Non-linear estimation is easy

Non-linear state estimation and some related topics, like parametric estimation, fault diagnosis, and perturbation attenuation, are tackled here via a new methodology in numerical differentiation. The corresponding basic system theoretic definitions and properties are presented within the framework of differential algebra, which permits to handle system variables and their derivatives of any order. Several academic examples and their computer simulations, with on-line estimations, are illustrating our viewpoint

Generating Series for Interconnected Nonlinear Systems and the Formal Laplace-Borel Transform

Formal power series methods provide effective tools for nonlinear system analysis. For a broad range of analytic nonlinear systems, their input-output mapping can be described by a Fliess operator associated with a formal power series. In this dissertation, the inter connection of two Fliess operators is characterized by the generating series of the composite system. In addition, the formal Laplace-Borel transform of a Fliess operator is defined and its fundamental properties are presented. The formal Laplace-Borel transform produces an elegant description of system interconnections in a purely algebraic context. Specifically, four basic interconnections of Fliess operators are addressed: the parallel, product, cascade and feedback connections. For each interconnection, the generating series of the overall system is given, and a growth condition is developed, which guarantees the convergence property of the output of the corresponding Fliess operator. Motivated by the relationship between operations on formal power series and system interconnections, and following the idea of the classical integral Laplace-Borel transform, a new formal Laplace-Borel transform of a Fliess operator is proposed. The properties of this Laplace-Borel transform are provided, and in particular, a fundamental semigroup isomorphism is identified between the set of all locally convergent power series and the set of all well-defined Fliess operators. A software package was produced in Maple based on the ACE package developed by the ACE group in Université de Marne-la Vallée led by Sébastien Veigneau. The ACE package provided the binary operations of addition, concatenation and shuffle product on the free monoid of formal polynomials. In this dissertation, the operations of composition, modified composition, chronological products and the evaluation of Fliess operators are implemented in software. The package was used to demonstrate various aspects of the new interconnection theory

The Magnus expansion and some of its applications

Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to built up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as Time-Dependent Exponential Perturbation Theory. Every Magnus approximant corresponds in Perturbation Theory to a partial re-summation of infinite terms with the important additional property of preserving at any order certain symmetries of the exact solution. The goal of this review is threefold. First, to collect a number of developments scattered through half a century of scientific literature on Magnus expansion. They concern the methods for the generation of terms in the expansion, estimates of the radius of convergence of the series, generalizations and related non-perturbative expansions. Second, to provide a bridge with its implementation as generator of especial purpose numerical integration methods, a field of intense activity during the last decade. Third, to illustrate with examples the kind of results one can expect from Magnus expansion in comparison with those from both perturbative schemes and standard numerical integrators. We buttress this issue with a revision of the wide range of physical applications found by Magnus expansion in the literature.Comment: Report on the Magnus expansion for differential equations and its applications to several physical problem

Interconnections of Nonlinear Systems Driven by L₂-ITÔ Stochastic Processes

Fliess operators have been an object of study in connection with nonlinear systems acting on deterministic inputs since the early 1970\u27s. They describe a broad class of nonlinear input-output maps using a type of functional series expansion, but in most applications, a system\u27s inputs have noise components. In such circumstances, new mathematical machinery is needed to properly describe the input-output map via the Chen-Fliess algebraic formalism. In this dissertation, a class of L2-Itô stochastic processes is introduced specifically for this purpose. Then, an extension of the Fliess operator theory is presented and sufficient conditions are given under which these operators are convergent in the mean-square sense. Next, three types of system interconnections are considered in this context: the parallel, product and cascade connections. This is done by first introducing the notion of a formal Fliess operator driven by a formal stochastic process. Then the generating series induced by each interconnection is derived. Finally, sufficient conditions are given under which the generating series of each composite system is convergent. This allows one to determine when an interconnection of Fliess operators driven by a class of L2-Itô stochastic processes is well-defined

Flatness, tangent systems and flat outputs

En esta tesis doctoral se presentan diversos métodos para la linealización de sistemas de control no lineales o para el estudio de la platitud. Se utilizan dos aproximaciones diferentes, en concreto: geometría diferencial y álgebra diferencial.En el marco de álgebra diferencial, se presenta un estudio de los sistemas lineales de control desde la perspectiva de la teoría de módulos. A pesar de que los resultados han sido establecidos previamente por otros autores, algunas demostraciones y ejemplos son originales.Entre las nuevas demostraciones cabe resaltar la que se refiere a la equivalencia entre sistemas de control lineales en representación de variables de estado, y los módulos sobre un anillo de operadores diferenciales. Los resultados de este estudio son ampliamente utilizados en el desarrollo de otros capítulos de la tesis en los que se usa el álgebra diferencial. En este contexto las principales contribuciones son:Una nueva demostración del hecho, bien conocido, que la linealización por realimentación estática y la linealización por realimentación dinámica son equivalentes en el caso de sistemas de entrada simple. Para la linealización de este tipo de sistemas, se desarrolla un nuevo algoritmo.Un procedimiento teórico para linealizar sistemas de entrada múltiple, basado en el cociente de módulos. También se ha hecho un paquete informático para llevar a cabo los cálculos necesarios. Debe mencionarse que este procedimiento es válido para linealizar sistemas mediante realimentación estática, así como para sistemas que sólo puedan linealizarse mediante realimentación dinámica.Una condición para comprobar si las salidas linealizantes encontradas pueden obtenerse mediante prolongaciones. Como aplicación, se muestran algunos ejemplos de sistemas linealizables por prolongaciones. Algunos de estos sistemas se creían que no eran linealizables mediante esta técnica.Postprint (published version

Regelungstheorie

The workshop “Regelungstheorie” (control theory) covered a broad variety of topics that were either concerned with fundamental mathematical aspects of control or with its strong impact in various fields of engineering

Flat systems, equivalence and trajectory generation

Flat systems, an important subclass of nonlinear control systems introduced via differential-algebraic methods, are defined in a differential geometric framework. We utilize the infinite dimensional geometry developed by Vinogradov and coworkers: a control system is a diffiety, or more precisely, an ordinary diffiety, i.e. a smooth infinite-dimensional manifold equipped with a privileged vector field. After recalling the definition of a Lie-Backlund mapping, we say that two systems are equivalent if they are related by a Lie-Backlund isomorphism. Flat systems are those systems which are equivalent to a controllable linear one. The interest of such an abstract setting relies mainly on the fact that the above system equivalence is interpreted in terms of endogenous dynamic feedback. The presentation is as elementary as possible and illustrated by the VTOL aircraft