Nonlinear state estimation with robust convergence

The problem of designing a dynamic measurement processor for the on-line estimation of the state of a multi-input multi-output nonlinear plant is addressed. The plant can be either observable or detectable, encompassing a broad range of cases in process systems engineering. On the basis of the structure of a suitable property of robust nonlinear estimability, an estimator is built and its convergence studied, yielding sufficient conditions for robust convergence, a systematic construction, and a tractable gain tuning procedure. The estimability property has a verifiable test, and the execution of the tuning procedure, as well as the interpretation of its convergence features can be achieved within a conventional control framework for linear single-input controllers or single-output filters. The estimation of a continuous polymer reactor is considered as an application example.     

Vehicle state estimation for anti-lock control with nonlinear observer

Vehicle state estimation during anti-lock braking is considered. A novel nonlinear observer based on a vehicle dynamics model and a simplified Pacejka tire model is introduced in order to provide estimates of longitudinal and lateral vehicle velocities and the tire-road friction coefficient for vehicle safety control systems, specifically anti-lock braking control. The approach differs from previous work on vehicle state estimation in two main respects. The first is the introduction of a switched nonlinear observer in order to deal with the fact that in some driving situations the information provided by the sensor is not sufficient to carry out state estimation (i.e., not all states are observable). This is shown through an observability analysis. The second contribution is the introduction of tire-road friction estimation depending on vehicle longitudinal motion. Stability properties of the observer are analyzed using a Lyapunov function based method. Practical applicability of the proposed nonlinear observer is shown by means of experimental results.     

On the convergence of an extended state observer for nonlinear systems with uncertainty

The extended state observer first proposed by Jingqing Han in [J.Q. Han, A class of extended state observers for uncertain systems, Control Decis. 10 (1) (1995) 85-88 (in Chinese)] is the key link toward the active disturbance rejection control that is taking off as a technology after numerous successful applications in engineering. Unfortunately, there is no rigorous proof of convergence to date. In this paper, we attempt to tackle this long unsolved extraordinary problem. The main idea is to transform the error equation of objective system with its extended state observer into a asymptotical stable system with a small disturbance, for which the effect of total disturbance error is eliminated by the high-gain.     

Extended state observer for uncertain lower triangular nonlinear systems

The extended state observer (ESO) is a key part of the active disturbance rejection control approach, a new control strategy in dealing with large uncertainty. In this paper, a nonlinear ESO is designed for a kind of lower triangular nonlinear systems with large uncertainty. The uncertainty may come from unmodeled system dynamics and external disturbance. We first investigate a nonlinear ESO with high constant gain and present a practical convergence. Two types of ESO are constructed with explicit error estimations. Secondly, a time varying gain ESO is proposed for reducing peaking value near the initial time caused by constant high gain approach. The numerical simulations are presented to show visually the peaking value reduction. The mechanism of peaking value reduction by time varying gain approach is analyzed.     

Neural networks for nonlinear state estimation

Estimating the state of a nonlinear stochastic system (observed through a nonlinear noisy measurement channel) has been the goal of considerable research to solve both filtering and control problems. In this paper, an original approach to the solution of the optimal state estimation problem by means of neural networks is proposed, which consists in constraining the state estimator to take on the structure of a multilayer feedforward network. Both non-recursive and recursive estimation schemes are considered, which enable one to reduce the original functional problem to a nonlinear programming one. As this reduction entails approximations for the optimal estimation strategy, quantitative results on the accuracy of such approximations are reported. Simulation results confirm the effectiveness of the proposed method.     

Magnetic sensor-based simultaneous state and parameter estimation using a nonlinear observer

This paper presents an observer design technique for a newly developed non-intrusive position estimation system based on magnetic sensors. Typically, the magnetic field of an object as a function of position needs to be represented by a highly nonlinear measurement equation. Previous results on observer design for nonlinear systems have mostly assumed that the measurement equation is linear, even if the process dynamics are nonlinear. Hence, a new nonlinear observer design method for a Wiener system composed of a linear process model together with a nonlinear measurement equation is developed in this paper. First, the design of a two degree-of-freedom nonlinear observer is proposed that relies on a Lure system representation of the observer error dynamics. To improve the performance in the presence of parametric uncertainty in the measurement model, the nonlinear observer is augmented to estimate both the state and unknown parameters simultaneously. A rigorous nonlinear observability analysis is also presented to show that a dual sensor configuration is a sufficient and necessary condition for simultaneous state and parameter estimation. Finally, the developed observer design technique is applied to non-intrusive position estimation of the piston inside a pneumatic cylinder. Experimental results show that both position and unknown parameters can be reliably estimated in this application.     

Applying the unscented Kalman filter for nonlinear state estimation

Based on presentation of the principles of the EKF and UKF for state estimation, we discuss the differences of the two approaches. Four rather different simulation cases are considered to compare the performance. A simple procedure to include state constraints in the UKF is proposed and tested. The overall impression is that the performance of the UKF is better than the EKF in terms of robustness and speed of convergence. The computational load in applying the UKF is comparable to the EKF.     

Nonlinear state estimation for rigid-body motion with low-pass sensors

In this paper, we consider the state estimation problem for the nonlinear kinematic equations of a rigid body observed under low-pass sensors. The problem is motivated from a walking robot application where inclinometers and gyros are the sensors used. We show that a non-local high gain observer exists for the nonlinear rigid-body kinematic equations and that it under a small angle assumption is possible to use one inclinometer only to estimate two angles.     

Distributed moving horizon state estimation for nonlinear systems with bounded uncertainties

In this work, we propose a distributed moving horizon state estimation (DMHE) design for a class of nonlinear systems with bounded output measurement noise and process disturbances. Specifically, we consider a class of nonlinear systems that are composed of several subsystems and the subsystems interact with each other via their subsystem states. First, a distributed estimation algorithm is designed which specifies the information exchange protocol between the subsystems and the implementation strategy of the DMHE. Subsequently, a local moving horizon estimation (MHE) scheme is designed for each subsystem. In the design of each subsystem MHE, an auxiliary nonlinear deterministic observer that can asymptotically track the corresponding nominal subsystem state when the subsystem interactions are absent is taken advantage of. For each subsystem, the nonlinear deterministic observer together with an error correction term is used to calculate a confidence region for the subsystem state every sampling time. Within the confidence region, the subsystem MHE is allowed to optimize its estimate. The proposed DMHE scheme is proved to give bounded estimation errors. It is also possible to tune the convergence rate of the state estimate given by the DMHE to the actual system state. The performance of the proposed DMHE is illustrated via the application to a reactor-separator process example.     

Noise modeling concepts in nonlinear state estimation

It is quite common to assume that uncertainty enters through additive white noise sources when using recursive state estimation algorithms. Also unknown and time-varying parameters are often modeled similarly by augmenting the states with a parameter vector. Further, it is common to reflect initial model uncertainty through the choice of the initial covariance matrices for the states and parameters.In this paper we study noise modeling based on a hypothesis that it is important to model noise correctly. In practice this implies a critical view on the dominating ‘additive noise paradigm’ as a means to model uncertainty. Alternative concepts of modeling the noise are investigated, and it is shown that modeling noise by introducing it in the system auxiliary variables and control inputs may have a positive impact on estimation performance.     

Nonlinear bounded-error state estimation of continuous-time systems

This paper presents a first study on the application of interval analysis and consistency techniques to state estimation of continuous-time systems described by nonlinear ordinary differential equations. The approach is presented in a bounded-error context and the resulting methodology is illustrated by an example.     

Sampled-data extended state observer for uncertain nonlinear systems

In this paper, we present a sampled-data nonlinear extended state observer (NLESO) design method for a class of nonlinear systems with uncertainties and discrete time output measurement. To accommodate the inter-sample dynamics, an inter-sample output predictor is employed in the structure of the NLESO to estimate the system output in the sampling intervals, where the prediction is used in the proposed observer instead of the system output. The exponential convergence of the sampled-data NLESO is also discussed and a sufficient condition is given by the Lyapunov method. A numerical example is provided to illustrate the performance of the proposed observer.     

Nonlinear observer design for state and disturbance estimation

A new systematic framework for nonlinear observer design that allows the concurrent estimation of the process state variables together with key unknown process or sensor disturbances is proposed. The nonlinear observer design problem is addressed within a similar methodological framework as the one introduced in [N. Kazantzis, C. Kravaris, Nonlinear observer design using Lyapunov's auxiliary theorem, Systems Control Lett. 34 (1998) 241; A.J. Krener, M. Xiao, Nonlinear observer design in the Siegel domain, SIAM J. Control Optim. 41 (2002) 932.] for state estimation purposes only. From a mathematical standpoint, the problem under consideration is addressed through a system of first-order singular PDEs for which a rather general set of solvability conditions is derived. A nonlinear observer is then designed with a state-dependent gain that is computed from the solution of the system of singular PDEs. Under the aforementioned conditions, both state and disturbance estimation errors converge to zero with assignable rates. The convergence properties of the proposed nonlinear observer are tested through simulation studies in an illustrative example involving a biological reactor.     

A hybrid nonlinear state observer for concentration profiles reconstruction in nonlinear simulated moving bed

In this paper, a state observer is proposed for the reconstruction of the concentration profiles in a simulated moving bed. The approach is based on a simple Luenberger-like correction term. The observer is used under the assumption that the flow rates are constant during each switching period. The matrix gain of the correction term may then be re-computed at the beginning of each switching period corresponding to flow rates being changed by the controller. Validating simulations are proposed to assess the efficiency of the proposed profile reconstruction and its robustness against uncertainties on modelling parameters. Comparisons are also done with open-loop simulation based-observer in order to strengthen the relevance of the correction term.     

Advanced point-mass method for nonlinear state estimation

State estimation of discrete-time nonlinear non-Gaussian stochastic systems by point-mass approach, which is based on discretization of state space by a regular grid and numerical solution of Bayesian recursive relations, is treated. The stress is laid to grid design which is crucial for estimator quality and significantly affects the computational demands of the estimator. Boundary-based grid design, thrifty convolution, and multigrid design with grid splitting and merging are proposed. The main advantages of these techniques are nonnegligible support delimitation, time-saving computation of convolution, and effective processing of multimodal probability density functions, respectively. The techniques are involved into the basic point-mass approach and a new general-purpose, more sophisticated point-mass algorithm is designed. Computational demands and estimation quality of the designed algorithm are presented and compared with the particle filter in a numerical example.     

Simultaneous input and state estimation for nonlinear systems with applications to flow field estimation

This paper studies the problem of simultaneous input and state estimation (SISE) for nonlinear dynamical systems with and without direct input–output feedthrough. We take a Bayesian perspective to develop a sequential joint input and state estimation approach. Our scheme gives rise to a nonlinear Maximum a Posteriori optimization problem, which we solve using the classical Gauss–Newton method. The proposed approach generalizes a number of SISE methods presented in the literature. We illustrate the effectiveness of the proposed scheme for nonlinear systems with direct feedthrough in an oceanographic flow field estimation problem involving submersible drogues that measure position intermittently and acceleration continuously.     

Robust discrete‐time nonlinear sliding mode state estimation of uncertain nonlinear systems

In this paper, we propose a discrete‐time nonlinear sliding mode observer for state and unknown input estimations of a class of single‐input/single‐output nonlinear uncertain systems. The uncertainties are characterized by a state‐dependent vector and a scalar disturbance/unknown input. The discrete‐time model is derived through Taylor series expansion together with nonlinear state transformation. A design methodology that combines the discrete‐time sliding mode (DSM) and a nonlinear observer design is adopted, and a strategy is developed to guarantee the convergence of the estimation error to a bound within the specified boundary layer. A relation between sliding mode gain and boundary layer is established for the existence of DSM, and the estimation is made robust to external disturbances and uncertainties. The unknown input or disturbance can also be estimated through the sliding mode. The conditions for the asymptotical stability of the estimation error are analysed. Application to a bioreactor is given and the simulation results demonstrate the effectiveness of the proposed scheme. Copyright © 2006 John Wiley & Sons, Ltd.     

A survey on cyber attacks against nonlinear state estimation in power systems of ubiquitous cities

It is well-known that critical infrastructures would be targets for cyber attacks. In this paper, we focus on the power systems (i.e. smart grids) in ubiquitous cities, where every meter is linked to an information network through wireless networking. In a smart grid system, information from smart meters would be used to perform a state estimation in real time to maintain the stability of the system. A wrong estimation may lead to disastrous consequences (e.g. suspension of electricity supply or a big financial loss). Unfortunately, quite a number of recent results showed that attacks on this estimation process are feasible by manipulating readings of only a few meters. In this paper, we focus on nonlinear state estimation which is a more realistic model and widely employed in a real power grid environment. We category cyber attacks against nonlinear state estimation, and review the mechanisms behind. State-of-the-art security measures to detect these attacks are discussed via sensor protection. Hope that the community would be able to come up with a secure system architecture for ubiquitous cities.     

Design of a nonlinear distributed parameter observer for a pressure swing adsorption plant

As an integral part of an advanced process monitoring and control scheme, the reconstruction of the spatial concentration, temperature, and pressure profiles of an oxygen production plant is studied by the design of a nonlinear observer with distributed parameters. The considered pressure swing adsorption process consists of two adsorbers and each adsorber of two layers. The design of the nonlinear observer with distributed parameters is illustrated for a single adsorption layer, which is described by six quasilinear partial differential equations and related boundary conditions. Thereby, a late lumping approach is used in order to design the injected correction functions in the observer equations and to introduce tuning parameters with a physical meaning. The observer is extended to the entire adsorption process and its simulation shows an excellent convergence behavior.     

Observer-Based Nonlinear Control of A Torque Motor with Perturbation Estimation

This paper presents an observer-based nonlinear control method that was developed and implemented to provide accurate tracking control of a limited angle torque motor following a 50Hz reference waveform. The method is based on a robust nonlinear observer, which is used to estimate system states and perturbations and then employ input-output feedback linearization to compensate for the system nonlinearities and uncertainties. The estimation of system states and perturbations allows input-output linearization of the nonlinear system without an accurate mathematical model of nominal plant. The simulation results show that the observer-based nonlinear control method is superior in comparison with the conventional model-based state feedback linearizing controller.