Boundary-value descriptor systems: well-posedness, reachability and observability

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 reachability 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 t...

[1]  D. Luenberger.  Dynamic equations in descriptor form , 1977 .

[2]  T. Kailath,et al.  Generalized state-space systems , 1978, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[3]  David G. Luenberger,et al.  Time-invariant descriptor systems , 1978, Autom..

[4]  P. Dooren.  The Computation of Kronecker's Canonical Form of a Singular Pencil , 1979 .

[5]  O. Bosgra,et al.  The determination of structural properties of a linear multivariable system by operations of system similarity 2. Non-proper systems in generalized state-space form† , 1980 .

[6]  Thomas Kailath,et al.  Linear Systems , 1980 .

[7]  R. F. Sincovec,et al.  Solvability, controllability, and observability of continuous descriptor systems , 1981 .

[8]  D. Cobb.  Feedback and pole placement in descriptor variable Systems , 1981 .

[9]  T. Kailath,et al.  A generalized state-space for singular systems , 1981 .

[10]  S. Liberty,et al.  Linear Systems , 2010, Scientific Parallel Computing.

[11]  F. Lewis.  Adjoint matrix, Bezout theorem, Cayley-Hamilton theorem, and Fadeev's method for the matrix pencil(sE-A) , 1983, The 22nd IEEE Conference on Decision and Control.

[12]  Milton B. Adams,et al.  Linear estimation of boundary value stochastic processes , 1983 .

[13]  F. Lewis.  Descriptor systems: Expanded descriptor equation and Markov parameters , 1983 .

[14]  D. Cobb,et al.  Descriptor variable systems and optimal state regulation , 1983 .

[15]  Alan Willsky,et al.  Linear estimation of boundary value processes , 1983, The 22nd IEEE Conference on Decision and Control.

[16]  Alan Willsky,et al.  Linear smoothing for descriptor systems , 1984, The 23rd IEEE Conference on Decision and Control.

[17]  D. Cobb.  Controllability, observability, and duality in singular systems , 1984 .

[18]  F. Lewis.  Descriptor systems: Decomposition into forward and backward subsystems , 1984 .

[19]  Israel Gohberg,et al.  Time varying linear systems with boundary conditions and integral operators. I. The transfer operator and its properties , 1984 .

[20]  A. Willsky,et al.  Linear estimation of boundary value stochastic processes-- Part I: The role and construction of complementary models , 1984 .

[21]  Alan S. Willsky,et al.  Linear estimation of boundary value stochastic processes--Part II: 1-D smoothing problems , 1984 .

[22]  F. Lewis.  Fundamental, reachability, and observability matrices for discrete descriptor systems , 1985 .

[23]  Ramine Nikoukhah,et al.  System theory for two point boundary value descriptor systems , 1986 .

[24]  Basil G. Mertzios,et al.  On the generalized Cayley-Hamilton theorem , 1986 .

[25]  Arthur J. Krener,et al.  A causal realization theory, part I; linear deterministic systems , 1987 .

[26]  D. Bender.  Lyapunov-like equations and reachability/observabiliy Gramians for descriptor systems , 1987 .

[27]  A. Willsky,et al.  Estimation for boundary-value descriptor systems , 1989 .