Proportional
Changes the output in proportion to the current error. A larger value speeds up the response, but too much can increase oscillation and overshoot.
Tools / Control Engineering
PID Playground
Adjust the PID parameters and explore disturbances, noise, A/B comparison and cascade control.
Controls
Configure the PID and process
PID settings
Standard form: the sliders cover Ti from 0 to 50 s and Td from 0 to 10 s. Enter up to 200 s and 100 s directly. Ti = 0 disables integral action in this tool.
Set point and first-order process
ProcessG(s) = K / (T s + 1)
Disturbance
Add an input disturbance
A step load disturbance \(d(t)\) is added immediately before the process. A negative value temporarily lowers the process value, and the PID acts to restore the set point.
Actuator Limits
Output limits and anti-windup
Compare the requested controller output with the output applied after limiting. Anti-windup stops integration only when it would drive the controller farther into saturation.
Measurement
Measurement noise and derivative filter
Noise is added to the sensor measurement. A larger filter time constant makes the output smoother but slows the derivative response.
Cascade Control
Add an inner PID loop
The current PID becomes the outer loop and its output becomes the set point of the inner PID loop. With Td₂ = 0, the inner loop acts as a PI controller. The inner loop should generally be tuned faster than the outer loop.
A / B Comparison
Save and compare settings
Setting A has not been saved yet.
The simulation updates as soon as a value changes. For extreme combinations, it stops updating the result to keep the calculation safe.
Block Diagram
PID control block diagram
Let \(u_c(t)\) be the controller-requested output and \(u_a(t)\) the applied output. When noise is enabled, the measured value \(y_m(t)\), formed by adding \(n(t)\) to the true process value \(y(t)\), is fed back.
The outer PID generates the inner set point \(SP_2\), and the inner PID makes the inner process value \(PV_2\) respond quickly. The final process value \(y(t)\) is fed back to the outer PID.
Response
Simulation results
- Rise time
10% → 90% - Overshoot
- Settling time
±2% - Steady-state error
at the end
- Maximum deviation after disturbance
|PV − SP| - Recovery time after disturbance
±2%
- Time in output saturation
- Maximum absolute requested output
Setting A and current setting B →
| Metric | Setting A | Setting B |
|---|---|---|
| Rise time | ||
| Overshoot | ||
| Settling time | ||
| Steady-state error | ||
| Late-stage output range | ||
| Time in saturation |
Guide
What P, I and D do
Integral
Accumulates error over time to remove steady-state offset. Too much integral action can increase overshoot and oscillation.
Derivative
Responds to the rate of error change and damps rapid motion. It can reduce overshoot, but real systems require care with measurement noise.
Equations
PID equations and variables
PID control uses the difference between the set point and the sensor measurement as the error.
1. Error definition
\[ e(t)=r(t)-y_m(t) \]
2a. Standard (ideal) form
Selected\[
u_c(t)=K_p\left[
e(t)+\frac{1}{T_i}\int_{0}^{t}e(\tau)\,d\tau+
T_d\frac{de(t)}{dt}
\right]
\]
\(K_p\) multiplies all three actions, while \(T_i\) and \(T_d\) are entered as time values. In this tool, \(T_i=0\) disables integral action.
2b. Parallel form
Selected\[
u_c(t)=
\underbrace{K_p e(t)}_{\text{proportional P}}+
\underbrace{K_i\int_{0}^{t}e(\tau)\,d\tau}_{\text{integral I}}+
\underbrace{K_d\frac{de(t)}{dt}}_{\text{derivative D}}
\]
\(K_p\), \(K_i\) and \(K_d\) are entered as independent coefficients. It corresponds to the standard form through \(K_i=K_p/T_i\) and \(K_d=K_pT_d\).
3. First-order process used by this tool
Transfer function\[ G(s)=\frac{K}{Ts+1},\quad Y(s)=G(s)\{U_a(s)+D(s)\} \]
Time domain\[ T\frac{dy(t)}{dt}+y(t)=K\{u_a(t)+d(t)\} \]
The sum of limited applied output \(u_a(t)\) and input disturbance \(d(t)\) enters the process, and the simulator calculates how \(y(t)\) changes over time.
\[ d(t)=\begin{cases}0&(t\lt t_{\mathrm{dist}})\\d_0&(t\ge t_{\mathrm{dist}})\end{cases} \]
4. Output limits and anti-windup
\[u_a(t)=\operatorname{clip}\!\left(u_c(t),u_{\min},u_{\max}\right)\]
When output limiting is enabled, the requested controller output \(u_c(t)\) is constrained and the applied output \(u_a(t)\) enters the process.
\[\frac{dI}{dt}=\begin{cases}0&(u_c\gt u_{\max}\land e\gt 0)\ \text{or}\ (u_c\lt u_{\min}\land e\lt 0)\\e(t)&\text{otherwise}\end{cases}\]
With anti-windup enabled, integration stops only when it would drive the controller farther into saturation. It resumes when the error acts toward leaving saturation.
5. Measurement noise and derivative filter
Sensor measurement\[y_m(t)=y(t)+n(t)\]
First-order derivative filter\[D_f(s)=\frac{s}{T_f s+1}\,Y_m(s)\]
The derivative term strongly responds to high-frequency changes in the measurement. With the filter enabled, this tool uses \(D_f\) for derivative action. A larger \(T_f\) suppresses noise more strongly but also slows the response to rapid changes.
6. Cascade control
Outer loop\[SP_2(t)=C_1(s)\{r(t)-y_m(t)\}\]
Inner PID loop\[u_c(t)=K_{p2}\left[e_2(t)+\frac{1}{T_{i2}}\int e_2(t)dt-T_{d2}\frac{dPV_2(t)}{dt}\right]\]
The outer PID output becomes the inner-loop set point \(SP_2\), and the inner PID corrects \(e_2=SP_2-PV_2\). Its derivative action uses changes in \(PV_2\), avoiding derivative kick when \(SP_2\) changes. With \(T_{d2}=0\), it acts as a PI controller.
Symbols, meanings and units
| Symbol | Meaning | Unit / dimension |
|---|---|---|
| \(r(t)\) | Set point (SP) | \([\mathrm{Q}]\) |
| \(y(t)\) | Process value (PV) | \([\mathrm{Q}]\) |
| \(y_m(t)\) | Sensor measurement including noise | \([\mathrm{Q}]\) |
| \(n(t)\) | Measurement noise | \([\mathrm{Q}]\) |
| \(e(t)\) | Error | \([\mathrm{Q}]\) |
| \(u_c(t)\) | Controller-requested output | \([\mathrm{M}]\) |
| \(u_a(t)\) | Limited applied output (MV) | \([\mathrm{M}]\) |
| \(u_{\min},u_{\max}\) | Minimum and maximum output | \([\mathrm{M}]\) |
| \(I(t)\) | Integrated error state | \([\mathrm{Q}\cdot\mathrm{s}]\) |
| \(d(t),d_0\) | Input disturbance and its configured magnitude | \([\mathrm{M}]\) |
| \(t_{\mathrm{dist}}\) | Disturbance start time | \([\mathrm{s}]\) |
| \(t,\tau\) | Time | \([\mathrm{s}]\) |
| \(K_p\) | Proportional gain | \([\mathrm{M}/\mathrm{Q}]\) |
| \(T_i\) | Integral time in standard form | \([\mathrm{s}]\) |
| \(T_d\) | Derivative time in standard form | \([\mathrm{s}]\) |
| \(T_f\) | Derivative-filter time constant | \([\mathrm{s}]\) |
| \(SP_2,PV_2,e_2\) | Inner-loop set point, process value and error | arbitrary unit |
| \(K_{p2},T_{i2},T_{d2}\) | Inner PID proportional gain, integral time and derivative time | gain, \([\mathrm{s}]\), \([\mathrm{s}]\) |
| \(K_i\) | Integral gain in parallel form | \([\mathrm{M}/(\mathrm{Q}\cdot\mathrm{s})]\) |
| \(K_d\) | Derivative gain in parallel form | \([\mathrm{M}\cdot\mathrm{s}/\mathrm{Q}]\) |
| \(K\) | Steady-state process gain | \([\mathrm{Q}/\mathrm{M}]\) |
| \(T\) | Process time constant | \([\mathrm{s}]\) |
| \(s\) | Laplace variable | \([\mathrm{s}^{-1}]\) |
How to read the units: \(\mathrm{Q}\) denotes the unit of the process value and \(\mathrm{M}\) the unit of the manipulated value. For temperature control with \(y\) in °C and applied heater output \(u_a\) in %, \(K_p\) is in %/°C, \(K_i\) in %/(°C·s), and \(K_d\) in %·s/°C. The vertical axes in this simulator use arbitrary units because no particular physical process is specified.
A constant set point is applied at time zero and simulated numerically in your browser. The same deterministic noise waveform is used each time so settings can be compared under identical conditions.