Roots in the left half plane represent the stable system
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Jan 20, 2024 03:23 AM
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In control system theory and the analysis of dynamic systems, the stability of a system is often determined by the location of the roots of its characteristic equation in the complex plane. These roots are also known as the poles of the system and are typically derived from the denominator of the system's transfer function.
Here's why roots in the left half of the complex plane represent stability:
- Exponential Solutions: The solutions to the differential equations that describe linear time-invariant (LTI) systems are exponential in nature. These solutions can be written in terms of the system's poles. For a given pole , the time-domain solution will include a term of the form .
- Real Part of Poles: The real part of the pole determines whether the exponential term will grow or decay over time. If the real part is positive , the exponential term will grow without bounds as time increases, which indicates an unstable system. Conversely, if the real part is negative , the exponential term will decay to zero as time goes on, indicating a stable system.
- Left Half Plane (LHP): The complex plane is divided into two halves by the imaginary axis. The left half-plane consists of all points (or complex numbers) with a negative real part. Therefore, poles located in the LHP have negative real parts, and the associated exponential terms in the system's response will decay over time, which is a sign of stability.
- Bounded Input, Bounded Output (BIBO) Stability: For a system to be BIBO stable, every bounded input must produce a bounded output. This condition is met when all the poles of the transfer function have negative real parts, ensuring that the system's response will not grow unbounded over time.
- Physical Interpretation: In physical systems, energy dissipation mechanisms (like resistance in electrical circuits, or damping in mechanical systems) often result in poles being in the LHP. These mechanisms ensure that the system naturally dissipates energy over time, leading to stable behavior.
In summary, for a system to be considered stable, all of its poles must lie in the left half-plane of the complex plane. This ensures that all modes of the system decay over time, and the system's response does not grow to infinity. If any pole lies on the right half-plane, or even on the imaginary axis (except for certain repeated pole conditions on the axis), the system is considered unstable or marginally stable, respectively.
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