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The root locus reaches infinity at the poles and zeros of the open-loop transfer function. The root locus will start at open-loop poles and end at open-loop zeros as the gain increases from zero to infinity.

Root Locus is used in designing PID controllers by observing how the addition of Proportional, Integral, and Derivative terms affect the system's poles and zeros, thus shaping the root locus and the system's response.

Breakaway points occur where multiple root locus branches meet and separate from the real axis. It represents a value of gain where the system’s poles move off the real axis and can impact the stability of the system.

Asymptotes indicate the directions in which the branches of a root locus diagram will travel to infinity. The angles of the asymptotes are determined by the formula $\theta = (2k+1)\frac{180^\circ}{p-z}$ where $p$ is the number of poles, $z$ is the number of zeros, and $k$ is an integer.

Adding poles or zeros to a system changes the root locus and therefore the system's stability. Poles pull the root locus to the right (towards instability), while zeros pull it to the left (towards stability).

The root locus is a graphical representation in the s-plane of the roots of a transfer function's characteristic equation as a system parameter is varied, typically the gain. It is used to analyze and predict the stability and transient behavior of a control system.

The centroid is the point on the real axis where the asymptotes intersect. It is calculated by the formula $\sigma = \frac{\sum poles - \sum zeros}{p-z}$, providing insights regarding the root locus' overall direction.

Natural frequency refers to the frequency at which a system will oscillate in absence of damping. Lines of constant natural frequency are also plotted on the root locus, providing insight into the oscillatory behavior of the system as the gain changes.

Sensitivity of a control system to parametric changes is depicted on the root locus plot. The plot shows how sensitive the system's poles are to changes in system parameters like the gain.

The closed-loop poles of a control system are the poles of the transfer function when feedback is applied. The root locus method directly indicates the movement of these poles for different system gains, illustrating the effect on closed-loop system stability.

The root locus helps determine system stability. If the locus is in the left half of the s-plane (negative real parts), the system is stable; if it crosses into the right half (positive real parts), the system can become unstable.

Gain margin is the amount of gain increase or decrease required to make the system unstable. It can be determined from the root locus by finding the gain at which the root locus crosses the imaginary axis.

The angle of departure is the angle at which a root locus branch leaves a complex pole, and the angle of arrival is the angle at which a branch approaches a complex zero. These angles are calculated using the principle of phase angle equality at the locations of poles and zeros.

Phase margin is the amount of additional phase lag at the gain crossover frequency that can be tolerated before the system becomes unstable. It is represented on the root locus by the angle to the critical point at the -1 position on the Nyquist plot.