数式処理による線形代数と制御系の正準分解への応用

抄録

A method is proposed for dealing with geometrical concepts of linear algebra by symbolic manipulation. A linear subspace can be represented by the vector expression, i.e., the linear combination of its bases with arbitrary parameters. Algorithms are shown to describe several operations of subspaces by the vector expression. The package of functions LINALG is developed on REDUCE 3.2, which can carry out the operations of linear subspaces: the sum and the intersection of subspaces, the complementary subspace, and the image, the range, the inverse image, the kernel and the invariant subspaces of a linear transformation. The package also includes the functions to compute the rank and the degeneration condition of a symbolic matrix and to solve simultaneous linear equations with symbolic coefficients.
As an application, the function is defined which obtains the Kalman's canonical form of a linear control system with model parameters. Using the functions of LINALG, it is easy to determine the four subspaces distinguished by the controllability and the observability. The function outputs the dimension, the bases and the vector expression of each subspace, the controllability and the observability matrices, and the canonical form.