123k views
5 votes
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.

User Inkyu
by
8.5k points

1 Answer

7 votes

Final answer:

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.

User Imelda
by
8.7k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.