Conditioned Invariant Subspaces Research Articles

A structural approach to unknown inputs observation for switching linear systems

The problem of devising an asymptotic observer for a given function of the state of a switching linear system in the presence of unknown inputs is considered. Solvability is studied both in the case of sufficiently large dwell time and in that of dwell time greater than a fixed threshold. A complete characterization of solvability in terms of necessary and sufficient conditions is given in both cases. It is shown that the necessary and sufficient conditions can be checked in practice in the first case and, under slightly more restrictive hypotheses, also in the second case by means of algorithmic procedures, which also provide a method to synthesize the observer sought for. The employed methodology makes use of geometric concepts to reveal the structural aspects of the problem and to derive its solutions. In particular, a key role is played by the novel notion of robust conditioned invariant subspace that is minimal with respect to the properties of containing a given subspace and of being externally stabilizable.

Fault detection and isolation of two-dimensional (2D) systems

In this work, we develop a novel fault detection and isolation (FDI) scheme for discrete-time two-dimensional (2D) systems that are represented by the Fornasini–Marchesini model II (FMII). This is accomplished by generalizing the basic invariant subspaces including unobservable, conditioned invariant and unobservability subspaces of 1D systems to 2D models. These extensions have been achieved and facilitated by representing a 2D model as an infinite dimensional (Inf-D) system on a Banach vector space, and by particularly constructing algorithms that compute these subspaces in a finite and known number of steps. By utilizing the introduced subspaces, the FDI problem is formulated and necessary and sufficient conditions for its solvability are provided. Sufficient conditions for solvability of the FDI problem for 2D systems using both deadbeat and LMI-based filters have also been developed. Moreover, the capabilities and advantages of our proposed approach are demonstrated by performing an analytical comparison with the currently available 2D geometric methods in the literature.

A geometric approach to fault detection and isolation of multi-dimensional (n-D) systems

In this work, we develop a novel fault detection and isolation (FDI) scheme for discrete-time multi-dimensional (n-D) systems for the first time in the literature. These systems represent as generalization of the Fornasini---Marchesini model II two- and three-dimensional (2-D and 3-D) systems. This is accomplished by extending the geometric FDI approach of one-dimensional (1-D) systems to n-D systems. The basic invariant subspaces including unobservable, conditioned invariant and unobservability subspaces of 1-D systems are generalized to n-D models. These extensions have been achieved and facilitated by representing an n-D model as an infinite dimensional system, and by particularly constructing algorithms that compute these subspaces in a finite and known number of steps. By utilizing the introduced subspaces the FDI problem is formulated and necessary and sufficient conditions for its solvability are provided. Sufficient conditions for solvability of the FDI problem for n-D systems using LMI filters are also developed. Moreover, the capabilities and advantages of our proposed approach are demonstrated by performing an analytical comparison with the only currently available 3-D geometric methods in the literature.

Perturbations preserving conditioned invariant subspaces

Given the set of matrix pairs keeping a subspace invariant, we obtain a miniversal deformation of a pair belonging to an open dense subset of . It generalizes the known results when S is a supplementary subspace of the unobservable one. Copyright © 2011 John Wiley & Sons, Ltd.

Stratification and bundle structure of the set of conditioned invariant subspaces in the general case

We extend some known results on the smooth stratification of the set of conditioned invariant subspaces to a general pair (C,A)∈ C (p+n)×n without any assumption on the observability. More precisely, we prove that the set of ( C, A)-conditioned invariant subspaces having a fixed Brunovsky–Kronecker structure is a submanifold of the corresponding Grassmann manifold, with a fiber bundle structure relating the observable and nonobservable part, and we then compute its dimension. We also prove that the set of all ( C, A)-conditioned invariant subspaces having a fixed dimension is connected, provided that the nonobservable part of ( C, A) has at most one eigenvalue (this condition is in general necessary).

Towards a compactification of the set of conditioned invariant subspaces

A compactification of the set of conditioned invariant subspaces of fixed dimension for an observable pair ( C, A) is proposed. It contains the almost conditioned invariant subspaces of the same dimension. In certain cases the compactification is shown to be smooth and a complete geometric description is given in the case of a single-output system.

A cellular decomposition of the manifold of observable conditioned invariant subspaces

Given an observable system defined by a pair of matrices ( C, A) we obtain a cellular decomposition of the manifold of ( C, A)-conditioned invariant subspaces with fixed observability indices of the restricted system and we apply this decomposition to compute their homology groups.

On the parametrization of conditioned invariant subspaces and observer theory

The present paper is an in depth analysis of the set of conditioned invariant subspaces of a given observable pair ( C, A) . We do this analysis in two different ways, one based on polynomial models starting with a characterization obtained in [P.A. Fuhrmann, Linear Operators and Systems in Hilbert Space, 1981; IEEE Trans. Automat. Control AC-26 (1981) 284], the other being a state space approach. Toeplitz operators, projections in polynomial and rational models, Wiener–Hopf factorizations and factorization indices all appear and are tools in the characterizations. We single out an important subclass of conditioned invariant subspaces, namely the tight ones which already made an appearance in [P.A. Fuhrmann, U. Helmke, Systems Control Lett. 30 (1997) 217], a precursor of the present paper. Of particular importance for the study of the parametrization of the set of conditioned invariant subspaces of an observable pair (C,A) is the structural map that associates with any reachable pair, with only the input dimension constrained, a uniquely determined conditioned invariant subspace. The construction of this map uses polynomial models and the shift realization. New objects, the partial observability and reachability matrices are introduced which are needed for the state space characterizations. Kernel and image representations for conditioned invariant subspaces are derived. Uniqueness of a kernel representation of a conditioned invariant subspace is shown to be equivalent to tightness. We pass on to an analysis and derivation of the Kronecker–Hermite canonical form for full column rank, rectangular polynomial matrices. This extends the work of A.E. Eckberg (A characterization of linear systems via polynomial matrices and module theory, Ph.D. Thesis, MIT, Cambridge, MA, 1974), G.D. Forney [SIAM J. Control Optim. 13 (1973) 493] and D. Hinrichsen H.F. Münzner and D. Prätzel-Wolters [Systems Control Lett. 1 (1981) 192]. We proceed to give a parametrization of such matrices. Based on this and utilizing insights from D. Hinrichsen et al. [Systems Control Lett. 1 (1981) 192], we parametrize the set of conditioned invariant subspaces. We relate this to image representations, making contact with the work of J. Ferrer et al. [Linear Algebra Appl. 275/276 (1998) 161; Stratification of the set of general (A,B)-invariant subspaces (1999)]. We add a new angle by being able to parametrize the set of all reachable pairs in a kernel representation, this via an embedding result for rectangular polynomial matrices in square ones. As a by product we redo observer theory in a unified way, giving a new insight into the connection to geometric control and to the stable partial realization problem.

The topology of the set of conditioned invariant subspaces

We consider the topology of the set of conditioned invariant subspaces of an observable pair ( C, A) of a fixed dimension. By fixing the observability indices of the restricted system, a stratification by finitely many smooth manifolds is obtained, termed Brunovsky strata. It is shown that each Brunovsky stratum is homotopy equivalent to a generalized flag manifold. From this description an effective formula for the Betti numbers of the Brunovsky strata can be derived.

A homeomorphism between observable pairs and conditioned invariant subspaces

A bijective correspondence between similarity classes of observable systems ( C, A) and n-codimensional conditioned invariant subspaces of a pair ( C, A) is constructed that leads to a homeomorphism of the spaces. This is applied to the parametrization of inner functions of fixed McMillan degree. Proofs using state space methods as well as using polynomial models are given.

Disturbance Localization with Dead-beat Control by Measurement Feedback for Linear Periodic Discrete-time Systems

The new notions of inner controllable, outer controllable and outer reconstructible subspace, together with the extensions of the classical notions of controlled invariant and conditioned invariant subspaces to the class of linear periodic discrete-time systems, are the main tools used in this paper in order to study the disturbance localization problem by measurement feedback for the same class of systems. By means of these concepts the necessary and sufficient solvability conditions of such a problem and synthesis procedures of the solutions are found for three different cases, i.e. with and without an additional requirement of output or state dead-beat control.

Linear function dead-beat observers with disturbance localization for linear periodic discrete-time systems

Abstract In this paper, the classical notion of a conditioned invariant subspace is extended to the case of linear periodic discrete-time systems, properly characterized and suitably specialized to the notion of an outer reconstructible subspace. These ideas are used in order to find necessary and sufficient conditions for the existence of a linear function dead-beat observer for such systems under unknown and unmeasurable disturbances and a synthesis procedure for such an observer.

Comments on ‘Conditioned invariant subspaces, disturbance decoupling and solutions of rational matrix equations’

Comments on ‘Conditioned invariant subspaces, disturbance decoupling and solutions of rational matrix equations’

Self-bounded controlled invariants versus stabilizability

Self-bounded controlled and self-hidden conditioned invariant subspaces, recently introduced by the authors for a more direct and neat handling of some fundamental concepts of the geometric approach, such as controllability subspaces, are proved in this paper to be very useful tools also in dealing with synthesis problems with stability requirements.

Conditioned invariant subspaces, disturbance decoupling and solutions of rational matrix equations

Conditioned invariant subspaces are introduced both in terms of output injection and in terms of state estimation. Various properties of these subspaces are explored and the problem of disturbance decoupling by output injection (OIP) is defined. It is then shown that OIP is equivalent to the problem of disturbance decoupled estimation as introduced in Willems (1982) and Willems and Commault (1980). Both solvability conditions and a description of solutions for a class of rational matrix equations of the form X(s)M(s) = Q(s) on several ways are given in state-space form. Finally, the problem of output stabilization with respect to a disturbance is briefly addressed.