Root Locus of Second-Order Feedback
Follow feedback poles through a repeated root and connect them to step-response damping.
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Theory of Second-Order Feedback
A unity negative feedback loop with a plant transfer function L(s) = K / [s(s+2)] has a closed-loop characteristic equation of s² + 2s + K = 0. The roots of this equation dictate the stability and transient behavior of the system as the gain K varies.
Example: Calculating Poles and Damping
Let's trace the poles and the step response when Gain K = 2 and Time = 1.
- Given: K = 2, t = 1. The characteristic equation is s² + 2s + 2 = 0.
- The roots are s = -1 ± √(1 - 2) = -1 ± i. The expected roots are a conjugate pair with a real part of -1 and imaginary magnitude of 1.0.
- The damping ratio (ζ) is 1 / √K = 1 / √2 ≈ 0.707106.
- The natural frequency (ωn) is √K = √2.
- The damped natural frequency (ωd) is ωn√(1 - ζ²) = √2√(1 - 0.5) = 1.
- The step response y(t) formula for K > 1 is 1 - e^(-t) [cos(ωd*t) + (ζ*ωn/ωd)*sin(ωd*t)].
- At t = 1, y(1) = 1 - e^(-1) [cos(1) + sin(1)].
- Calculating the expected result: y(1) = 1 - 0.367879 * (0.540302 + 0.841470) = 1 - 0.508326 = 0.491674.
- The expected overshoot fraction is e^(-π*ζ/√(1-ζ²)) = e^(-π) = 0.043213.
The step response reaches 0.491674 at t = 1, while the roots are a conjugate pair at -1 ± i. This indicates an underdamped system.
Experiment: Tuning the Gain
Use the interactive module to observe how the poles move and the step response changes.
- Set Gain K to 0.5 and observe that the roots are real and distinct (-0.293 and -1.707). The response is overdamped.
- Set Gain K to 1. The roots converge to a repeated pole at -1. The system is critically damped.
- Set Gain K to 2. The roots split vertically into a conjugate pair (-1 ± i). The response becomes underdamped and exhibits overshoot.
- Adjust Time to see how the transient response evolves over time.
Limitations
This model assumes an ideal, linear, time-invariant plant L(s) = K / [s(s+2)] and unity negative feedback. In physical systems, there might be higher-order dynamics, actuator saturation, or time delays that are not captured by this simple second-order model. For K = 0, the transfer function yields a zero response, but the internal plant modes retain a marginal pole at the origin, meaning the system is internally only marginally stable, a detail hidden if only looking at the input-output transfer function.
References
The underlying principles of root locus and second-order system response are fundamental topics in classical control theory.
- MIT OCW 6.302 Feedback Systems, Recitation 7 - Provides background on root locus techniques and their application to system analysis.
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Root Locus of Second-Order Feedback module.
Try Root Locus of Second-Order Feedback on Riano →