Observability for bilinear systems

Since the observability for bilinear systems is affected by the input, this paper defines three kinds of observability for bilinear systems ; namely (1) observability with some input, (2) observability with any input, (3) observability with unknown input. The necessary and sufficient condition for the observability with some input and sufficient conditions for the observability with any input and with unknown input for discrete bilinear systems are derived respectively and the effects of the input are clarified     

Observability analysis for structured bilinear systems: A graph-theoretic approach

This paper is devoted to the generic observability analysis for structured bilinear systems using a graph-theoretic approach. On the basis of a digraph representation, we express in graphic terms the necessary and sufficient conditions for the generic observability of structured bilinear systems. These conditions have an intuitive interpretation and are easy to check by hand for small systems and by means of well-known combinatorial techniques for large-scale systems.     

Approximate observability of infinite dimensional bilinear systems using a Volterra series expansion

In this paper, the approximate observability of a class of infinite dimensional bilinear systems is investigated. The observability conditions are discussed based on a Volterra series representation of this class of systems. A testable observability criterion is derived for the case where the infinitesimal generator is self-adjoint or Riesz-spectral operator. The theoretical results are illustrated by examples.     

Abstract bilinear systems: The forward shift approach

A bilinear realization theory for a Volterra series input-output map is given. The approach involves a special transform representation for a Volterra series and certain shift operators on a Fock space. The approach yields in a very simple manner a theory of span reachability, observability and minimality for bilinear systems.     

On gramians and balanced truncation of discrete-time bilinear systems

This paper obtains a balanced truncation model reduction method for discrete-time bilinear systems. After deriving the sufficient conditions for the discrete-time bilinear systems to lead to bounded outputs for some bounded inputs, we study the properties of the reachability and observability gramians. The gramians can be computed by solving two generalized Lyapunov equations. By balancing the reachability and observability gramians, a model reduction method is obtained. Some propositions of the reduced order models are given. A numerical example is given to illustrate the effectiveness of the proposed method.     

Duality between reachability and observability for adjoint systems

The Kalman duality between the reachability and observability of finite-dimensional linear systems is generalized to adjoint systems (in the sense of Arbib-Manes). The theory includes previous results on infinite-dimensional linear systems and linear systems over rings, and yields new results for classes of nonlinear systems such as bilinear and polynomial systems.     

An observable canonical form of discrete-time bilinear systems

A canonical form of discrete-time bilinear systems is investigated. A new concept of sequential observability is introduced and it is shown that the discrete bilinear system is sequentially observable if and only if it can be transformed into some canonical form. One of the distinctions between discrete-time and continuous-time bilinear systems is discussed from uniform observability and sequential observability.     

On the structure of energy functions of singularly perturbed bilinear systems

Two balanced model reduction schemes for bilinear systems have been proposed in the literature. The first schema is based on the algebraic controllability and observability matrices Gramians. The second is based on the notion of the controllability and observability energy functions already introduced for nonlinear systems. The aim of this paper is to present some interesting structures of the energy functions and algebraic Gramians of bilinear singularly perturbed systems. These results permit to avoid the ill conditioning and the high dimensionality in the computation of the energy functions of bilinear singularly perturbed systems. In the second part of the paper, closed‐loop balancing is considered and similar development leads to the so‐called past and future energy functions. Copyright © 2005 John Wiley & Sons, Ltd.     

Generic uniform observability analysis for bilinear systems

In this paper, we study the property of generic uniform observability for structured bilinear systems. More precisely, to check whether or not a structured bilinear system is generically uniformly observable, we provide a group of necessary conditions and a second group of sufficient ones. These conditions are expressed in quite simple graphic-terms. On one hand, all the given conditions are easy to check because they deal with finding paths in a digraph. On the other hand, the proposed method is based on a graph-theoretic approach and assumes only a knowledge of the system’s structure. This makes the suggested approach particularly well suited to study large scale systems or systems with unknown parameters, as may be the case during a conception stage.     

Observation of bilinear systems with application to biological control

In this paper a system of bilinear equations is derived which under appropriate conditions model the microbial cellgrowth and product formation of various waste treatment and fermentation systems. Given the available measurements, the problem of obtaining an asymptotic estimate of the state of the system is considered. Necessary and sufficient conditions for the observability of a general system of bilinear equations are also derived. As accurate biological sensors do not often exist the results are significant for the on-line control of these biological processes.     

Realization theory of discrete-time linear switched systems

The paper presents realization theory of discrete-time linear switched systems. We present necessary and sufficient conditions for an input–output map to admit a discrete-time linear switched system realization. In addition, we present a characterization of minimality of discrete-time linear switched systems in terms of reachability and observability. Further, we prove that minimal realizations are unique up to isomorphism. We also discuss algorithms for converting a linear switched system to a minimal one and for constructing a state-space representation from input–output data. The paper uses the theory of rational formal power series in non-commutative variables.     

Partial stabilisation of non-homogeneous bilinear systems

In this work, we study in a Hilbert state space, the partial stabilisation of non-homogeneous bilinear systems using a bounded control. Necessary and sufficient conditions for weak and strong stabilisation are formulated in term of approximate observability like assumptions. Applications to parabolic and hyperbolic equations are presented.     

Discrete-time observability for distributed parameter systems

The aim of this paper is to study the observability for systems described by first-order evolution equations and for those described by second-order evolution equations in the case of discrete-time observations. For the systems with a finite number of sensors we present necessary and sufficient conditions for observability. We show that these distributed parameter systems are never finite-step observable. We give the restricted sets of the initial-state spaces whose elements are N-step observable. We also investigate the relations between the systems with discrete-time observations and the systems with continuous-time observations from the viewpoint of observability Moreover, we see the essential difference between the parabolic case and the hyperbolic case.     

Observability analysis of conewise linear systems via directional derivative and positive invariance techniques

Belonging to the broad framework of hybrid systems, conewise linear systems (CLSs) form a class of Lipschitz piecewise linear systems subject to state triggered mode switchings. Motivated by state estimation of nonsmooth switched systems, this paper exploits directional derivative and positive invariance techniques to characterize finite-time and long-time local observability of a general CLS. For the former observability notion, directional derivative results are developed from the simple switching property, and they yield improved observability conditions. For the latter notion, we focus on the case where a nominal trajectory has finitely many switchings. In order to characterize long-time behaviors of the CLS, necessary and sufficient conditions are obtained for the interior of a positively invariant cone. By employing these conditions, we establish connections between finite-time and long-time local observability; underlying positive invariance properties are unveiled.     

Exact observability and exponential stability of infinite-dimensional bilinear systems

This paper deals with exact observability and stability for a class of infinite-dimensional bilinear systems. First we show that the bilinear systems are exactly observable for any positive bounded input signal. Second we prove that given such an input signal the open-loop bilinear systems (or time-varying linear systems) are exponentially stable. A strong motivation of our study comes from chemical engineering processes described by partial differential equations.     

Observability and ε-observability for distributed-parameter systems

The aim of this article is to study the observability and state reconstruction of parabolic systems. It is shown that observability can be deduced from the coercivity of a bilinear form associated with a quadratic criterion, and that the solution depends continuously on the data. The regularization of the criterion enables the notion of ε-observability of parabolic systems to be introduced, and the relationships between the two notions are studied. The practical case of systems of finite dimension obtained by discretization of continuous equations is also studied, from the point of view of observability as well as ε-observability and the results obtained, and numerical examples are provided to indicate why it may be of interest to search a priori for an ε-observation in applications.     

On H2 model reduction of bilinear systems

The H₂ model reduction problem for continuous-time bilinear systems is studied in this paper. By defining the H₂ norm of bilinear systems in terms of the state-space matrices, the H₂ model reduction error is computed via the reachability or observability gramian. Necessary conditions for the reduced order bilinear models to be H₂ optimal are given. The gradient flow approach is used to obtain the solution of the H₂ model reduction problem. The formulation allows certain properties of the original models to be preserved in the reduced order models. The model reduction procedure developed can also be applied to finite-dimensional linear time-invariant systems. A numerical example is employed to illustrate the effectiveness of the proposed method.     

Observability Under Sampling for Nonlinear Systems

A previous article has investigated the problem of the preservation of the observability under sampling in the framework of compact manifolds. A natural generalization of the cited result would be to extend it to the systems globally Lipschitzian defined on non compact manifolds. A simple representative of this class of systems is constituted by the set of bilinear systems, for this particular class of systems the preservation of the property of observability has been shown in a previous work. In this paper, we deal with the preservation of the observability under sampling for globally Lipschitzian systems defined on R ⁿ.     

Local observability of invariant dynamics on compact Lie groups with square integrable output map functions

In this work, we give a sufficient algebraic condition for the local observability problem of invariant control systems on compact Lie groups such that the output map is not differentiable. In particular, the usual techniques involving Lie derivatives do not work. Our approach comes from the representation theory. We use the regular representation to construct a bilinear system on the Hilbert space of the square integrable function defined on the group to a finite-dimensional vector space. If this bilinear system is observable, then we prove that the invariant control system is locally observable.     

Some generalized notions of observability

Generalized state-space systems are a more natural way of describing dynamical behavior than standard state-space systems. Such systems exhibit a much richer observability structure than standard state-space systems. Three notions of observability are defined for generalized state-space systems. In contrast to previous work, regularity is not assumed. Geometric characterization of each type of observability and matrix rank tests for each type of observability are derived