Transfer function and state space model are developed for system shown below.
| (input) | f(t) | = | external force applied on mass | |
| (output) | z(t) | = | position | unknown |
| m | = | mass | ||
| k | = | spring constant | ||
| b | = | damping coefficient |
1 unknown so 1 equation is needed.
Newton's 2nd law:
Laplace transform (eq. 1):
Solve for Z(s)/F(s):
When there is a mass in a system, its position and velocity are commonly chosen as state variables. Also, position, velocity, and force (input) are sufficient to determine this system's future position (output). For these reasons, position and velocity are chosen as state variables.
State vector:
Input vector:
Output vector:
Rewrite (eq. 1) in these new notations:
Rearrange equations to express αΊ‹(t) and y(t) in terms of x(t) and u(t):
Organize into matrix format:
