# Interdisciplinary Applied Mathematics

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Hz = z + VT AzT VT WVz + VT Abu(t),    y = lT Vz.

The system can be expressed in the original form (equation (17.13)) by left-multiplying by H1 to obtain

z = H^z + H-1VT AzT VT WVz + H-1VT Abu(t),    y = lT Vz.

(17.16)

Setting

J’ = H-1,    b = H-1VT b,    l’ = VT l,

where H-1VTAzTVTWVz is quadratic in z and can be written in the form zTW’z for some W’. Then, equation (17.16) can be reduced to a quadratic system of the form

z = J’z + zT W’z + b’u(t),    y’ = l’T z,

where y’ is an approximation to y. The key step in this approach is the use of Arnoldi projection to reduce the large quadratic tensor to a small quadratic tensor.

The merits of this approach are: (i) This method is much more accurate than the linearized models and can be automated. (ii) It is computationally very effective. The problem with this approach is that it is not very accurate for highly nonlinear systems even though it has a quadratic nonlinear term. If higher-order terms are included, the cost of the reduced order model increases as O(n4), and the number of coefficients to be evaluated is very large.

The key idea in the trajectory piecewise-linear approach is based on representing the nonlinear system with a piecewise-linear system and then reducing each of these pieces with Krylov subspace projection methods (Rewien-ski and White, 2001). Instead of approximating the individual components as piecewise-linear and then composing hundreds of components to make a system with exponentially many different linear regions, a small set of linearizations is generated about the state trajectory, which is the response to a “training input.” Introducing the change of variables x = Vz in equation (17.13) and multiplying the resulting equation by VT yields

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