ME 3600 Control Systems
Linearity Principles
o The analysis of nonlinear systems is, in general, quite complicated – solutions to
nonlinear equations are not easily obtained.
o However, if these systems remain close to some nominal operating conditions, then it
may be appropriate to analyze their behavior using a model that has been linearized
about the nominal conditions. This simplifies the analysis of the system behavior.
o Two properties must be satisfied by a system (or model) for it to be linear. These
properties are defined in terms of the system input and output as follows
1. Principle of Superposition
If,
System
System
then,
System
2. Principle of Homogeneity (  is a scalar)
If,
System
then,
System
Kamman – ME 3600 – page: 1/2
Examples
1.
System
 x  m( x)   (mx)   y … homogeneity satisfied
x1  x2  m( x1  x2 )  mx1  mx2  y1  y2 … superposition satisfied
System is linear.
2.
System
 x  m( x)  b   (mx)  b   y … homogeneity not satisfied
x1  x2  m( x1  x2 )  b  mx1  mx2  b  y1  y2 … superposition not satisfied
System is nonlinear.
3.
System
 x  ( x)2   2 x2   2 y   y … homogeneity not satisfied
x1  x2  ( x1  x2 )2  x12  x22  2 x1x2  y1  y2 … superposition not satisfied
System is nonlinear.
4.
System
 x  d ( x) dt   (dx dt )   y … homogeneity satisfied
x1  x2  d ( x1  x2 ) dt  (dx1 dt )  (dx2 dt )  y1  y2 … superposition satisfied
System is linear.
5.
System
 x  a( x)  b (d ( x) dt )   (ax  b(dx dt ))   y … homogeneity satisfied
x1  x2  a( x1  x2 )  b(d ( x1  x2 ) dt )

 … superposition satisfied
 (ax1  b(dx1 dt ))  (ax2  b(dx2 dt ))  y1  y2 
System is linear.
Kamman – ME 3600 – page: 2/2