Correct Answer - Option 1 : integrator, 0.25
Concept:
In type – 0 system, the steady-state error (ess) is defined for the step-input signal as:
\({e_{ss}} = \frac{A}{{1 + {k_p}}}\)
Where,
A = magnitude of the step-input signal
Kp = positional error constant
Kp = \(\mathop {\lim }\limits_{s \to 0} G\left( s \right)H\left( s \right)\)
Calculation:
Given that, Unit – step signal A = 1 and ess = 0.2.
\({e_{ss}} = \frac{1}{{1 + {K_p}}}\)
\(\therefore {K_p} = \frac{1}{{0.2}} - 1\)
= 5 -1
Kp = 4
So, the open-loop Transfer function will be:
\(= \frac{4}{{{s^0}}}\) type – 0 system.
\(\therefore {\rm{\;}}\mathop {\lim }\limits_{s \to 0} G\left( S \right)H\left( S \right) = {K_p} = 4\)
Now, the error will be removed from the system if we increase the type of system form type – 0 to type – 1 system, i.e.
Type – 0 means \(\frac{1}{{{s^0}}}\) term in the open-loop Transfer function.
And type – 1 means \(\frac{1}{{{s^1}}}\) term in the open-loop Transfer function.
Hence we say that unity –gain block must be an integrator i.e \(\frac{1}{s}\) in the open-loop Transfer function.
Now, Open-loop Transfer function will be:
\(G'\left( s \right) = \frac{{G\left( S \right).H\left( S \right)}}{S}\)
It is a type – 1 system.
Since, given that Unit ramp signal is applying to the system,
So, \(e_{ss}^1 = \frac{1}{{{K_v}}};\) for type – 1 system.
And Kv = velocity error Constant
\(= \mathop {\lim }\limits_{s \to 0} G\left( s \right)H\left( s \right)\)
\(= \mathop {\lim }\limits_{s \to 0} G\left( s \right)H\left( s \right) \times \frac{{G\left( s \right)H\left( S \right)}}{s}\)
\(= \mathop {\lim }\limits_{s \to 0} G\left( s \right)H\left( s \right)\)
= Kp
Kv = 4
New steady-state error will be:
\(e_{ss}^1 = \frac{1}{4} = 0.25\)
Important Points:
Steady-state Error for different types of System is shown in the table:
Type
|
Step input
|
Ramp input
|
Parabolic input
|
Type – 0
|
\(\frac{A}{{1 + {k_p}}}\)
|
∞
|
∞
|
Type – 1
|
0
|
\(\frac{A}{{{K_v}}}\)
|
∞
|
Type – 2
|
0
|
0
|
\(A/{K_a}\)
|
ess is only valid for a closed-loop stable system and calculated for a closed-loop system by using the open-loop transfer function.