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

Some nonlinear effects Aerodynamic drag on a vehicle Rotodynamic pump Nonlinear spring effect Nonlinear geometry Check valve modeling

Aerodynamic drag effect Typical aerodynamic drag on a passenger vehicle can be modeled by a drag force, Ra, given by where ρ is mass density of air, C D is drag factor due to vehicle shape, A f is frontal (projected) area, and Vr is vehicle speed

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.

Hardening spring A typical relation for the characteristic of a hardening spring is given by where F is the spring force and δ is the spring deflection from free length.

Nonlinear geometry Fmagnetic mg θ L Mass, m

Examples to illustrate nonlinear methods Pendulum with magnetic force applied Spring-loaded pendulum Hanging sign problem Print-head mechanism

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.)

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. )

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.

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.