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I'm not sure what differentiates discrete from continuous systems in trying to prove certain properties of controllability. I have a chain of proofs from class that prove each other for a continuous system:

We'll call our system $ \Sigma = (A,B,C)$

1) $ \Sigma $ is controllable if $(A,B)$ is controllable

2) The Gramian control matrix $W_c(t) $ is positive definite

3) There does not exist $ x=Ty $ with det $\space T\neq0 $ such that

$T^{-1}AT=\begin{bmatrix} A_{11} \ A_{12} \\ 0 \ A_{22} \end{bmatrix}, \space T^{-1}B=\begin{bmatrix} B_1 \\ 0 \end{bmatrix}$

4) The Kalman rank condition is satisfied.

5) The PBH test is satisfied for rank$[A-\lambda I \space B]=n$ for all $\lambda \in \sigma(A)$ and for all $\lambda\in \mathbb{C}$

How is this applied to the discrete case?

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I believe, not sure, this is all more or less the same discrete case. First all the system matrices need to be discretized [2].

  1. Controllability, exactly the same [1].

  2. I believe this still holds. The matrix equation stays the same $A W_c + W_c A^T = -BB^T$, only the integral gramian equation changes see [3].

  3. I believe also still the same. Since we are now only working with the discretized system matrices.

  4. Same, since we are now only working with the discretized system matrices.

  5. Same, since we are now only working with the discretized system matrices.

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Actually, it is more easy to prove these for discrete time case. Consider this:

Let $x_{k+1} = F x_k + G u_k$. Then we can write the following:

$x_1 = F x_0 + G u_0$

$x_2 = F x_1 + G u_1 = F^2 x_0 + FG u_0 + G u_1$

...

$x_n = F x_{n-1} + G u_{n-1} = F^n x_0 + F^{n-1}G u_0 + F^{n-2}G u_1 + \dots + G u_{n-1}$

$x_n - F^n x_0 = [F^{n-1}G ~~~ F^{n-2}G ~~~ \dots ~~~ G] [u_0 ~~~ u_1 ~~~ \dots ~~~ u_{n-1}]^T$

Remember the definition of controllability (strictly speaking the following one is reachability). We must find a finite $u_k$ such that the state of the system can be transferred to an arbitrary state $x_f$ from any initial state $x_0$. Now, let $x_f = x_n - F^n x_0$. Then we can find such $u_k$ if and only if the Kalman rank condition holds (follows from the last equation above).

Other conditions are equivalent to Kalman rank condition regardless of the system being continuous or discrete (except Gramian condition, here discrete Gramian should be used). They just follow from linear algebra.

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