Find the number of poles in the left half plane (LHP), the right half plane (RHP) and on the jω-axis for the feedback control system as shown. Is the system stable?

Explanation:

Feedback Control System Stability Analysis

Understanding the Problem: To determine the stability of the given feedback control system, we need to analyze the location of the system poles. The poles of the system are the roots of the characteristic equation, which is derived from the closed-loop transfer function. Stability is determined based on the following criteria:

• Left Half Plane (LHP) Poles: Poles in the LHP correspond to stable components of the system.
• Right Half Plane (RHP) Poles: Poles in the RHP indicate unstable components of the system.
• jω-Axis Poles: Poles on the imaginary axis may suggest marginal stability or oscillatory behavior, depending on their multiplicity.

Analysis of the Correct Option (Option 2):

Given: The system has:

• 2 LHP Poles
• 2 RHP Poles
• 0 Poles on the jω-Axis

Step-by-Step Analysis:

1. Determine the Characteristic Equation: The characteristic equation of the system is derived from the closed-loop transfer function, which is typically given as:

T(s) = G(s) / [1 + G(s)H(s)]

Where G(s) is the open-loop transfer function, and H(s) is the feedback transfer function. The characteristic equation is obtained by setting the denominator to zero:

1 + G(s)H(s) = 0

2. Locate the Poles: Solving the characteristic equation provides the locations of the poles in the complex plane. Based on the problem statement, the poles are distributed as follows:

• 2 poles in the Left Half Plane (LHP): These poles indicate stable components of the system.
• 2 poles in the Right Half Plane (RHP): These poles indicate unstable components of the system.
• 0 poles on the jω-axis: There are no marginally stable or purely oscillatory components.

Note: The presence of RHP poles means the system is inherently unstable, regardless of the LHP poles.

3. Determine System Stability: A system is stable if all poles are located in the LHP. In this case, the presence of 2 RHP poles makes the system unstable. The 2 LHP poles do not compensate for the instability caused by the RHP poles.

Conclusion: The system is unstable because it has 2 poles in the RHP. This aligns with the correct answer, which is Option 2.

Additional Information

To further understand the analysis, let’s evaluate why the other options are incorrect:

Option 1: 1 LHP pole, 3 RHP poles, 0 jω poles, system is unstable.

This option suggests that there is only 1 LHP pole and 3 RHP poles. While this configuration also makes the system unstable (due to the RHP poles), it does not match the given system's pole distribution (2 LHP poles and 2 RHP poles). Therefore, this option is incorrect.

Option 3: 2 LHP poles, 2 RHP poles, 0 jω poles, system is stable.

This option claims that the system is stable despite having 2 RHP poles. This is incorrect because any RHP poles render the system unstable. Stability requires all poles to be in the LHP. Hence, this option is incorrect.

Option 4: 1 LHP pole, 3 RHP poles, 0 jω poles, system is stable.

This option is inconsistent because it claims stability despite having 3 RHP poles. A stable system cannot have any RHP poles. Thus, this option is also incorrect.

Conclusion:

The correct analysis of the pole locations reveals that the system has 2 LHP poles, 2 RHP poles, and 0 jω-axis poles, making the system unstable. This matches Option 2. The other options misrepresent the pole distribution or incorrectly assess system stability, leading to their elimination.