Second-order

A second-order transfer function ωn² / (s² + 2ζωn s + ωn²).

signal
Gradara equations
2 ports
2 parameters

BlockSecond-orderKindsecondOrderLibraryContinuous

Description

A second-order low-pass system with unity DC gain, set by natural frequency ωn and damping ratio ζ. Use it to model a sensor, actuator, or loop with an oscillatory response.

Example

Step responses A unit step through a first-order filter, a second-order system, an integrator, and a delay; a ramp through a derivative.

Step at 0.1 sFirst orderSecond orderIntegratorDelay 0.3 sRamp, slope 2Derivative

In Gradara, select a Second-order block and press F1, then choose Open example. It opens as a new model in My models, ready to run.

Also in this example:Limited integrator Step Low-pass filter Ramp Derivative Transport delay

Ports

Inputs 1

usignal

Input signal.

Outputs 1

ysignal

Output signal.

Parameters

Natural frequencywn10 rad/s

≥ 0.01

Natural frequency ωn, in rad/s. At least 0.01.
Damping ratiozeta0.7

≥ 0

Damping ratio ζ. 0 or more: below 1 the step response overshoots, 1 is critically damped, above 1 is overdamped, and 0 oscillates without decay.

Equations

Modelica

dx1/dt = x2
dx2/dt = wn² · (u − x1) − 2 · zeta · wn · x2
y = x1, with x1 = x2 = 0 at time 0
Transfer function: wn² / (s² + 2 · zeta · wn · s + wn²)

Implementation

The block’s Modelica model

Real x1(start=0, fixed=true);
Real x2(start=0, fixed=true);
der(x1) = x2;
der(x2) = wn*wn*(u - x1) - 2*zeta*wn*x2;
y = x1;

Assumptions and limitations

Both states start at 0.