If the poles of a closed-loop are complex with positive real parts, then the system is: 

Explanation:

Correct Option Analysis:

The correct option is:

Option 2: Exponential and oscillatory increase.

This option correctly describes the behavior of a system whose poles are complex with positive real parts. In control systems and signal processing, the poles of a system determine its stability and response characteristics. The presence of complex poles indicates that the system will exhibit oscillatory behavior, while positive real parts of the poles indicate that the amplitude of these oscillations will grow exponentially over time, leading to an unstable system with increasing oscillatory output.

Additional Information

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

Option 1: Exponential decay.

This option is incorrect because exponential decay occurs when the poles of the system have negative real parts. In such a case, the system's response diminishes over time, leading to a stable system with decreasing amplitude. Since the poles in the given scenario have positive real parts, this option does not apply.

Option 3: Decaying and oscillatory.

This option is incorrect because decaying oscillations occur when the poles are complex with negative real parts. This results in an exponentially decaying amplitude of the oscillatory response, leading to a stable system. However, the given scenario describes poles with positive real parts, which leads to increasing, not decreasing, oscillations.

Option 4: Exponential increase.

This option is partially correct but incomplete. Exponential increase describes the growth of the system's response over time due to positive real parts of the poles. However, it does not account for the oscillatory nature of the response indicated by the complex poles. Therefore, "exponential increase" alone does not fully capture the system's behavior.

Conclusion:

Understanding the implications of pole locations in the complex plane is crucial for analyzing system stability and response characteristics. Poles with positive real parts lead to an unstable system with an exponentially growing response, while complex poles introduce oscillations. The correct description for a system with complex poles and positive real parts is "exponential and oscillatory increase," highlighting the dual aspects of growth and oscillation in the system's behavior.