Geometric theory for the singular Roesser model

Abstract

1and differentiating, we get Cr = li -€PV -EPV, ) = -Q i C -Q u + U .Substituting for li and U from the original system, and simplifying, we get where and A different viewpoint is taken in developing the decoupling transformations for singularly perturbed linear systems. The proposed transformations have the advantage over the previous ones since they also decouple the transformation equations (13) and (14), enabling us to perform the computations in parallel. This is numeri-cally very efficient for the case of time-varying systems where the corresponding differential equations are stiff. REFERENCES Gajic, D. Petkovski, and X. Shen, Singularly Perturbed and Weakly Coupled Linear Control Systems-A Recursive Approach. Abstract-( A , E , B)-invariant and ( E , A , B)-invariant subspaces forthe two-dimensional singular Roesser model are investigated. These subspaces are related to the existence of the solutions when the boundary conditions are in these subspaces. Also existence of a solution sequence in certain subspaces derived from the invariant subspaces is shown. The boundary conditions that appear in the solution when some semistates in the solution are restricted to zero are also investigated.

A geometric approach with stability for two-dimensional systems

“…In this case, the LMI (20) is feasible, which implies internal stabilisability of V , and its solution yields K = −5.8697 0.1389 −0.0031 . By using (15) we find that now the pair (Ξ 1 , Ξ 2 ) is asymptotically stable, as it satisfies the stability condition (9). With this choice , did not change after the introduction of Λ, so that the internal stabilisation previously performed has not been affected; on the other hand, V has been externally stabilised since the pair (0, −0.1225) is now asymptotically stable.…”

“…In the last two decades, many valuable results have been achieved in the attempt to develop a geometric theory for 2-D systems, [3], [8], [9], [12]. In particular, in [3] a definition of controlled invariance was proposed for Fornasini-Marchesini (FM) models.…”

Geometric techniques for implicit two-dimensional systems

“…In [17], it was shown that the definition of controlled invariance in [12] can be extended to models in the form (5). This definition retains the fundamental properties listed as (a) and (b) in the discussion of the 1-D case above.…”

“…By contrast with the model considered in [12], the FM model (6-7) is closed under the feedback u i,j = F x i,j , which gives rise to the closed-loop local state update equation…”

“…A second definition of controlled invariance for 2-D systems was provided in [12] for the implicit 2-D model…”