Interdisciplinary Applied Mathematics

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In the piecewise-linear approach, s linearized models of the nonlinear system with expansion points x0,… ,xs_i are considered, i.e.,


X = f(xj) + Aj(x — Xi) + bu(t),


where x0 is the initial state of the system and Ai are the Jacobians of f (•) evaluated at the states xi. Considering a weighted combination form,


s—1


x = £ H(x)f(xi) + Wj(x)Aj(x — xj)] + bu(t),


i=0


where w, (x) are weights that sum to 1. Assuming that a qth-order basis V has already been generated, the following reduced-order model is obtained:


z = (Ar • w'(z)T)z + у • w'(z)T + bru(t),    y = lrz,


where y = [VT(f(xo) — Aoxo),…, VT(f(xs— i) — As—ixs—i)], br = VTb, lr = lTV, Ar = [A0r A1r … A(s—i)^ , w’ = [w0, wi,…, ws—i] is a vector of weights, and Air = VTA,V. The weights are computed in the following manner.


1. For i = 0,…, (s — 1) compute di = ||z — zi^2.


2. Compute m = min[di : i = 1,…, (s — 1)].


3. For i = 0,…, (s — 1) compute wi = (exp(di)/m))—25.


4.    Normalize wi.


This implies that the linearized point that is closest to the current position gets the maximum weight. Instead of finding linearized models covering the entire N-dimensional state space, a collection of models is generated along a single trajectory of the system. This is done by simulating the system at the initial point and moving ahead by a very small interval from that point to get a new point, and the process is repeated for each point. However, this method requires performing simulation of the initial nonlinear system, which may be very costly due to the initial size of the problem. This problem is avoided by using the Arnoldi-based Krylov subspace method, instead of the full simulation, to simulate the nonlinear system and obtain approximate trajectory and linearization points, making the process faster.

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