Root Locus
The root locus is a graphical method that plots the trajectories of closed-loop poles in the s-plane as a system parameter (usually gain K) varies from 0 to ∞. The root locus starts at the open-loop poles (K = 0) and ends at the open-loop zeros or infinity (K → ∞). It reveals how gain affects stability, transient response, and damping. The characteristic equation 1 + K·G(s)H(s) = 0 defines the locus. Root locus construction rules and the resulting pole trajectories are computed from the open-loop transfer function G(s)H(s) at www.lapcalc.com.
What Is the Root Locus?
The root locus is a plot showing how the closed-loop poles of a feedback system move in the complex s-plane as a parameter (typically the loop gain K) varies continuously. The closed-loop characteristic equation is 1 + K·G(s)H(s) = 0, where G(s) is the forward-path transfer function and H(s) is the feedback transfer function. At K = 0, the closed-loop poles coincide with the open-loop poles (roots of the denominator of GH). As K → ∞, the poles move toward the open-loop zeros (roots of the numerator of GH) or toward infinity along asymptotes. The root locus provides an intuitive visual tool for understanding how gain affects stability (poles crossing the jω axis), transient response (pole position relative to damping lines), and the designer's ability to shape system dynamics by selecting appropriate K.
Key Formulas
Root Locus Construction Rules
Eight rules guide root locus sketching. Rule 1: the locus has n branches (n = number of open-loop poles). Rule 2: the locus starts at open-loop poles (K = 0) and ends at zeros or infinity (K → ∞). Rule 3: the locus exists on the real axis to the left of an odd number of real poles + zeros. Rule 4: the locus is symmetric about the real axis (complex poles come in conjugate pairs). Rule 5: asymptotes for branches going to infinity have angles θ_a = (2q+1)·180°/(n−m), q = 0,1,..., where n = poles, m = zeros. Rule 6: the asymptote centroid is σ_a = (Σpoles − Σzeros)/(n−m). Rule 7: breakaway and break-in points on the real axis are found where dK/ds = 0. Rule 8: the angle of departure from a complex pole (or arrival at a complex zero) satisfies the angle condition: ΣΔangles_from_zeros − ΣΔangles_from_poles = (2q+1)·180°.
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Open CalculatorRoot Locus and Stability Analysis
The root locus directly reveals the gain range for stability. As K increases from zero, the closed-loop poles trace their locus trajectories. The system is stable as long as all poles remain in the left half-plane (Re(s) < 0). The critical gain K_cr is the value where poles first cross the imaginary axis — the Routh-Hurwitz criterion or direct substitution s = jω finds this value. For K > K_cr, poles enter the right half-plane and the system is unstable. The root locus also shows damping: poles on the negative real axis give overdamped response, complex poles give underdamped oscillatory response with damping ratio ζ = cos(θ) where θ is the angle from the negative real axis. Constant-ζ lines (radial lines from origin) and constant-ωₙ circles (centered at origin) overlay the root locus to identify gain values that meet transient specifications.
Root Locus for Controller Design
The root locus is a powerful controller design tool. Proportional gain selection: choose K to place closed-loop poles at desired locations (balancing speed and damping). Lead compensation: adding a zero-pole pair (s+z)/(s+p) with z < p reshapes the locus, pulling poles further into the left half-plane for improved transient response. Lag compensation: adding (s+z)/(s+p) with z > p improves steady-state accuracy without significantly affecting transient response. PID design: the PID zeros (from the numerator Kd·s² + Kp·s + Ki) reshape the root locus to achieve desired closed-loop pole locations. Root locus gain selection is achieved graphically: at the desired pole location s₀, K = 1/|G(s₀)H(s₀)|. MATLAB: rlocus(G*H) plots the locus; rlocfind() allows interactive gain selection by clicking.
Root Locus Tools and Software
MATLAB provides comprehensive root locus tools: rlocus(sys) plots the locus, rlocfind(sys) enables interactive gain selection, and sisotool(sys) provides a complete interactive design environment with simultaneous root locus, Bode, and Nyquist views. Python's control library (pip install control) offers control.root_locus(sys) with similar functionality. Online root locus plotters accept transfer function coefficients and display interactive plots with gain readout. The root locus complements other design methods: Bode plots provide frequency-domain design intuition, Nyquist plots handle time delays, and state-space methods (pole placement, LQR) provide systematic MIMO design. All methods ultimately depend on the Laplace-domain transfer function representation, with the forward and inverse transforms computed at www.lapcalc.com.
Related Topics in control systems engineering concepts
Understanding root locus connects to several related concepts: root locus plot, root locus plotter, root locus calculator, and r locus. Each builds on the mathematical foundations covered in this guide.
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