Hybrid Identification Toolbox

•  Regression problem: reconstruct a PWA map from noisy samples. In this case, one is not dealing with a dynamical system (with inputs and outputs) but just with a static map that is sampled. HIT computes a data-based PWA approximation of the map (see the examples ex_approx_1d.m and ex_cake1.m. - the second example shows how to approximate discontinuous PWA maps). The approximation has idmodes.s modes (in HITs jargon, a "mode" is an affine hyperplane + the region where is valid).
• Identification problem: reconstruct a PWARX hybrid system from noisy inputs and outputs (see for instance ex_pwarx_2d_3modes.m). PWARX systems are multi-input/single-output descriptions of hybrid systems. Thus no state appears. But they can be re-written as PWA systems pretty much the same way ARX models can be re-written as linear systems in the state-space form.
  
  The PWARX systems used in the HIT toolbox are in the form
  
  y(k)=idmodes.par{i}* [x(k)' 1]' if x(k) \in \idmodes.regions(i)
  
  where x(k)=[y(k-1) ... y(k-na) u'(k-1) ... u'(k-nb)] is the vector of regressors and the integers na and nb are the system orders. If you want to write a PWARX model in the PWA form you have to interpret x(k) as a state, y(k) as an output and find the matrices A_i, b_i, f_i, C_i, D_i, g_i that given a sequence u(k) produce the same output of the PWARX model.

@MISC{hit,
  author  = {G. Ferrari-Trecate},
  title   = {{Hybrid Identification Toolbox (HIT)}},
  year    = {2005},
}