The closed loop transfer function of a system is \(T\left( s \right) = \frac{4}{{\left( {{s^2} + 0.4s + 4} \right)}}\). The steady state error due to unit step input is ________.
Concept:
KP = position error constant = \(\mathop {\lim }\limits_{s \to 0} G\left( s \right)H\left( s \right)\)
Kv = velocity error constant = \(\mathop {\lim }\limits_{s \to 0} sG\left( s \right)H\left( s \right)\)
Ka = acceleration error constant = \(\mathop {\lim }\limits_{s \to 0} {s^2}G\left( s \right)H\left( s \right)\)
Steady state error for different inputs is given by
Input | Type -0 | Type - 1 | Type -2 |
Unit step | \(\frac{1}{{1 + {K_p}}}\) | 0 | 0 |
Unit ramp | ∞ | \(\frac{1}{{{K_v}}}\) | 0 |
Unit parabolic | ∞ | ∞ | \(\frac{1}{{{K_a}}}\) |
From the above table, it is clear that for type – 1 system, a system shows zero steady-state error for step-input, finite steady-state error for Ramp-input and \(\infty \) steady-state error for parabolic-input.
Calculation:
The given closed-loop transfer function is \(T\left( s \right) = \frac{4}{{\left( {{s^2} + 0.4s + 4} \right)}}\)
To find a steady state error, we require an open loop transfer function.
The open-loop transfer function \( = \frac{{CLTF}}{{CLTF - 1}}\)
\( = \frac{{\frac{4}{{\left( {{s^2} + 0.4s + 4} \right)}}}}{{\frac{4}{{\left( {{s^2} + 0.4s + 4} \right)}} - 1}}\)
\( = \frac{4}{{s\left( {s + 0.4} \right)}}\)
Error constant \({K_V} = \mathop {\lim }\limits_{s \to 0} \frac{4}{{s\left( {s + 0.4} \right)}} = \infty \)
Steady-state error \( = \frac{1}{{\infty }} = 0\)
Important Questions from Steady State Error
The steady-state error due to unit step input to a type-1 system is:
With reference to the error analysis of the control systems, the term 'acceleration error constant' stands for:
Which one of the following coefficient is associated with Unit Ramp function?
If the output of the system at steady state does not agree with the input, then the system is said to have _________ which determines the _________ of the system.
A unity negative feedback closed loop system has a plant with the transfer function \(G(s) = \dfrac{1}{s^2 + 2s + 2}\) and a controller Ge(s) in the feedforward path. For a unit step input, the transfer function of the controller that gives minimum steady slate error is
