First-order step response
User-supplied measurements and assumptions; Zero-initial-state delayed first-order unit-step response K(1−e^(−(t−delay)/τ)); settle at 2% relative to nonzero final magnitude.
Control systems and dynamics ↗Control systems and dynamics calculators
25 calculators. Search or choose one below.
Calculators
Second-order damping response
User-supplied measurements and assumptions; Zero-state unit-step response of Kωn²/(s²+2ζωns+ωn²), with analytic underdamped, critical and overdamped branches.
Control systems and dynamics ↗Second-order poles from damping
User-supplied measurements and assumptions; Roots of s²+2ζωns+ωn², sorted by real then imaginary part.
Control systems and dynamics ↗Transfer-function pole-zero map
User-supplied measurements and assumptions; Numerical polynomial roots, degrees ≤6; flag near pole-zero pairs without canceling them or claiming internal stability.
Control systems and dynamics ↗Control DC gain
User-supplied measurements and assumptions; Evaluate numerator constant/denominator constant; denominator zero explicitly undefined without symbolic cancellation.
Control systems and dynamics ↗Bode frequency response
User-supplied measurements and assumptions; Evaluate G(jω); output magnitude 20log10|G| and phase unwrapped over increasing positive frequency samples.
Control systems and dynamics ↗Nyquist response coordinates
User-supplied measurements and assumptions; Evaluate transfer function on positive imaginary axis; no Nyquist encirclement or stability conclusion from finite samples.
Control systems and dynamics ↗Sampled gain and phase crossover margins
User-supplied measurements and assumptions; Detect bracketed 0 dB and odd-180° crossings; interpolate dB and unwrapped phase against log frequency.
Control systems and dynamics ↗Root-locus sampled gains
User-supplied measurements and assumptions; Negative unity-feedback characteristic polynomial D(s)+K N(s); evaluate numerical roots at supplied gains, not continuous branch tracking.
Control systems and dynamics ↗Closed-loop feedback composition
User-supplied measurements and assumptions; Output/reference = PC/(1−sign×PCH); multiply rational polynomial coefficients without unstable pole-zero cancellation.
Control systems and dynamics ↗State-space eigenvalue stability
User-supplied measurements and assumptions; Continuous: all eigenvalue real parts <0; discrete: all magnitudes <1.
Control systems and dynamics ↗State-space controllability rank
User-supplied measurements and assumptions; Form [B,AB,...,A^(n−1)B]; Gaussian elimination with supplied absolute pivot tolerance; ratio of retained elimination pivots is a scale diagnostic, not a condition number.
Control systems and dynamics ↗State-space observability rank
User-supplied measurements and assumptions; Form vertical [C;CA;...;CA^(n−1)]; Gaussian rank with supplied absolute pivot tolerance; pivot-scale ratio is diagnostic only.
Control systems and dynamics ↗Discrete state-space time response
User-supplied measurements and assumptions; Supplied discrete linear model: y[k]=Cx[k]+Du[k], then x[k+1]=Ax[k]+Bu[k]; no implicit continuous-time discretization.
Control systems and dynamics ↗Continuous-to-discrete model
User-supplied measurements and assumptions; Zero-order hold via exponential of augmented block [[A,B],[0,0]]×sample time; works with singular A.
Control systems and dynamics ↗Forward-Euler discrete-to-continuous approximation
User-supplied measurements and assumptions; Invert forward-Euler approximation: Ac≈(Ad−I)/Ts, Bc≈Bd/Ts; explicitly not exact matrix-logarithm conversion.
Control systems and dynamics ↗Positive-gain PID series-parallel conversion
User-supplied measurements and assumptions; Series K(1+1/(Ti s))(1+Td s) expands to parallel Kp+Ki/s+Kd s.
Control systems and dynamics ↗PID sampled first-order plant response
User-supplied measurements and assumptions; Sampled parallel PID with backward rectangular integration and derivative on measured output; exact zero-order-hold first-order plant update; output saturation, no anti-windup.
Control systems and dynamics ↗PID first-order integral windup comparison
User-supplied measurements and assumptions; Compare identical saturated sampled first-order PID simulations with and without conditional integration; freeze integration when its update pushes farther into saturation.
Control systems and dynamics ↗Control sensor low-pass tradeoff
User-supplied measurements and assumptions; Compare analog H=1/(1+j f/fc) with matched-pole sampled H(z)=(1−α)/(1−αz⁻¹), α=e^(−2πfc/fs); require both cutoff and signal below Nyquist.
Control systems and dynamics ↗Control sample-delay phase loss
Loop frequency, sample-transport delay.
Control systems and dynamics ↗Control deadband quantization model
Memoryless symmetric dead-zone followed by uniform quantization.
Control systems and dynamics ↗Control stable-loop ramp error
Unity negative feedback rational open-loop transfer function with ascending-power coefficients.
Control systems and dynamics ↗Control disturbance rejection
Plant-controller models, disturbance location.
Control systems and dynamics ↗Control actuator saturation duty
Command trace, actuator bounds.
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