State and parameter estimation in chemical and biochemical processes: a tutorial

This paper aims at giving an overview of available results of state and parameter approaches for chemical and biochemical processes. It is largely organized as a tutorial and starts with a brief reminder concerning the design of extended Luenberger (ELO) and Kalman (EKO) observers, followed by an illustrative nonlinear observer algorithm. Evaluation of the performance of classical observers in presence of model uncertainties will serve as a basis for the motivation of designing asymptotic and interval observers, that do not require the knowledge of the process kinetics. The design of state observers with known kinetic models but uncertain kinetic parameters will then be considered via suggestions of improvements of the EKO and the introduction of two other types of observers (observers where the unknown parameters are used as design parameters; adaptive observers). Finally, the design of on-line parameter estimation schemes will be introduced. One of the objectives of the present survey is also to suggest new research directions.     

How Do Two Observers Pool Their Knowledge About a Quantum System?

In the theory of classical statistical inference one can derive a simple rule by which two or more observers may combine independently obtained states of knowledge together to form a new state of knowledge, which is the state which would be possessed by someone having the combined information of both observers. Moreover, this combined state of knowledge can be found without reference to the manner in which the respective observers obtained their information. However, we show that in general this is not possible for quantum states of knowledge; in order to combine two quantum states of knowledge to obtain the state resulting from the combined information of both observers, these observers must also possess information about how their respective states of knowledge were obtained. Nevertheless, we emphasize this does not preclude the possibility that a unique, well motivated rule for combining quantum states of knowledge without reference to a measurement history could be found. We examine both the direct quantum analog of the classical problem, and that of quantum state-estimation, which corresponds to a variant in which the observers share a specific kind of prior information. PACS: 03.67.-a, 02.50.-r, 03.65.Bz     

Results of a User Study on 2D Hurricane Visualization

We present the results from a user study looking at the ability of observers to mentally integrate wind direction and magnitude over a vector field. The data set chosen for the study is an MM5 (PSU/NCAR Mesoscale Model) simulation of Hurricane Lili over the Gulf of Mexico as it approaches the southeastern United States. Nine observers participated in the study. This study investigates the effect of layering on the observer's ability to determine the magnitude and direction of a vector field. We found a tendency for observers to underestimate the magnitude of the vectors and a counter‐clockwise bias when determining the average direction of a vector field. We completed an additional study with two observers to try to uncover the source of the counter‐clockwise bias. These results have direct implications to atmospheric scientists, but may also be able to be applied to other fields that use 2D vector fields.     

Quasi-ISS/ISDS observers for interconnected systems and applications

This paper considers interconnected nonlinear dynamical systems and studies observers for such systems. For single systems the notion of quasi-input-to-state dynamical stability (quasi-ISDS) for reduced-order observers is introduced and observers are investigated using error Lyapunov functions. It combines the main advantage of ISDS over input-to-state stability (ISS), namely the memory fading effect, with reduced-order observers to obtain quantitative information about the state estimate error. Considering interconnections quasi-ISS/ISDS reduced-order observers for each subsystem are derived, where suitable error Lyapunov functions for the subsystems are used. Furthermore, a quasi-ISS/ISDS reduced-order observer for the whole system is designed under a small-gain condition, where the observers for the subsystems are used. As an application, we prove that quantized output feedback stabilization for each subsystem and the overall system is achievable, when the systems possess a quasi-ISS/ISDS reduced-order observer and a state feedback law that yields ISS/ISDS for each subsystem and therefor the overall system with respect to measurement errors. Using dynamic quantizers it is shown that under the mentioned conditions asymptotic stability can be achieved for each subsystem and for the whole system.     

Adaptive observers for nonlinearly parameterized class of nonlinear systems

In this paper, one proposes adaptive observers for a class of uniformly observable MIMO nonlinear systems with general nonlinear parameterizations. The state and the unknown parameters of the considered systems are supposed to lie in bounded domains which size can be arbitrarily large and the exponential convergence of the observers is shown to result under a well-defined persistent excitation condition. Moreover, the gain of the observers involves a design function that has to satisfy a simple condition which is given. Different expressions of such a function are proposed and it is shown that adaptive high gain like observers and adaptive sliding mode like observers can be derived by considering particular expressions of the design function. The theory is supported by simulation results related to the estimation of the biomass concentration and the Contois model parameters in a bioreactor.     

Piecewise-continuous observers for linear systems with sampled and delayed output

The paper presents a new class of state observers for linear systems with sampled and delayed output measurements. These observers are derived using the theory of a particular class of hybrid systems called piecewise-continuous systems, and can be easily implemented. The performances of the piecewise-continuous observers are compared with the performances of state observers designed using the Lyapunov–Krasovskii techniques. A piecewise-continuous observer is designed and implemented to an experimental visual servoing platform.     

Asymptotic H∞ disturbance attenuationbased on perfect observation

A design method of controllers which ensure internal stability and attain asymptotically H^∞ disturbance attenuation is presented. The design procedure consists of two steps: (1) to design an H^∞ state feedback control via an algebraic Riccati equation approach; and (2) under a certain minimum-phase condition, to recover the achievable performance asymptotically by applying high-gain observers. It makes use of the perfect observation for the design of high-gain observers. It is shown that the asymptotic recovery can be attained by using reduced-order observers, provided there is no direct feedthrough of controls and disturbances in observations     

Positive state‐bounding observer for positive interval continuous‐time systems with time delay

This paper is concerned with the problem of positive observer synthesis for positive systems with both interval parameter uncertainties and time delay. Conventional observers may no longer be applicable for such kind of systems due to the positivity constraint on the observers, and they only provide an estimate of the system state in an asymptotic way. A pair of positive observers with state‐bounding feature is proposed to estimate the state of positive systems at all times in this paper. A necessary and sufficient condition for the existence of desired observers is first established, and the observer matrices can be obtained easily through the solutions of a set of linear matrix inequalities (LMIs). Then, to reduce the error signal between the system state and its estimates, an iterative LMI algorithm is developed to compute the optimized state‐bounding observer matrices. Finally, a numerical example is presented to show the effectiveness and applicability of the theoretical results. Copyright © 2011 John Wiley & Sons, Ltd.     

On the dimensionality of face space

The dimensionality of face space is measured objectively in a psychophysical study. Within this framework, we obtain a measurement of the dimension for the human visual system. Using an eigenface basis, evidence is presented that talented human observers are able to identify familiar faces that lie in a space of roughly 100 dimensions and the average observer requires a space of between 100 and 200 dimensions. This is below most current estimates. It is further argued that these estimates give an upper bound for face space dimension and this might be lowered by better constructed "eigenfaces" and by talented observers.     

Modal consensus observers for distributed parameter systems

This article proposes a new type of a consensus protocol for the synchronization of distributed observers in systems governed by parabolic partial differential equations. Addressing the goal of sharing useful information among distributed observers, it delves into the details governing the modal decompositions of distributed parameter systems. Assuming that two different groups of sensors are available to provide process information to the two distributed observers, the proposed modal consensus design ensures that only useful information is transmitted to the requisite modal components of each of the observers. Without any consensus protocol, the observers capture different frequency content of the spatial process in differing degrees, as it relates to the concept of modal observability. Their modal components exhibit different learning behavior toward the process state. In the extreme case, it turns out that certain modal components of the distributed observers occasionally behave as naïve observers. To ensure that, both collectively and modal componentwise, the observers agree both with the process state and with each other, a modal component consensus protocol is proposed. Such a consensus protocol is mono-directional and provides only useful information necessary to the appropriate modal component of the distributed filters that behaves as a naïve modal observer. This protocol, when abstracted and applied to different state decompositions can be viewed as mono-directional projections of information transmitted and received by the participating distributed observers. Detailed numerical studies of advection PDE in one and two spatial dimensions are included to elucidate the details of the proposed modal consensus observers.     

On fault-tolerant observers

The problem of designing asymptotic observers for multioutput systems with one unreliable output measurement is addressed. It is shown that observers designed on the basis of observability indexes are not fault tolerant, and an observer for linear systems which tolerates failures in one a priori known output is given. The results are extended to nonlinear systems with no inputs: it is pointed out that the proposed design procedure may enlarge the class of nonlinear systems for which a nonlinear observer can be obtained     

A robust estimation approach for uncertain systems with perturbed measurements

This paper deals with state estimation problem for uncertain continuous‐time systems. A numerical treatment is proposed for designing interval observers that ensures guaranteed upper and lower bounds on the estimated states. In order to take into account possible perturbations on the system and its outputs, a new type of interval observers is introduced. Such interval observers consist of two coupled general Luenberger‐type observers that involve dilatation functions. In addition, we provide an optimality criterion in order to find optimal interval observers that lead to tight interval error estimation. The proposed existence and optimality conditions are expressed in terms of linear programming. Also, some illustrative examples are given. Copyright © 2015 John Wiley & Sons, Ltd.     

Uniformly Observable and Globally Lipschitzian Nonlinear Systems Admit Global Finite-Time Observers

There exist semi-global and finite-time converging observers for nonlinear systems that are uniformly observable and globally Lipschitzian. This was achieved with local finite-time stability theory, together with application of the technique of high-gain observers (the gain is constant). Under the same conditions, it is shown that these systems admit global finite-time observers. The proposed finite-time observers are based on a modified version of high-gain observer (now the gain is an exponential function with arbitrary growth rate). Design procedures are worked out for such observers, and a numerical example is presented to show the effectiveness of the proposed method.      

Closed loop observers bundle for uncertain biotechnological models

We propose nonlinear observers for a class of biotechnological processes. These observers are an extension of the open loop asymptotic observers (observers with unknown inputs) devoted to biotechnological systems for which some parts of the model are unknown. We take benefit of the additional outputs which are (nonlinear) functions of the state to design a closed loop observer. The global convergence of these nonlinear observers is proven. We use these observers to design interval based observers which predict guaranteed intervals in which the state is lying. We run simultaneously a broad set of interval observers and we select the best ones. The method is illustrated with a model describing the bioconversion of a substrate using micro-organisms in a bioreactor.     

Design of observers for non-linear time-varying systems

Novel stable observers are proposed for discrete, time-varying systems having time-varying, sector-bounded non-linearities. First, the form of the non-linearity is assumed to be known for all time and a stable non-linear filter is presented. Then assuming that the form of the non-linearity is not known but only a sector bound is given, we propose a linear filter and an upper bound on the sector constant to obtain a stable observer. The convergence conditions for the proposed observers are given directly in terms of the design parameters and are, therefore, very easy to check.     

非线性时变系统基于状态观测器的输出跟踪

研究了一类仿射非射线时变系统基于状态观测器的输出反馈跟踪问题。首先,设计出该系统的状态观测器,然后,给出了一种用估计状态实现渐近跟踪期望信号的控制器设计方案,并证明了相应闭环系统的指数稳定性,其中系统的控制器和观测器可以分离独立进行设计。最后,给出了数值仿真研究,结果表明了该方法的有效性。     

Fault reconstruction for Takagi–Sugeno fuzzy systems via learning observers

This paper addresses the problem of observer-based fault reconstruction for Takagi–Sugeno fuzzy systems. Two types of fuzzy learning observers are constructed to achieve simultaneous reconstruction of system states and actuator faults. Stability and convergence of the proposed observers are proved using Lyapunov stability theory, and necessary conditions for the existence of the observers are further discussed. The design of fuzzy learning observers can be formulated in terms of a series of linear matrix inequalities that can be conveniently solved using convex optimisation technique. A single-link flexible manipulator is employed to verify the effectiveness of the proposed fault-reconstructing approaches.     

Adaptive Lipschitz Observer Design for a Mammalian Model

This paper deals with observer designs for a proposed mathematical model of circadian rhythms which exist in almost every living organism. A 7th order model for mammalian circadian rhythms which captures the main dynamic features is considered in this paper. A recent result of one‐sided Lipschitz observer design in the literature is applied to this mammalian model to show a possibility of reducing measurements for circadian models in system biology. The mammalian model presented may contain an uncertainty parameter. An adaptive design of the Lipschitz observer is then applied to deal with this case. Besides detailed designs of both observers, detailed analysis is also performed for nonlinear functions in the mammalian model to show that the Lipschitz observers can indeed be applied. Several simulation studies of the proposed observers are carried out with the results shown in this paper.