Automation & Control: Stability of Linear Systems

The roots of the numerator of a transfer function, representing the values of 's' that make the numerator of the transfer function equal to zero.

The roots of the denominator of a transfer function, representing the values of 's' that make the denominator of the transfer function equal to zero.

A visual representation of the poles and zeros of a transfer function in the complex 's' plane. Poles are typically marked as crosses and zeros as circles.

A system's ability to return to a stable equilibrium point after being disturbed. A stable system eventually settles down after an initial disturbance.

A system that, after an initial disturbance, exhibits oscillations that continue indefinitely without decaying.

A system that, after an initial disturbance, diverges from its equilibrium point and continues to grow without bound.

A mathematical tool used to analyze and design control systems. It converts time-domain signals into frequency-domain representations. Allows us to analyze system stability and frequency response.

The process of finding the roots of a polynomial, often used to determine the poles and zeros of a transfer function.

A system's stability is determined by the location of its closed-loop poles on the s-plane. If all poles are in the left half of the s-plane, the system is stable.

The characteristic equation of a closed-loop system is used to determine the system's stability. The roots of this equation correspond to the poles of the system.

The Root Locus is a graphical tool that visualizes the movement of a system's poles as the gain (k) of a compensator changes.

A compensator is a device that is added to a feedback control system to modify its performance and achieve desired characteristics.

The closed-loop transfer function represents the overall behavior of a feedback control system, taking into account the effects of the plant, feedback, and compensator.

A system is considered stable if all the coefficients of its characteristic equation are positive. However, even if all coefficients are positive, it doesn't guarantee stability. The roots of the characteristic equation must all be in the left half of the s-plane for the system to be stable.

A stable system will return to its equilibrium state after a disturbance.

An unstable system's output will diverge and increase indefinitely after a disturbance.

The point on the real axis of the root locus where the closed-loop poles transition from real to complex values. This is often the point where the system becomes underdamped.

The responsiveness of a system to an input. It is often measured by the time it takes for a system to reach a steady state after a disturbance.

A measure of damping in a system. It determines how quickly oscillations decay after an input.

The time it takes for a system's response to settle to within a certain percentage of its final value.

The time it takes for a system to reach its first peak after an input.

The percentage overshoot is the difference between the peak value and the steady state value, expressed as a percentage of the steady state value.

The frequency of oscillations in an underdamped system. It is important to note that this not the natural frequency of the system.

The condition where the denominator of the closed-loop transfer function becomes zero, indicating the location of a closed-loop pole.

The gain factor that is multiplied by the open-loop transfer function, 'k', affecting the location of the closed-loop poles on the root locus.

The factor by which the gain factor 'k' can be multiplied before the closed-loop system becomes unstable. It is found from the root locus where it crosses the imaginary axis (jω axis).

The process of plotting the root locus, where the gain factor 'k' is varied and the closed-loop poles are plotted on the complex plane, helping analyze system stability.