An Algebraic Approach to the Realization of Lossless Negative Imaginary Behaviors

We consider a behavior with an equal number of inputs and outputs for which there exists a quadratic differential form which is positive for nonzero trajectories of the behavior and whose derivative is equal to the scalar product of the input vector and the derivative of the output vector. Such systems occur, for example, when considering conservative mechanical systems. We give a method of computation of a state space realization from an observable image representation of such a behavior. We apply the insights derived from this realization procedure to the synthesis of lossless mechanical systems.