Nyquist Stability Criterion

The Nyquist Stability Criterion is a key graphical tool in control theory for assessing system stability. It connects the open-loop transfer function's frequency response to stability, using encirclements of the -1 point to reveal insights about system behavior.

1. Definition of the Nyquist Stability Criterion

   • A graphical method used to determine the stability of a control system.
   • Relies on the frequency response of the open-loop transfer function.
   • Provides a relationship between the number of encirclements of the -1 point and the number of right-half plane poles.

2. Open-loop transfer function

   • Represents the system's response without feedback.
   • Typically denoted as ( G(s)H(s) ), where ( G(s) ) is the plant and ( H(s) ) is the controller.
   • Essential for analyzing system behavior in the frequency domain.

3. Nyquist contour

   • A closed path in the complex plane used to evaluate the open-loop transfer function.
   • Typically consists of a semicircular arc in the right-half plane and a straight line along the imaginary axis.
   • Ensures that all frequencies are accounted for in the stability analysis.

4. Nyquist plot construction

   • Involves plotting the open-loop transfer function ( G(j\omega)H(j\omega) ) as ( \omega ) varies from (-\infty) to (+\infty).
   • The plot shows how the gain and phase of the system change with frequency.
   • Important for visualizing the system's stability characteristics.

5. Encirclements of the -1 point

   • The number of times the Nyquist plot encircles the point (-1, 0) in the complex plane indicates stability.
   • Each clockwise encirclement corresponds to a right-half plane pole.
   • The Nyquist criterion states that the number of encirclements must equal the number of right-half plane poles for stability.

6. Right-half plane poles

   • Poles of the open-loop transfer function located in the right-half of the complex plane indicate instability.
   • The presence of these poles affects the stability of the closed-loop system.
   • The Nyquist criterion accounts for these poles when determining overall system stability.

7. Gain and phase margins

   • Gain margin is the amount of gain increase that can be tolerated before the system becomes unstable.
   • Phase margin is the additional phase lag at the gain crossover frequency before instability occurs.
   • Both margins provide insight into the robustness of the control system.

8. Stability determination

   • Stability is determined by analyzing the Nyquist plot and the number of encirclements of the -1 point.
   • A system is stable if the number of encirclements equals the number of right-half plane poles.
   • The analysis helps predict how the system will respond to disturbances.

9. Closed-loop system stability

   • Refers to the stability of the system when feedback is applied.
   • The Nyquist criterion helps assess how feedback affects the overall system stability.
   • A stable closed-loop system will return to equilibrium after a disturbance.

10. Limitations and assumptions

    • Assumes linear time-invariant (LTI) systems for analysis.
    • Requires that the open-loop transfer function is rational and does not have infinite gain at any frequency.
    • The Nyquist criterion may not apply to systems with non-minimum phase behavior or significant time delays.