Understanding system stability is crucial in control systems. The stability of a system is primarily determined by the location of its poles in the complex s-plane.

System Stability and Pole Locations

The behavior of a dynamic system over time, particularly its response to inputs, is categorized by its stability. We classify system stability based on where the system's poles are located in the s-plane:

• Stable System: A system is considered stable if all its poles are located strictly in the left half of the s-plane (LHP). In a stable system, the natural response will decay to zero over time.
• Unstable System: A system is unstable if at least one pole is located in the right half of the s-plane (RHP), or if there are repeated poles (poles with multiplicity > 1) located precisely on the imaginary axis. An unstable system's natural response typically grows unbounded over time.
• Marginally Stable System: A system is marginally stable if all its poles are located in the left half of the s-plane, and there are one or more simple poles (poles with multiplicity = 1) located precisely on the imaginary axis. It is critical that there are no poles in the right half of the s-plane for marginal stability. The natural response of a marginally stable system will oscillate continuously without decaying or growing.

Analyzing System Conditions

The question provides specific conditions regarding the system's poles:

• "simple poles lying on the imaginary axis": This condition indicates that the system has poles exactly on the imaginary axis, and they are not repeated (i.e., their multiplicity is one).
• "no poles to its right": This condition implies that there are no poles in the right half of the s-plane, which would otherwise lead to an unstable system.

When these two conditions are met – that is, a system possesses simple poles lying on the imaginary axis and has no poles in the right half of the s-plane – it perfectly aligns with the definition of a marginally stable system. Such a system will exhibit sustained, undamped oscillations in its response.