A linear time invariant (LTI) system has a transfer function $$G(s) = \frac{10(s + 1)}{s(s^2 + 2s + 5)}$$ The system is placed in unity negative feedback configuration. For a unit ramp reference, the steady state error is _______. (rounded off to two decimal places)
Steady State Error Analysis for Ramp Input
The problem asks for the steady-state error ($e_{ss}$) of a linear time-invariant (LTI) system with a given transfer function $G(s)$ in a unity negative feedback configuration when subjected to a unit ramp input.
Transfer Function and Feedback Configuration
The system's transfer function is given as:
$ G(s) = \frac{10(s + 1)}{s(s^2 + 2s + 5)} $The system is configured in unity negative feedback. For a unit ramp input $r(t) = t$, the Laplace transform is $R(s) = \frac{1}{s^2}$.
Calculating Velocity Error Constant ($K_v$)
For a ramp input, the steady-state error is determined by the velocity error constant ($K_v$). The formula is:
$ e_{ss} = \frac{1}{K_v} $The velocity error constant $K_v$ is calculated as:
$ K_v = \lim_{s \to 0} s G(s) $Substitute the given transfer function $G(s)$:
$ K_v = \lim_{s \to 0} s \left( \frac{10(s + 1)}{s(s^2 + 2s + 5)} \right) $Cancel out the '$s$' term:
$ K_v = \lim_{s \to 0} \frac{10(s + 1)}{s^2 + 2s + 5} $Evaluate the limit by substituting $s = 0$:
$ K_v = \frac{10(0 + 1)}{0^2 + 2(0) + 5} = \frac{10(1)}{5} = \frac{10}{5} = 2 $So, the velocity error constant $K_v = 2$.
Determining Steady State Error ($e_{ss}$)
Now, calculate the steady-state error using the formula $e_{ss} = \frac{1}{K_v}$:
$ e_{ss} = \frac{1}{2} = 0.5 $The calculated steady-state error is $0.5$. This value lies within the provided range of 0.49 to 0.51.
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
