In control systems, stability refers to the ability of a system to return to equilibrium or remain at a desired steady state after being disturbed by an input or external disturbance. Stability is crucial for the reliable operation of systems such as motors, airplanes, industrial processes, and electrical circuits.
For linear time-invariant (LTI) systems, stability can be analyzed in both the time domain and the frequency domain. There are several criteria used to assess the stability of these systems, including:
1. Routh-Hurwitz Criterion (time-domain)
2. Nyquist Criterion (frequency-domain)
3. Bode Plot Method (frequency-domain)
This chapter will explore these methods in detail and explain how to apply them to evaluate system stability.

The Routh-Hurwitz Criterion is a time-domain method for determining the stability of a system by examining the location of the poles of its characteristic equation. A system is stable if all poles of its transfer function have negative real parts (i.e., they lie in the left half of the complex plane). The criterion provides a way to check if a system has poles in the right half of the complex plane (which would indicate instability) without solving for the poles explicitly.
Steps to Apply the Routh-Hurwitz Criterion:
1. Write the characteristic equation of the system (usually the denominator of the closed-loop transfer function).
2. Construct the Routh array:
○ The Routh array is constructed from the coefficients of the characteristic polynomial.
○ The first two rows of the array are filled with the coefficients of the even and odd powers of s, respectively.
○ Each subsequent row is calculated using the determinant of the two elements directly above it.
3. Determine the number of sign changes in the first column of the Routh array:
○ The number of sign changes corresponds to the number of poles in the right half-plane.
○ If there are no sign changes, the system is stable.
○ If there are sign changes, the system is unstable.

The Nyquist Criterion is a frequency-domain stability criterion used to determine the stability of a feedback system. It involves plotting the Nyquist plot, which is the plot of the system's frequency response G(jω) as ω varies from −∞ to +∞. The criterion helps determine if the closed-loop system will be stable based on the open-loop frequency response.
Nyquist Criterion Steps:
1. Construct the Nyquist plot:
○ Calculate the frequency response G(jω) for various frequencies ω, where ω varies from −∞ to +∞.
○ Plot G(jω) in the complex plane.
2. Count the number of encirclements of the point −1 in the Nyquist plot:
○ The system is stable if the Nyquist plot does not encircle the point −1 in the complex plane.
○ The system is marginally stable if the plot encircles −1 once in a counter-clockwise direction.
○ The system is unstable if the plot encircles −1 in a clockwise direction or does not encircle −1 as needed.

A Bode plot is a graphical method for analyzing the frequency response of a system. It consists of two plots:
1. Magnitude plot: Shows how the gain of the system varies with frequency.
2. Phase plot: Shows how the phase shift of the system varies with frequency.
The Bode plot method can be used to assess the stability and performance of the system, particularly the gain margin and phase margin.
Bode Plot Steps:
1. Plot the Magnitude Response:
○ The magnitude of the transfer function |G(jω)| is plotted on a logarithmic scale against frequency ω.
2. Plot the Phase Response:
○ The phase of the transfer function arg (G(jω)) is plotted on a logarithmic scale against frequency ω.
3. Identify the Gain Margin and Phase Margin:
○ Gain Margin: The amount by which the system’s gain can be increased before the system becomes unstable. It is found by determining the gain at the frequency where the phase is −180°.
○ Phase Margin: The additional phase lag required to bring the system to instability. It is found by determining the phase at the frequency where the magnitude is 1 (0 dB).

Each stability criterion provides valuable insights into system behavior:
● Routh-Hurwitz: Works directly with the characteristic equation and is very effective for determining stability without having to solve for the poles.
● Nyquist: Useful for analyzing closed-loop systems, particularly when you are concerned with the effects of feedback on stability.
● Bode Plot: Ideal for frequency-domain analysis and provides a direct way to visualize the system's gain and phase at various frequencies, making it useful for tuning controllers.
In many cases, engineers will use a combination of these methods to analyze system stability from different perspectives.