Inputs

sys

LTI system created by tf(), ss(), dss(), etc.

Outputs

K

DC gain matrix. For a system with m inputs and p outputs, the array k has dimensions [p, m].

See @tf/dcgain for the method used for systems represented by a transfer function.

For a continuous-time state space system $(A,B,C,D)$, the gain is given by $$ K = G(s=0) = \left. C \, (s\,I - A)^{-1} B + D\,\right|_{s=0} = C \, A^{-1} B + D $$

In the discrete-time case the gain of a state space system $(A,B,C,D,T)$ is given by $$ K = G(z=1) = \left. C \, (z\,I - A)^{-1} B + D\,\right|_{z=1} = C \, (I-A)^{-1} B + D $$

Example

 A = [0,1,0;0,0,1;-1,-3,-3];
 B = [0 0;0,1;1,1];
 C = [1,0 0;0,1,1];
 D = [2,0;0,0];
 K = dcgain (ss (A,B,C,D))

 K =
   3   4
   0  -1

See also: @tf/dcgain, @lti/freqresp, @tf/tf, @ss/ss, dss

Source Code: @lti/dcgain