Question

Convert each of the following differential equations into the state-variable form:

x¨+x˙=−u

Answer 1

To convert the differential equation x¨ + x© = -u into state-variable form, introduce state variables for displacement (x1) and velocity (x2). The equations in the state-variable form are: dx1/dt = x2 and dx2/dt = -x1 - u.

Step-by-step explanation:

The differential equation x¨ + x© = -u can be converted into the state-variable form by introducing state variables to represent the system's outputs and their derivatives. Let's define the state variables as follows: State variable x1 equals displacement x, and the State variable x2 equals velocity x©. Hence, the first derivative of x1 is x2, and the first derivative of x2 is the original equation's second derivative x¨. Using these state variables, the original equation can be rewritten in the state-variable form as:

• dx1/dt = x2
• dx2/dt = -x1 - u

This system of first-order differential equations represents the state-space form of the given second-order differential equation.