Keep your face always toward the sunshine - and shadows will fall behind you. Walt Whitman
Idea Transcript
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.