Stability criteria are essential in Control Theory, helping us determine if systems behave predictably. Various methods, like Routh-Hurwitz and Nyquist, analyze system responses to ensure stability, guiding engineers in designing reliable control systems.
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Routh-Hurwitz Stability Criterion
- Determines the stability of a linear time-invariant (LTI) system by analyzing the characteristic polynomial.
- Requires the construction of the Routh array to assess the number of roots with positive real parts.
- A system is stable if all elements in the first column of the Routh array are positive.
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Nyquist Stability Criterion
- Utilizes the Nyquist plot to assess the stability of a feedback system based on its open-loop frequency response.
- Relates the number of encirclements of the critical point (-1,0) in the complex plane to the number of poles in the right half-plane.
- Provides insight into gain and phase margins, indicating how much gain or phase can change before instability occurs.
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Bode Stability Criterion
- Analyzes the frequency response of a system using Bode plots to determine stability margins.
- Phase margin and gain margin are key indicators derived from the Bode plot, indicating how close the system is to instability.
- A positive phase margin and gain margin suggest a stable system, while negative values indicate potential instability.
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Root Locus Method
- A graphical technique for analyzing how the roots of a system change with varying feedback gain.
- Provides insight into system stability by showing the trajectory of poles in the complex plane as gain is adjusted.
- Helps in designing controllers by visualizing the effect of pole placement on system behavior.
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Lyapunov Stability Theory
- Focuses on the concept of Lyapunov functions to assess stability without solving differential equations.
- A system is stable if a suitable Lyapunov function can be found that decreases over time.
- Provides a method for proving stability in nonlinear systems, expanding the applicability of stability analysis.
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Phase Margin and Gain Margin
- Phase margin is the additional phase lag at which the system becomes unstable, measured at the gain crossover frequency.
- Gain margin is the amount of gain increase that can be tolerated before the system becomes unstable, measured at the phase crossover frequency.
- Both margins are critical for assessing robustness and stability in control systems.
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Pole-Zero Analysis
- Involves examining the locations of poles and zeros in the complex plane to determine system behavior.
- The relative positions of poles and zeros influence the stability and transient response of the system.
- A system is stable if all poles are in the left half-plane; zeros can affect the system's response but do not directly determine stability.
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State-Space Stability Analysis
- Uses state-space representation to analyze the stability of dynamic systems through eigenvalues of the system matrix.
- Stability is determined by the location of eigenvalues in the complex plane; all eigenvalues must have negative real parts for stability.
- Provides a comprehensive framework for analyzing multi-input multi-output (MIMO) systems.
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Jury Stability Test (for Discrete-Time Systems)
- A method for determining the stability of discrete-time systems by analyzing the characteristic polynomial's roots.
- Involves constructing the Jury array and checking the conditions for stability based on the array's elements.
- A system is stable if all roots lie within the unit circle in the z-plane.
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Circle Criterion
- A graphical method for assessing the stability of nonlinear control systems using the Nyquist plot.
- Involves drawing a circle in the complex plane to determine the stability region for feedback systems.
- Provides a visual representation of how the system's gain and phase affect stability, particularly in the presence of nonlinearities.