Find the Laplace transform of the following deep state space model:
\begin{align}
    x^{(1)}_t&=A^{(1)}x^{(1)}_{t-1}+u_t\\
    x^{(2)}_t&=A^{(2)}x^{(2)}_{t-1}+x^{(1)}_t
\end{align}

To find the Laplace transform of the given deep state space model, let's first represent the system in a form suitable for transformation. The model is defined as:

\[
\begin{align*}
    x^{(1)}_t &= A^{(1)}x^{(1)}_{t-1} + u_t \tag{1}\\
    x^{(2)}_t &= A^{(2)}x^{(2)}_{t-1} + x^{(1)}_t \tag{2}
\end{align*}
\]

### Step 1: Express the system in a continuous-time difference equation form.

We assume that \( x^{(1)}_t \) and \( x^{(2)}_t \) are sequences in discrete time. We can convert these to the continuous-time domain by shifting indices by one and considering the Laplace transform.

Rewriting the equations as:
\[
\begin{align*}
    x^{(1)}_t - A^{(1)}x^{(1)}_{t-1} &= u_t \tag{3}\\
    x^{(2)}_t - A^{(2)}x^{(2)}_{t-1} &= x^{(1)}_t \tag{4}
\end{align*}
\]

Now, taking the Laplace transform of both sides of these equations:

### Step 2: Apply the Laplace Transform

Taking the Laplace transform of equation (3):

\[
\mathcal{L}\{x^{(1)}_t\} - A^{(1)}\mathcal{L}\{x^{(1)}_{t-1}\} = \mathcal{L}\{u_t\}
\]

Let \( X^{(1)}(s) \), \( X^{(2)}(s) \), and \( U(s) \) denote the Laplace transforms of \( x^{(1)}_t \), \( x^{(2)}_t \), and \( u_t \), respectively. Then:

\[
X^{(1)}(s) - A^{(1)}X^{(1)}(s) = U(s)
\]

This simplifies to:

\[
X^{(1)}(s) (1 - A^{(1)}) = U(s)
\]

So:

\[
X^{(1)}(s) = \frac{U(s)}{1 - A^{(1)}}
\]

Taking the Laplace transform of equation (4):

\[
\mathcal{L}\{x^{(2)}_t\} - A^{(2)}\mathcal{L}\{x^{(2)}_{t-1}\} = \mathcal{L}\{x^{(1)}_t\}
\]

Substituting the Laplace transforms:

\[
X^{(2)}(s) - A^{(2)}X^{(2)}(s) = X^{(1)}(s)
\]

This simplifies to:

\[
X^{(2)}(s)(1 - A^{(2)}) = X^{(1)}(s)
\]

Substituting the expression for \( X^{(1)}(s) \) from earlier:

\[
X^{(2)}(s)(1 - A^{(2)}) = \frac{U(s)}{1 - A^{(1)}}
\]

So:

\[
X^{(2)}(s) = \frac{U(s)}{(1 - A^{(1)})(1 - A^{(2)})}
\]

### Summary of Results
- The Laplace transform of \( x^{(1)}_t \) is given by:
  \[
  X^{(1)}(s) = \frac{U(s)}{1 - A^{(1)}}
  \]
- The Laplace transform of \( x^{(2)}_t \) is given by:
  \[
  X^{(2)}(s) = \frac{U(s)}{(1 - A^{(1)})(1 - A^{(2)})}
  \]

These are the Laplace transforms of the state variables \( x^{(1)}_t \) and \( x^{(2)}_t \) in the given deep state space model.

Would you like more details or have any questions?

Here are 5 related questions to further explore the topic:
1. How would the Laplace transform change if the matrices \( A^{(1)} \) and \( A^{(2)} \) were not scalars but instead matrices?
2. What physical systems might be modeled using such a deep state space model?
3. How would you interpret the poles of the transfer function \( X^{(1)}(s) \) or \( X^{(2)}(s) \)?
4. How does the input \( u_t \) affect the stability of the system?
5. Can this model be extended to a more complex multi-layer state space representation?

**Tip:** When dealing with state space models, it's important to check the eigenvalues of the system matrices to understand the system's stability and behavior.

Explore the Laplace transform of a deep state space model including equations and step-by-step solutions. Learn how to apply Laplace transforms to continuous-time systems with matrices. Suitable for advanced undergraduate students and researchers.

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