Final answer:
The question pertains to finding the output Y(s) using the state-space representation and Laplace transforms. The process involves applying the Laplace transform to the state-space equations and solving them to find Y(s).
Step-by-step explanation:
The question concerns the state-space representation of a dynamic system in control theory. Typically, a state-space model is given in the form of a set of linear differential equations that define the system dynamics. Although the original question does not provide the full state-space representation, it hints at a system with a single input represented by a sine function, sin(3t), and the matrices that could represent the state equation in the form:
№ x' = Ax + Bu,
where A is the state matrix, B is the input matrix, x is the state vector (comprising values like position and velocity), and u is the input to the system.
To find the output Y(s) in the Laplace domain, one would typically apply the Laplace transform to the state-space equations and solve for Y(s), incorporating initial conditions if provided.
For example, if the state-space model with example matrices as shown:
№ x' = [1 2; -1 3]x + [1; 1]sin(3t)
With the Laplace transform and assuming zero initial conditions, the output would be obtained by:
Y(s) = C(sI - A)^-1 B + D
where C is the output matrix, I is the identity matrix, A and B are from the state-space model, and D is the direct transmission matrix (if any).