On some aspects of polynomial dynamical systems

The aim of this work is to study exact algebraic criteria local/global observability ([HK77], [Ino77]) for polynomial dynamical system by means of algebraic geometry and computational commutative algebra in the vein of [SR76], [Son79a], [Son79b], [Bai80], [Bai81], [Bar95], [Bar99], [Nes98], [Tib04], [KO13], [Bar16]. A key point in this topic is to work with polynomials with real coefficients and their real roots instead of their complex roots, as it is usually the case ([CLO15], [KR00]). A central concept is then the real radical of an ideal [BN93], [Neu98], [LLM+13], along with the Krivine- Dubois-Risler real nullstellensatz for polynomial rings [Kri64], [Dub70], [Ris70], [BCR98]. Underestimating this point leads to incorrect results (see, e.g. [Bar16] remark on [KO13]). This thesis is therefore devoted to set the necessary algebraic tools in the right context and level of generality (i.e. real algebra and real algebraic geometry) for applications to our dynamical systems and to further develop their exploit in this context. The first two chapters set the algebraic and algebraic geometry preliminaries. The third chapter is devoted to the applications of the previous algebraic concepts to the study of the ob- servability of polynomial dynamical systems. In the last chapter an approach to the construction of Lyapunov funtions to prove stability in estimation problems is presented

A symbolic network-based nonlinear theory for dynamical systems observability

EBM and MSB acknowledge the Engineering and Physical Sciences Research Council (EPSRC), grant Ref. EP/I032608/1. ISN acknowledges partial support from the Ministerio de Economía y Competitividad of Spain under project FIS2013-41057-P and from the Group of Research Excelence URJC-Banco de Santander.Peer reviewedPublisher PD

On the observability of embedded polynomial dynamical systems

Testing a system for observability is of great practical relevance in technical applications. For linear systems, this problem was solved decades ago. The observability of nonlinear systems can be formally defined, but the actual verification is extremely difficult. For the subclass of poynomial systems, the observability can be decided in a finite number of calculation steps. In this paper, we provide an observability test for embedded polynomial systems. The observability test uses methods of algebraic geometry

Global Identifiability of Differential Models

Many real-world processes and phenomena are modeled using systems of ordinary differential equations with parameters. Given such a system, we say that a parameter is globally identifiable if it can be uniquely recovered from input and output data. The main contribution of this paper is to provide theory, an algorithm, and software for deciding global identifiability. First, we rigorously derive an algebraic criterion for global identifiability (this is an analytic property), which yields a deterministic algorithm. Second, we improve the efficiency by randomizing the algorithm while guaranteeing the probability of correctness. With our new algorithm, we can tackle problems that could not be tackled before. A software based on the algorithm (called SIAN) is available at https://github.com/pogudingleb/SIAN

Stability of uniformly bounded switched systems and Observability

This paper mainly deals with switched linear systems defined by a pair of Hurwitz matrices that share a common but not strict quadratic Lyapunov function. Its aim is to give sufficient conditions for such a system to be GUAS.We show that this property of being GUAS is equivalent to the uniform observability on $[0,+\infty)$ of a bilinear system defined on a subspace whose dimension is in most cases much smaller than the dimension of the switched system.Some sufficient conditions of uniform asymptotic stability are then deduced from the equivalence theorem, and illustrated by examples.The results are partially extended to nonlinear analytic systems

Approximate zero polynomials of polynomial matrices and linear systems

This paper introduces the notions of approximate and optimal approximate zero polynomial of a polynomial matrix by deploying recent results on the approximate GCD of a set of polynomials Karcaniaset al. (2006) 1 and the exterior algebra Karcanias and Giannakopoulos (1984) 4 representation of polynomial matrices. The results provide a new definition for the "approximate", or "almost" zeros of polynomial matrices and provide the means for computing the distance from non-coprimeness of a polynomial matrix. The computational framework is expressed as a distance problem in a projective space. The general framework defined for polynomial matrices provides a new characterization of approximate zeros and decoupling zeros Karcanias et al. (1983) 2 and Karcanias and Giannakopoulos (1984) 4 of linear systems and a process leading to computation of their optimal versions. The use of restriction pencils provides the means for defining the distance of state feedback (output injection) orbits from uncontrollable (unobservable) families of systems, as well as the invariant versions of the "approximate decoupling polynomials". The overall framework that is introduced provides the means for introducing measures for the distance of a system from different families of uncontrollable, or unobservable systems, which may be feedback dependent, or feedback invariant as well as the notion of "approximate decoupling polynomials"

Approximate zero polynomials of polynomial matrices and linear systems

This paper introduces the notions of approximate and optimal approximate zero polynomial of a polynomial matrix by deploying recent results on the approximate GCD of a set of polynomials [1] and the exterior algebra [4] representation of polynomial matrices. The results provide a new definition for the "approximate", or "almost" zeros of polynomial matrices and provide the means for computing the distance from non-coprimeness of a polynomial matrix. The computational framework is expressed as a distance problem in a projective space. The general framework defined for polynomial matrices provides a new characterization of approximate zeros and decoupling zeros [2], [4] of linear systems and a process leading to computation of their optimal versions. The use of restriction pencils provides the means for defining the distance of state feedback (output injection) orbits from uncontrollable (unobservable) families of systems, as well as the invariant versions of the "approximate decoupling polynomials"