Set open-loop poles and zeros, sweep gain K, and watch closed-loop poles trace the s-plane in real time. Asymptotes, stability margin, and step response computed automatically.
System Definition
Presets
Open-Loop Poles (up to 4)
Open-Loop Zeros (up to 3)
Gain K
Results
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Damping Ratio ζ
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Natural Freq. ωn [rad/s]
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Overshoot [%]
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Settling Time Ts [s]
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Critical Gain K_crit
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Dominant Pole (Re)
Rlocus
Step
CAE Note
Root locus is the first step in designing active vibration control or servo control for mechanical systems. Follow up with MATLAB Control System Toolbox or the open-source Python-control library for detailed design including Bode plots and robust stability margins.
Theory & Key Formulas
Start: K=0 → open-loop poles
End: K→∞ → open-loop zeros (remainder to ∞ along asymptotes)
Stability: all closed-loop poles in left half-plane
What is Root Locus?
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What exactly is a "root locus" plot? It sounds complicated.
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Basically, it's a map showing how the poles of a closed-loop system move as we change the controller's gain. In this simulator, when you move the Gain K slider, you're watching those poles (the 'roots') trace out paths on the complex plane. Where they end up tells us if the system will be stable and how fast it responds.
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Wait, really? So the lines I see are the possible locations for the system's poles? What do the starting and ending points mean?
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Exactly! The rules are simple: when K=0, the closed-loop poles start at the open-loop poles (the red 'X's). When K→∞, they end at the open-loop zeros (the blue 'O's). If there are more poles than zeros, the extra branches shoot off to infinity along asymptotes. Try setting a system with three poles and no zeros in the simulator and watch this happen.
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That makes sense. But how do I know if the system becomes unstable? Is it when the poles cross that dashed imaginary axis?
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You've got it! Stability is determined by pole location. A pole in the left-half plane (negative real part) is stable. The moment any branch crosses the imaginary axis into the right-half plane, the system becomes unstable for that gain. In practice, like in tuning a drone's autopilot, you use the root locus to find the maximum safe gain before instability. Watch the crosshairs on the plot in the simulator as you increase K.
Physical Model & Key Equations
The root locus plots the roots of the closed-loop characteristic equation as the gain K varies from 0 to infinity. This equation comes from the basic feedback loop.
Here, G(s) is the open-loop transfer function, K is the scalar gain you adjust in the simulator, and s is the complex frequency variable. The roots (s) that satisfy this equation for each K are the closed-loop poles.
When the number of poles (n) exceeds zeros (m), (n-m) branches go to infinity. Their direction is governed by asymptote angles and a centroid on the real axis.
θk are the asymptote angles, σa is the centroid, pi are the open-loop poles, and zj are the open-loop zeros. These are the dashed lines and the star you see on the simulator plot for systems like G(s)=1/[s(s+2)(s+4)].
Real-World Applications
Active Vibration Control: In precision manufacturing or aerospace structures, unwanted vibrations can ruin a process. Engineers use root locus to design a controller that places closed-loop poles in locations that actively dampen these vibrations, which you can experiment with by placing a pair of complex poles in the simulator.
Servo Motor Positioning: From robotic arms to CNC machines, precise motor control is critical. The root locus helps balance the trade-off between fast response (poles far left) and overshoot (poles with low damping). Try adding a zero in the simulator to "bend" the locus left for better performance.
Automotive Cruise Control: The system must maintain speed despite hills and wind disturbances. Root locus analysis is used to select a gain that provides steady tracking without oscillatory or sluggish behavior, ensuring a comfortable ride.
Flight Control Systems (Fly-by-Wire): Modern aircraft rely on electronic stability augmentation. Engineers use root locus to ensure the aircraft's pitch and roll responses remain stable and well-damped across all flight conditions, a life-critical application of the stability boundary you see in the simulator.
Common Misconceptions and Points to Note
There are a few key points you should be aware of when starting to use this tool. First, limiting yourself to placing poles and zeros only on the real axis is a missed opportunity. With poles only on the real axis, the response becomes non-oscillatory and lacks overshoot. However, in practice, when you need a fast response, you often have to tolerate some oscillation (damped oscillation) and aim for complex conjugate poles. For example, try using this tool to compare how the step response changes between placing poles at -2 and -3 on the real axis versus placing them at -2 ± 2j.
Next, the misconception that "any zero improves stability". While zeros in the left-half plane (stable zeros) do have the effect of speeding up the response, zeros in the right-half plane (unstable zeros) are very tricky to handle. For instance, with poles at -1 and -5, if you place a zero at +2 (right-half plane) and increase K, you can observe how the locus quickly enters the right-half plane, leading to instability. This is called a non-minimum phase system and is the cause of phenomena like inverse response in real systems.
Finally, avoid thinking that "the root locus tells you everything". The root locus is excellent for visualizing transient response characteristics (stability, frequency, damping), but it doesn't directly tell you about performance aspects like disturbance rejection or robustness (strength against model errors). In practice, the standard approach is to determine a rough gain using the root locus and then verify gain margin and phase margin using frequency response analysis with Bode plots.