0
$\begingroup$

How can I calculate z transfer function of the state space model when model includes vectors. For example what is z transfer function of following model when we assume that first state is the system output.

$x(k+1)=\begin{bmatrix} \ -a&A\\-b&0\end{bmatrix}x(k)+\begin{bmatrix} \ c\\d\end{bmatrix}u(k)$

$\endgroup$
0
$\begingroup$

You are starting of with a discrete state space model of the form

$$ \begin{array}{c} x(k+1) = A\,x(k) + B\,u(k) \\ y(k) = C\,x(k) + D\,u(k) \end{array} \tag{1} $$

The z-transfer function of that state space model can be obtained using

$$ G(z) = C\,(z\,I-A)^{-1}B + D. \tag{2} $$

You can derive this using that $x(k+1) = z\,x(k)$, so the first line from $(1)$ can be written as $(z\,I-A)\,x(k) = B\,u(k)$. Solving that for $x(k)$ gives $x(k) = (z\,I-A)^{-1}B\,u(k)$. Equation $(2)$ van now be obtained by substituting this for $x(k)$ into the expression for $y(k)$ in $(1)$.

What the $A$ and $B$ matrices should be in your example should be clear from your question. Since you stated that the output $y(k)$ is the first component of $x(k)$, then $C = \begin{bmatrix}1 & 0 & \cdots & 0\end{bmatrix}$ and $D=0$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.