A nonlinear canonical form for reduced order observer design

International audienceThis paper presents a nonlinear canonical form which is used for the design of a reduced order observer. Sufficient and necessary geometric conditions are given in order to transform a special class of nonlinear systems to the proposed nonlinear canonical form and the corresponding reduced order observer is analyzed

On the observer canonical form for Nonlinear Time-Delay Systems

6 pagesInternational audienceNecessary 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

A High-Gain Nonlinear Observer With Limited Gain Power

International audienceIn this note we deal with a new observer for nonlinear systems of dimension n in canonical observability form. We follow the standard high-gain paradigm, but instead of having an observer of dimension n with a gain that grows up to power n, we design an observer of dimension 2n − 2 with a gain that grows up only to power 2

Finite time observers: application to secure communication

International audienceIn this paper, control theory is used to formalize finite time chaos synchronization as a nonlinear finite time observer design issue. This paper introduces a finite time observer for nonlinear systems that can be put into a linear canonical form up to output injection. The finite time convergence relies on the homogeneity properties of nonlinear systems. The observer is then applied to the problem of secure data transmission based on finite time chaos synchronization and the two-channel transmission method

Observers for canonic models of neural oscillators

We consider the problem of state and parameter estimation for a wide class of nonlinear oscillators. Observable variables are limited to a few components of state vector and an input signal. The problem of state and parameter reconstruction is viewed within the classical framework of observer design. This framework offers computationally-efficient solutions to the problem of state and parameter reconstruction of a system of nonlinear differential equations, provided that these equations are in the so-called adaptive observer canonic form. We show that despite typical neural oscillators being locally observable they are not in the adaptive canonic observer form. Furthermore, we show that no parameter-independent diffeomorphism exists such that the original equations of these models can be transformed into the adaptive canonic observer form. We demonstrate, however, that for the class of Hindmarsh-Rose and FitzHugh-Nagumo models, parameter-dependent coordinate transformations can be used to render these systems into the adaptive observer canonical form. This allows reconstruction, at least partially and up to a (bi)linear transformation, of unknown state and parameter values with exponential rate of convergence. In order to avoid the problem of only partial reconstruction and to deal with more general nonlinear models in which the unknown parameters enter the system nonlinearly, we present a new method for state and parameter reconstruction for these systems. The method combines advantages of standard Lyapunov-based design with more flexible design and analysis techniques based on the non-uniform small-gain theorems. Effectiveness of the method is illustrated with simple numerical examples

Modulating function based algebraic observer coupled with stable output predictor for LTV and sampled-data systems

This paper proposes an algebraic observer-based modulating function approach for linear time-variant systems and a class of nonlinear systems with discrete measurements. The underlying idea lies in constructing an observability transformation that infers some properties of the modulating function approach for designing such algebraic observers. First, we investigate the algebraic observer design for linear time-variant systems under an observable canonical form for continuous-time measurements. Then, we provide the convergence of the observation error in an L2-gain stability sense. Next, we develop an exponentially stable sampled-data observer which relies on the design of the algebraic observer and an output predictor to achieve state estimation from available measurements and under small inter-sampling periods. Using a trajectory-based approach, we prove the convergence of the observation error within a convergence rate that can be adjusted through the fixed time-horizon length of the modulating function and the upper bound of the sampling period. Furthermore, robustness of the sampled-data algebraic observer, which yields input-to-state stability, is inherited by the modulating kernel and the closed-loop output predictor design. Finally, we discuss the implementation procedure of the MF-based observer realization, demonstrate the applicability of the algebraic observer, and illustrate its performance through two examples given by linear time-invariant and linear time-variant systems with nonlinear input-output injection terms.Comment: 15 pages, 9 figures, submitted to Automatic

High gain observer for structured multi-output nonlinear systems

In this note, we present two system structures that characterize classes of multi-input multi-output uniformly observable systems. The first structure is decomposable into a linear and a nonlinear part while the second takes a more general form. It is shown that the second system structure, being more general, contains several system structures that are available in the literature. Two high gain observer design methodologies are presented for both structures and their distinct features are highlighted

Observer design for a schistosomiasis model

International audienceThis paper deals with the state estimation for a schistosomiasis infection dynamical model described by a continuous nonlinear system when only the infected human population is measured. The central idea is studied following two major angles. On the one hand, when all the parameters of the model are supposed to be well known, we construct a simple observer and a high-gain Luenberger observer based on a canonical controller form and conceived for the nonlinear dynamics where it is implemented. On the other hand, when the nonlinear uncertain continuous-time system is in a bounded-error context, we introduce a method for designing a guaranteed interval observer. Numerical simulations are included in order to test the behavior and the performance of the given observers