Claudia Califano

On the observer design in discrete-time

Abstract The paper deals with the equivalence, under coordinates change and output transformation, of a discrete-time nonlinear system to observer canonical forms. Necessary and sufficient conditions for local equivalence to these forms are given.

From Chronological Calculus to Exponential Representations of Continuous and Discrete-Time Dynamics: A Lie-Algebraic Approach

In this paper, formal exponential representations of the solutions to nonautonomous nonlinear differential equations are derived. It is shown that the chronological exponential admits an ordinary exponential representation, the exponent being given by an explicitly computable Lie series expansion. The results are then used to describe controlled dynamics, dynamics under sampling and forced discrete-time dynamics. The study emphasizes the role of Lie algebra techniques in nonlinear control theory and specifies structural similarities between nonautonomous differential equations, dynamics under sampling and forced discrete-time dynamics up to hybrid ones.

Extended Lie Brackets for Nonlinear Time-Delay Systems

In this note the Extended Lie bracket operator is introduced for the analysis and control of nonlinear time-delay systems (NLTDS). This tool is used to characterize the integrability conditions of a given submodule. The obtained results have two fundamental outcomes. First, they define the necessary and sufficient conditions under which a given set of nonlinear one-forms in the n-dimensional delayed variables x(t),...,x(t-sD) , with D constant but unknown, are integrable, thus generalizing the well known fundamental Frobenius Theorem to delay systems. Secondly, they set the basis for the extension to this context of the geometric approach used for delay-free systems. The effectiveness of the results is shown by solving the problem of the equivalence of a NLTDS to an accessible Linear Time-Delay System (LTDS) by bicausal change of coordinates.

Canonical observer forms for multi-output systems up to coordinate and output transformations in discrete time

In the present paper the problem of equivalence under coordinate changes and output transformations to observer canonical forms is addressed in discrete time for multi-output systems. Necessary and sufficient conditions are given for local equivalence to this form which yields a straightforward observer design with linear error dynamics.

On the problem of feedback linearization

The paper studies and solves in a geometric framework the problem of partial feedback linearization for discrete-time dynamics. An algorithm for computing the largest linearizable subsystem is proposed. This approach can be considered as dual to the one already proposed in literature in an algebraic context.

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.

On linear equivalence for time-delay systems

The aim of the present paper is to introduce new mathematical tools for the analysis and control of nonlinear time-delay systems (NLTDS). An Extended Lie bracket operation equivalent to the Lie bracket operation for system without delays is introduced. It will be shown that this operation, which generalizes that introduced in, helps to characterize certain properties of a given submodule, such as nilpotency. This basic property is then used to define the conditions under which a given unimodular matrix represents a bicausal change of coordinates. The effectiveness of the proposed approach will be shown by solving an important basic problem: to characterize if a NLTDS is equivalent or not, to a Linear Time-Delay System by bicausal change of coordinates.

The Observer Error Linearization Problem via Dynamic Compensation

Linearization by output injection has played a key role in the observer design for nonlinear control systems for almost three decades. In this technical note, following some recent works, geometric necessary and sufficient conditions are derived for the existence of a dynamic compensator solving the problem under regular output transformation. An algorithm which computes a compensator of minimal order is given.

Characterization of accessibility for a class of nonlinear time-delay systems

The accessibility of the class of driftless single-input nonlinear time-delay systems is fully characterized for the first time. This result is obtained through the introduction of new tools within a geometric approach recently introduced in the literature. Moreover, all those possible autonomous elements, which can depend on the variables with time-delay, are also characterized when the system is not accessible and in consequence, a canonical form of those systems is deduced.

On the Observer Canonical Form for Nonlinear Time--Delay Systems

Abstract Necessary and sufficient geometric conditions for the equivalence of a nonlinear time–delay system with one output, under bicausal change of coordinates and output transformation, to a linear weakly observable time–delay system up to output injection are given. These conditions are derived through the use of the Extended Lie Bracket operator recently introduced in the literature for dealing with time–delay systems. The results presented show how this operator is useful in the analysis of this class of nonlinear systems.