An algebraic representation of parameter sensitivity in linear time-invariant systems

Abstract This paper presents a new algebraic representation of trajectory parameter sensitivities for linear time-invariant ordinary differential equation systems. By working from first principles, the parameter sensitivities are obtained from the partial derivatives of the system matrices and state transition matrix. The resulting matrix-operator form allows one to compute the complete set of parameter sensitivities with at most 2nr quadrature integrals where n is the state dimension and r is the control dimension. Additionally, this form provides considerable geometric-insight into the sensitivity system and, in particular, some of the properties related to controllability are discussed. Finally, the results concerning the partial derivatives of the state transition matrix are interesting in their own right, and they allow us to extend some previously reported ( 15 , 16 ) structural properties of the sensitivity system to a more general case.

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