A polynomial approach to the realization of J-lossless behaviours

In this paper, a class of behaviours known as J-lossless behaviours is introduced, where J is a symmetric two-variable polynomial matrix. For a certain J, it is shown that the resulting set of J-lossless behaviours are SISO behaviours such that for each of such behaviours, there exists a quadratic differential form which is positive for nonzero trajectories of the behaviour and whose derivative is equal to the product of the input variable and the derivative of the output variable. We then give a method of computation of a state space realization from a transfer function of such a behaviour using polynomial algebraic methods. We also show that our method of realization has application in the synthesis of lossless mechanical systems with a given transfer function using springs and masses