Question

Which of the following is a condition for a stable feedback control system?

• A. Roots of the characteristic equation should be real
• B. Poles of the closed loop transfer function should lie in the left half of the complex plane
• C. Bode plots of the corresponding open loop transfer function should monotonically decrease
• D. Poles of the closed loop transfer function should lie in the right half of the complex plane

Hint:

Poles do not need to be real to be stable; they can be complex numbers (which cause oscillatory behaviors). The lone prerequisite for absolute stability is that their real part must be strictly negative (Left-Half Plane placement).

Answer 1

The Correct Option is B

Concept: The stability of a closed-loop system is dictated by its transient behavior. If a system's response to a temporary disturbance grows continuously over time, the system is unstable. If the transient response decays back to a stable baseline, the system is stable.

Step 1: Linking pole location to transient responses.
The denominator of a closed-loop transfer function is known as the characteristic equation. The roots of this characteristic equation are the closed-loop poles. If a pole is located at a complex coordinates position \( s = \sigma + j\omega \), its corresponding time-domain response profile contains an exponential multiplier term: \[ y(t) \propto e^{\sigma t} \] Where \(\sigma\) represents the real part of the pole position on the complex \(s\)-plane.

Step 2: Evaluating the stability threshold boundary.

• If \(\sigma \gt 0\) (the pole lies in the Right-Half Plane), then \(e^{\sigma t}\) grows exponentially toward infinity as \(t \rightarrow \infty\), resulting in an unstable system.

• If \(\sigma \lt 0\) (the pole lies in the Left-Half Plane), then \(e^{\sigma t}\) decays exponentially toward zero as \(t \rightarrow \infty\), allowing the system to stabilize.
Thus, a control system is stable if and only if all poles of the closed-loop transfer function have negative real parts, meaning they must lie entirely within the Left-Half Plane (LHP).