Nonlinear grey-box modeling of vehicle dynamics. Many new vehicle features (like Electronic Stability Programs (ESP), indirect Tire Pressure Monitoring Systems (TPMS), road-tire friction monitoring systems, and so forth) rely on models of the underlying vehicle dynamics. The so-called bicycle vehicle model is a rather simple model structure that is frequently being used in the vehicle dynamics literature. In this example we will start-off with this model structure and try to estimate the longitudinal and the lateral stiffness of a tire. The actual modeling work was originally carried out by Erik Narby in his MSc work at NIRA Dynamics AB, Sweden.

The grey-box modeling of a large and complex nonlinear system. The purpose is to show the ability to use the IDNLGREY model to estimate a large number of parameters (16) in a system having many inputs (10) and outputs (5). The system is an aerodynamic body. We create a model that predicts the acceleration and velocity of the body using the measurements of its velocities (translational and angular) and various angles related to its control surfaces.

Grey-box modeling of the dynamics of an industrial robot arm. The robot arm is described by a nonlinear three-mass flexible model according to Figure 1. This model is idealized in the sense that the movements are assumed to be around an axis not affected by gravity. For simplicity, the modeling is also performed with gear ratio r = 1 and the true physical parameters are afterwards obtained through a straightforward scaling with the true gear ratio. The modeling and identification experiments detailed below is based on the work published in

Construct, estimate and analyze nonlinear grey-box models.

Perform IDNLGREY modeling based on C MEX model files. It uses a simple system where nonlinear state space modeling really pays off.

Create nonlinear grey box time series models. Time series models are models not using any measured inputs. Three idealized ecological systems are studied where two species either:

Identify the parameters of a complex yet artificial nonlinear discrete-time system with one input and one output. The system was originally proposed and discussed by Narendra and Li in the article

Grey-box modeling of a static single-input, single-output system using a MATLAB function as the ODE file.

Provide optional input arguments to IDNLGREY models. The discussion concentrates on how to do this for C-MEX types of model files, yet to some minor extent we will also address the most relevant parallels to MATLAB file modeling.

Estimate parameters of a nonlinear grey box model using multiple experiment data. A system exhibiting dry friction between two solid bodies will be used as the basis for the discussion. In this system, one body is fixed, while the other body moves forward and backward over the fixed body due to an exogenous force according to Figure 1.

Design C-MEX model files that involve scalar, vector as well as matrix parameters. As a modeling basis, we will use a somewhat idealized industrial robot, where the left-hand sides of the derived state space equations are not explicitly given. To make it a little bit more illustrative, we will also employ multiple experiment data in the identification part.

Include and simulate an IDNLGREY model in Simulink®. We use a chemical reaction system as a modeling basis. The first modeling and identification part of the example can be run without Simulink.

How the estimation algorithm choices may impact the results for a nonlinear grey box model estimation. We use data produced by a nonlinear pendulum system, which is schematically shown in Figure 1. We show in particular how the choice of differential equation solver impacts the results.