# dcgain

Low-frequency (DC) gain of LTI system

k = dcgain(sys)

## Description

k = dcgain(sys) computes the DC gain k of the LTI model sys.

### Continuous Time

The continuous-time DC gain is the transfer function value at the frequency s = 0. For state-space models with matrices (ABCD), this value is

K = D – CA–1B

### Discrete Time

The discrete-time DC gain is the transfer function value at z = 1. For state-space models with matrices (ABCD), this value is

K = D + C(I – A)–1B

## Examples

collapse all

Create the following 2-input 2-output continuous-time transfer function.

$H\left(s\right)=\left[\begin{array}{cc}1& \frac{s-1}{{s}^{2}+s+3}\\ \frac{1}{s+1}& \frac{s+2}{s-3}\end{array}\right]$

H = [1 tf([1 -1],[1 1 3]) ; tf(1,[1 1]) tf([1 2],[1 -3])];

Compute the DC gain of the transfer function. For continuous-time models, the DC gain is the transfer function value at the frequency s = 0.

K = dcgain(H)
K = 2×2

1.0000   -0.3333
1.0000   -0.6667

The DC gain for each input-output pair is returned. K(i,j) is the DC gain from input j to output i.

z1 is an iddata object containing the input-output estimation data.

Estimate a process model from the data. Specify that the model has one pole and a time delay term.

sys = procest(z1,'P1D')
sys =
Process model with transfer function:
Kp
G(s) = ---------- * exp(-Td*s)
1+Tp1*s

Kp = 9.0754
Tp1 = 0.25655
Td = 0.068

Parameterization:
{'P1D'}
Number of free coefficients: 3
Use "getpvec", "getcov" for parameters and their uncertainties.

Status:
Estimated using PROCEST on time domain data "z1".
Fit to estimation data: 44.85%
FPE: 6.02, MSE: 5.901

Compute the DC gain of the model.

K = dcgain(sys)
K = 9.0754

This DC gain value is stored in the Kp property of sys.

sys.Kp
ans = 9.0754

## Tips

The DC gain is infinite for systems with integrators.