01.2 DC Gain Explained
DC gain is the steady-state output of a system when you apply a constant (DC) input. “DC” comes from electronics-direct current means zero frequency.
Physically:
If you apply a constant force to a mass-spring-damper and wait forever, how far does it move? That ratio (output/input at steady state) is the DC gain.
Mathematically:
Set $s = 0$ in the transfer function:
\[\text{DC gain} = P(0) = \lim_{s \to 0} P(s)\]For your system 01.1 Basics of Open Loop: \(P(0) = \frac{\omega^2}{0 + 0 + \omega^2} = 1\)
This means a unit step input eventually produces a unit step output (no amplification or attenuation at steady state).
Why It Matters
- DC gain = 1: Output tracks input perfectly at steady state
- DC gain > 1: System amplifies constant inputs
- DC gain < 1: System attenuates constant inputs
- DC gain = infinity: System has an integrator (type 1 or higher), output grows unbounded for step input
MATLAB
Method 1
MATLAB has a function
dcgain(). Just pass a LTI object and it will give you a DC gain.dcgain(P1_ss)and it will output 18 03.1 Creating a LTI Object in MATLAB - Part 1 (P1_ss) is taken from here
Method 2
You can also use freqresp function to get DC Gain. freqresp gives you the output at a particular frequency or for an array of frequency if you give it an LTI Object
DC_gain = freqresp(P1_ss, 2*pi*0) %always convert the input to radian.