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. Earlier, Van der Schaft and Oeloff had considered a specific form of realization for such behaviours that plays an important role in their model reduction procedure. In our paper, we give a method of computation of a state space realization from a transfer function of such a behaviour in the same form as considered by Van der Schaft and Oeloff, using polynomial algebraic methods. Apart from being useful in enlarging the scope of the model reduction procedure of Van der Schaft and Oeloff, we show that our method of realization also has application in the synthesis of lossless mechanical systems with given transfer functions using springs and masses.