Calculator · Dynamics & Control
PID Controller Calculator
Enter the plant G(s) = N(s)/D(s) in descending powers of s and the parallel-form PID gains. The result is the unity negative-feedback closed loop with C(s) = Kp + Ki/s + Kd·s — evaluated, not tuned.
Plant
Controller gains
Result
| Closed-loop verdict | Unstable |
|---|---|
| Closed-loop poles | -20.01, -2.122, -1.057, 0.09283 + 0.6609j, 0.09283 − 0.6609j |
| Percent overshoot | The final-value theorem requires every pole of s·Y(s) in the open left half-plane; this system is not asymptotically stable, so no finite steady state exists. |
| Settling time (2%) | The final-value theorem requires every pole of s·Y(s) in the open left half-plane; this system is not asymptotically stable, so no finite steady state exists. |
| Steady-state step error | The closed loop is not asymptotically stable, so no finite steady-state error exists. |
| Phase margin | -17.39 ° |
| Gain margin | No phase crossover |
Technical reference
C_ideal(s) = Kp + Ki/s + Kd·s (parallel form)
C_filtered(s) = Kp + Ki/s + Kd·s/(1 + s/N_f), N_f > 0
L(s) = C(s)·G(s), T(s) = C(s)G(s) / [1 + C(s)G(s)] for unity negative feedback
- Kp
- proportional gain[–]
- Ki
- integral gain[1/s]
- Kd
- derivative gain[s]
- N_f
- derivative filter coefficient (filter time constant 1/N_f); not a plant numerator[1/s]
- G(s)
- plant transfer function entered above
This is the parallel (Kp, Ki, Kd) form, not the standard/interacting Ti, Td form; Anemodia never converts between the two.
The displayed units Kp [–], Ki [1/s], Kd [s] are the normalized transfer-function units this calculator uses. In a physical loop, controller-gain units follow from the units of the error and control signals.
Integral action adds a pole at the origin and, for a stable unity-feedback loop under the usual error-constant conditions, can eliminate steady-state step error. It also adds phase lag and may reduce margins or worsen transient response, depending on the plant and gains. Derivative action can add phase lead and improve damping or transient behavior, but that effect is also plant- and gain-dependent; ideal derivative amplifies high-frequency noise, motivating the filtered form Kd·s/(1 + s/N_f).
This calculator evaluates the closed loop for the exact Kp, Ki and Kd you enter — it does not search for or suggest gains. Automatic tuning is not performed anywhere on this page.
Continue
- · Continuous-time, single-input single-output, linear time-invariant plant with real coefficients.
- · Unity negative feedback: H(s) = 1. Non-unity measurement paths are available in the Control Design Lab.
- · Parallel (ideal) PID form C(s) = Kp + Ki/s + Kd·s, optionally with the realizable derivative Kd·s/(1 + s/N_f).
- · No actuator saturation, anti-windup, time delay or discretization is modelled, and no set-point weighting is applied.
- · Overshoot and settling time are measured from the simulated step response over an automatically chosen horizon; a quantity not observed within that horizon is reported as such rather than guessed.
Diagnostics
- ⚠ Note 2 pole(s) lie in the open right half-plane: the response grows without bound.
- ⚠ Note 2 pole(s) lie in the open right half-plane: the response grows without bound.
- ⚠ 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.