Control Response Lab
Explore one precisely defined continuous-time control family: a first-order plant P(s)=K/(τs+1), proportional controller C(s)=Kc and constant feedback gain h. The open-plant reference response and closed-loop reference response are computed from exact formulas with zero initial conditions. Inspect the pole and Bode plots before treating a design as stable; this is an educational model, not a physical safety certificate.
Key features
- Exact open-plant and closed-loop unit-step or scaled-step traces over a chosen horizon
- Frequency magnitude in dB and phase in degrees using s=j2πf, with frequency in hertz
- Closed-loop pole, stability classification and algebraic DC transfer value
- Explicit stable, marginal and unstable examples, including a right-half-plane pole
- Editable plant, proportional controller, feedback sensor and reference amplitude
- Local standalone SVG charts and numeric CSV traces
How to use
- Start with the stable preset or enter a first-order plant gain K and positive time constant τ in seconds.
- Enter proportional gain Kc, nonnegative feedback gain h and the reference amplitude. Choose the time horizon and positive frequency bounds in hertz.
- Run analysis to derive the open and closed transfer functions, exact step responses and complex frequency responses.
- Read the closed-loop pole: a negative value is asymptotically stable in this model, zero is marginal and positive is unstable. Compare the open and closed charts.
- Download the SVG charts and CSV samples if you need a reproducible record of the configured experiment.
Use cases
- See how proportional feedback moves the pole of a first-order process
- Compare transient speed against the changed closed-loop DC value
- Recognize a right-half-plane pole caused by a chosen negative controller gain
- Check how magnitude and phase change across a specified frequency band
Frequently asked questions
Can I enter an arbitrary transfer function or a PID controller?
No. This version analyzes P(s)=K/(τs+1), C(s)=Kc and a constant feedback gain h only. It assumes continuous time, linear time-invariant behavior and zero initial conditions; it does not solve arbitrary higher-order or delayed systems.
What exactly is shown in the two response traces?
The blue trace treats the reference as the direct input to P(s). The orange trace is reference-to-output T(s)=KcK/(τs+1+KcKh) in a negative-feedback connection. The controller changes the forward gain, so the traces are compared as two explicit configurations rather than pretending they share identical actuation.
How is stability decided?
The sole closed-loop pole is -(1+KcKh)/τ in reciprocal seconds. A negative pole is asymptotically stable, zero is marginal, and a positive pole is unstable for this modeled state. The algebraic DC transfer value of an unstable case is not a settled output.
Why can the step plot stop before settling?
The plot ends at the time horizon you entered. The displayed 2% settling estimate is derived from the stable exponential pole, not from the last sampled point. It is undefined for marginal or unstable settings.
Is this suitable for a real plant or safety decision?
No. It omits delay, saturation, disturbance, noise, uncertainty, higher-order dynamics and actuator limits. Validate any physical controller with measured plant data, independent engineering review and appropriate safeguards.
Are the parameters uploaded?
No. Analysis and file creation run locally in this browser tab. The parameters are not submitted to a tool API.
Privacy
The entered parameters and calculated traces remain in browser memory. This tool has no parameter-upload endpoint; SVG and CSV files are created only when you request a download.
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