# Documentation

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# dcgain

Low-frequency (DC) gain of LTI system

## Syntax

`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

### Example 1

To compute the DC gain of the MIMO transfer function

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

type

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

to get the result:

```ans = 1.0000 -0.3333 1.0000 -0.6667 ```

### Example 2

To compute the DC gain of an identified process model, type;

```load iddata1 sys = idproc('p1d'); syse = procest(z1, sys) dcgain(syse) ```

The DC gain is stored same as `syse.Kp`.

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### Tips

The DC gain is infinite for systems with integrators.