Dynamics & Control · Lab
Control Design Lab
Put a controller in front of a plant and see exactly what it costs and what it buys. Every number here comes from the same time, frequency, stability and root-locus engines used elsewhere — nothing is tuned automatically, and nothing is declared optimal.
Plant G(s) = N(s) / D(s)
Controller C(s)
The slider is logarithmic from 10⁻³ to 10³; the field accepts any value.
The slider is logarithmic from 10⁻³ to 10³; the field accepts any value.
The slider is logarithmic from 10⁻³ to 10³; the field accepts any value.
Term Kd·s/(1 + s/N). Filter time constant τ = 1/N = 0.05 s; limiting high-frequency derivative gain Kd·N = 2.
Design requirements
Requirements are treated as constraints, not as a promise. Each enabled requirement is checked against the numbers the engines produced for the controller you chose. Several requirements can be mutually unsatisfiable; nothing here searches for a design.
Closed-loop verdict
| Closed-loop poles | -20.01, -2.122, -1.057, 0.09283 + 0.6609j, 0.09283 − 0.6609j |
|---|---|
| Right-half-plane poles | 2 |
| Dominant pole(s) | 0.09283 + 0.6609j, 0.09283 − 0.6609j |
| Steady-state step error | Not defined |
| Phase margin | -17.39 ° |
| Gain margin | No phase crossover |
The verdict is the direct classification of the roots of 1 + C(s)G(s)H(s) = 0 for this exact configuration — not an inference from the root-locus crossing scan.
The closed loop is not asymptotically stable, so no finite steady-state error exists.
Controller
C(s) = Kp + Ki/s + Kd·s/(1 + s/N)
| Controller numerator | 0.15s² + 1.05s + 1 |
|---|---|
| Controller denominator | 0.05s² + s |
| Form | Biproper — direct feedthrough |
| Controller poles | 0, -20 |
| Controller zeros | -1.137, -5.863 |
| Derivative filter τ = 1/N | 0.05 s |
| High-frequency derivative gain Kd·N | 2 |
Loop channels
(0.15s² + 1.05s + 1) / (0.05s⁵ + 1.15s⁴ + 3.1s³ + 2s²)
(0.15s² + 1.05s + 1) / (0.05s⁵ + 1.15s⁴ + 3.1s³ + 2.15s² + 1.05s + 1)
(0.05s⁵ + 1.15s⁴ + 3.1s³ + 2s²) / (0.05s⁵ + 1.15s⁴ + 3.1s³ + 2.15s² + 1.05s + 1)
(0.15s² + 1.05s + 1) / (0.05s⁵ + 1.15s⁴ + 3.1s³ + 2.15s² + 1.05s + 1)
(0.15s⁵ + 1.5s⁴ + 4.45s³ + 5.1s² + 2s) / (0.05s⁵ + 1.15s⁴ + 3.1s³ + 2.15s² + 1.05s + 1)
(0.05s² + s) / (0.05s⁵ + 1.15s⁴ + 3.1s³ + 2.15s² + 1.05s + 1)
(− 0.15s² − 1.05s − 1) / (0.05s⁵ + 1.15s⁴ + 3.1s³ + 2.15s² + 1.05s + 1)
All seven channels share the closed-loop characteristic polynomial Dc·Dg·Dh + Nc·Ng·Nh, so S + T = 1 holds identically. Y/R and T coincide only when H(s) = 1. The disturbance d is additive at the PLANT INPUT, before G(s); the noise n is added to the measurement inside the feedback path. U/R is linear control effort — no actuator limit is modelled.
Uncontrolled versus controlled
| Quantity | Baseline C = 1 | Controlled | Change |
|---|---|---|---|
| Percent overshoot[%] | 14.46 | — | not comparable |
| Settling time (2%)[s] | 12.5 | — | not comparable |
| Rise time (10%–90%)[s] | 2.68 | — | not comparable |
| Steady-state step error[–] | 0 | — | not comparable |
| Phase margin[°] | 53.41 | -17.39 | -70.81 |
| Gain margin[dB] | 15.56 | — | not comparable |
| Closed-loop bandwidth[rad/s] | 0.7966 | — | not comparable |
| Peak control effort |u|[–] | — | — | not comparable |
The baseline is the SAME plant and the SAME feedback path with C(s) = 1, measured over the SAME simulated horizon. A change is not automatically an improvement.
What this controller traded
- · The controller makes the closed loop unstable; the baseline loop was asymptotically stable.
- · Phase margin drops from 53.41° to -17.39° — less damping in reserve.
Suggested starting point (optional)
Ziegler–Nichols, ultimate-cycle (continuous-cycling) rule
Sustained oscillation occurs at Ku = 6.000 with ω = 1.414 rad/s, so Tu = 4.443 s.
| Ultimate gain Ku | 6 |
|---|---|
| Oscillation frequency ω | 1.414 rad/s |
| Ultimate period Tu = 2π/ω | 4.443 s |
- · Ziegler–Nichols ultimate-cycle rule, applied to the parallel form C(s) = Kp + Ki/s + Kd·s.
- · Ku and Tu are read from the imaginary-axis crossing of the proportional-only unity-feedback loop, not estimated from a test.
- · The rule is empirical and was derived for self-regulating process plants; it is not an optimum and offers no margin guarantee.
- · The classical PID row is tuned for roughly quarter-amplitude decay and typically gives a large overshoot.
- · Any proposal must be verified with the actual time, frequency and stability results before it is trusted.
Applying a row only writes the parameters. It does not verify anything: the resulting loop is then evaluated by the same time, frequency and stability engines as any hand design, and it is not claimed to be optimal.
Closed-loop poles and zeros
| Root | Re(s) | Im(s) | Multiplicity |
|---|---|---|---|
| Pole × | -20.006 | 0 | 1 |
| Pole × | -2.1223 | 0 | 1 |
| Pole × | -1.0575 | 0 | 1 |
| Pole × | 0.09283 | 0.66092 | 1 |
| Pole × | 0.09283 | -0.66092 | 1 |
Continue with this system
The current model travels with the link; nothing is re-entered and nothing opens on its own.
The plant travels; the controller you tuned here does not, because none of these workspaces has a controller to receive it.
Notes and limitations
Diagnostics
- ⚠ Note 2 pole(s) lie in the open right half-plane: the response grows without bound.
- · Info The loop phase never reaches -180° (modulo 360°) inside the evaluated range: no finite gain margin is defined here.
- ⚠ Note Repeated poles on the imaginary axis produce a response growing like t: this is unstable, not marginally stable.
Continuous-time, linear, time-invariant, single-input single-output. No time delay, actuator saturation, anti-windup, discretization, plant uncertainty model or automatic optimization is included. The controller you see is the controller that was evaluated.
What this Lab models
A continuous-time SISO feedback loop built from a plant G(s), a classical controller C(s) — P, PI, PD, PID with optional filtered derivative, or lead, lag and lead-lag compensation — and an optional non-unity measurement path H(s), evaluated across time response, frequency response, root locus, sensitivity S and complementary sensitivity T, and explicit design requirements.