Dynamics & Control · Lab

Transfer Function Lab

Enter any continuous-time SISO transfer function in descending powers of s and read its poles, zeros, stability and time response. Roots come from the numerical solver, and the response from the state-space realization.

The same G(s), seen in the time domain, the frequency domain or through the algebraic stability tests.

G(s) = N(s) / D(s)

Numerator N(s) · degree 0

Denominator D(s) · degree 2

Examples

Excitation

Excitation u(t)

Step amplitude A

Numerator effect

Overlay a second numerator against the same denominator: identical poles, a different transient.

System summary

Numerator N(s)	1
Denominator D(s)	s² + 2s
System order (deg D)	2
Numerator order (deg N)	0
Properness	Strictly proper — deg(N) < deg(D)
DC gain G(0)	Not defined — D(0) = 0 (pole at the origin)
Poles	2
Zeros	0
Stability	Marginally stable — simple pole(s) on the imaginary axis

Diagnostics

· Info Simple poles on the imaginary axis give a bounded, non-decaying mode: the system is marginally stable, not asymptotically stable.

Response y(t)

Time view

Window [s]

output solid · input dashed grey · horizon 3.5 s

Computed numerically (RK4) from the controllable canonical realization.

Measured response metrics

Final value y∞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.	Not applicable
Peak value	1.5
Peak time	3.5 s
Percent overshootThe 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.	Not applicable
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.	Not applicable
Settling time (5%)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.	Not applicable
Rise time (10%–90%)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.	Not applicable

Continue with this system

The current model travels with the link; nothing is re-entered and nothing opens on its own.

Root LocusSame G(s) as the open-loop plant · study how the closed-loop poles move as the loop gain varies.Control DesignSame G(s) as the plant · add and compare P, PI, PD and PID controllers around it.State SpaceSame G(s) as an equivalent controllable-canonical A, B, C, D realisation.

What this Lab models

Any continuous-time SISO G(s) = N(s)/D(s), entered as coefficients: zeros are the roots of the numerator N(s), poles the roots of the denominator D(s), and the stability verdict follows from the relevant pole locations. Responses to standard inputs use the same G(s). With loop analysis on, margins and Nyquist encirclements apply to the loop transfer function L(s); algebraic closed-loop stability uses the characteristic polynomial of the selected context, D(s) + N(s) for the unity-feedback loop.