ON STATE MODELS OF LINEAR TIME-VARIANT DISCRETE-TIME-SYSTEMS

A decoupled full-dimension form is introduced for state models. In this form, the dimension of the state vector is time-invariant, which is equal to the dimension of the state-space at each time instant. Therefore, the dimension of the state vector is minimized from the time-variant point of view and all the state variables are independent. Moreover, the state vector in a state model of the decoupled full-dimension form is partitioned into two decoupled subvectors, each of which alone forms a subsystem. One of the subsystems is an FIR system whose state vector is dependent only on inputs at (finitely) past instants. In addition, this subsystem is completely reachable. The other subsystem is an IIR system whose state-space is equal to the force-free solution set of the state vector of the subsystem. It is proved that every state model in the conventional form can be mapped by a linear one-to-one map into an equivalent state model in the decoupled full-dimension form. In comparison with a state model in the conventional form, the one in the decoupled full-dimension form has the advantages that the dimension of its state vector clearly shows the state-space and the set of all possible states for prescribed past inputs. This avoids erroneous conclusions resulting from including or selecting a state that the system does not own. The proposed decoupled full-dimension form, which suggests a new realization form of the input-output map, is consequently a convenient tool for studying LTV systems.