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 (A, B, C, D), 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 (A, B, C, D), this value is

K = D + C(I – A)^–1B

Examples

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Compute the DC Gain of a MIMO Transfer Function

Open Live Script

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

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

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.

Compute DC Gain of Identified Model

This example uses:

System Identification Toolbox

Open Live Script

Load the estimation data.

load iddata1 z1

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.

Documentation