Symbolic and numerical software tools for LFT-based low order uncertainty modeling

Abstract

In this paper we give a theoretical and a computational solution to the most general inner-outer factorization problem formulated for a discrete-time system G. Our method is based on descriptor state-space computations and relies on an efficient dislocation of the minimal indices and of the "unstable" zeros of G by left multiplication with all-pass factors. The minimal indices are dislocated by solving for the stabilizing solution an algebraic Riccati equation of order n (the sum of left minimal indices) while the n b unstable zeros are dislocated by solving a Lyapunov equation of order n b. The results reported here are a nontrivial extension of a recently developed approach to the continuous-time inner-outer factorization problem.

Generalized LFT-Based Representation of Parametric Uncertain Models

“…Together with reliable numerical tools for handling descriptor systems available in the MATLAB descriptor system toolbox [14] and with symbolic preprocessing techniques for parametric system matrices of [4,15], we have a very promising approach to efficiently generate low order LFTrepresentations of uncertain physical systems. …”

“…Additional symbolic manipulations can contribute significantly in obtaining lower order LFT-realizations. For example, by using Horner or optimal evaluation schemes of polynomials [15], important reduction of orders can be occasionally achieved.…”

“…(1) Construct a descriptor LFT-representation, such that The main advantage of this simpler LFT-realization procedure is that we can apply the symbolic preprocessing techniques of [4,15] to the concatenated symbolic matrix [N()jD()] (see step 1), which contains all matrices of the system. Hence, it is expected that the resulting LFT-realization is of lower order than an LFT-representation, which is realized using the more general procedure of Section 5.1, where each system matrix is realized separately.…”

Exact Nonlinear Modelling Using Symbolic Linear Fractional Transformations

“…Several approaches can be used to obtain the uncertain parameterized LTI LFT model, see (Varga, A. et al, 1998;Bates, D. and Postlethwaite, I., 2002) for a review. One of the first approaches to use symbolic LFT modelling for linear systems was proposed in (Varga, A. et al, 1998;Varga, A. and Looye, G., 1999). These two references propose a two-step modelling approach for the nonlinear system given by equations (4-6).…”

Model Reduction Software in the SLICOT Library

“…Since most of LFT descriptions are basically assimilated to discretetime systems, model reduction techniques able to handle non-minimal 1-D discrete-time systems can be employed not only to perform exact reductions but also to compute lower order approximations. In [26], several LFTmodels have been generated starting from a parametric linear state space model of a civil aircraft. The order of initial LFT models were up to 300 and reductions employing the mex -function sysred led to manageable low order exact and approximate LFT models.…”