Question

Represent the following differential equation in state space form: x(t) = Ax(t) + Bu(t)

5i(t) + 7i(t) + 9i""(t) + 3∫i(t)dt=e ₁(t) - e ₂(t)
Represent the system in observable form as well.

Answer 1

The state-space representation of the given differential equation is:

`\[ \dot{x}(t) = \begin{bmatrix} 7 & -9 & -5 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix} x(t) + \begin{bmatrix} 3 \\ 0 \\ 0 \end{bmatrix} u(t) + \begin{bmatrix} -1 \\ 0 \\ 1 \end{bmatrix} e_1(t) - \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} e_2(t) \]`

The system in observable form is:

`\[ y(t) = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix} x(t) \]`

Step-by-step explanation:

To represent the given differential equation in state-space form, we first identify the coefficients of each term and organize them into matrices. The state matrix (A) is formed by coefficients of x(t), the input matrix ( B) is formed by coefficients of u(t), and the output matrix ( C) is chosen to extract the desired output variable. In this case, the state-space representation is obtained as shown in the final answer.

To express the system in observable form, we select ( C) such that the system is observable. In this specific case,
`\( C = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix} \)` is chosen, making the system observable. Observable form ensures that the internal state of the system can be reconstructed from the output measurements.