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.     

Robust adaptive observer for nonlinear systems with unmodeled dynamics

Unmodeled dynamics exist in almost all applications of observers due to the impossibility of using exact and detailed models. It is highly desired that the observers can dominate the effects of unmodeled dynamics independently to prevent the state estimations from diverging and to get the precise estimations. Based on adaptive nonlinear damping, this paper presents a robust adaptive observer for multiple-input multiple-output nonlinear systems with unknown parameters, uncertain nonlinearities, disturbances and unmodeled dynamics. The observer only has one adaptive parameter no matter how high the order of the system is and how many unknown parameters there are. With the proposed observer, neither estimating the unknown parameters or solving linear matrix inequalities is needed. It is shown that the state estimation error is uniformly bounded and can be made arbitrarily small.     

Adaptive observers with arbitrary exponential rate of convergencefor nonlinear systems

Concerns the same class of linearly parameterized single-output nonlinear systems that the authors previously identified in (1992) in terms of differential geometric conditions. When persistency of excitation conditions are satisfied, the adaptive observers presented in this note guarantee arbitrarily fast exponential convergence both of parameter and state estimates to actual parameters and states, while previous adaptive observers guarantee only exponential (not arbitrarily fast) convergence. This extends earlier results for linear systems     

Nonlinear observers for a class of continuous-time uncertain state delayed systems

This paper is concerned with the problem of observer design for a class of time-delay nonlinear systems with parameter uncertainties. The purpose of this problem is to design the gain-scheduled state observers such that, for the addressed nonlinearities as well as all admissible parameter uncertainties in state and output equations, the observation process remains globally exponentially stable, independently of the time delay. The nonlinearities are assumed to satisfy the global L ipschitz conditions, and the parameter uncertainties are allowed to be time varying, unstructured and norm bounded. An effective matrix inequality methodology is developed to solve the proposed problem. W e derive the conditions for the existence of the desired robust nonlinear observers, and then characterize the analytical expression of these observers. Two numerical examples demonstrate the validity and applicability of the present approach.     

Robust adaptive observers for nonlinear systems with boundeddisturbances

Since existing adaptive observers for nonlinear systems may generate unbounded parameter estimates in the presence of bounded disturbances, robust adaptive observers are presented which prevent parameter estimate drift. In addition the input-to-state stability of the error dynamics with respect to disturbances and parameter time-derivatives is guaranteed by generalizing a persistency of excitation result. Asymptotic convergence of state estimation errors is still achieved for systems in adaptive observer form when disturbances are not present, by a suitable extension of Barbalat's lemma     

Uncertainty observer-based robust tracking control for high-order nonlinear systems using forwarding technique

Based on uncertainty observers, we develop a robust tracking control approach for high-order nonlinear systems subject to uncertainties satisfying mismatched condition. In this study, by combining with forwarding technique, uncertainty observers are designed to estimate uncertainties including parameter perturbations and external disturbances in a bottom-up way; at the same time, the convergence rate of estimate error of uncertainty observers is analysed in detail. The insensitivity to uncertainties for each manifold, no need of a prior knowledge on the bound of the uncertainty and the feasibility of practical implementation are three advantages of the proposed scheme. Moreover, approximate error of uncertainty observer can be made sufficiently small by tuning the parameters. The design procedures are elaborated and the boundedness of all trajectories of the closed-loop systems is guaranteed. Two examples are illustrated, highlighting the superiority of the proposed methodology via comparison with other control strategies.     

Redesign of adaptive observers for improved parameter identification in nonlinear systems

We propose a method for redesigning adaptive observers for nonlinear systems. The redesign uses an adaptive law that is based on delayed observers. This increases the computational burden, but gives significantly better parameter identification and robustness properties. In particular, given that a special persistency of excitation condition is satisfied, we prove uniform global asymptotic stability and semi-global exponential stability of the origin of the state and parameter estimation error, and give explicit lower bounds on the convergence rate of both the state and parameter estimation error dynamics. For initial conditions with a known upper bound, we prove tunable exponential convergence rate. To illustrate the use of the proposed method, we apply it to estimate the unmeasured flow rate and the uncertain friction parameters in a model of a managed pressure drilling system. The simulation results clearly show the improved performance of the redesigned adaptive observer compared to a traditional design.     

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.     

Nonlinear observer for simultaneous states and unknown parameter estimation

This paper presents the design of an observer for the simultaneous estimation of states and unknown parameter for a class of nonlinear systems whose nonlinearity satisfies a bounded Jacobian condition. The paper presents two alternate observers based on the structure of the system. The conditions for the existence of these observers can be expressed as a linear matrix inequality and solved using standard solvers. The case of time-varying parameter and multiple unknown parameter have also been investigated. The use of the developed methodology is demonstrated through illustrative examples.     

An adaptive observer incorporating A Priori system data

It is shown, for a class of adaptive observers which estimate system state and parameters from the scalar input and measurement data of a linear system, that certain a priori knowledge of the parameters can be represented by linear equations involving the parameters, and that the observers are readily modified so that their parameter estimates reflect this knowledge.     

A class of nonlinear observers for discrete-time systems with parametric uncertainty

In this paper the problem of observer design is considered for a class of nonlinear discrete-time systems with parametric uncertainty. The problem addressed aims at designing the gain-scheduled state observers such that, for all admissible nonlinearities and time-varying parameter uncertainties in the state equation, the observation process is asymptotically stable. An effective, purely algebraic methodology is developed to solve the proposed problem for discrete-time systems. It is shown that the solution is related to a generalized Riccati-like matrix equation. Specifically, by using the generalized inverse theory and singular value decomposition technique, we obtain the conditions for the existence of desired robust nonlinear observers and then characterize the explicit expression of these observers. Two numerical examples are used to demonstrate the applicability and flexibility of the present approach.     

Adaptive nonlinear observer for state and unknown parameter estimation in noisy systems

This paper proposes a novel adaptive observer for Lipschitz nonlinear systems and dissipative nonlinear systems in the presence of disturbances and sensor noise. The observer is based on an H_∞ observer that can estimate both the system states and unknown parameters by minimising a cost function consisting of the sum of the square integrals of the estimation errors in the states and unknown parameters. The paper presents necessary and sufficient conditions for the existence of the observer, and the equations for determining observer gains are formulated as linear matrix inequalities (LMIs) that can be solved offline using commercially available LMI solvers. The observer design has also been extended to the case of time-varying unknown parameters. The use of the observer is demonstrated through illustrative examples and the performance is compared with extended Kalman filtering. Compared to previous results on nonlinear observers, the proposed observer is more computationally efficient, and guarantees state and parameter estimation for two very broad classes of nonlinear systems (Lipschitz and dissipative nonlinear systems) in the presence of input disturbances and sensor noise. In addition, the proposed observer does not require online computation of the observer gain.     

Sliding mode observers for nonlinear models with unbounded noise and measurement uncertainties

This work extends the applicability of variable structure observers designed for nonlinear systems in two ways. First, it is proved that these observers using a boundary-layer scheme can be applied to system models described by Ito differential equations, resulting in almost sure and mean square exponential estimation error. Second, the use of variable structure observers is extended to nonlinear measurement models containing disturbance effects. Also, a novel approach for obtaining the required parameters in the observer design is provided. Finally, two examples are given to illustrate the application and favorable convergence properties of these generalizations.     

Adaptive Impulsive Observers for a Class of Switched Nonlinear Systems with Unknown Parameter

This work investigates and solves the design of adaptive impulsive observers for a class of uncertain switched nonlinear systems with unknown parameter. Sufficient conditions are derived for designing such observers for each subsystem to reconstruct asymptotically and update system states in real time. The state observer is represented in terms of impulsive differential equations. The parameter estimation law is modelled by an impulse‐free, time‐varying differential equation associated with the impulse time sequence in order to determine when the observer estimated state is updated. The asymptotic convergence to zero of the observation errors is established by applying the method of multiple time‐varying Lyapunov functions. Sufficient conditions are derived that guarantee the convergence of parameter estimation. An example of switched Lorenz system along with numeric and simulation results is presented to demonstrate the effectiveness of the proposed method.     

Global adaptive output-feedback control of nonlinear systems. I.Linear parameterization

The problem of designing global adaptive output-feedback tracking controls for single-input single-output nonlinear systems which are linear with respect to the input and an unknown constant parameter vector is addressed. A class of systems which can be globally controlled by adaptive observer-based output-feedback compensators is identified by geometric coordinate-free conditions. The nonlinearities depend on the output only: growth conditions are not required. Each system in the class admits observers with linear error dynamics and is minimum phase, i.e., it has linear asymptotically stable zero dynamics. When the parameters are known, new sufficient conditions for global output-feedback tracking control are obtained as a special case. For linear systems the result recovers a well-known fundamental adaptive result. Three examples are discussed     

State Estimation For An Agonistic‐Antagonistic Muscle System

Research on assistive technology, rehabilitation, and prosthetics requires the understanding of human machine interaction, in which human muscular properties play a pivotal role. This paper studies a nonlinear agonistic‐antagonistic muscle system based on the Hill muscle model. To investigate the characteristics of the muscle model, the problem of estimating the state variables and activation signals of the dual muscle system is considered. In this work, parameter uncertainty and unknown inputs are taken into account for the estimation problem. Three observers are presented: a high gain observer, a sliding mode observer, and an adaptive sliding mode observer. Theoretical analysis shows the convergence of the three observers. Numerical simulations reveal that the three observers are comparable and provide reliable estimates.     

Modeling and simulation of nonlinear and adaptive observers

This paper is concerned with methods of designing nonlinear state observers for polynomic systems, which can be modeled by state equations with linear, bilinear, and quadratic terms. The general structure of polynomic observers is introduced and simulation serves as a useful tool to determine the final observer parameters. In the case of polynomic systems and linear systems, where some of the parameters are unknown, the polynomic observer theory can also be applied for a combined estimation of states and parameters. Several examples are given to demonstrate the necessity of modeling and simulation for choosing the observer structure as well as for adjusting the observer parameters.     

Asymptotic attenuation of a class of nonlinear systems with unknown sinusoidal disturbances

In this paper, nonlinear observers are incorporated into the adaptive control to synthesize controllers for a class of uncertain nonlinear systems with unknown sinusoidal disturbances which are presented in matched and unmatched forms. In addition to magnitudes and phases, frequencies of the sinusoidal disturbances need not be known as well, so long as the overall order is known. Nonlinear observers are constructed to eliminate the effect of unknown sinusoidal disturbances to improve the steady-state output tracking performance-asymptotic output tracking is achieved. The adaptation law is used to obtain the estimate of all unknown parameters. The presented disturbance decoupling algorithms can deal with matched and unmatched unknown sinusoidal disturbances.     

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.