Stratification of Controllability and Observability Pairs—Theory and Use in Applications

Abstract

Abstract. Cover relations for orbits and bundles of controllability and observability pairs associated with linear time-invariant systems are derived. The cover relations are combinatorial rules acting on integer sequences, each representing a subset of the Jordan and singular Kronecker structures of the corresponding system pencil. By representing these integer sequences as coin piles, the derived stratification rules are expressed as minimal coin moves between and within these piles, which satisfy and preserve certain monotonicity properties. The stratification theory is illustrated with two examples from systems and control applications, a mechanical system consisting of a thin uniform platform supported at both ends by springs, and a linearized Boeing 747 model. For both examples, nearby uncontrollable systems are identified as subsets of the complete closure hierarchy for the associated system pencils.Key words. Stratification, matrix pairs, controllability, observability, robustness, Kronecker structures, orbit, bundle, closure hierarchy, cover relations, StratiGraph.1. Introduction. Computing the canonical structure of a linear time-invariant (LTI) system,ẋ(t) = Ax(t) + Bu(t) with states x(t) and inputs u(t), is an ill-posed problem, i.e., small changes in the input data matrices A and B may drastically change the computed canonical structure of the associated system pencil A − λI B (e.g., see [13]). Besides knowing the canonical structure, it is equally important to be able to identify nearby canonical structures in order to explain the behavior and possibly determining the robustness of a state-space system under small perturbations. For example, a state-space system which is found to be controllable may be very close to an uncontrollable one, and can therefore by only a small change in some data, e.g., due to round-off or measurement errors, become uncontrollable. If the LTI system considered and all nearby systems in a given neighborhood are controllable, the system is called robustly controllable (e.g., see [46]).The qualitative information about nearby linear systems is revealed by the theory of stratification for the corresponding system pencil. A stratification shows which canonical structures are near to each other (in the sense of small perturbations) and their relation to other structures, i.e., the theory reveals the closure hierarchy of orbits and bundles of canonical structures. A cover relation guarantees that two canonical structures are nearest neighbours in the closure hierarchy.For square matrices, Arnold [1] examined nearby structures by small perturbations using versal deformations. For matrix pencils, Elmroth and Kågström [23] first investigated the set of 2-by-3 matrix pencils and later extended the theory, in collaboration with Edelman, to general matrices and matrix pencils [17,18]. In line of this work, the theory has further been developed in [21], and for matrix pairs in [20,42]. Several other people have worked on the theory of stratifications and similar topics, and we refer...

Orbit Closure Hierarchies of Skew-symmetric Matrix Pencils

“…Therefore it is important to get knowledge about the canonical forms (or canonical structure information) of the pencils that are close to A − λB. One way to investigate this problem is to construct the stratification (i.e., the closure hierarchy) of orbits and bundles of the pencils [17].The stratification of matrix pencils under strict equivalence transformations [16,17,18] as well as the stratification of controllability and observability pairs [19] are known. StratiGraph [21,24] is a software tool for computing and visualization of such stratifications.…”

“…The stratification of matrix pencils under strict equivalence transformations [16,17,18] as well as the stratification of controllability and observability pairs [19] are known. StratiGraph [21,24] is a software tool for computing and visualization of such stratifications.…”

Canonical Structure Transitions of System Pencils

“…The bundle stratifications were derived for general matrix pencils [16], skewsymmetric matrix pencils and polynomials [8,12], controllability and observability pairs [17], and linearizations of polynomial matrices [11,23]. Here we present stratification of system pencil bundles.…”

“…If S is a minimal realization then the finite zeros are transmission zeros, otherwise further analysis is needed to determine the types of the zeros. For example, the inputand output-decoupling zeros (uncontrollable and unobservable modes) can be analyzed by studying the controllability and observability system pencils, respectively [17].…”

“…In line of previous work on matrix pencils [16], and especially controllability pencils A − sI B , observability pencils [ A−sI C ] [17] and related works on system pencils [5,7,19], we present in Sections 4 and 5 the stratification theory of S and S 0 , which reveals how the canonical forms and associated system characteristics can change under arbitrarily small perturbations of the original system pencil. The stratification shows the closure hierarchy of orbits (and bundles) of canonical structures, where an orbit is a manifold of system pencils with the same canonical structure.…”