Question

Draw a Nyquist locus for the unity-feedback control systems with the open-loop transfer functions given below:

G(s) = K/(s-1)(s+1)²

Answer 1

A Nyquist plot for a specific transfer function in a control system, but the provided reference information is not relevant to the creation of such a plot.

Step-by-step explanation:

A Nyquist locus for a unity-feedback control system with a specified open-loop transfer function. Analyzing and drawing a Nyquist plot involves understanding concepts from control systems, a field within electrical engineering. Unfortunately, the provided reference information does not directly pertain to the Nyquist plot or control systems, so we cannot use it to assist with this question.

To construct a Nyquist locus for the given open-loop transfer function G(s) = K/(s-1)(s+1)², you would need to plot the frequency response of G(s) as s varies along the contour of the right-half plane in the complex plane, including the jω-axis from ∞ to -∞ and a semicircle at infinity. It is crucial to note the location of poles and zeros of G(s), as poles on the right-half plane indicate instability in the control system.

Answer 2

The Nyquist plot for the given open-loop transfer function
`\(G(s) = (K)/((s-1)(s+1)^2)\)` includes a clockwise encirclement around the point s = 1 indicating a right-half plane pole and a repeated counter-clockwise encirclement around s = -1 showing a left-half plane pole with magnitude and phase information for different frequencies.

Step-by-step explanation:

To sketch the Nyquist plot, follow these steps:

1. Open-loop transfer function:
`\(G(s) = (K)/((s-1)(s+1)^2)\)`

2. Stability analysis: First, determine the critical points and poles of the transfer function.

Critical points: s = 1 pole, s = -1 pole, and s = -1 (repeated pole)

The open-loop transfer function has poles at s = 1 and a repeated pole at s = -1.

3. Nyquist plot:

• For s = j
  `\(\omega\)` where j is the imaginary unit and
  `\(\omega\)` is the frequency), substitute s into the transfer function to get the magnitude and phase.
• Plot these points in the complex plane for different values of
  `\(\omega\)`, ranging from 0 to
  `\(\infty\)`.
• As
  `\(\omega\)` approaches infinity, evaluate the behavior of the Nyquist plot concerning the critical points and poles.

Remember:

• For every clockwise encirclement of the critical point by the Nyquist plot, the system has N = -1 encirclement, indicating a right-half plane pole.
• For every counter-clockwise encirclement, the system has N = +1 encirclement, implying a left-half plane pole.
• - The point -1 + j0 is important since it represents the complex conjugate poles.

With this information, you can sketch the Nyquist plot showing the system's stability concerning the critical points and poles. If you have a graphing tool or software that can plot Nyquist diagrams, you can input the system's transfer function and visualize the Nyquist plot there.