Disturbance rejection for systems over rings

It is demonstrated how the spaces V* and V−, known in the geometric theory of linear systems can be generalized to systems over rings and how one can check whether these spaces have the feedback property. Based on these results, the solution of the disturbance-rejection problem is given, with and without internal stability

Necessary conditions for multiple constraint optimization problems

In this paper a necessary condition is given for a real-valued function f to attain a maximum at a point b subject to the condition XES, where S is given as an intersection of a finite number of sets in an n-dimensional Euclidean space. It is shown that well-known necessary conditions in mathematical programming, like the Lagrange multipliers theorem and results ofF. John, Mangasarian and Fromovitz, are immediate consequences of this general condition. The result is also used to derive ageneral necessary condition for discrete-time optimal control problems, which contains the results of Halkin (discrete maximum principle), Jordan and Polak, and Canon, Cullum and Polak as special cases. As a final application of the necessary condition a simple proof of the Pontryagin maximum principle for continuous-time control problems is given

Controlled invariance in systems over rings

The definition of controlled invariant (i.e. (A,B)-invariant) subspaces of a linear system is extended to systems over rings. It is observed that in this more general setting, the equivalence of the geometric and the feedback characterization is no longer true. Particular attention is paid to the weakly unobservable space V*, and conditions are given for this space to satisfy the feedback characterization. These conditions have the form of the existence of a factorization of the transfer function. An application to the disturbance rejection problem is given

Stabilizability subspaces for systems over rings

Results obtained previously for controlled invariant subspaces for systems over rings are generalized to stabilizability subspaces. Stability is defined based on an axiomatically introduced concept of convergence. The results are applied to the problem of disturbance decoupling with internal stability for systems over rings

The formal Laplace transform for smooth linear systems

A class of time invariant linear systems is introduced. For systems in this class a formal Laplace transform is defined and invertibility properties are studied using this transform. The results are related to known results in literature

(A,B)-invariant and stabilizability subspaces : a frequency domain description with applications

In a paper of E. Emre and the author a polynomial characterization for (A,B-invariant subspaces is given. The characterization is used to give a frequency domain criterion for the solvability of the disturbance decoupling problem. In this paper a more elementary and simpler treatment is given. Furthermore, stabilizability subspaces are introduced, are given a frequency domain characterization and are used to solve a variety of problems