Consider the stable closed-loop system shown in the figure. The asymptotic Bode magnitude plot of $G(s)$ has a constant slope of -20 dB/decade at least till 100 rad/sec with the gain crossover frequency being 10 rad/sec. The asymptotic Bode phase plot remains constant at -90° at least till $\omega = 10$ rad/sec. The steady-state error of the closed-loop system for a unit ramp input is ______________(rounded off to 2 decimal places).
To find the steady-state error for a unit ramp input, we need to determine the type of system from the transfer function \(G(s)\). Given a slope of -20 dB/decade, it indicates a single pole at the origin, implying a type 1 system.
The steady-state error, \(e_{ss}\), for a type 1 system with a unit ramp input is given by:
\(e_{ss} = \frac{1}{K_v}\)
where \(K_v\) is the velocity error constant.
The velocity error constant is defined as:
\(K_v = \lim_{s \to 0} sG(s)\)
The gain crossover frequency is given as 10 rad/sec. The magnitude at this frequency for the Bode plot satisfies:
\(|G(j\omega_{gc})| = 1\)
Thus, at \(\omega = 10\),
\(|G(j10)| = 1\)
Since the system has a -20 dB/decade slope, \(G(s)\) is proportional to \(\frac{1}{s}\), leading to:
\(G(s) = \frac{K}{s}\)
Here, \(K\) is the gain. At the gain crossover frequency:
\(|G(j10)| = \frac{K}{10} = 1 \Rightarrow K = 10\)
Thus, \(G(s) = \frac{10}{s}\).
Calculating the velocity error constant:
\(K_v = \lim_{s \to 0} s \times \frac{10}{s} = 10\)
Therefore, the steady-state error is:
\(e_{ss} = \frac{1}{K_v} = \frac{1}{10} = 0.10\)
This computed error of 0.10 is within the given range of 0.09 to 0.11.
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
