Sette Diop

On regularized numerical observers

Diop et al. previously (1994) proposed a general observer design methodology based upon numerical differentiation and the interpretation of observability of a system as the solvability of the systems dynamical equations for the state vector in terms of a finite number of derivatives of the output and input. Numerical differentiation-base observers are alternatives to asymptotic observers for nonlinear systems. Various techniques are known to be efficient for the estimation of the few first derivatives from data with low frequency content, such as polynomial- and spline-based least squares, and averaged central differences. The main advantages of such observers are intuitiveness, flexibility and speed. However, as is the case of many inverse problems, differentiation is an ill-posed operator. This communication proposes the use of regularization to partially overcome the noise sensitivity that is inherent in the standard numerical differentiation. Regularization of numerical derivatives from experimental data consists of two operations: filtering and differentiation. Mollification is a method of filter design that is fairly well amenable to a mathematical analysis, including computation of estimation error bounds. In this method experimental data is projected onto Sobolev spaces of signals with less high frequency content, which may then be differentiated stably. The filters in question are infinite dimensional. They can be implemented approximately by means of digital Fourier transformation on finite moving windows of the data.

A global exponential observer based on numerical differentiation

The extended Kalman filter is known to have excellent filtering characteristics, but its convergence is guaranteed only if it is initialized close enough to the true state value. Numerical differentiation based observers, on the other hand, may be designed to be globally convergent to a neighborhood of the true state value, but when the measurements are corrupted by significant uncertain signals their state estimates are delayed. In a previous work we proposed an observer design scheme for nonlinear systems which combines these two techniques to yield a globally exponentially converging observer. The necessity to implement an extended Kalman filter has two shortcomings. One is it does not apply when we face a system with an observability condition weaker than that the linearized dynamics about the estimated trajectory is completely uniformly observable. The other limitation is the computation burden. In this paper we propose an alternative which is still a global exponential observer. While the computation burden may not have been significantly reduced, definitely, it applies to a wider class of nonlinear systems. Noteworthy we prove that this global exponential observer is also bounded which is a feature the nonlinear observers in the literature rarely possess. The observer scheme is illustrated through a bioprocess example whose linearization is not completely uniformly observable.

Two numerical differentiation techniques for nonlinear state estimation

Quite successfully regularization methods have been used in the numerical analysis literature in approaches to the ill-posed problem of numerically differentiating a signal from its discrete, potentially uncertain, samples. One of these approaches proposed an algorithm for the computation of an optimal spline whose first derivatives are estimates of the first derivatives of the signal. These algorithms suffer from a large amount of computation they imply. We propose two versions of this smoothing spline computation algorithm which reduce the computation burden, and thus, yield two potentially valuable tools to the design problem of online nonlinear state estimators.

The algebraic theory of nonlinear observability revisited

The algebraic theory of nonlinear observability is revisited by first enlarging the class of systems previously considered by the author (1991). We also provide new insights in the singularity of the notion of observability. The basic geometric picture of observability is that of a projection map of the system trajectories onto the data trajectories. Observability is then seen as the condition for this map to be finite in the sense that its fibres are generically finite. It is this definition which easily passes to differential algebraic geometric characterizations. Next, most of the picture is kept by defining the notion of singular observation data, i.e. the special data for which the generic observability property is lost. In this paper, we present as many elements of this theory as possible.

Optimization of the interval approach for Chlorella vulgaris biomass estimation

As a result of different environmental issues, especially global warming and the greenhouse effect, biotechnology using microalgae has become a very promising alternative for carbon dioxide mitigation. Indeed, these unicellular microorganisms reduce efficiently carbon dioxide emissions through their photosynthetic activity. In order to maximize the efficacy of this biological process, one of the challenges is the efficient on-line estimation of the microalgae biomass for control strategies. In this context, several studies have established the performance and robustness of the interval observer for biomass estimation. This paper proposes a method of optimization of the gains tuning of the interval observer for the biomass concentration of Chlorella vulgaris culture in a continuous photobioreactor, using Total Inorganic Carbon measurements. This study provides two procedures for choosing the gains of the estimation strategy under a specific operating condition. The optimization methodology is validated by numerical simulations in the presence of uncertain model parameters and noisy measurements.

Robust state reconstruction of linear neutral–type delay systems with application to lossless transmission lines: a convex optimization approach

This paper is concerned with the problem of robust observer design for linear systems with neutral-type time-delays. Delay-independent and delay-dependent conditions are presented to solve the observation/filtering issue under noisy output measurements. Stated as linear matrix inequalities conditions, these sufficient conditions enable the determination of the observer gains that guarantee both asymptotic convergence of the observer in case of noiseless measurements and robust filtering in case of presence of measurements errors. The proposed linear matrix inequality conditions are derived without any major approximation or assumption on the neutral type time-delay system which make the observer design straightforward and less conservative.

On differential algebraic decision methods for the estimation of anaerobic digestion models

Monitoring and control of anaerobic digestion of organic wastes by microorganisms are parts of actual world efforts to preserve environment. The anaerobic digestion is a biochemical process in which microorganisms (or bacteria) biodegrade organic matters into biogas (methane and carbon dioxide). Given the complexity of biochemical processes going on in such a bioreactor, control models are almost exclusively written in terms of mass balances of various species of interest. Such models are highly nonlinear and may contain many parameters which need to be identified. But the most challenging part of this estimation work concerns the online estimation of a key variable named the specific growth rates of microorganisms. It is invoked in most mass balance models. There is no devices to measure it so as techniques of estimation are very welcome in this field. The communication presents how differential algebraic decision methods can help find partial answers to the problem of online estimation of biomass specific growth rates based upon easily available measurements.

A differential algebraic approach to anaerobic digestion estimation problems

The paper deals with identifiability and observability of methane fermentation (MF) processes. In such kind of processes, generally carried out in continuously stirred tank bioreactors, the organic matter is depolluted by microorganisms into biogas and compost in the absence of oxygen. The biogas is an additional energy source, which can replace fossil fuel sources. The differential algebraic approach of general observation problems has been applied to investigate the identification and observation of a simple MF model. The major discovery is that the biomass specific growth rate can be stably estimated from easily measured quantities: the dilution rate and the biogas flow rate. Next if the yield coefficients are assumed known then, of course, the biomass concentration is observable. Unfortunately, even under the latter strongest assumption the substrate concentration is not observable. This concentration becomes observable if an additional model, say the Monod model, is assumed for the specific growth rate. Illustrative simulations are presented. The experimental validation is under investigation

A model based numerical differentiation observer for linear systems

Abstract Most known observers suffer from a significant drawback, the peaking phenomenon. This is actually inherent to the nature of the error signal which is fed back to the duplicate of the plant model in the structure of these observers. The error signal is of output type in the sense that it is the error between the plant measurements and the prediction of these measurements based on the current estimation of the plant state. In this paper we investigate an observer structure in which the error signal which is fed back to the plant model duplicate is of state type. The generation of the state error feedback for the observer necessitates a pre-estimation of the plant state through some differentiation and inversion ingredients. In this paper we concentrate on the constant linear theory of this observer structure. In particular we show that, under a mild assumption that the uncertainty on the state pre-estimate is bounded with supposedly known bounds, the observer gain may be chosen to yield a no-peaking feature. This more complex observer structure seems to be the price to pay for the latter no-peaking advantage. An additional benefit of the new observer structure is a considerable simplification in the computation of the observer gain. The so-called principle of separation is also shown to be preserved.

Nonlinear Predictive Control for continuous Chlorella vulgaris culture in a photobioreactor

In the framework of environment preservation, microalgae biotechnology appears to be a promising alternative for CO2 mitigation. Implementation of advanced control strategies can be further considered to improve potential performances of these organisms. In this context, this paper proposes the implementation of predictive control combined with an on-line estimation of the Chlorella vulgaris biomass, using Total Inorganic Carbon measurements. The elaboration of interval observers for biomass estimation is first detailed. This estimation is further included in a Nonlinear Model Predictive Control (NMPC) framework considered to regulate the biomass concentration. Finally experimental results are presented which validate the proposed theoretical developments.