HYS2PWA Transformation of HYSDEL model into PWA model
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Title: hys2pwa
Project: Transformation of HYSDEL model into PWA model
Input: S: structure containing MLD model generated by HYSDEL compiler
verbose: =0 silent
1 verbose important information
2 verbose detailed information
constr: In hysdel, bounds need to be specified for real states
and inputs.
=0 don't use these bounds as additional constraints
=1 use these bounds as additional constraints. Check them
a priori (and not a posteriori as constraints in the
must section) thus speeding up the transformation. Do
not add them to Hi{i}, Ki{i} in order to have an
efficient representation of the polyhedra.
=2 as 1, but add the bounds to Hi{i}, Ki{i}
plot: =0: don't plot anything (=default)
=1: plot resulting polyhedral partition over real state-
input space with bounds as specified in HYSDEL
=2: plot resulting polyhedral partition as in 1 and
additionally the PWA output functions
Plot 1 and 2 work only for dimensions 1 and 2.
Remark: The plotting feature is not contained in the publicly
distributed version. However, the files are available from the
authors on request (see below).
Output: P: structure containing the PWA model, where P{k} is the k-th
PWA submodel with binary state xb = P{k}.xb and binary input
ub = P{k}.ub.
Let's consider now the PWA dynamics for a given xb and ub:
The real state/input space is partitioned into a polyhedral
partition with N polyhedra
Hi{i}*[xr; ur] <= Ki{i}, i=1,...,N.
The PWA dynamics are defined on the polyhedral partition by
x' = fx(i) * xr + fu(i) * ur + f0(i)
y = gx(k) * xr + gu(i) * ur + g0(i)
Additionally, the z and d-variables (as defined in HYSDEL)
are given by
z = Zx(i) * xr + Zu(i) * ur + Z0(i)
d = delta(i)
P{k}.delta_AD denotes the delta variables, that are defined in the
AD section, i.e. that is part of a hyperplane arrangement (in
contrast to delta variables defined in the logic section which
are not part of an hyperplane arrangement as they depend via a
boolean expression on other delta variables).
If P{k}.delta(i) > 0, then the i-th delta variable has been defined
in the AD section and it has computational order 'P{k}.delta(i)'.
The delta variables of the i-th region are given by P{k}.delta{i}.
According to Ziegler, we define the markings in the following way:
Example: delta = [-1 -1 +1;
-1 +1 +1]
implies the existence of 2 hyperplanes generating 3 polyhedra:
1. A x <= B;
2. A(1,:) <= B(1), A(2,:) >= B(2);
3. A(1,:) >= B(1), A(2,:) >= B(2);
This is according to the definition of the paper, but contrary to
the way the deltas are internally processed in hys2pwa! In the
very end of hys2pwa, delta is multiplied with (-1) to get the
proper definition.
In general, the polyhedral partition is not bounded. Bounds
resulting from the bounds defined in HYSDEL are given by
Authors: Tobias Geyer <geyer@control.ee.ethz.ch>, Fabio D. Torrisi
Papers: 'Efficient Mode Enumeration of Compositional Hybrid Systems',
in A. Pnueli & O. Maler, eds. 'Hybrid Systems: Computation
and Control', Vol. 2623 of Lecture Notes in Computer Science,
Springer Verlag, pp. 216-232.
see also the extended version:
'Efficient Mode Enumeration of Compositional Hybrid Systems',
Technical Report, AUT03-01, ETH Zurich, 2003