Network Stability and Pole-Zero Concepts Quiz Quiz

Explore key concepts in network stability by analyzing pole-zero locations and their effects on system behavior. This quiz assesses your understanding of stability criteria, transfer function dynamics, and practical scenarios related to poles and zeros in system design.

1. Pole Locations and Stability

   Which of the following statements best describes the requirement for a continuous-time linear system to be stable in terms of its pole locations?

   1. At least one pole must be in the right half of the s-plane.
   2. Poles can be in either half of the s-plane if zeros are in the left half.
   3. All system poles must lie on the real axis regardless of sign.
   4. All system poles must lie strictly in the left half of the complex s-plane.

   Explanation: For continuous-time linear systems, stability requires that all poles of the transfer function are strictly in the left half of the complex s-plane, ensuring all natural responses decay over time. If poles are on the right half, the system exhibits growing exponentials and becomes unstable. Poles on the real axis must have negative real parts to be stable, so simply being on the real axis is not enough. The location of zeros does not compensate for unstable pole locations; therefore, having zeros in the left half does not ensure stability if poles are elsewhere.

2. Zero Locations and System Response

   How does the placement of a system zero near the origin of the s-plane affect the step response of a stable system?

   1. It reduces the initial value of the step response and may introduce a zero crossing.
   2. It causes the response to become unstable even if the poles are stable.
   3. It slows down the response and increases the settling time.
   4. It increases the initial overshoot without changing the speed of the response.

   Explanation: A zero near the origin typically reduces the initial value of the step response, potentially causing the output to cross zero (zero crossing) due to phase inversion. While zeros can alter the transient characteristics, they do not override stability provided by poles; thus, having the zero near the origin will not make a system unstable. Zeros near the origin can momentarily suppress the initial output without increasing overshoot or settling time substantially. The main effect is on the shape of the initial response, not its speed.

3. Stability of Discrete-Time Systems

   For a discrete-time system, under what condition on the pole locations is the system considered stable?

   1. All poles must be real and non-negative.
   2. All poles must have magnitudes greater than one.
   3. All poles must be located strictly inside the unit circle on the z-plane.
   4. Zeros must be inside the unit circle regardless of pole locations.

   Explanation: A discrete-time system is stable if all of its poles lie strictly inside the unit circle in the complex z-plane, which ensures the output will not grow unbounded. Poles with magnitudes greater than one cause instability by producing increasing responses. The requirement for poles to be real and non-negative does not guarantee stability. Zeros do not determine stability; they affect the response but not the boundedness of the output for bounded input.

4. Pole-Zero Cancellation Consequences

   What potential risk arises from designing a system in which a pole and a zero cancel each other exactly at the same location in the s-plane?

   1. It always improves the transient response without drawbacks.
   2. It guarantees noise immunity for the system.
   3. It may cause hidden unstable modes to exist, leading to practical instability.
   4. It causes the frequency response to remain unchanged at all frequencies.

   Explanation: Exact pole-zero cancellation can mask unstable behavior if the cancellation occurs at an unstable pole, because the pole remains present in the overall system's internal response even if it is not visible in the transfer function. This can lead to instability in real-world systems due to imperfection or noise. It does not provide noise immunity and, instead, increases risk if not implemented perfectly. Although the frequency response at the cancelled frequency disappears, the underlying dynamics may not be properly controlled, and transient response is not always improved.

5. Interpreting Poles in System Design

   If a system designer observes that all the system poles are located close to the imaginary axis in the left half of the s-plane, what characteristic is the designer likely to notice in the system's time-domain response?

   1. Immediate instability and divergent output.
   2. A very fast settling response with minimal oscillation.
   3. Constant steady-state error regardless of input.
   4. A prolonged, slowly decaying oscillatory response.

   Explanation: Poles near the imaginary axis in the left half of the s-plane result in a system that is weakly stable, with slowly decaying natural modes, often exhibiting oscillatory behavior that persists for a long time before dying out. This means the response does not settle quickly. Rapid settling and minimal oscillation are associated with poles farther left in the s-plane. Instability only occurs with poles in the right half, not the left. Steady-state error is unrelated to pole location and depends more on system type and input.