Concept of Stability in Control Systems

 

Concept of Stability in Control Systems

Stability in control systems refers to the ability of the system to return to equilibrium after a disturbance. A stable system produces bounded output for bounded input.

Root Locations in the S-Plane

  • Left Half of S-Plane (LHP): Poles in LHP → system is stable.

  • Right Half of S-Plane (RHP): Poles in RHP → system is unstable.

  • On the Imaginary Axis:

    • Single pole on imaginary axis → critically stable (oscillatory but not diverging).

    • Repeated poles on imaginary axis → unstable.

Types of Stability

  • Stable System: All poles in LHP. Output settles to a finite value.

  • Unstable System: Any pole in RHP or repeated poles on imaginary axis. Output diverges.

  • Critically Stable System: Poles exactly on the imaginary axis (non-repeated). Output oscillates indefinitely.

  • Conditionally Stable System: Stability depends on parameter values (e.g., gain).

Absolute vs. Relative Stability

  • Absolute Stability: Determines whether the system is stable or unstable (yes/no).

  • Relative Stability: Measures how stable the system is, i.e., the degree of stability.

    • Depends on distance of poles from the imaginary axis.

    • Greater distance → faster decay of transients → more stable.

Example Analysis

  1. System with poles at -2, -5

    • All poles in LHP → Stable.

  2. System with poles at +3, -4

    • One pole in RHP → Unstable.

  3. System with poles at ±j2

    • Poles on imaginary axis → Critically stable.

  4. System with poles at -2 ± j3

    • Poles in LHP but close to imaginary axis → Stable but relatively less stable compared to poles at -10.



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