Reachability, observability, and minimality for shift-invariant two-point boundary-value descriptor systems (original) (raw)
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Boundary-value descriptor systems: well-posedness, reachability and observability
International Journal of Control, 1987
In this paper we introduce the class of two-point boundary-value descriptor systems (TPBVDS), discrete-time systems described by possibly linear dynamics and a set of boundary conditions constraining the values of the system 'state' at the two endpoints of the system's interval of definition. By introducing a standard form for regular pencils we obtain a new and simple generalized Cayley-Hamilton theorem that simplifies our investigation of well-posedness, Green's function solutions, and reachability and observability for TPBVDS. There are two distinct notions of reach ability and observability that one can define for TPBVDS, associated with processes that propagate inward from and outward toward the boundaries. We investigate each of these in detail, obtaining, among other things, far simpler forms for the reachability and observability results than found previously in the literature. In addition, we describe several methods for the efficient solution of TPBVDS, one involving recursions from each end of the interval toward the other and two others involving recursions that proceed outward toward and inward from the boundaries.
Realization Theory for Deterministic Boundary-Value Descriptor Systems
Realization and Modelling in System Theory, 1990
This paper examines the realization of acausal weighting patterns with two-point boundary-value descriptor systems (TPBVDSs). We restrict our attention to the subclass of TPBVDSs which are extendible, i.e., whose input-output weighting pattern can be extended outwards indefinitely, and stationary, so that their weighting pattern is shift-invariant. Then, given an infinite acausal shift-invariant weighting pattern, the realization problem consists in constructing a minimal TPBVDS over a fixed interval, whose extended weighting pattern matches the given pattern. The realization method which is proposed relies on a new transform, the (s,t) transform, which is used to determine the dimension of a minimal realization, and to construct a minimal realization by factoring two homogeneous rational matrices in the variables s and t.
Some invariants of discrete-time descriptor systems
Applied Mathematics and Computation, 2002
Many papers analyze the role of controllability and observability indices of normal linear systems in control theory and their use in the study of structural properties. This work introduces these indices for linear discrete-time descriptor systems. Such indices are studied in two different ways, by means of the state-space approach and using its transfer matrix. A canonical form of the state-space system and a special decomposition of the nonproper transfer matrix are considered in order to discuss the reachability and observability indices.
Generalized Shift-Invariant Systems
Constructive Approximation, 2005
A countable collection X of functions in L 2 (IR d) is said to be a Bessel system if the associated analysis operator T * X : L 2 (IR d) → 2 (X) : f → (f, x) x∈X is well-defined and bounded. A Bessel system is a fundamental frame if T * X is injective and its range is closed. This paper considers the above two properties for a generalized shift-invariant system X. By definition, such a system has the form X = ∪ j∈J Y j , where each Y j is a shift-invariant system (i.e., is comprised of lattice translates of some function(s)) and J is a countable (or finite) index set. The definition is general enough to include wavelet systems, shift-invariant systems, Gabor systems, and many variations of wavelet systems such as quasi-affine ones and non-stationary ones. The main theme of this paper is the 'fiberization' of T * X , which allows one to study the frame and Bessel properties of X via the spectral properties of a collection of finite-order Hermitian non-negative matrices.
Reachability, Observability and Minimality for a Class of 2D Continuous-Discrete Systems
2007
Reachability and observability criteria are obtained for 2D continuous-discrete time-variable Attasi type systems by using suitable 2D reachability and observability Gramians. Necessary and sufficient conditions of reachability and observability are derived for time-invariant systems. The duality between the two concepts is emphasized as well as their connection with the minimality of these systems.
Reachability and observability indices of a discrete-time periodic descriptor system
Applied Mathematics and Computation, 2004
This work introduces the indices of reachability and observability of a discrete-time periodic descriptor linear system. The reachability and observability indices at time s, s 2 Z, of a periodic collection of nonproper transfer matrices are characterized. Using a special kind of coprime decompositions, the relationship between the indices at consecutive times is analyzed and an algorithm is constructed.
IEEE Transactions on Automatic Control, 1989
In this paper, we introduce the concept of internal stability for two-point boundary-value descriptor systems (TPBVDSs). Since TPBVDSs are defined only over a finite interval, the concept of stability is not easy to formulate for these systems. The definition which is used here consists in requiring that as the length of the interval of definition increases, the effect of boundary conditions on states located close to the center of the interval should go to zero. Stochastic TPBVDSs are studied, and the property of stochastic stationarity is characterized in terms of a generalized Lyapunov equation satisfied by the variance of the boundary vector. A second generalized Lyapunov equation satisfied by the state variance of a stochastically stationary TPBVDS is also introduced, and the existence and uniqueness of positive definite solutions to this equation is then used to characterize the property of internal stability.