Coherency and Model Reduction: A State Space Point of View

Relationship Between Coherency, Controllability and Observability In state space analysis, controllability is a notion that measures the effect of an input to a particular state, while observability measures the effect of a state to the norm of the system´s output vector. Uncontrollability and unobservability represent redundancy in the linear state space model. This paper presents the analytical connections between coherency of generators and redundant states in the linear state model. As discussed in Section 2, every pair of perfectly coherent generators results in uncontrollable states in the linear model. These states are exhibited in the form of zero singular values in the model´s controllability gramian (W2??). If a reduced order model is formed by aggregating the coherent generators, then the zero singular values are removed from the controllability gramian of the reduced order model. Parallel results are then presented for the observability gramian (W2??). The combined results on the connection between coherency and the two gramians provide the stimulus to investigate several topics relative to the coherent behavior of generators.