New interpretation of balancing state space representations as an input-output energy minimization problem

Given a linear dynamical system, there exists an infinite number of state space models that realize this system. These models are related by a similarity transformation. Even though controllability and observability properties of minimal realizations are invariant, the more general notions of degrees of controllability and observability are very dependent on the particular state space representation. It is shown that there exists a realization (A, B, C) that is termed ‘internally balanced’ and that has a natural basis when dealing with the input-output energies distribution. The energies of the balanced model are equally distributed between input-state and slate-output maps. An energy storage efficiency is then defined naturally and maximization of this quantity is shown to be desirable. It is demonstrated that the energy storage efficiency is at its maximum when the state space representation (A, B, C) is chosen in its balanced form.