State-variable and output equations

This set of system equations can now be manipulated into state-variable form. Knowing state-variables are X1 and X2 and the input is F, state-variable equations will look like the following:

Let the output of the system be the velocity of the engine. Then the output equation will become:

1. Transfer function

To find the transfer funciton of the system, first, take Laplace transforms of above state-variable and output equations.

Using these equations, derive the transfer function Y(s)/F(s) in terms of constants. When finding the transfer function, zero initial conditions must be assumed. The transfer function should look like the one shown below.

2. State-space

Another method to solve the problem is to use the state-space form. Four matrices A, B, C, and D characterize the system behavior, and will be used to solve the problem. The state-space form that were manipulated from the state-variable and the output equations is shown below.