State isomorphism approach to global identifiability of nonlinearsystems

Global deterministic identifiability of nonlinear systems is studied by constructing the family of local state isomorphisms that preserve the structure of the parametric system. The method is simplified for homogeneous systems, where such isomorphisms are shown to be linear, thereby reducing the identifiability problem to solving a set of algebraic equations. The known conditions for global identifiability in linear and bilinear systems are special cases of these results     

A generic topology library

Requirements in scientific computing emerge from various areas such as algebraic topology, geometrical algebra, and differential topology with different notations. Cell and complex properties are introduced in order to derive a common specification for properties of data structures. Only topological properties are used, thereby separating the actual data storage structure from the stored data. Several theoretical topological properties are introduced, and traversal capabilities which excel current implementations are presented and accompanied by selected examples.This work focuses on extracting these necessary mathematical concepts and introduces generic programming concepts necessary to fully transfer the mathematical concepts. Not only theoretical contributions are presented, but they are also demonstrated by means of applications in scientific computing.     

Structural identifiability of dynamical systems

Deterministic identifiability of finite dimensional dynamical systems is analysed using a generalized concept of structural invariants. Results applied to time-invariant bilinear and time-variable linear systems yield algebraic conditions for identifiability which are natural generalizations of the well known identifiability criteria for time-invariant linear systems.     

Minimality and identifiability of discrete‐time switched autoregressive exogenous systems

This article studies the problem of minimality and identifiability for switched autoregressive exogenous (SARX) systems. We propose formal definitions of the concepts of identifiability and minimality for SARX models. Based on these formalizations, we derive conditions for minimality and identifiability of SARX systems. In particular, we show that polynomially parameterized SARX systems are generically identifiable.     

Equivalence and identifiability analysis of uncontrolled nonlinear dynamical systems

The problem of parameter identifiability has been considered from different points of view in the case of nonlinear dynamical systems. For analytic systems the standard approach for uncontrolled systems is the Taylor series approach (Pohjanpalo, Math. Biosciences 41 (1978) 21), or the approaches based on differential algebra for polynomial and rational systems. The similarity transformation approach, based on the local state isomorphism theorem, gives a sufficient and necessary condition for global identifiability of nonlinear controlled systems. But it leads only to a necessary condition for identifiability in the case of some uncontrolled systems. Our contribution consists in using the equivalence of systems, based on the straightening out theorem, to analyse the identifiability of uncontrolled systems. From this theory, we state the necessary or sufficient identifiability conditions, some of them depending on the state variable dimension.     

New results for identifiability of nonlinear systems

The parameter identifiability problem of deterministic non-linear control systems is studied. Relations between nonlinear observability, nonlinear functional expansions, the uniqueness theorem of nonlinear realization theory, and identifiability are investigated. By using these relations, necessary and sufficient conditions for identifiability are obtained for the first time. These results provide insight into the identifiability problem for nonlinear systems.     

Structural identifiability analyses of candidate models for in vitro Pitavastatin hepatic uptake

In this paper a review of the application of four different techniques (a version of the similarity transformation approach for autonomous uncontrolled systems, a non-differential input/output observable normal form approach, the characteristic set differential algebra and a recent algebraic input/output relationship approach) to determine the structural identifiability of certain in vitro nonlinear pharmacokinetic models is provided. The Organic Anion Transporting Polypeptide (OATP) substrate, Pitavastatin, is used as a probe on freshly isolated animal and human hepatocytes. Candidate pharmacokinetic non-linear compartmental models have been derived to characterise the uptake process of Pitavastatin. As a prerequisite to parameter estimation, structural identifiability analyses are performed to establish that all unknown parameters can be identified from the experimental observations available.     

Direct equivalence between geometric conditions and algebraic conditions for discrete-time nonlinear observer

Unlike the continuous-time case, algebraic necessary and sufficient conditions for a single output discrete-time system to be state equivalent to a nonlinear observer canonical form have been found and are easier to verify for those who are not accustomed to differential geometry. The geometric conditions look very different from the algebraic conditions. In this paper, we show direct equivalence of the geometric conditions and the algebraic conditions in order to enhance the understanding of the geometric conditions.     

On the discrete-time normal form

Necessary and sufficient conditions under which the discrete-time normal form exists are set both in geometric and in algebraic frameworks, leading to equivalent conditions. The results are therefore extended to generic nonlinear nonaffine continuous-time systems     

Difference algebra and system identification

The framework of differential algebra, especially Ritt’s algorithm, has turned out to be a useful tool when analyzing the identifiability of certain nonlinear continuous-time model structures. This framework provides conceptually interesting means to analyze complex nonlinear model structures via the much simpler linear regression models. One difficulty when working with continuous-time signals is dealing with white noise in nonlinear systems. In this paper, difference algebraic techniques, which mimic the differential-algebraic techniques, are presented. Besides making it possible to analyze discrete-time model structures, this opens up the possibility of dealing with noise. Unfortunately, the corresponding discrete-time identifiability results are not as conclusive as in continuous time. In addition, an alternative elimination scheme to Ritt’s algorithm will be formalized and the resulting algorithm is analyzed when applied to a special form of the nfir model structure.     

An easy to check criterion for (un)indentifiability of uncontrolledsystems and its applications

In this paper we give a necessary condition for structural identifiability of uncontrolled autonomous systems. This condition only turns on the identifiability of the right-hand side of the nonlinear differential system. We prove that this necessary condition becomes sufficient when the state is one-dimensional. To the best of our knowledge, the theoretical results obtained in this paper are new although the proofs are trivial. But they give an easy to check condition as it is shown by the study of some typical examples, in which we do much less computation than the involved literature to prove identifiability properties     

DAISY: a new software tool to test global identifiability of biological and physiological systems

A priori global identifiability is a structural property of biological and physiological models. It is considered a prerequisite for well-posed estimation, since it concerns the possibility of recovering uniquely the unknown model parameters from measured input-output data, under ideal conditions (noise-free observations and error-free model structure). Of course, determining if the parameters can be uniquely recovered from observed data is essential before investing resources, time and effort in performing actual biomedical experiments. Many interesting biological models are nonlinear but identifiability analysis for nonlinear system turns out to be a difficult mathematical problem. Different methods have been proposed in the literature to test identifiability of nonlinear models but, to the best of our knowledge, so far no software tools have been proposed for automatically checking identifiability of nonlinear models. In this paper, we describe a software tool implementing a differential algebra algorithm to perform parameter identifiability analysis for (linear and) nonlinear dynamic models described by polynomial or rational equations. Our goal is to provide the biological investigator a completely automatized software, requiring minimum prior knowledge of mathematical modelling and no in-depth understanding of the mathematical tools. The DAISY (Differential Algebra for Identifiability of SYstems) software will potentially be useful in biological modelling studies, especially in physiology and clinical medicine, where research experiments are particularly expensive and/or difficult to perform. Practical examples of use of the software tool DAISY are presented. DAISY is available at the web site http://www.dei.unipd.it/~pia/.     

On decentralized dynamic pole placement and feedback stabilization

In this paper we study the feedback control problem using an r-channel decentralized dynamic feedback control scheme. We will develop the theory in the behavioral framework. Using this framework we introduce an algebraic parameterization of the space of all possible feedback compensators having a bounded McMillan degree, and we show that this parameterization has the structure of an algebraic variety. We define the pole-placement map for this problem, and we give exact conditions when this map is onto, and almost onto. Finally we provide new necessary and sufficient conditions which guarantee that the set of stabilizable plants is a generic set     

Switched nonlinear differential algebraic equations: Solution theory,Lyapunov functions,and stability

We study switched nonlinear differential algebraic equations (DAEs) with respect to existence and nature of solutions as well as stability. We utilize piecewise-smooth distributions introduced in earlier work for linear switched DAEs to establish a solution framework for switched nonlinear DAEs. In particular, we allow induced jumps in the solutions. To study stability, we first generalize Lyapunov’s direct method to non-switched DAEs and afterwards obtain Lyapunov criteria for asymptotic stability of switched DAEs. Developing appropriate generalizations of the concepts of a common Lyapunov function and multiple Lyapunov functions for DAEs, we derive sufficient conditions for asymptotic stability under arbitrary switching and under sufficiently slow average dwell-time switching, respectively.     

Nonlinear Finite Element Analysis using an Object-Oriented Philosophy – Application to Beam Elements and to the Cosserat Continuum

An Object-Oriented Programming (OOP) frame-work is presented for solving nonlinear structural mechanics problems by means of the Finite Element Method (FEM). Emphasis is placed on engineering applications (geometrically nonlinear beam model, and elastoplastic Cosserat continuum), and OOP is employed as an effective tool, which plays an important role in the FEM treatment of such applications. The implementation is based on computational abstractions of both mathematical and physical concepts associated to structural mechanics problems involving geometrical and material nonlinearities. The overall class organization for nonlinear mechanics modeling is discussed in detail. All the analyses rely on a generic control class where several classical and modern nonlinear solution schemes are available. Examples which explore, demonstrate and validate the main features of the overall computational system are presented and discussed.     

On impulsive modes of linear singular systems subject to decentralized output feedback

Some algebraic properties of linear singular systems at infinity under static decentralized output feedback are studied in this note. New concepts of algebraic multiplicity and geometric multiplicity of the impulsive decentralized fixed modes, and the impulsive decentralized cycle index of the singular system are defined. These concepts indicate how much a singular system can be made close to being impulse-free by decentralized output feedback, and the results are shown to be generic. The geometric multiplicity and the impulsive decentralized cycle index are determined in terms of the system matrices explicitly. The number of impulsive modes that can be eliminated is given in terms of these indexes. The impulsive decentralized cycle index is shown to characterize generic properties on controllability and observability of the closed-loop system through an individual channel (or an external channel). Illustrative examples are provided.     

Identifiability of homogeneous systems using the state isomorphism approach

New results are presented concerning the state isomorphism approach to global identifiability analysis of parameterized classes of nonlinear state space systems with specified initial states. In particular we study the class of homogeneous systems, for which, under certain conditions, the local state isomorphism for a pair of indistinguishable parameter vectors is shown to be homogeneous of degree one. For homogeneous polynomial systems, conditions are given under which the local state isomorphism becomes linear. Here, the issue of whether or not the observability rank condition holds at the origin is shown to be of key importance. The scope of the results, which extend to the multivariable case, is discussed and illustrated by a number of worked examples. This demonstrates how the developed theory can be put to use to investigate the global identifiability properties of parameterized model classes.     

Parameter identifiability of nonlinear systems: the role of initial conditions

Identifiability is a fundamental prerequisite for model identification; it concerns uniqueness of the model parameters determined from the input-output data, under ideal conditions of noise-free observations and error-free model structure. In the late 1980s concepts of differential algebra have been introduced in control and system theory. Recently, differential algebra tools have been applied to study the identifiability of dynamic systems described by polynomial equations. These methods all exploit the characteristic set of the differential ideal generated by the polynomials defining the system. In this paper, it will be shown that the identifiability test procedures based on differential algebra may fail for systems which are started at specific initial conditions and that this problem is strictly related to the accessibility of the system from the given initial conditions. In particular, when the system is not accessible from the given initial conditions, the ideal I having as generators the polynomials defining the dynamic system may not correctly describe the manifold of the solution. In this case a new ideal that includes all differential polynomials vanishing at the solution of the dynamic system started from the initial conditions should be calculated. An identifiability test is proposed which works, under certain technical hypothesis, also for systems with specific initial conditions.     

Topographical Properties of Generic Images

Topographical curves in images are defined by certain extremality conditions involving the gradient of the greyvalue function. Curves extracted by some edge operators and watersheds studied in geography are both examples of such topographical curves.In the first part of this article we list the possible geometrical features of certain topographical curves—such as intersection points, singular points, endpoints, curvature extrema and inflections—for generic images and for generic 1-parameter families of, linearly or non-linearly, diffused images (intuitively, generic phenomena arise with probability 1).In a second experimental part the same topographical curves are computed in discrete images and in Gaussian blurred families of images. It turns out that the geometrical features of the smooth classification also figure in these discrete approximations. The consequences of these results for edge-linking algorithms are briefly discussed.     

Parameter identifiability for distributed parameter systems of hyperbolic type

In this paper we study questions regarding parameter identifiability for distributed parameter systems of hyperbolic type. The unknown parameters are input distribution functions. We consider the systems with continuous-time input-output data and the systems with discrete-time inputs-output data. For these systems we present the necessary and sufficient conditions for identifiability, in the case of a finite number of sensors. We investigate the relations between the continuous-time input-output systems and the discrete-time input-output systems from the viewpoint of identifiability. Moreover we give the sets of input distribution functions which are N-step identifiable, for the discrete-time input-output systems.