ(๐ง)
๐ โ = lim
๐งโ1
๐ง โ 1
๐ง
1
1 + ๐บ ๐๐ด๐ ๐ง ๐บ(๐ง)
๐๐ง
๐ง โ 1 2
๐ ๐ง =
๐๐ง
๐ง โ 1 2
๐ โ = lim
๐งโ1
๐
๐ง โ 1 [1 + ๐บ ๐๐ด๐ ๐ง ๐บ ๐ง ]
19. Velocity Error Constant ๐พ๐ฃ
โข ๐พ๐ฃ is given as
โข Steady state error due to sampled ramp input is given as
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๐ โ = lim
๐งโ1
๐
๐ง โ 1 [1 + ๐บ ๐๐ด๐ ๐ง ๐บ ๐ง ]
๐พ๐ฃ =
1
๐
lim
๐งโ1
๐ง โ 1 ๐บ ๐๐ด๐ ๐ง ๐บ ๐ง
๐ โ =
1
๐พ๐ฃ
20. Example-5
โข Find the steady-state position error for the digital position
control system with unity feedback and with the transfer
functions
1. For a sampled unit step input.
2. For a sampled unit ramp input
โข ๐พ๐ and ๐พ๐ฃ are given as
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๐บ ๐๐ด๐ ๐ง =
๐พ(๐ง + ๐)
(๐ง โ 1)(๐ง โ ๐)
๐ถ ๐ง =
๐พ๐(๐ง โ ๐)
๐ง โ ๐
,0 < ๐, ๐, ๐ < 1
Solution
๐พ๐ฃ =
1
๐
lim
๐งโ1
๐ง โ 1 ๐บ ๐๐ด๐ ๐ง ๐บ ๐ง๐พ ๐ = lim
๐งโ1
๐บ ๐๐ด๐ ๐ง ๐บ(๐ง)