The state transition matrix for the system is given by a specified equation. What is the state transition matrix?

Steps to Solve

1. Identify the system representation

The given system is represented as follows:

$$ \begin{bmatrix} \dot{x}_1 \ \dot{x}_2 \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 1 & 1 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix} + \begin{bmatrix} 1 \ 1 \end{bmatrix} u $$

Here, the matrix from the coefficients of the state variables is the system matrix.

2. Extract the system matrix

The matrix $A$, which is the system matrix, is obtained directly as:

$$ A = \begin{bmatrix} 1 & 0 \ 1 & 1 \end{bmatrix} $$

3. Calculate the state transition matrix

The state transition matrix $\Phi(t)$ for a linear time-invariant system can be calculated using the matrix exponential:

$$ \Phi(t) = e^{At} $$

4. Compute the matrix exponential

To compute $e^{At}$, we first find the eigenvalues and eigenvectors of the matrix $A$. The eigenvalues $\lambda$ are found by solving:

$$ \text{det}(A - \lambda I) = 0 $$

where $I$ is the identity matrix.

Calculating:

$$ \text{det}\left(\begin{bmatrix} 1 - \lambda & 0 \ 1 & 1 - \lambda \end{bmatrix}\right) = (1 - \lambda)^2 = 0 $$

This gives us $\lambda_1 = 1$ (double root).

5. Find eigenvectors

To find the eigenvector corresponding to $\lambda_1 = 1$, solve:

$$ (A - I) \vec{v} = 0 $$

This gives:

$$ \begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix} \begin{bmatrix} v_1 \ v_2 \end{bmatrix} = 0 $$

From which we can get any vector of form $\begin{bmatrix} 1 \ 0 \end{bmatrix}$.

6. Compute Jordan form and exponential

Since we have a repeated eigenvalue, we construct the Jordan block, which gives a unique form:

$$ J = \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix} $$

The matrix exponential in this case results in:

$$ e^{Jt} = e^t \begin{bmatrix} 1 & t \ 0 & 1 \end{bmatrix} $$

Thus, we find:

$$ \Phi(t) = e^{At} = e^t \begin{bmatrix} 1 & t \ 0 & 1 \end{bmatrix} $$

7. Final state transition matrix

The final state transition matrix is:

$$ \Phi(t) = e^t \begin{bmatrix} 1 & t \ 0 & 1 \end{bmatrix} $$