Final answer:
In this case, the state-transition matrix is Φ(t) = [eᵗ 5e⁽⁻ᵗ⁾; 0 e⁽⁻ᵗ⁾], the state vector is x(t) = [eᵗ; 0], and the output of the system is y(t) = eᵗ.
Step-by-step explanation:
To solve for the state-transition matrix, state vector, and output of the system represented here, we'll follow these steps:
1. Identify the given system:
- The system can be represented by the state-space equations:
ẋ = Ax,
y = Cx,
where x represents the state vector, A represents the system matrix, and C represents the output matrix.
2. Determine the system matrix A:
- The system matrix A is given as:
A = [-1 -5; 0 1].
3. Calculate the state-transition matrix Φ(t):
- The state-transition matrix Φ(t) is given by:
Φ(t) = e⁽ᴬᵗ⁾,
where e⁽ᴬᵗ⁾ represents the matrix exponential of At.
- - To calculate e⁽ᴬᵗ⁾, we'll first calculate the eigenvalues and eigenvectors of A.
- - The eigenvalues of A can be found by solving the characteristic equation det(A - λI) = 0, where I is the identity matrix.
- - Solving det(A - λI) = 0, we find the eigenvalues to be -1 and 1.
- - Next, we calculate the eigenvectors corresponding to each eigenvalue.
- - For the eigenvalue -1, we solve the equation (A - (-1)I)x = 0 and find the eigenvector to be [1; 0].
- - For the eigenvalue 1, we solve the equation (A - 1I)x = 0 and find the eigenvector to be [5; 1].
- - Now, we can form the matrix P using the eigenvectors as columns: P = [1 5; 0 1].
- - The matrix P will be used to diagonalize A.
- - We also calculate the inverse of P, denoted as P^(-1).
- Using P and P⁽⁻¹⁾, we can diagonalize A as:
D = P⁽⁻¹⁾AP.
- Substituting the values, we get:
D = [1 0; 0 -1].
- Now, we can calculate e⁽ᴬᵗ⁾ using the diagonalized form D:
e⁽ᴬᵗ⁾ = Pe⁽ᴰᵗ⁾P⁽⁻¹⁾
- Substituting the values, we get:
e⁽ᴬᵗ⁾ = [1 5; 0 1] * e⁽[¹ ⁰; ⁰ ⁻¹]ᵗ⁾ * [1 -5; 0 1].
- The resulting state-transition matrix Φ(t) is:
Φ(t) = [eᵗ 5e⁽⁻ᵗ⁾; 0 e⁽⁻ᵗ⁾].
4. Determine the state vector x(0):
- The initial state vector x(0) is given as:
x(0) = [1; 0].
5. Calculate the state vector x(t):
- The state vector x(t) can be calculated using the state-transition matrix Φ(t) and the initial state vector x(0):
x(t) = Φ(t) * x(0).
- Substituting the values, we get:
x(t) = [eᵗ 5e⁽⁻ᵗ⁾; 0 e⁽⁻ᵗ⁾] * [1; 0].
- Simplifying further, we get:
x(t) = [eᵗ; 0].
6. Determine the output matrix C:
- The output matrix C is given as:
C = [1 2].
7. Calculate the output y(t):
- The output y(t) can be calculated using the output matrix C and the state vector x(t):
y(t) = C * x(t).
- Substituting the values, we get:
y(t) = [1 2] * [eᵗ; 0].
- Simplifying further, we get:
y(t) = eᵗ.
Therefore, the state-transition matrix is Φ(t) = [eᵗ 5e⁽⁻ᵗ⁾; 0 e⁽⁻ᵗ⁾], the state vector is x(t) = [eᵗ; 0], and the output of the system is y(t) = eᵗ.