Root Locus

Introduction :

The root locus is a graphical method for analyzing the behavior of a control system as a parameter, typically a proportional gain (K), varies. It provides insights into the closed-loop system's stability and transient response. The root locus technique is an essential tool for control system design and analysis.

Key Definitions

• Open-Loop Transfer Function (OLTF) : The transfer function of the system without any feedback, typically denoted as G(s).
• Closed-Loop Transfer Function (CLTF) : The transfer function of the system with feedback, typically denoted as T(s).
• Root Locus : A graphical representation of the possible locations of the closed-loop poles as the parameter (usually the proportional gain) varies.
• Pole : A point in the complex plane where the denominator of the transfer function becomes zero. Poles determine the system's stability and transient response.

The Basics of Root Locus

$1+KG(s)H(s)=0$

• K is the proportional gain.
• G(s) is the OLTF.
• H(s) is the feedback transfer function.

Rules and Conventions

1. The number of branches in the root locus is equal to the number of poles of G(s)H(s).
2. The root locus starts at the poles of G(s)H(s) and ends at the zeros of G(s)H(s).
3. The root locus branches move towards the finite zeros of G(s)H(s).
4. The root locus approaches asymptotes that intersect at a point in the complex plane known as the centroid.
5. The angles of asymptotes can be calculated as:

​ θa = (2n+1)Π/N Where:

• • θa is the angle of an asymptote.
  • N is the number of poles (or zeros) to the left of the point where the asymptote starts.
  • n is an integer ranging from 0 to N-1.

6. The location of breakaway and re-entry points can be found by differentiating the characteristic equation with respect to K.

Interpreting the Root Locus

Stability

• If all the poles of the closed-loop system lie in the left-half of the complex plane (real parts are negative), the system is stable.
• If any pole crosses the imaginary axis into the right-half plane (real part is positive), the system becomes unstable.

Performance

• The root locus provides information about the system's transient response. Branches closer to the origin indicate faster response.
• The damping ratio and natural frequency of the system can be estimated based on the root locus.

Conclusion

The root locus method is a powerful tool for understanding and designing control systems. By examining the behavior of the system as a parameter changes, engineers can make informed decisions about control system design and performance. The graphical representation of root locus allows for quick and intuitive analysis, making it an essential technique in control engineering.