PID Controller Tuning Simulator

The process models use $G(s)=k e^{-\theta s}/(a_2s^2+a_1s+1)$. For first order, $a_2=0$ and $a_1=T$. For second order, $a_2=1/\omega_n^2$ and $a_1=2\zeta/\omega_n$. The DC motor follows Newton's and Kirchhoff's laws: $G_{\omega V}(s)=K_t/[(Js+b)(Ls+R)+K_tK_e]$. The voltage and speed bases convert this to a normalized gain. The motor equations and sample constants follow CTMS; the equations are also derived directly here.

The parallel feedback controller is $C(s)=K_p+K_i/s+K_ds/(1+T_fs)$. In the time simulation, derivative acts on output, so $u_*=K_p(r-y)+I-K_d\dot y_f$. A backward-Euler derivative filter and trapezoidal error integration run once per controller sample. Saturation clips the command. Conditional anti-windup rejects an integral increment only when its error direction would push a saturated command farther into its limit. It resumes integration when the error reverses. The input disturbance is added after saturation and before the transport delay.

Plant state transitions under each held input are evaluated analytically. A fractional delay splits a sample into two exact held-input transitions. This treats the delay exactly for the simulated sampled command; controller sampling still requires a convergence check. Initial states are zero. The time normalization of 1 s keeps mixed numerical coefficient vectors dimensionless without concealing physical units.

SIMC PI for a first-order model uses $K_p=T/[k(\lambda+\theta)]$, $T_i=\min[T,4(\lambda+\theta)]$, $K_i=K_p/T_i$, $K_d=0$ (Skogestad 2003, Eqs. 23–24, attached PDF p. 6). Ziegler–Nichols reaction-curve PID uses $K_p=1.2T/(k\theta)$, $T_i=2\theta$, $T_d=\theta/2$, with $K_i=K_p/T_i$ and $K_d=K_pT_d$ (Åström–Murray, Table 11.1a). Filtering changes the ideal Ziegler–Nichols response.

The continuous frequency response includes exact transport delay. Phase is calculated as the controller's analytic atan2 phase minus the stable plant-denominator phase and $\omega\theta$, anchored to the model's low-frequency branch. It therefore does not depend on the sweep's starting frequency. The unity-gain frequencies come from the real polynomial in $\omega^2$ obtained from $|C(j\omega)G(j\omega)|^2=1$.

For gain margin, zeros of the analytic phase derivative partition frequency into monotonic intervals. Every negative-real-axis level $-180^\circ+360^\circ m$ is checked in both directions within the selected range. Each crossing is refined by 34 log-frequency bisections. The signed gain margin closest to 0 dB and signed phase margin closest to 0° are returned with their crossover frequencies, following the MathWorks margin definition. Phase margin is the signed angular distance to the nearest odd-180° branch. These scalar margins do not establish internal stability. An absent crossing is reported as unavailable in the selected range. A range containing 1000 or more phase-crossing candidates is rejected rather than reporting a partial search. Peak sensitivity still uses the 601-point frequency grid.

The phase-view selector provides phase-crossover detail, gain-crossover detail, and the full unwrapped range. Focused views restrict the neighborhood of the selected crossover, including no more than roughly 180° of transport-delay phase variation, and draw the crossover markers and phase-margin arrow against the appropriate odd-180° guide. A crossover outside the displayed neighborhood remains reported in the header.

The simulated controller sample period is $h=T_\mathrm{span}/N$. Changing the interval count or duration changes the actual sampled-controller model, not just a plant integration mesh. The plant transition between commands is analytical. Resolution warnings check plant poles, the derivative filter, and $h\omega_\mathrm{control}>0.2$, where the control frequency scale is the largest unity-gain crossover (including outside the plot) or largest rational closed-loop pole magnitude. This last threshold is a conservative numerical-resolution guideline for comparison with a continuous design, not a sampled-loop stability theorem.

Root locus solves $D(s)+\alpha N(s)=0$ for 61 multipliers from 0 to 10, keeping the filter time fixed and scaling every gain. A first-order Padé factor $(1-\theta s/2)/(1+\theta s/2)$ approximates delay only for this pole analysis. Uncancelled modes are retained. Dots show all sampled roots without implying branch connectivity; rings mark the current gain, and grey rings mark open-loop poles. Roots outside the chosen square are omitted.

Signals are deviations from a zero operating point. Plant gains and setpoints are positive; feedback is unity and negative. All process signals use per-unit scaling. For the motor, actual voltage equals command × voltage base, and actual speed equals output × speed base. Motor back EMF and torque constants are independently entered in SI units. Radians are dimensionless in the mechanical equations.

The models are linear and time invariant. The only actuator nonlinearity included is command saturation. The disturbance has the same units as normalized command and represents an additive plant-input load, not a motor shaft-torque load. There is no derivative action on setpoint. Anti-windup is conditional integration, with no hidden back-calculation gain.

Only first-order plants receive automatic tuning. Second-order and DC motor choices use the visible manual gains. The controller is parallel; SIMC is PI here, so no series-to-parallel PID conversion is needed. Time is always in seconds, angular frequency in rad/s; the motor mechanical-unit selector converts compound SI and US units explicitly. Other compatible fields use the catalog's normal unit display.

Use positive, stable first- or second-order self-regulating plants and the stated rigid-rotor DC motor model. This does not model integrating plants, unstable plants, inverse response, current-limited drives or cascaded control loops.

Continuous Bode margins and Padé poles describe the unsaturated continuous controller. The sampled, saturated transient can behave differently. Positive displayed margins alone do not certify stability or robustness to unmodeled dynamics. First-order Padé poles can be inaccurate near delay-dominated instability; use exact-delay frequency analysis and the sampled response together. No-crossing messages apply only to the chosen frequency range. Peak sensitivity may miss a very narrow peak.

Resolve the fastest plant, controller and closed-loop time scales, as well as the derivative filter. Warnings request a smaller controller sample period. Increasing the interval count or shortening the duration changes that sample period and therefore changes the simulated controller. Repeat with twice as many intervals when checking convergence toward the continuous design. A sample-and-hold command is assumed, with zero-order-hold plant evolution; continuous-controller reference values differ slightly at finite sampling. The simulation stops filling new dynamic states if the output exceeds 10⁸ p.u., and explicitly warns before that display limit.

Overshoot, rise and settling are measured relative to the requested step, not the observed final value. A −1 s timing value in the calculation details means the threshold or settling criterion was not achieved within the observation interval. Metrics before a disturbance cannot show later disturbance rejection. The final error and IAE cover the whole run.

The CTMS university mirror labels inertia with an erroneous /s² in one list. This tool uses kg·m², consistent with inertia dimensions and the source's MATLAB model. Settings are starting points for model studies and commissioning analysis; validate against the actual plant and controller implementation.

The exact-model scalar margin search is limited to 999 phase-crossing candidates per selected frequency range; narrow the range if prompted. Positive scalar margins may coexist with an unstable feedback loop, especially with multiple crossings. No Nyquist winding count or disk-margin analysis is performed. The full phase plot can span many delay rotations; choose a crossover-detail view to inspect a particular margin.