Derivative in discrete version of state-space definition

[WasabiFan]

Section 5.3.3, Definition 5.3.1, shows a continuous and discrete version of the same equation:

In the continuous case, the LHS is the derivative of x, but in the discrete case the derivative is dropped and no delta takes its place: the LHS simply reads x_{k+1}. Why is this?

[calcmogul]

Although it isn't clear from the notation, the continuous and discrete versions have different A and B matrices. The discrete equation returns the next state rather than a change in state because while you could have x_{k+1} - x_k = Ax_k + Bu_k, you have to do an extra addition by x_k to propagate the model forward. Also note that the poles would be given by the eigenvalues of A + I rather than the eigenvalues of A. I know a mathematics guy that's annoyed by the deviation in notation from the continuous version, but 🤷‍♂️. Practicality beats purity in this case.

Section 6.5 "Matrix exponential" through 6.7 go into more detail on what the discrete matrices are defined as, and G.3 "Zero-order hold for state-space" shows a derivation.

[WasabiFan]

Thanks for the clarification! I appreciate the quick response. It might be good to have a note about that near the definition.