How this tool works
Nineteen modules cover the core undergraduate control-systems workflow, from first- and second-order transient response through classical frequency-domain design, state-space analysis, and discrete-time stability.
Charts are selected to match the engineering quantity being studied: pole, root-locus, Nyquist, and z-plane modules use complex-plane plots; Bode and margin modules use logarithmic frequency axes; time-response modules use time histories; modules without a meaningful chart report results directly instead of forcing unrelated quantities into a graph.
The PI module evaluates the actual PI-controlled first-order transfer function, including the controller zero, rather than substituting a standard second-order numerator.
Calculators and topics covered
- control systems
- feedback
- stability
- PID
- root locus
- Bode
- Nyquist
- state space
- digital control
- step response
- time constant
- damping ratio
- natural frequency
- overshoot
Core equations
first-order lag: G(s)=K/(τs+1), y(t)=KU(1−e^(−t/τ))second-order form: G(s)=ωₙ²/(s²+2ζωₙs+ωₙ²)pole-zero model: G(s)=K(s+z)/(s²+a₁s+a₀)negative feedback: T=G/(1+GH), S=1/(1+GH)Routh criterion: first-column sign changes equal the number of right-half-plane rootssteady-state error: e_ss=A/(1+Kp), A/Kv, or A/Ka for step, ramp, or parabola inputsroot-locus example: 1+K/[s(s+a)(s+b)]=0Ziegler–Nichols PID: Kp=0.6Ku, Ti=Tu/2, Td=Tu/8PI closed loop: T(s)=KpK(Ti s+1)/[τTi s²+Ti(1+KpK)s+KpK]lead/lag: C(s)=K(1+sTz)/(1+sTp)first-order Bode: |G|=K/√(1+(ωτ)²), ∠G=−atan(ωτ)margins: GM=1/|L(jωpc)|, PM=180°+∠L(jωgc)state space: ẋ=Ax+Bu, y=Cx; controllability=[B AB], observability=[C;CA]canonical state feedback: K=[ωₙ²−a₀, 2ζωₙ−a₁]sampling: z=e^(sT), fN=fs/2discrete first order: y[k+1]=a y[k]+b u[k]second-order Jury conditions: |a₀|<1, 1+a₁+a₀>0, 1−a₁+a₀>0
Worked examples
PI control of a first-order thermal plant
A heater behaves as a first-order plant with a DC gain of 2 °C per unit of controller output and a time constant of 4 s. Open loop it takes about 12 s to reach 95% of a setpoint change and always leaves a steady-state offset. A PI controller is added to remove the offset and speed the loop up.
Inputs- Plant: K = 2, τ = 4 s, so G(s) = 2/(4s + 1)
- Controller: K_p = 8, T_i = 0.2 s
- Unit step setpoint change
Method- The loop gain is K_p·K = 16. Closing a PI loop on a first-order plant gives the second-order characteristic polynomial τT_i s² + T_i(1 + K_pK)s + K_pK.
- That maps onto the standard form with ωₙ = √(K_pK/(τT_i)) = 4.472 rad/s and ζ = 0.4752.
- Because ζ < 1 the response is underdamped, with a damped frequency of ω_d = ωₙ√(1 − ζ²) = 3.935 rad/s.
- Overshoot follows M_p = exp(−πζ/√(1 − ζ²)) = 18.33%, peaking at t_p = π/ω_d = 0.798 s.
- The 2% settling time is approximately 4/(ζωₙ) = 1.882 s.
- The integrator drives steady-state error to zero for a step, so the loop settles exactly on the setpoint.
Result: The PI loop settles within 2% in 1.88 s with 18.3% overshoot and no steady-state offset, against roughly 12 s and a permanent offset for the plant alone.
Interpretation: The offset is gone because the integrator adds a pole at the origin, making this a type 1 loop. The 18.3% overshoot is the price of the aggressive integral time; lengthening T_i to 0.3 s at the same K_p raises ζ to 0.582 and cuts overshoot to 10.6% while leaving the settling time essentially unchanged at 1.88 s, which is usually the better trade for a thermal process. Note this treatment ignores actuator saturation — a real heater cannot deliver negative power, so a loop tuned this tightly will integrator-wind-up on a large setpoint step unless anti-windup is implemented.
Method and assumptions
Assumptions
- Models are linear and time invariant unless a module explicitly describes a sampled recursion; actuator saturation, dead zones, backlash, quantisation, and rate limits are not represented.
- Time-response calculations assume zero initial conditions and ideal step inputs.
- The Routh module accepts real polynomials up to eighth order and handles a zero pivot with an epsilon substitution and a complete zero row with the auxiliary-polynomial derivative method.
- The root-locus module uses the specific unity-feedback open-loop model K/[s(s+a)(s+b)] so the plotted branches have an unambiguous physical interpretation.
- The lead/lag and Bode modules evaluate ideal rational transfer functions without transport delay.
- Gain/phase margins and the Nyquist module use L(s)=K/[s(τ₁s+1)(τ₂s+1)].
- The state-space controllability and observability module is restricted to a two-state SISO model.
- State-feedback pole placement uses controllable canonical form; arbitrary A and B matrices require a general pole-placement algorithm.
- Digital modules assume uniform periodic sampling and ideal arithmetic.
Limitations and design boundaries
- This workbench is intended for teaching and preliminary analysis, not final safety-critical controller validation.
- MIMO loop-shaping, observers/Kalman filters, LQR/LQG, H-infinity synthesis, nonlinear control, adaptive control, and model-predictive control are not yet implemented.
- The Nyquist module focuses on a representative stable-plant loop form and does not automatically count arbitrary open-loop right-half-plane poles.
- Controller tuning must still be validated against actuator limits, sensor noise, plant uncertainty, delays, and operating-point changes.
- Numerical plots use finite sweeps and double-precision arithmetic; they are not substitutes for a dedicated simulation environment when high-order or stiff models are involved.
Sources and references
Primary sources are preferred for ratings, standards, manufacturer data, and externally defined constants.
- Ogata, Modern Control EngineeringTransient response, Routh–Hurwitz stability, root locus, state-space analysis, and pole placement.
- Nise, Control Systems EngineeringPole-zero interpretation, steady-state error, frequency response, gain/phase margins, and compensator design.
- Franklin, Powell and Emami-Naeini, Feedback Control of Dynamic SystemsSensitivity, state-space control, digital control, and frequency-domain design.
- Åström and Murray, Feedback SystemsFeedback principles, robustness, loop shaping, state-space models, and digital implementation.
- Åström and Hägglund, PID Controllers: Theory, Design, and TuningPID structure, practical tuning, and limitations of Ziegler–Nichols rules.
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