Discrete time analogs of descriptor systems with derivative in output

Question

For continuous time, Descriptor systems of the form

$$ Ex'(t)=Ax(t)+Bu(t)$$ $$ y(t)=C_1x(t)+C_2x'(t) $$ are called Descriptor systems with derivative in the output.

The discrete time analog of such a system would be $$ Ex(k+1)=Ax(k)+Bu(k) $$ $$ y(k)=C_1x(k)+C_2x(k+1) $$ What is such a system called? Are there results and research on such systems? For the continuous case, I know that concepts like observability and controllability are studied. What about the discrete time case? Can anyone help me with references to such works?

Answer 1

The equation you wrote describes a non-causal system. For the obvious reason that the output depend on future states, such systems are rarely studied in control theory. In general derivatives on the right hand side of differential equations tend to be frowned upon; when they are necessary, and discrete-time equivalents are used, the backward approximation $\dot{y}(t_i) \approx (y(t_i) - y(t_{i-1}))/(t_i-t_{i-1})$ is preferred over the forward-looking one.