Page 63 - Parimad teadustööd 2013/2014

TALLINNA ÜLIKOOLI ÜLIÕPILASTE 2013/2014. ÕPPEAASTA PARIMAD TEADUSTÖÖD / ARTIKLITE KOgUMIK LOODUSTEADUSED
a linear operator :E1→E2, which is a × matrix, transforms one-forms into two-forms. its (i,j)-th entry reads
where ∈E1 and deg :=max {,,=1,...,}.
The exterior derivative of operator is such that its (i,j)-th entry is given by
The procedure for computing the flat outputs, whenever they exist, is adapted from [2]. It consists of the following steps:
iii. Find the operator that satisfies the property .
iV. among the possible ’s, only those satisfying
V. Compute such that () = −.
Vi. only unimodular matrices are kept. A flat output is obtained by integration of d = .
Implementation.
Procedure.
i. Compute () as in (4).
ii. Compute the -dimensional vector of 1-forms , defined by (5).
are kept.
In order to find a flat output for nonlinear system, the following functions were programmed in mathe- matica:
»» EliminateInput finds an implicit representation (2) of a given system in the explicit form (1).
»» PolynomialMatrix finds a polynomial matrix () according to (4).
»» Flatness checks if there exists a flat output for any deg =,0≤≤stop.
»» LinearizingOneForms1 and LinearizingOneForms2 compute the vector of one-forms (5).
»» FlatOutput computes a flat output of the given system as well as finds how system states and inputs can be expressed as functions of the flat output and its forward shifts.
Example.
Let us find a flat output for the following discrete-time nonlinear system using the program FlatOutput
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