Root Locus

The root locus technique, developed by Walter R. Evans in 1948, is a powerful graphical tool that illustrates the movement of system poles in the complex plane as a parameter varies. This method is crucial for stability analysis and control system design.

Fundamental Concepts

The root locus plot shows all possible locations of closed-loop poles for a system with the transfer function:

T(s) = K·G(s)/(1 + K·G(s))

where:

• K is the variable gain parameter
• G(s) is the transfer function
• T(s) is the closed-loop transfer function

Construction Rules

The root locus follows several key rules:

1. Starting Points: Branches begin at poles
2. Ending Points: Branches terminate at zeros or infinity
3. Symmetry: The plot is symmetric about the real axis
4. Real Axis Segments: Portions of the real axis where an odd number of poles and zeros exist to the right
5. Asymptotes: For systems with more poles than zeros, branches extend to infinity along asymptotic lines

Analysis Features

Stability Assessment

• Points in the left half-plane indicate stable systems
• Crossing points on the imaginary axis mark stability boundaries
• Right half-plane locations represent unstable systems

Performance Indicators

1. Natural Frequency: Distance from origin
2. Damping Ratio: Cosine of angle from negative real axis
3. Settling Time: Related to real part of poles
4. overshoot: Determined by damping ratio

Applications

Control System Design

• PID controller tuning
• gain selection optimization
• compensation design

Industrial Uses

• process control
• servo systems design
• aircraft control
• robot manipulators

Digital Implementation

Modern analysis often employs computer-aided tools:

• MATLAB Control System Toolbox
• Simulink modeling
• Python control analysis packages

Limitations and Considerations

1. Assumptions

   • linear time-invariant systems behavior
   • Single parameter variation
   • causality system requirements

2. Practical Constraints

   • numerical accuracy
   • visualization limits
   • model uncertainty

Advanced Topics

• multivariable root locus
• discrete root locus
• parameter space methods
• robust control assessment

See Also

• Nyquist stability criterion
• Routh-Hurwitz stability
• frequency response
• state feedback

The root locus method remains a fundamental tool in control system analysis and design, bridging classical control theory with modern computational approaches. Its visual nature makes it particularly valuable for understanding system behavior and making informed design decisions.