Nonlinear Systems

1
Nonlinear Systems

• Modeling some nonlinear effects
• Standard state equation description
• Equilibrium points
• Linearization about EPs
• Simulation and insight
• Equilibrium point design

2
Some nonlinear effects

• Aerodynamic drag on a vehicle
• Rotodynamic pump
• Nonlinear spring effect
• Nonlinear geometry
• Check valve modeling

3
Aerodynamic drag effect
Typical aerodynamic drag on a passenger vehicle
can be modeled by a drag force, Ra, given by
where ? is mass density of air, CD is drag factor
due to vehicle shape, Af is frontal (projected)
area, and Vr is vehicle speed
4
Rotodynamic pump
A typical model for the outlet port of a
rotodynamic pump is given by
where P is outlet port pressure, N is shaft
speed, and Q is outlet port flow.
5
Hardening spring
A typical relation for the characteristic of a
hardening spring is given by
where F is the spring force and d is the spring
deflection from free length.
6
Nonlinear geometry
Mass, m
L
Fmagnetic
?
mg
7
Examples to illustrate nonlinear methods

• Pendulum with magnetic force applied
• Spring-loaded pendulum
• Hanging sign problem
• Print-head mechanism

8
Formulation of standard equations

• Identify the state and input vectors X(t) and
  U(t).
• Formulate a set of system equations.
• Reduce the equations to the form

(If this is not possible then a different
approach needs to be taken which we will not
discuss here.)
9
Equilibrium points

• We seek equilibrium points (EPs) under the
  following conditions
• Assume all inputs are constant, U(t) Uc.
• Assume all derivatives go to zero simultaneously.
  
• The equations become
• Solve the EP equations for Xep, given Uc.

(There may be zero, one or many EPs. Finding them
can be a daunting task on occasion. )
10
Linearization about an EP

• To gain insight about the nature of a given EP we
  can linearize the model about the EP.
• We use a Taylor series method, expanding about
  the EP.
• The resulting linearized model can be written as
  

See Linearization a la Taylor notes.
11
Simulation for insight

• To locate a stable EP in a difficult problem we
  can sometimes simulate the dynamic response and
  watch it go to the EP.
• Once such an EP has been located we can simulate
  the behavior in the vicinity of the EP to get a
  feeling for the local behavior.
• It is also possible to numerically approximate
  the linearized A, B matrices at an EP.

See Hanging Sign example and Numerical
linearization notes.