Approximating Nonlinear Behavior Using an Array of LTI
Systems
This example shows how to approximate the nonlinear behavior of a system as an array of
interconnected LTI models.The example describes linear approximation of pitch axis dynamics of an airframe over a
range of operating conditions. The array of linear systems thus obtained is used to create
a Linear Parameter Varying (LPV) representation of the dynamics. The LPV model serves
as an approximation of the nonlinear pitch dynamics.About Linear Parameter Varying (LPV) ModelsIn many situations the nonlinear dynamics of a system need to be approximated using
simpler linear systems. A single linear system provides a reasonable model for behavior
limited to a small neighborhood around an operating point of the nonlinear system. When
the nonlinear behavior needs to be approximated over a range of operating conditions, we
can use an array of linear models that are interconnected by suitable interpolation rules.
Such a model is called an LPV model.To generate an LPV model, the nonlinear model is trimmed and linearized over a grid of
operating points. For this purpose, the operating space is parameterized by a small
number of scheduling parameters. These parameters are often a subset of the inputs,
states, and output variables of the nonlinear system. An important consideration in the
creation of LPV models is the identification of a scheduling parameter set and selection of
a range of parameter values at which to linearize the model.We illustrate this approach for approximating the pitch dynamics of an airframe.Pitch Dynamics of an AirframeConsider a three-degree-of-freedom model of the pitch axis dynamics of an airframe. The
states are the Earth coordinates , the body coordinates , the pitch angle ,
and the pitch rate. Figure 1 summarizes the relationship between the inertial and
body frames, the flight path angle , the incidence angle , and the pitch angle.Approximating Nonlinear Behavior Using an Array of LTI Systems