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Local Modeling

A similar treatment of finite state-space modeling may be found in [Johansen and Foss(1993b)]. A model of general state-space system may be written as:

$\begin{displaymath}\frac{dx}{dt}=f(x,u,v)\end{displaymath}$

y=g(x,u,v)

where $x\in R^{n}$ is the state vector, $u\in R^{m}$ is the input vector, $y\in R^{s}$ is the output vector, and $v\in R^{r}$ is the input disturbance vector. The local model M_i may then be described by

$\begin{displaymath}\frac{dx_{i}}{dt}=f(x_{i},u,v_{i})\end{displaymath}$

y_i=g(x_i,u,v_i)

where the state and disturbance vectors of model i may not necessarily be the same as the state and disturbance vectors of the other models or the actual system being modeled. The operating point $\phi$ at any time t is a single point in the operating space $\Phi$ , which is made up of different operating regimes $\Phi _{i}\in \Phi$ . The operating point $\phi$ may be written as:

$\begin{displaymath}\phi (t)=h(y(t),u(t),x(t))\end{displaymath}$

Typically, $\phi$ can be parameterized as a function of the input u(t) or the output y(t). Assume that for each local model there are model validity functions $\rho _{i}$ that map the entire operating space $\Phi$ to [0,1]. The overall model for N different model regimes becomes

$\begin{displaymath}w_{i}(\phi (t))=\frac{\rho _{i}(\phi (t))}{\sum _{i=1}^{N}\rho _{i}(\phi (t))}\end{displaymath}$

$\begin{displaymath}\frac{dx_{i}}{dt}=\sum _{i=1}^{N}f(x_{i},u,v_{i})\end{displaymath}$

$\begin{displaymath}y=\sum _{i=1}^{N}g(x_{i},u,v_{i})\, w_{i}(\phi (t))\end{displaymath}$

The weighting function $w_{i}(\phi )$ is normalized so that at any operating point, the local model weighting functions sum to unity.

Next: Methodology Up: Multiple Model Approach for Previous: Introduction and Motivation

Edward Price Gatzke
1999-07-12