Short Notes: Time Response Analysis | Control Systems - Electrical Engineering (EE) PDF Download

 Page 1


Time Response Analysis Notes
Introduction
Time response analysis examines how a control system responds to inputs over
time. It is critical in evaluating system performance, stability , and tr ansient be-
havior in control systems engineering.
K ey Concepts
• Time Response : The output y(t) of a system to a given input, typically di-
vided into tr ansient and steady-state responses.
• Tr ansient Response : T empor ary behavior as the system tr ansitions to equi-
librium.
• Steady-State Response : Long-term behavior after tr ansients deca y .
• Applications : Used to design and analyze systems in control engineering,
such as roboti cs, aerospace, and industrial automation.
S ystem Classification
• First-Order S ystems : Described b y a single pole, tr ansfer functionT(s) =
K
ts+1
, whereK is the DC gain andt is the tim e constant.
• Second-Order S ystems : Described b y two poles, tr ansfer function T(s) =
?
2
n
s
2
+2??ns+?
2
n
, where?
n
is the natur al frequ ency and? is the damping r atio.
• Higher-Order S ystems : Modeled as combinations of first- and second-order
systems, a nalyzed b y dominant poles.
Time Response of First-Order S ystems
• Step Re sponse : F or a unit step input, the output is:
y(t) =K
(
1-e
-
t
t
)
, t= 0
• Time Constant (t ) : Time for the response to reach 63.2 % of its final value
(y(t) = 0.632K ).
• Settling Time : Approximatelyt
s
˜ 4t for the response to reach within 2 %
of the final value.
Time Response of Second-Order S ystems
• Step Res ponse : F or a unit step input, the output depends on? :
y(t) = 1-
e
-??nt
v
1-?
2
sin
(
?
d
t+ tan
-1
v
1-?
2
?
)
, ?
d
=?
n
v
1-?
2
1
Page 2


Time Response Analysis Notes
Introduction
Time response analysis examines how a control system responds to inputs over
time. It is critical in evaluating system performance, stability , and tr ansient be-
havior in control systems engineering.
K ey Concepts
• Time Response : The output y(t) of a system to a given input, typically di-
vided into tr ansient and steady-state responses.
• Tr ansient Response : T empor ary behavior as the system tr ansitions to equi-
librium.
• Steady-State Response : Long-term behavior after tr ansients deca y .
• Applications : Used to design and analyze systems in control engineering,
such as roboti cs, aerospace, and industrial automation.
S ystem Classification
• First-Order S ystems : Described b y a single pole, tr ansfer functionT(s) =
K
ts+1
, whereK is the DC gain andt is the tim e constant.
• Second-Order S ystems : Described b y two poles, tr ansfer function T(s) =
?
2
n
s
2
+2??ns+?
2
n
, where?
n
is the natur al frequ ency and? is the damping r atio.
• Higher-Order S ystems : Modeled as combinations of first- and second-order
systems, a nalyzed b y dominant poles.
Time Response of First-Order S ystems
• Step Re sponse : F or a unit step input, the output is:
y(t) =K
(
1-e
-
t
t
)
, t= 0
• Time Constant (t ) : Time for the response to reach 63.2 % of its final value
(y(t) = 0.632K ).
• Settling Time : Approximatelyt
s
˜ 4t for the response to reach within 2 %
of the final value.
Time Response of Second-Order S ystems
• Step Res ponse : F or a unit step input, the output depends on? :
y(t) = 1-
e
-??nt
v
1-?
2
sin
(
?
d
t+ tan
-1
v
1-?
2
?
)
, ?
d
=?
n
v
1-?
2
1
where?
d
is the damped natur al frequency .
• Damping C ases :
– Under damped (0 < ? < 1 ) : Oscillatory response with deca ying ampli-
tude.
– Critically Damped (? = 1 ) : F astest non-oscillatory response.
– Over damped (? > 1 ) : Slow , non-oscillatory response.
– Undamped (? = 0 ) : Sustained oscillations.
Performance Metrics
• Rise Time (t
r
) : Time to go from 10 % to 90 % of the final value. F or second-
order s ystems:
t
r
˜
1.8
?
n
(underdamped, approximate)
• Peak Time (t
p
) : Time to reach the first peak:
t
p
=
p
?
n
v
1-?
2
• Percent O vershoot ( %OS ) : Maximum overshoot relative to final value:
%OS = 100·e
-
?p
v
1-?
2
• Settling Time (t
s
) : Time to sta y within±2 % (or 5 %) of the final value:
t
s
˜
4
??
n
(for 2 % criterion)
• Steady-State Error (e
ss
) : Difference between desired and actual output as
t?8 , calculated using the final value theorem:
e
ss
= lim
s?0
s·
R(s)
1+G(s)H(s)
Pr a ctical Consider ations
• Stability : Determined b y pole locations; all poles must be in the left half-
plane for stabi lity .
• Input Types : Common inputs include step, r amp, and impulse for testing
system respon se.
• S ystem Tuning : A djust? and?
n
to balance speed, overshoot, and stability .
• T ools : MA TLAB, Simulink, or Python used for simulation and analysis of
time respons e.
2