Interdisciplinary Applied Mathematics

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z = VT f (Vz) + VT bu(t) and y = lT Vz.


The two key issues here are first, selecting a reduced basis V such that the system provides a good approximation of the original system. This has already been addressed in the previous sections. The second issue, which makes the Taylor-series-expansion-based reduced-models inefficient, is the computation of the term VTf (Vz). For linear and quadratic reduced-order models, the linear and the quadratic terms from the Taylor expansion about an equilibrium point x0 are considered, and all higher-order terms are neglected, i.e.,


f(x) « f(x0) + Ao(x — x0) + ^w0(x — x0) <8> (x — x0),


where <g> is    the    Kronecker    product    and    A0    and W0    are    the    Jacobian and


the Hessian of f(•). For the linear case, the reduced-order model becomes


z = VT f (x0)+A0r z + VT bu(t)    and y = lT Vz,


while for the quadratic case, the reduced order model becomes


z = VTf(xo) + Aorz + -Wor(z (g) z) + VTbu(t) and у = lTVz,


where A0r = VTA0V and W0r = VTW0(V <g) V) are q x q and q x q2 matrices, respectively, which are typically dense and must be represented explicitly.    As    a result,    the    cost    of    computing VTf(Vz) and    the    cost    of


storing the matrices A0r (A0r and W0r for the quadratic case) are O(q2) in the linear case and O(q3) in the quadratic case. Hence, although the method based on Taylor expansion may be extended to higher orders of nonlinearities, this approach is limited in practice to cubic expansions due to exponentially growing memory and computational costs.

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