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ctranspose, '

Conjugate dynamic system model

Syntax

H = ctranspose(G)

Description

example

H = ctranspose(G) computes the conjugate of the dynamic system model or static model G. The ctranpose command is equivalent to the ' operator.

If G is a dynamic system model poles P and zeros Z, then the conjugate system H = ctranspose(G)= G' has the following properties.

  • In continuous time, the poles of H are –P and the zeros of H are –Z. The frequency response of H is the Hermitian transpose of the frequency response of G:

    H() = G()H.

  • In discrete time with sample time Ts, the poles of H are 1/P and the zeros of H are 1/Z. The frequency response of H is the Hermitian transpose of the frequency response of G:

    H(z) = G(z)H,

    where z = eiωTs.

If G is a static model, then H is the complex-conjugate transpose of the matrix.

Examples

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Compute the conjugate of a one-input, two-output transfer function model.

s = tf('s');
G1 = (s + 1)/(s^2 + 2*s + 1);
G2 = 1/s;
G = [G1;G2]
G =
 
  From input to output...
           s + 1
   1:  -------------
       s^2 + 2 s + 1
 
       1
   2:  -
       s
 
Continuous-time transfer function.

The conjugate has two outputs and one input, and takes s to –s.

H = ctranspose(G)
H =
 
  From input 1 to output:
     -s + 1
  -------------
  s^2 - 2 s + 1
 
  From input 2 to output:
  -1
  --
  s
 
Continuous-time transfer function.

Using the ' operator yields the same result as ctranspose.

H = G'
H =
 
  From input 1 to output:
     -s + 1
  -------------
  s^2 - 2 s + 1
 
  From input 2 to output:
  -1
  --
  s
 
Continuous-time transfer function.

Obtain the frequency response and the Hermitian transpose of the frequency response of a transfer function at one frequency.

G = [tf(1,[1 0]);tf([1 0],[1 1])];
w = 3;
resp = freqresp(G,w)
resp = 2×1 complex

   0.0000 - 0.3333i
   0.9000 + 0.3000i

The Hermitian transpose of the frequency response is the frequency response of the conjugate of G.

respH = freqresp(G',w)
respH = 1×2 complex

   0.0000 + 0.3333i   0.9000 - 0.3000i

Input Arguments

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Input system, specified as a dynamic system model or static model such as a tf, zpk, or ss model. G can be a generalized model provided that its control design blocks are scalar and nondynamic, such as a scalar ureal parameter.

Output Arguments

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Conjugate model of G, returned as a dynamic system model or static model of the same type as G.

Introduced before R2006a