Question

Use Laplace transforms to solve the following stage-space equation for y(t) when the input u(t) is a unit step.

x=-[5 0]x+[3]u(t)
[-1 -2] [1]
y = [1 0]x; x(0) = [1]
[0]

Answer 1

To solve the given stage-space equation using Laplace transforms, we first need to transform the equation into the Laplace domain. Then, we can use inverse Laplace transform to find the expression for y(t) in the time domain.

Step-by-step explanation:

In order to solve this problem using Laplace transforms, we need to transform the given differential equation into the Laplace domain. The given system can be represented as:

X(s) = (-[5 0]sI - [3])U(s)

Y(s) = [1 0]X(s)

Where X(s) and U(s) are the Laplace transforms of x(t) and u(t) respectively, and Y(s) is the Laplace transform of y(t).

By substituting the given values, we get:

X(s) = (-[5 0]sI - [3])(1/s)

Simplifying the equation, we get:

X(s) = [-5/s 0](1/s) - [3](1/s)

X(s) = [-5/s² 0] - [3/s]

Finally, using the inverse Laplace transform, we can find the expression for y(t) in the time domain.