Fixed poles of disturbance rejection by dynamic measurement feedback: a geometric approach

Self-Bounded Subspaces for Nonstrictly Proper Systems and Their Application to the Disturbance Decoupling With Direct Feedthrough Matrices

“…On the other hand, the latter have important advantages over those expressed by (9). First, the conditions in Theorems 3 and 4 are split into a so-called structural condition (which is the solvability condition of the same decoupling problem considered without stability requirements) and a stability condition (which is independent of the signal to be decoupled being either unaccessible, measurable or even known in advance with finite preview [8]).…”

“…Hence, the independent role of each condition in the solution of different decoupling problems is made more explicit when these are expressed as shown in Theorems 3 and 4. Second, from a computational point of view, checking the set of conditions of Theorems 3 and 4 is much simpler than checking (9). In fact, R * Σ can be simply computed as the intersection V * Σ ∩ S * Σ = V * Σ ∩ S * Σ , 1 , whereas the determination of V * g requires eigenspace computations, which is rather critical for high-order systems [4], [12], [15].…”

“…Indeed, the order of the compensator solving these problems is, in general, equal to the dimension of the output-nulling where the state trajectory lies. Hence, since the solvability conditions given in Theorems 3 and 4 are constructive, the solution of the MSDPs based on this set of conditions leads to compensators whose order is smaller than that of compensators designed by resorting to conditions (9). As a result, the use of R * Σ allows us to reduce the state dimension of the decoupling filter.…”

“…In the last two decades, many papers have been written on the solution of different control problems involving the concept of selfboundedness (see [7]- [9], and the references therein). Moreover, much effort has been devoted to the generalization of these classic results to nonstrictly proper systems.…”

Structural analysis of sensor location for disturbance rejection by measurement feedback

“…In the case where the state is not available for measurement, the problem of disturbance rejection by measurement feedback is more complex and has been solved in an elegant way in geometric terms, see Schumacher (1980) and Willems and Commault (1981). The characterization of the solutions with internal stability and the set of fixed modes have been also characterized geometrically, see del Muro Cuellar and Malabre (2001) and Eldem and Ozguler (1988). In the framework of structured systems introduced by Lin (1974), the DRMF problem has been solved graphically in Commault, Dion, and Hovelaque (1997) and van der Woude (1993).…”