Questions: Transfer Functions and System Modeling

Question 1

A transfer function has poles at s = -2 and s = 3 + 2j. What can you conclude about the system's stability?

• A. The system is stable because one pole is in the left half-plane
• B. The system is unstable because one pole has a positive real part
• C. The system is marginally stable because the imaginary part of one pole is non-zero
• D. Stability cannot be determined from pole locations alone

Answer: (B)

For BIBO stability of an LTI system, ALL poles must lie in the left half-plane (negative real part). The pole at s = 3 + 2j has real part +3, which is positive, placing it in the right half-plane. This guarantees an unstable (exponentially growing) response mode, regardless of where the other pole is located.

Question 2

A transfer function with a pole at s = 0 (the origin of the s-plane) represents an unstable system.

• T. True
• F. False

Answer: False

A pole at s = 0 corresponds to a marginally stable integrating mode — the response neither grows nor decays, but stays bounded (for bounded input) or grows as a ramp for a step input, depending on how 'stable' is defined. It is only poles with strictly positive real parts that cause exponentially growing (truly unstable) responses. The distinction between marginally stable and unstable is important in control design.

Question 3

What is the physical meaning of a pole of a transfer function?

Model answer: A pole is a value of s at which the transfer function G(s) goes to infinity. Physically, each pole corresponds to a natural mode of the system — a frequency at which the system will respond spontaneously without sustained input. The pole location in the complex s-plane encodes both the damping (real part) and the oscillation frequency (imaginary part) of that natural mode.

Poles arise as the roots of the denominator polynomial, which corresponds to the characteristic equation of the underlying differential equation. The natural modes they represent are the system's intrinsic behaviors: exponential decay, oscillation, or growth. Control design often consists of choosing where to place poles (via feedback) to get desired natural behavior.