I have the following system: \begin{align*} \dot{x} & =Ax+Bu\\ y & = Cx+f \end{align*} Where $x$ and $y$ are vectors, $A$, $B$, $C$ are matrices, and $f$ a constant term (same dimension as $y$). The system is expressed in a "pseudo" state space form, what is troubling me is the constant term "$f$", I don't know how to handle it, how can I make it fit in the usual state space form? Which should be like this: \begin{align*} \dot{x} & =Ax+Bu\\ y & = Cx+Du \end{align*} from a practical point of view, since $f$ is constant, I could define $\bar{y}$ as \begin{align*} \bar{y}=y-f \end{align*} however, from a theoretical point of view, $\bar{y}$ no longer represents the real output of the system, which is $y$, that represents the physical variables I can measure. I can't overcome this impasse. The final goal is to compute the transfer matrix of the system.
State-Space model with constant term in the output equation
2 Answers
You can augment the state vector with $f$ such that $$x_{ag}:=\left[\matrix{x\\ f}\right]$$ Then we can write equivalently $$\dot{x}_{ag}=\left[\matrix{\dot{x}\\ \dot{f}}\right]=\left[\matrix{Ax+Bu\\ 0}\right]=\left[\matrix{A & 0\\ 0 & 0 }\right]x_{ag}+\left[\matrix{B\\ 0}\right]u$$ and $$y=Cx+f=\left[\matrix{C & I}\right]x_{ag}$$ which is now in the standard state-space form.
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0A "smaller" option would be: $x_{ag} := [x^T\ 1]^T$ and $C_{ag} := [C\ f]$. – 2017-02-19
You can also augment $u$ like
$$\begin{align*} \dot{x} &= Ax + \begin{bmatrix}B & 0\end{bmatrix} \begin{bmatrix}u \\ f\end{bmatrix} \\ y &= Cx + \begin{bmatrix}0 & I\end{bmatrix} \begin{bmatrix}u \\ f\end{bmatrix} \end{align*}$$
Then you can calculate the transfer function matrix as usual like $$G(s)=C(sI - A)^{-1}\tilde{B} + \tilde{D}$$ where $\tilde{B} := \begin{bmatrix}B & 0\end{bmatrix}$ and $\tilde{D} := \begin{bmatrix}0 & I\end{bmatrix}$.