Which systems that are controlled in real applications are modeled as linear systems $$ \dot x = Ax + Cu\\ y = Cx + Du $$ with a nonzero $D$ matrix?

I myself have only been stumbling over systems that don't need a $D$ matrix and was wondering about where these kind of other systems pop up.

  • $\begingroup$ Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. $\endgroup$
    – Community Bot
    Nov 25, 2022 at 18:02
  • $\begingroup$ What type of systems have you been stumbling over? Electrical, mechanical, chemical, other? Please add more details. $\endgroup$
    – AJN
    Nov 26, 2022 at 3:00
  • $\begingroup$ I am looking for any examples. I am sure there must be plenty in all of these areas. I am just not creative enough to come up with one. In principle, a nonzero $D$ implies a static+dynamic system behaviour and I just don't know any systems that behave in such a way. $\endgroup$
    – fhchl
    Nov 28, 2022 at 12:09
  • $\begingroup$ Most systems are linear on a small scale, but nonlinear on a large scale. Some common examples are linear oscillations of mechanical systems (linear elasticity is sufficiently accurate for small strains and often finite degrees of freedom sufficient), also are discrete approximations of electrical circuits good enough for real world applications, mixing problems (concentration, temperature), ... $\endgroup$ Nov 28, 2022 at 12:54
  • $\begingroup$ Maybe my question is not clear enough but i am just asking about such systems with a special structure, namely with nonzero D. $\endgroup$
    – fhchl
    Nov 28, 2022 at 16:09

1 Answer 1


An example of a system in which D is non-zero is a (point) mass on a flat surface, if we have a horizontal force F on mass M as input u, and as output both an accelerometer measurement and the position of the mass , our system could look something like:

$\dot{x} = A x + B u \\ y = Cx + D u$


$\dot{x} = \begin{bmatrix} \dot{v} \\ \dot{p}\end{bmatrix} = \begin{bmatrix} 0 & 0\\ 1 & 0 \end{bmatrix} \begin{bmatrix} v \\ p\end{bmatrix} + \begin{bmatrix} 1/M\\ 0\end{bmatrix} F$

$y = \begin{bmatrix} 0 & 0 \\ 0 & 1\end{bmatrix}\begin{bmatrix} v \\ p\end{bmatrix} + \begin{bmatrix} 1/M \\ 0 \end{bmatrix}F$

in which $p$ is the positon of the mass, and $v$ the velocity.

The places where I encounter non-zero D matrices are often from sensor measurements that basicly measure the input. This is not always useful, as when you know your input, you don't need to measure it anymore. However, you don't always know your input. In such cases, writing your system as above may help you uncover the input.

  • $\begingroup$ This examples fits the schema, but isn't it quite contrived? One could just assume that the input u is known and model the problem without D. $\endgroup$
    – fhchl
    Feb 1, 2023 at 8:40
  • $\begingroup$ u is not necessarily known. (This might become clearer if you add noise terms.) And yes, whether you assume u is known or not, you could write the system differently, but that is the case for any model you make. The point of writing it like this is that this is a standard form, for which many people have made extensions: e.g. it is clear from literature how you would use this system in a Kalman filter, while that may not be clear if you write the system down differently. $\endgroup$
    – Chris_abc
    Feb 1, 2023 at 9:20
  • $\begingroup$ True and great point! Your answer even explains this. I should have read it more carefully. $\endgroup$
    – fhchl
    Feb 9, 2023 at 8:28

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