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Orthogonal Polynomials Approximation and Balanced Truncation for a Lowpass Filter

Abstract This paper considers the problem of orthogonal polynomial approximation based balanced truncation for a lowpass filter. The proposed method combines the system properties of balanced truncation, the computational effectiveness of proper orthogonal decomposition and the approximation capability of the orthogonal polynomials approximation. Orthogonal polynomials series expansion of the reachability and observability gramians is used in order to avoid solving large-scale Lyapunov equations and thus, significantly reducing the computational effort for obtaining the balancing transformation. The proposed method is applied for model reduction of a lowpass analog filter. Different sets of orthonormal functions are obtained from Legendre, Laguerre and Chebyshev orthogonal polynomials and the corresponding reduced order models are compared. The approximation precision is measured by the relative mean square error between the outputs of the full order model and the obtained reduced order models.