State estimation for a class of nonlinear systems

Abstract

We propose a new type of Proportional Integral (PI) state observer for a class of nonlinear systems in continuous time which ensures an asymptotic stable convergence of the state estimates. Approximations of nonlinearity are not necessary to obtain such results, but the functions must be, at least locally, of the Lipschitz type. The obtained state variables are exact and robust against noise. Naslin's damping criterion permits synthesizing gains in an algebraically simple and efficient way. Both the speed and damping of the observer response are controlled in this way. Model simulations based on a Sprott strange attractor are discussed as an example.

State estimation for MISO non–linear systems in controller canonical form

“…The structure of the matrices (30c)-(30e) makes it possible to deduce the unit static gain of the observer. The system (30) describing the dynamics of convergence of the observation errors is close to that which has previously been proposed (Schwaller et al, 2013), and the conditioning of the system proposed in (3) may be used with advantage.…”

“…In a precedent study (Schwaller et al, 2013), we dealt with a specific class of non-linear SISO (single input single output) systems, described by Fliess (1990), called the generalized controller canonical form (Zeitz, 1985). In principle, every uniformly observable (Hermann and Krener, 1977;Gauthier and Bornard, 1981) smooth enough SISO system with vector input uÔtÕ and output yÔtÕ can be transformed into this normal form, and extended to the following MISO (multiple input single output) systems (Glumineau and Lôpez-Morales, 1999): We previously limited the field of application of the proposed observer because of the decomposition of the function ΨÔtÕ into two distinct parts which had the effect of limiting the functions ΨÔtÕ to the third order, and necessitated the integration of all or part of the latter.…”

“…One thus overcomes a common weak point of high gain observations, i.e., their sensitivity to measurement noise. The second advantage comes from the non-linear function Ö Ψ ÖvÔτ Õ, U Ôτ Õ×, which is no longer subjected to the restrictive conditions used by Schwaller et al (2013), and covers the ensemble of the systems described by Fliess (1990). The vector Ö f Ôτ Õ (11d), of dimension n ¡ 1, compensates the effects of f Ôτ Õ and of possible external exogenous disturbance of (1) using the integral component I 0 Ôτ Õ (11e).…”

Resistance to Noise of Non-linear Observers in Canonical Form Application to a Sludge Activation Model

“…their sensitivity to measurement noise. The second advantage comes from the non-linear function r Ψ rvpτq, Upτqs which is no longer subjected to the restrictive conditions used in (Schwaller, Ensminger, Dresp-Langley, & Ragot, 2013), and covers the ensemble of the systems described by (Fliess, 1990). The vector r f pτq (14d), of dimension n´1, compensates the effects of f pτq, and of possible external exogenous disturbance of (2) using the integral component I 0 pτq (14e).…”

Feedback design of differential equations of reconstruction for second–order distributed parameter systems

“…The problem described above belongs to the class of problems of dynamical inversion (Lavrentiev et al, 1980;Schwaller et al, 2013;Banks and Kunisch, 1989;Lasiecka et al, 1999;Mordukhovich and Zhang, 1997;Mordukhovich, 2008;2011). The methodology of solving this problem suggested below uses an approach described, e.g., by Kryazhimskii and Osipov (1995), Maksimov (2002;1996;1995), Maksimov and Pandolfi (2001), Maksimov and Tröltzsch (2006), Kryazhimskii and Maksimov (2010), or Kapustyan and Maksimov (2014).…”