Rules for Combining Transfer Functions

  
Figure 1.2:

Figure 1.3 shows a serial connection of two systems: .

  
Figure 1.3:

Figure1.4 shows a parallel connection of two systems: .

  
Figure 1.5: Feedback Connection

Figure 1.5 shows a feedback connection of two systems:

Also

so

Proof:

for

Suppose , then

In general we may have non-linear systems i.e.,

To solve this system we need initial conditions . If we know and , this is necessary and sufficient to determine on . is the state of the system.

Example .

Orbiting Satellite

 
Figure 1.6: Orbiting Satellite

Lagrangian . Euler-Lagrange equations give:

We introduce

Then linearize the equations to obtain a linear approximation of the system.